Springer Monographs in Mathematics
For further volumes: www.springer.com/series/3733
Atsushi Yagi
Abstract Parabol...

Author:
Atsushi Yagi

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Springer Monographs in Mathematics

For further volumes: www.springer.com/series/3733

Atsushi Yagi

Abstract Parabolic Evolution Equations and their Applications

Atsushi Yagi Department of Applied Physics Graduate School of Engineering Osaka University Suita, Osaka 565-0871 Japan [email protected]

ISSN 1439-7382 ISBN 978-3-642-04630-8 e-ISBN 978-3-642-04631-5 DOI 10.1007/978-3-642-04631-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009939692 Mathematics Subject Classification (2000): 35K90, 35K57, 37L30, 92D25, 92D40 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Contents

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Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 1 Banach Spaces . . . . . . . . . . . . . . . . . . . . . 1.1 Distance of Subsets . . . . . . . . . . . . . . 1.2 Fractal Dimension . . . . . . . . . . . . . . . 2 Function Spaces with Values in a Banach Space . . . 2.1 Uniformly Bounded Function Spaces . . . . . 2.2 Continuously Differentiable Function Spaces . 2.3 Hölder Continuous Function Spaces . . . . . 2.4 Weighted Hölder Continuous Function Spaces 2.5 Space of Analytic Functions . . . . . . . . . . 3 Linear Operators . . . . . . . . . . . . . . . . . . . . 3.1 Bounded Linear Operators . . . . . . . . . . . 3.2 Part of Linear Operator . . . . . . . . . . . . 3.3 Multivalued Linear Operators . . . . . . . . . 3.4 Resolvent Equation and Pseudo-Resolvent . . 3.5 Spectra of Operators . . . . . . . . . . . . . . 4 Nonlinear Operators . . . . . . . . . . . . . . . . . . 4.1 Contraction Mappings . . . . . . . . . . . . . 4.2 Fréchet Differentiation . . . . . . . . . . . . . 5 Interpolation of Banach Spaces . . . . . . . . . . . . 5.1 Interpolation Spaces . . . . . . . . . . . . . . 5.2 Interpolation Theorem . . . . . . . . . . . . . 6 Adjoint Spaces and Operators . . . . . . . . . . . . . 6.1 Dual Spaces . . . . . . . . . . . . . . . . . . 6.2 Adjoint Spaces . . . . . . . . . . . . . . . . . 6.3 Adjoint Operators . . . . . . . . . . . . . . . 7 Extrapolation of Hilbert Spaces . . . . . . . . . . . . 7.1 Triplets of Spaces . . . . . . . . . . . . . . . 7.2 Extrapolation Theorem . . . . . . . . . . . . 8 Linear Operators Associated with Sesquilinear Forms 8.1 Sesquilinear Forms and Associated Operators 8.2 Adjoint Forms and Adjoint Operators . . . . .

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Integral Equations and Inequalities of Volterra Type . . . . . . 9.1 Generalized Inequality of Gronwall Type . . . . . . . . 9.2 Integral Inequality of Volterra Type . . . . . . . . . . . 9.3 Integral Equations of Volterra Type . . . . . . . . . . . 9.4 Dominate Convergence Theorem for Integral Equations Differential Inequalities . . . . . . . . . . . . . . . . . . . . . Sobolev–Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . 11.1 Boundaries of Domains . . . . . . . . . . . . . . . . . 11.2 Sobolev Spaces of Integral Order . . . . . . . . . . . . 11.3 Sobolev–Lebesgue Spaces in Rn . . . . . . . . . . . . 11.4 Sobolev–Lebesgue Spaces in Rn+ or Bounded Domains 11.5 Interpolation Property . . . . . . . . . . . . . . . . . . 11.6 Embedding Theorems . . . . . . . . . . . . . . . . . . 11.7 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Spaces H˚ ps (Ω) and Hp−s (Ω) . . . . . . . . . . . . . . 11.9 Product Spaces . . . . . . . . . . . . . . . . . . . . . . 11.10 Miscellaneous Results . . . . . . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . . . . . . . . . .

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Sectorial Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Sectorial Operators in Hilbert Spaces . . . . . . . . . . . . . . 1.1 Sectorial Operators Associated with Sesquilinear Form 1.2 Sectorial Operators in L2 Spaces . . . . . . . . . . . . 1.3 Shift Property in L2 . . . . . . . . . . . . . . . . . . . 1.4 Sectorial Operators in Product Spaces of L2 . . . . . . 2 Sectorial Operators in Banach Spaces . . . . . . . . . . . . . . 2.1 Sectorial Operators in Lp (1 < p < ∞) Spaces . . . . . 2.2 Extreme Cases p = 1 and p = ∞ . . . . . . . . . . . . 2.3 Adjoint Operators . . . . . . . . . . . . . . . . . . . . 2.4 Shift Property in Lp Spaces . . . . . . . . . . . . . . . 3 Sectorial Operators in Product Spaces . . . . . . . . . . . . . . 4 Yosida Approximation . . . . . . . . . . . . . . . . . . . . . . 4.1 Uniform Estimates . . . . . . . . . . . . . . . . . . . . 4.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . 5 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 5.1 Analytic Semigroups . . . . . . . . . . . . . . . . . . 5.2 Contraction Semigroups . . . . . . . . . . . . . . . . . 5.3 Adjoint Semigroups . . . . . . . . . . . . . . . . . . . 5.4 Approximate Semigroups . . . . . . . . . . . . . . . . 6 Generation of Analytic Semigroups . . . . . . . . . . . . . . . 6.1 Generation Results in Hilbert Spaces . . . . . . . . . . 6.2 Generation Results in Lp Spaces . . . . . . . . . . . . 7 Fractional Powers of Linear Operators . . . . . . . . . . . . . 7.1 Integral Formula . . . . . . . . . . . . . . . . . . . . . 7.2 Law of Exponent . . . . . . . . . . . . . . . . . . . . . 7.3 Law of Exponent, II . . . . . . . . . . . . . . . . . . .

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7.4 Moment Inequality . . . . . . . . . . . . . . . . . . 7.5 Comparison of Domains of Fractional Powers . . . 7.6 Fractional Powers of Adjoint Operators . . . . . . . 7.7 Fractional Powers and Analytic Semigroups . . . . 7.8 Generation Theorem of Continuous Semigroups . . Purely Imaginary Powers and Square Root Problem . . . . 8.1 Bounded Purely Imaginary Powers . . . . . . . . . 8.2 Imaginary Powers for Maximal Accretive Operators 8.3 Heinz and Kato Inequality . . . . . . . . . . . . . . 8.4 Square Root Problem . . . . . . . . . . . . . . . . 8.5 Symmetric Forms and Square Root Problem . . . . Notes and Further Researches . . . . . . . . . . . . . . . .

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Linear Evolution Equations . . . . . . . . . . . . . . . . . . . Part I. Autonomous Equations . . . . . . . . . . . . . . . . 1 Analytic Semigroups . . . . . . . . . . . . . . . . . . . . . 1.1 Generation of Analytic Semigroups . . . . . . . . . 1.2 Generator of Analytic Semigroup . . . . . . . . . . 1.3 Perturbation for Generators of Analytic Semigroups 2 Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . 2.1 Construction of Solutions . . . . . . . . . . . . . . 2.2 Maximal Regularity . . . . . . . . . . . . . . . . . 2.3 Higher-Order Maximal Regularity . . . . . . . . . 2.4 Spatial Regularity . . . . . . . . . . . . . . . . . . 3 Autonomous Parabolic Equations . . . . . . . . . . . . . . 3.1 Problems in Hilbert Spaces . . . . . . . . . . . . . 3.2 Problems in Lp Spaces . . . . . . . . . . . . . . . Part II. Nonautonomous Equations . . . . . . . . . . . . . 4 Nonautonomous Abstract Evolution Equations . . . . . . . 4.1 Structural Assumptions . . . . . . . . . . . . . . . 4.2 Some Consequences from (3.30) . . . . . . . . . . 4.3 Structural Assumptions and Yosida Approximation . 5 Evolution Operators . . . . . . . . . . . . . . . . . . . . . 5.1 Integral Equations . . . . . . . . . . . . . . . . . . 5.2 Convergence of Un (t, s) . . . . . . . . . . . . . . . 5.3 Basic Properties of U (t, s) . . . . . . . . . . . . . 6 Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . 7 Conditions (3.29) and (3.30) . . . . . . . . . . . . . . . . . 7.1 Case of Hilbert Spaces . . . . . . . . . . . . . . . . 7.2 Case of Lp Spaces . . . . . . . . . . . . . . . . . . 8 Maximal Regularity . . . . . . . . . . . . . . . . . . . . . 8.1 Refined Properties of U (t, s) . . . . . . . . . . . . 8.2 Maximal Regularity . . . . . . . . . . . . . . . . . 8.3 Spatial Regularity . . . . . . . . . . . . . . . . . . 9 Variational Methods . . . . . . . . . . . . . . . . . . . . .

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Semilinear Evolution Equations . . . . . . . . . . . . . . . . 1 Semilinear Abstract Evolution Equations . . . . . . . . . 1.1 Cauchy Problems . . . . . . . . . . . . . . . . . 1.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . 1.3 Regularity for More Regular Initial Data . . . . . 1.4 Global Existence . . . . . . . . . . . . . . . . . . 1.5 Lipschitz Continuity of Solutions in Initial Data . 1.6 Case β = 0 . . . . . . . . . . . . . . . . . . . . . 2 Variational Methods . . . . . . . . . . . . . . . . . . . . 3 Semilinear Parabolic Equations . . . . . . . . . . . . . . 4 Competition System with Diffusions . . . . . . . . . . . 4.1 Construction of Local Solutions . . . . . . . . . . 4.2 Nonnegativity of Local Solutions . . . . . . . . . 4.3 Global Solutions . . . . . . . . . . . . . . . . . . 5 Some Model in Immunology . . . . . . . . . . . . . . . 5.1 Construction of Local Solutions . . . . . . . . . . 5.2 Nonnegativity of Local Solutions . . . . . . . . . 5.3 Global Solutions . . . . . . . . . . . . . . . . . . 6 Nonautonomous Semilinear Abstract Evolution Equations Notes and Further Researches . . . . . . . . . . . . . . .

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Quasilinear Evolution Equations . . . . . . . . . . 1 Quasilinear Abstract Evolution Equations . . . . 1.1 Structural Assumptions . . . . . . . . . 1.2 Proof of Theorem 5.1 . . . . . . . . . . 1.3 Stronger Results for More Regular Data 1.4 Case ν ≤ β . . . . . . . . . . . . . . . . 2 Lipschitz Continuity of Solutions in Initial Data 3 Equations Including Semilinear Terms . . . . . 3.1 Existence Theorem . . . . . . . . . . . 3.2 Proof of Theorem 5.5 . . . . . . . . . . 3.3 Stronger Results for More Regular Data 3.4 Case ν ≤ β . . . . . . . . . . . . . . . . 4 Lipschitz Continuity of Solutions in Initial Data 5 Variational Methods . . . . . . . . . . . . . . . 6 Quasilinear Parabolic Equations . . . . . . . . . 6.1 Abstract Formulation . . . . . . . . . . 6.2 Construction of Local Solutions . . . . .

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Nonautonomous Parabolic Equations . . . . . . 10.1 Problems in Hilbert Spaces . . . . . . . 10.2 Problems in Lp Spaces . . . . . . . . . 10.3 Some Example of the Critical Case ν = 0 Perturbed Problems . . . . . . . . . . . . . . . 11.1 Evolution Operator . . . . . . . . . . . 11.2 Cauchy Problem . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . .

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Some Model for Honeybee Colonies . . . . . . . . . . . . 7.1 Abstract Formulation . . . . . . . . . . . . . . . . 7.2 Construction of Local Solutions . . . . . . . . . . . 7.3 Nonnegativity of Local Solutions . . . . . . . . . . 7.4 A Priori Estimates . . . . . . . . . . . . . . . . . . 7.5 Global Solutions . . . . . . . . . . . . . . . . . . . Some Chemotaxis Model . . . . . . . . . . . . . . . . . . 8.1 Abstract Formulation . . . . . . . . . . . . . . . . 8.2 Basic Properties of Operators A(U ) . . . . . . . . . 8.3 Construction of Local Solutions . . . . . . . . . . . 8.4 Nonnegativity of Local Solutions . . . . . . . . . . 8.5 Global Solutions . . . . . . . . . . . . . . . . . . . Nonautonomous Quasilinear Abstract Evolution Equations Notes and Further Researches . . . . . . . . . . . . . . . .

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Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . 1 Basics of Dynamical Systems . . . . . . . . . . . . . . . . 1.1 Equilibria and Attractors . . . . . . . . . . . . . . . 1.2 ω-limit Sets and Global Attractor . . . . . . . . . . 1.3 Stable and Unstable Manifolds . . . . . . . . . . . 1.4 Exponential Attraction . . . . . . . . . . . . . . . . 2 Representation Theorem of Stable and Unstable Manifolds 2.1 Discrete Case . . . . . . . . . . . . . . . . . . . . 2.2 Continuous Case . . . . . . . . . . . . . . . . . . . 3 Exponentially Stable Equilibria . . . . . . . . . . . . . . . 4 Exponential Attractors . . . . . . . . . . . . . . . . . . . . 4.1 Contraction Semigroups . . . . . . . . . . . . . . . 4.2 Compact Perturbation of Contraction Operator, I . . 4.3 Compact Perturbation of Contraction Operator, II . 4.4 Squeezing Property . . . . . . . . . . . . . . . . . 4.5 Continuous Dynamical System . . . . . . . . . . . 4.6 Continuous Dependence on Parameter . . . . . . . 5 Dynamical Systems for Semilinear Evolution Equations . . 5.1 Dynamical System . . . . . . . . . . . . . . . . . . 5.2 Absorbing and Invariant Compact Set . . . . . . . . 5.3 Exponential Attractors . . . . . . . . . . . . . . . . 5.4 Squeezing Property of S(t) . . . . . . . . . . . . . 6 Stationary Solutions to Semilinear Equations . . . . . . . . 6.1 Equilibria of Dynamical System . . . . . . . . . . . 6.2 Exponential Stability of U . . . . . . . . . . . . . . 6.3 Unstable Manifold of U . . . . . . . . . . . . . . . 6.4 Reaction–Diffusion Systems . . . . . . . . . . . . . 7 Dynamical Systems for Quasilinear Evolution Equations . . 7.1 Structural Assumptions . . . . . . . . . . . . . . . 7.2 Dynamical System . . . . . . . . . . . . . . . . . .

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7.3 Absorbing and Invariant Compact Set . 7.4 Exponential Attractors . . . . . . . . . Stationary Solutions to Quasilinear Equations 8.1 Equilibria of Dynamical System . . . . 8.2 Exponential Stability of U . . . . . . . 8.3 Unstable Manifold of U . . . . . . . . 8.4 Proof of Proposition 6.10 . . . . . . . Notes and Further Researches . . . . . . . . .

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Numerical Analysis . . . . . . . . . . . . . . . . 1 Semilinear Evolution Equations . . . . . . . 1.1 Approximate Problems . . . . . . . 1.2 Local Estimates of Convergence . . 2 Quasilinear Evolution Equations . . . . . . 2.1 Approximate Problems . . . . . . . 2.2 Local Estimates of Convergence . . 2.3 Proof of Lemma 7.1 . . . . . . . . . 3 Two-Dimensional Finite Element Methods . 3.1 Approximate Operators Aξ . . . . . 3.2 Some properties of Cξ (Ω) and Πξ . 3.3 Projection Operator Pξ . . . . . . . 3.4 Ritz Operator Rξ . . . . . . . . . . . 3.5 Domains of Fractional Powers of Aξ 4 Convergence of Exponential Attractors . . . 4.1 Discrete Case . . . . . . . . . . . . 4.2 Continuous Case . . . . . . . . . . . Notes and Further Researches . . . . . . . .

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Semiconductor Models . . . . . . . . . . . . . . . 1 Prototype Semiconductor Equations . . . . . . 1.1 Model Equations . . . . . . . . . . . . 1.2 Some Preliminary Results . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . 2.1 Construction of Local Solutions . . . . 2.2 Nonnegativity of Solutions . . . . . . 3 Global Solutions . . . . . . . . . . . . . . . . 3.1 A Priori Estimates for Local Solutions 3.2 Global Solutions . . . . . . . . . . . . 3.3 Global Norm Estimates . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . 4.2 Exponential Attractors . . . . . . . . . 5 Three-Dimensional Problem . . . . . . . . . . Notes and Further Researches . . . . . . . . .

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Activator–Inhibitor Models . . . . . . . . . . . . . 1 Prototype Reaction–Diffusion Model . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . 2.1 Construction of Local Solutions . . . . . 2.2 Nonnegativity of Local Solutions . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates from Below . . . . . 3.2 A Priori Estimates for Sobolev Norms . 3.3 Global Solution . . . . . . . . . . . . . 3.4 Global Norm Estimates . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . 4.2 Exponential Attractors . . . . . . . . . . 4.3 Squeezing Property of S(t) . . . . . . . 5 Instability of Homogeneous Stationary Solution 5.1 Localized Problem . . . . . . . . . . . . 5.2 Complexified Dynamical System . . . . 5.3 Unstable Manifold of U . . . . . . . . . Notes and Further Researches . . . . . . . . . .

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357 358 359 359 361 362 362 363 365 366 366 366 367 367 367 368 368 369 370

10 Belousov–Zhabotinskii Reaction Models . . . . . . 1 Field–Noyes Model . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . 2.1 Construction of Local Solutions . . . . . 2.2 Nonnegativity of Local Solutions . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates of Local Solutions . . 3.2 Global Existence . . . . . . . . . . . . . 3.3 Global Norm Estimate . . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . 4.2 Exponential Attractors . . . . . . . . . . 5 Keener–Tyson Model . . . . . . . . . . . . . . 6 Local Solutions . . . . . . . . . . . . . . . . . 6.1 Construction of Local Solutions . . . . . 6.2 Nonnegativity of Local Solutions . . . . 7 Global Solutions . . . . . . . . . . . . . . . . . 7.1 A Priori Estimates of Local Solutions . . 7.2 Global Existence . . . . . . . . . . . . . 7.3 Global Norm Estimate . . . . . . . . . . 8 Dynamical System . . . . . . . . . . . . . . . . 8.1 Construction of Dynamical System . . . 8.2 Exponential Attractors . . . . . . . . . . 9 Instability of Homogeneous Stationary Solution 9.1 Localized Problem . . . . . . . . . . . . 9.2 Complexified Dynamical System . . . .

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373 374 374 374 376 377 377 379 379 379 379 379 380 382 382 383 383 383 384 385 385 385 386 386 386 387

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9.3 Case where fu (u, v) ≤ 0 . . . . . . . . . . . . . . . . . . . 387 9.4 Case where 0 < fu (u, v) < 1 . . . . . . . . . . . . . . . . 388 Notes and Further Researches . . . . . . . . . . . . . . . . . . . . 389 11 Forest Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . 1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . . . . . . . . . 2.2 Construction of Local Solutions . . . . . . . . . . . . . 2.3 Nonnegativity of Local Solutions . . . . . . . . . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates for Local Solutions . . . . . . . . . 3.2 Global Solutions . . . . . . . . . . . . . . . . . . . . . 3.3 Global Norm Estimates . . . . . . . . . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Lyapunov Function . . . . . . . . . . . . . . . . . . . 4.2 ω-limit Sets . . . . . . . . . . . . . . . . . . . . . . . 4.3 Constituents of L2 -ω-limit Sets . . . . . . . . . . . . . f αδ < h < ∞ and where ab2 < 3(c + f ) . 4.4 Cases where c+f 4.5 Numerical Examples . . . . . . . . . . . . . . . . . . . 5 Homogeneous Stationary Solution . . . . . . . . . . . . . . . 5.1 Localized Problem in a Neighborhood of O . . . . . . ˜ O . . . . . . . . . . . . . . . . . . . 5.2 Spectrum of S(t) 5.3 Stability and Instability of O . . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . . . . . . . . . .

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391 391 392 392 393 394 395 395 398 398 400 400 403 405 405 408 409 410 411 412 414

12 Chemotaxis Models . . . . . . . . . . . . . . . . . . . . . . . 1 Chemotaxis Model Without Proliferation . . . . . . . . . 1.1 Abstract Formulation . . . . . . . . . . . . . . . 1.2 Construction of Local Solutions . . . . . . . . . . 1.3 Nonnegativity of Local Solutions . . . . . . . . . 2 Case where χ(ρ) = ρ . . . . . . . . . . . . . . . . . . . 2.1 Global Solutions for Small Initial Functions . . . 2.2 Lyapunov Function . . . . . . . . . . . . . . . . 2.3 Blowup of Solutions . . . . . . . . . . . . . . . . 3 Chemotaxis Model with Proliferation . . . . . . . . . . . 3.1 Local Solutions . . . . . . . . . . . . . . . . . . 3.2 A Priori Estimates of Local Solutions . . . . . . . 3.3 Global Solutions . . . . . . . . . . . . . . . . . . 3.4 Dynamical System . . . . . . . . . . . . . . . . . 3.5 Exponential Attractors . . . . . . . . . . . . . . . 3.6 Numerical Examples . . . . . . . . . . . . . . . . 4 Instability of Homogeneous Stationary Solution of (12.24) 4.1 Localized Problem . . . . . . . . . . . . . . . . . 4.2 Complexified Dynamical System . . . . . . . . . 4.3 Differentiability of F˜ (U ) . . . . . . . . . . . . .

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417 418 419 420 422 423 423 426 427 427 427 428 433 433 434 434 437 437 438 438

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4.4 Spectrum Separation Condition of A . . . . . . . . . . . . 438 4.5 Stability or Instability Conditions . . . . . . . . . . . . . . 441 Notes and Further Researches . . . . . . . . . . . . . . . . . . . . 442 13 Termite Mound Building Model . . . . . . . . . . . 1 Model equations . . . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . 2.2 Construction of Local Solutions . . . . . 2.3 Nonnegativity of Local Solutions . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates for Local Solutions . 3.2 Construction of Global Solutions . . . . 3.3 Global Norm Estimates . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . 4.2 Bounded absorbing set . . . . . . . . . . 4.3 Global Attractor . . . . . . . . . . . . . 4.4 Exponential Attractors . . . . . . . . . . 5 Instability of Homogeneous Stationary Solution 5.1 Localized Problem . . . . . . . . . . . . 5.2 Complexified Dynamical System . . . . 5.3 Differentiability of F˜ (U ) . . . . . . . . 5.4 Spectrum Separation Condition of A . . 5.5 Stability and Instability Conditions . . . Notes and Further Researches . . . . . . . . . .

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445 445 446 446 447 449 450 450 455 455 457 457 458 458 460 463 464 464 465 465 469 470

14 Adsorbate-Induced Phase Transition Model . . . . . . 1 Model Equations . . . . . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . . . 2.2 Basic Properties of A(U ) . . . . . . . . . . 2.3 Construction of Local Solutions . . . . . . . 2.4 Lower and Upper Bound of Local Solutions 3 Global Solutions . . . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates . . . . . . . . . . . . . . 3.2 Global Solutions . . . . . . . . . . . . . . . 3.3 Global Norm Estimates . . . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . . . 4.2 Exponential Attractors . . . . . . . . . . . . 5 Homogeneous Stationary Solutions . . . . . . . . . Notes and Further Researches . . . . . . . . . . . .

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471 472 472 472 474 475 477 479 479 483 483 483 483 484 484 485

15 Lotka–Volterra Competition Model with Cross-Diffusion . . . . . . 487 1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

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489 489 490 492 493 494 495 495 505 511 511 512 513 513 514 514 514 517 521 521 524 524

16 Characterization of Domains of Fractional Powers . . . . . . . . . 1 Domains of Fractional Powers in Hilbert Spaces . . . . . . . . . 1.1 Case of Self-Adjoint Operators . . . . . . . . . . . . . . 1.2 Bounded H∞ Functional Calculus . . . . . . . . . . . . 1.3 Integrable Condition Along Contour . . . . . . . . . . . 1.4 Equivalence Theorem . . . . . . . . . . . . . . . . . . . 1.5 Counterexample . . . . . . . . . . . . . . . . . . . . . . 2 Some Result on Matrix Operators . . . . . . . . . . . . . . . . . 3 Domains of Fractional Powers in Banach Spaces . . . . . . . . . 3.1 Equivalence Theorem in Banach Spaces . . . . . . . . . 4 Domains of Fractional Powers of Elliptic Operators in L2 Spaces 4.1 Case of Domains with Lipschitz Boundary . . . . . . . . 4.2 Case of Domains with C2 Boundary . . . . . . . . . . . . 4.3 Case of Convex Domains . . . . . . . . . . . . . . . . . 5 Domains of Fractional Powers of Elliptic Operators in Lp Spaces 6 Domains of Fractional Powers Under Dirichlet Conditions . . . . 6.1 Domains in L2 Spaces . . . . . . . . . . . . . . . . . . . 6.2 Domains in Lp Spaces . . . . . . . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . . . . . . . . . . .

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527 527 527 529 532 532 537 540 542 542 545 545 546 552 552 558 558 560 561

3

4

5

6

Local Solutions . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . 2.2 Basic Properties of Operators A(U ) . . . 2.3 Construction of Local Solutions . . . . . 2.4 Some Regularity Properties . . . . . . . 2.5 Nonnegativity of Local Solutions . . . . A Priori Estimates for Local Solutions . . . . . 3.1 Case where (15.16) is Valid . . . . . . . 3.2 Case where (15.17) is Valid . . . . . . . Global Solutions . . . . . . . . . . . . . . . . . 4.1 Global Solutions . . . . . . . . . . . . . 4.2 Global Norm Estimates . . . . . . . . . Dynamical System . . . . . . . . . . . . . . . . 5.1 Construction of Dynamical System . . . 5.2 Exponential Attractors . . . . . . . . . . Instability of Homogeneous Stationary Solution 6.1 Localized Problem . . . . . . . . . . . . 6.2 Dynamical System of Localized Problem ˜ ) . . . . . . . . 6.3 Differentiability of A(U 6.4 Spectrum Separation Condition of A . . 6.5 Instability of U . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . .

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

Preface

This monograph is intended to present the fundamentals of the theory of abstract parabolic evolution equations and to show how to apply to various nonlinear diffusion equations and systems arising in science. The theory gives us a unified and systematic treatment for concrete nonlinear diffusion models. Three main approaches are known to the abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. In order to keep the volume of the monograph in reasonable length, we will focus on the semigroup methods. For other two approaches, see the related references in Bibliography. The semigroup methods, which go back to the invention of the analytic semigroups in the middle of the last century, are characterized by precise formulas representing the solutions of the Cauchy problem for evolution equations. The analytic semigroup e−tA generated by a linear operator −A provides directly a fundamental solution to the Cauchy problem for an autonomous linear evolution equation, dU dt + AU = F (t), 0 < t ≤ T , U (0) = U0 , and the solution is t given by the formula U (t) = e−tA U0 + 0 e−(t−s)A F (s) ds. The evolution opU (t, s) for linear operators A(t) gives the formula U (t) = U (t, 0)U0 + erator t U (t, s)F (s) ds representing a solution to a nonautonomous linear evolution 0 equation, dU dt + A(t)U = F (t), 0 < t ≤ T , U (0) = U0 , and the evolution operator U (t, s) can be obtained from the analytic semigroups e−sA(t) through an operator-valued integral equation of Volterra type. Furthermore, a solution to the Cauchy problem for semilinear evolution equation, dU dt + AU = F (U ), 0 < , can be found as a solution of the integral equation U (t) = t ≤ T , U (0) = U 0 t e−tA U0 + 0 e−(t−s)A F (U (s)) ds. Similarly, a solution to the quasilinear problem, dU = U , can be obtained as a solution of the dt + A(U )U = F (U ), 0 < t ≤ T , U (0) t 0 integral equation U (t) = UU (t, 0)U0 + 0 UU (t, s)F (U (s)) ds, where UU (t, s) denotes the evolution operator for linear operators AU (t) = A(U (t)). For these problems, the solution formulas provide us important information on solutions such as the uniqueness, maximal regularity, smoothing effect, and so forth. Especially, for nonlinear problems, one can derive the Lipschitz continuity of solutions with respect to the initial values, even their Fréchet differentiability. Such information on xv

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solutions then enable us develop the study to next stages. In fact, we can construct dynamical systems determined from the Cauchy problems, investigate asymptotic behavior of solutions, show the existence of attractors, investigate the stability and instability of stationary solutions, construct smooth stable and unstable manifolds, and even can analyze solutions numerically in a general framework. So far, a great number of nonlinear diffusion models have already been presented in order to describe diverse interesting phenomena in the real world. Our abstract results are available to these models. Among others, the second half of the present monograph is devoted to handling various diffusion systems in the science of self-organizations. The self-organizing process is usually described by a nonlinear dynamical system. When it is well abstractly formulated, the basic rules of selforganization, which are locally uniform in general, can be described by diffusions of constituents and suitable interactions or reactions among them, and consequently the process can be written down into a nonlinear diffusion system. We will pick up prototype diffusion systems arising in physics, chemistry, biology, and ecology. After constructing dynamical systems, we shall further study the asymptotic behavior of trajectories, existence of attractors, stability and instability of equilibria, construction of smooth stable and unstable manifolds. These results then give a clear meaning or a reasonable interpretation for many notions and ideas suggested in the science of self-organizations such as the emergent property, robustness of process, fluctuations, instabilization of homogeneous states, reduction of active degree of freedom, and so forth. They also seem to give us a certain standpoint for studying the self-organizations theoretically in the framework of infinitedimensional dynamical systems. Chapter 1 prepares all basic matters which are used as terminology or fundamental facts such as Banach and Hilbert spaces, interpolation of Banach spaces, extrapolation of Hilbert spaces, closed linear operators, contraction operators, linear operators associated with sesquilinear forms, integral equations and inequalities of Volterra type, Sobolev–Lebesgue spaces, etc. Chapter 2 reviews the basic properties and matters of sectorial operators in Banach spaces generally with their proof. Sectorial operators indeed play the principal role in the theory of abstract parabolic evolution equations. Especially, we shed light upon the two subject notions, the fractional powers of sectorial operators which are a complex interpolation of operator powers of integral order with the law of exponent and the analytic semigroups which are exponential functions generated by sectorial operators of angles less than π2 . In both we utilize the techniques for analytic functions in complex domains with values in a complex Banach space. We also present many concrete sectorial operators determined from strongly elliptic operators in Hilbert and Banach spaces. In applications, complete characterization or sharp estimation of the domains of fractional powers have critical importance. However, they require some more sophisticated techniques called H∞ functional calculus for sectorial operators. So, they are reviewed separately in Chap. 16. Chapter 3 is devoted to studying the Cauchy problems for linear evolution equations. We are concerned with construction of fundamental solutions which give us formulas representing the solutions to the Cauchy problem. For the autonomous problems, the fundamental solutions are given simply by analytic semi-

Preface

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groups. Meanwhile, for the nonautonomous problems, the fundamental solutions are given by evolution operators which are constructed by using operator-valued integral equations of Volterra type. Chapters 4 and 5 then handle the Cauchy problems for nonlinear evolution equations. In Chap. 4, we construct local solutions for the semilinear evolution equations and derive various properties of the local solutions from the solution formulas, uniqueness, maximal regularity, smoothing effect, Lipschitz continuity with respect to the initial values, etc. A sufficient condition for the global existence is also given. Chapter 5 is devoted to handling the quasilinear evolution equations. We similarly construct the local solutions and derive their significant properties from the solution formula as for the semilinear equations. Chapters 6 and 7 continue the study of the Cauchy problem for nonlinear evolution equations. In Chap. 6, first, we review basic matters and results of the theory of infinite-dimensional dynamical systems, generally with their proofs. Secondly, we consider applications of the theory to the dynamical systems determined from semilinear and quasilinear evolution equations; more precisely, we show general methods how to construct ω-limit sets, global attractors, exponential attractors, stable and unstable manifolds of equilibria, etc. In Chap. 7, we develop numerical analysis for semilinear and quasilinear evolution equations. Setting suitable approximate evolution equations in finite-dimensional spaces, we present general methods how to estimate the order of convergence of approximate solutions. All the subsequent Chaps. 8, 9, 10, 11, 12, 13, 14, and 15 are devoted to studying prototype nonlinear diffusion models which describe the kinetics of self-organizing processes. We are concerned with constructing dynamical systems, constructing attractors, investigating stability and instability of stationary solutions, and constructing the stable and unstable manifolds. Those chapters can be read independently. Chapter 8 handles Semiconductor models with drift-diffusions. The Semiconductor models seem to be the first nonlinear diffusion models presented for describing the kinetics of systems made of a number of constituents which interact each other from the macroscopic view. Chapters 9 and 10 handle Activator–Inhibitor models and Belousov–Zhabotinskii reaction models, respectively. They are both well known as classical examples of pattern formations. Their model equations are written down by reaction–diffusion systems. Chapter 11 treats some diffusion model describing the kinetics of a forest. The model equations are written down by a system of one linear diffusion equation and two ordinary differential equations, one is nonlinear, and the other linear. By the kinematic model, we can find a natural forest boundary. Chapters 12 and 13 are concerned with chemotaxis models. Chemotaxis is known as one of indispensable machinery which sustain the life. In Chap. 12, we handle a bacterium aggregating model by interaction between bacteria and chemoattractant in the manner of positive feedback. The model equations are written down by diffusion–advection–reaction equations. In Chap. 13, a termite mound building model is handled as well. Chapter 14 treats a phase transition model of a metallic surface adsorbing CO molecules. Remarkable adsorption patterns are self-organized on the surface by in-

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teraction between adsorbed molecules and a phase of the surface. The model equations are written down by diffusion–advection–reaction equations. Chapter 15 is finally devoted to handling a segregation model of competing biological species. We consider the species segregating process by using Lotka– Volterra competition model with self-diffusions and cross-diffusions. I would like to express my profound gratitude to the numerous collaborators and colleagues around the world who have made, for the long period, valuable comments and suggestions. In particular, I would like to thank Professors Alan McIntosh, Angelo Favini, Masayasu Mimura, Messoud Efendiev, Tohru Tsujikawa, and Shin-ichi Nakagiri. I would also like to express my hearty thanks to my past students, Drs. Etsushi Nakaguchi, Koichi Osaki, Sang-Uk Ryu, Masashi Aida, Yasuhiro Takei, Le Huy Chuan, Takanori Shirai, and Hideaki Fujimura, my present postgraduate student Doan Duy Hai and postdoctoral Gianluca Mola. The joint works with them were extremely important for presenting this monograph. They also kindly read the draft very carefully. Finally, I would like to express my appreciation to Springer for their efficient handling of the publication. Osaka, July 2009

Atsushi Yagi

Chapter 1

Preliminaries

1 Banach Spaces 1.1 Distance of Subsets Let X be a Banach space with norm · . Throughout book, unless others are notified, Banach and Hilbert spaces are always defined over the complex field C. As usual, X is a metric space with the distance d(U, V ) = U − V determined from · . For U ∈ X and 0 < r < ∞, the set B(U ; r) = {U ∈ X; d(U , U ) < r} is called the ball with center U and radius r. Similarly, B(U ; r) = {U ∈ X; d(U , U ) ≤ r} is called the closed ball with center U and radius r. For a point U ∈ X and a subset A of X, their distance is defined by d(U, A) = inf d(U, V ). V ∈A

For a subset A ⊂ X and 0 < r < ∞, the r-neighborhood of A is defined by Wr (A) = {U ∈ X; d(U, A) < r}. For two subsets A and B of X, the Hausdorff pseudo-distance is defined by h(A, B) = sup d(U, B) = sup inf d(U, V ). U ∈A

U ∈A V ∈B

(1.1)

Unless A is a bounded subset, h(A, B) may possibly be infinity. It is seen by definition that h(A1 , A3 ) ≤ h(A1 , A2 ) + h(A2 , A3 ). A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_1, © Springer-Verlag Berlin Heidelberg 2010

(1.2) 1

2

1 Preliminaries

For two subsets A and B, their distance is defined by d(A, B) = max{h(A, B), h(B, A)}.

(1.3)

Unless A and B are bounded subsets, d(A, B) may possibly be infinity.

1.2 Fractal Dimension Let M be a compact subset of a Banach space X. For any ε > 0, M can be covered by a finite number of closed balls having a radius ε. Let N (ε) be the minimal number of balls with radius ε which cover the set M. We then define d(M) = lim sup ε→0

log N (ε) log 1ε

.

This superior limit, 0 ≤ d(M) ≤ ∞, is called the fractal dimension of M. In particular, when d(M) < ∞, M is said to have a finite fractal dimension. The following result is easily verified from the definition. Theorem 1.1 Let M be a compact subset of X with finite fractal dimension. Let Φ be a Lipschitz-continuous operator from M into another Banach space Y , namely, ΦU − ΦV Y ≤ LU − V X ,

U, V ∈ M,

with some constant L. Then, Φ(M) also has a finite fractal dimension with the relation dY (Φ(M)) ≤ dX (M). If M has a finite fractal dimension, then M can be embedded into a finitedimensional space by Mañé–Hölder-type projection as follows. Theorem 1.2 Let M be a compact subset of a real Banach space X with fractal dimension d < ∞. Let N > 2d be an integer, and let α0 = NN−2d (1+d) . Then, for any 0 < α < α0 , we have that for almost every bounded linear operator Π : X → RN , CΠ(U − V )α ≥ U − V ,

U, V ∈ M,

(1.4)

with some constant C > 0. For details, see [BEFN93, FO96, HY99]. Condition (1.4) shows in particular that Π is injective on the subset M and the inverse of Π on Π(M) is Hölder continuous with exponent α. As N → ∞, α0 → 1; so, α can also be taken close to 1; we get by Theorem 1.1 that, if (1.4) is valid with α = 1, then d(M) = d(Π(M)). The term “almost every” is in the sense of prevalence. Prevalence is a generalized notion of “almost every” for the points in Lebesgue measurable sets in Euclidean spaces for the vectors in infinite-dimensional spaces.

2 Function Spaces with Values in a Banach Space

3

Consider two Banach spaces X and Y such that Y ⊂ X continuously. Let M be a compact subset of Y with dY (M) < ∞. By Theorem 1.1, we verify that M has a finite fractal dimension in X, too, with dX (M) ≤ dY (M). However, the following theorem gives some converse estimation. Theorem 1.3 Let three Banach spaces X, Y , and Z be such that Z ⊂ Y ⊂ X with 1−θ continuous embeddings, and let · Y ≤ D · θZ · X on Z with some exponent 0 < θ < 1. If M is a compact set of Y with dY (M) < ∞ and is a bounded set of Z, then dX (M) ≤ dY (M) ≤ (1 − θ )−1 dX (M). Proof Let ε > 0, and let N X (ε) be the minimal number of balls of X with radius ε which cover M. There exist N X (ε) points Ui ∈ X such that M⊂

X (ε) N

X

B (Ui ; ε).

i=1 X

X

For each i, since M ∩ B (Ui ; ε) = ∅, there is a point Vi ∈ M ∩ B (Ui ; ε). Then, 1−θ provides the relation · Y ≤ D · θZ · X X

X

M ∩ B (Ui ; ε) ⊂ M ∩ B (Vi ; 2ε) ⊂ B

Y

Vi ; D[δZ (M)]θ (2ε)1−θ ,

where δZ (M) denotes the diameter of M in Z. Therefore, M⊂

X (ε) N

B

Y

Vi ; D[δZ (M)]θ (2ε)1−θ .

i=0

This shows that M can be covered by at least N X (ε)-balls of Y with radius ε˜ = D[δZ (M)]θ (2ε)1−θ , that is, N Y (˜ε ) ≤ N X (ε). We hence obtain that dY (M) = lim sup ε˜ →0

log N Y (˜ε ) log

1 ε˜

= (1 − θ )−1 lim sup ε→0

≤ lim sup ε→0

log N X (ε) log

1 ε

log N X (ε) log D[δ

1 θ 1−θ Z (M)] 2

1 + log ε1−θ

= (1 − θ )−1 dX (M).

It is already known that dX (M) ≤ dY (M).

2 Function Spaces with Values in a Banach Space Let X be a Banach space with norm · . We introduce various function spaces consisting of functions defined on an interval of R or on a domain of C with values in X.

4

1 Preliminaries

2.1 Uniformly Bounded Function Spaces Let [a, b] be a bounded closed interval. By B([a, b]; X) we denote the space of uniformly bounded functions on [a, b] (not necessarily smooth or measurable). The space is a Banach space with the supremum norm F B = sup F (t). a≤t≤b

−η

By B{a} ((a, b]; X), where η > 0, we denote the space of X-valued functions which are uniformly bounded on (a, b] with weight function (t − a)η , namely, F ∈ −η B{a} ((a, b]; X) if and only if (t − a)η F ∈ B((a, b]; X). We equip the space with the norm F B−η = sup (t − a)η F (t) = (t − a)η F B . {a}

a 0 be fixed exponents. If a continuous function ϕ(t, s) ≥ 0 defined for 0 ≤ s < t ≤ T has a weak singularity at t = s and satisfies the integral inequality t (1.39) ϕ(t, s) ≤ a(t − s)μ−1 + b (t − τ )ν−1 ϕ(τ, s) dτ, 0 ≤ s < t ≤ T , s

then 1 ϕ(t, s) ≤ a(μ)(t − s)μ−1 Eμ,ν [b(ν)] ν (t − s) ,

0 ≤ s < t ≤ T,

(1.40)

where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35). Proof By induction, we will show the estimate ϕ(t, s) ≤

n

abk

k=0

+ bn+1

(μ)(ν)k (t − s)μ+kν−1 (μ + kν) (ν)n+1 ((n + 1)ν)

t

(t − τ )(n+1)ν−1 ϕ(τ, s) ds.

(1.41)

s

Indeed, it is obvious for n = 0. Assume that this is true for n. Then, ϕ(t, s) ≤

n

(μ)(ν)k (ν)n+1 (t − s)μ+kν−1 + bn+1 (μ + kν) ((n + 1)ν) k=0 t τ (n+1)ν−1 μ−1 ν−1 a(τ − s) × (t − τ ) +b (τ − σ ) ϕ(σ, s) dσ dτ. abk

s

s

Hence, by (1.38), t (μ)((n + 1)ν) . (t − τ )(n+1)ν−1 (τ − s)μ−1 dτ = (t − s)μ+(n+1)ν−1 (μ + (n + 1)ν) s In addition, by (1.38), t τ s

(t − τ )(n+1)ν−1 (τ − σ )ν−1 ϕ(σ, s) dσ dτ

s

=

t s

=

t

(t − τ )(n+1)ν−1 (τ − σ )ν−1 ϕ(σ, s) dτ dσ

σ

((n + 1)ν)(ν) ((n + 2)ν)

t

(t − σ )(n+2)ν−1 ϕ(σ, s) dσ.

s

Hence, (1.41) is obtained for n + 1, too. Letting n → ∞ in (1.41), we observe that ϕ(t, s) ≤

∞ n=0

abn

(μ)(ν)n (t − s)μ+nν−1 , (μ + nν)

0 ≤ s < t ≤ T.

9 Integral Equations and Inequalities of Volterra Type

31

By the definition (1.35), the desired estimate (1.40) is concluded.

In a quite analogous way, we can show the following result. The proof will be left to the reader. Theorem 1.28 Let a ≥ 0 and b > 0 be constants, and let μ > 0 and ν > 0 be fixed exponents. If a continuous function ψ(t, s) ≥ 0 defined for 0 ≤ s < t ≤ T has a weak singularity at t = s and satisfies the integral inequality t ψ(t, τ )(τ − s)ν−1 dτ, 0 ≤ s < t ≤ T , (1.42) ψ(t, s) ≤ a(t − s)μ−1 + b s

then 1 ψ(t, s) ≤ a(μ)(t − s)μ−1 Eμ,ν [b(ν)] ν (t − s) ,

0 ≤ s < t ≤ T,

where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35).

9.3 Integral Equations of Volterra Type In this subsection, we consider integral equations of Volterra type. Let X be a Banach space, and let L(X) be the Banach space of all bounded linear operators on X equipped with the uniform operator norm · . Let 0 < T < ∞ be a fixed number and put = {(t, s) ∈ R2 ; 0 ≤ s < t ≤ T }. We consider the integral equation of Volterra type t Φ(t, s) = P (t, s) + Q(t, τ )Φ(τ, s) dτ, 0 ≤ s < t ≤ T ,

(1.43)

s

in L(X). Here, P and Q are in C(; L(X)) and are assumed to satisfy the estimates P (t, s) ≤ a(t − s)μ−1 ,

0 ≤ s < t ≤ T,

(1.44)

Q(t, s) ≤ b(t − s)ν−1 ,

0 ≤ s < t ≤ T,

(1.45)

with some fixed exponents μ > 0 and ν > 0 and some constants a > 0 and b > 0, respectively. These mean that P and Q have weak singularities at t = s. We can construct a unique solution to (1.43). Theorem 1.29 Under (1.44) and (1.45), there exists a unique function Φ ∈ C(; L(X)) such that Φ has a weak singularity at t = s and satisfies (1.43). Moreover, Φ is estimated by 1 Φ(t, s) ≤ a(μ)(t − s)μ−1 Eμ.ν [b(ν)] ν (t − s) , 0 ≤ s < t ≤ T , (1.46) where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35).

32

1 Preliminaries

Proof Before proving the assertion of theorem, we present a lemma. ˜ ∈ C(; L(X)) satisfy Lemma 1.3 Let P˜ , Q ˜ P˜ (t, s) ≤ a(t ˜ − s)μ−1 ,

0 ≤ s < t ≤ T,

(1.47)

˜ s) ≤ b(t ˜ − s)ν˜ −1 , Q(t,

0 ≤ s < t ≤ T,

(1.48)

with some exponents μ˜ > 0 and ν˜ > 0 and some constants a˜ > 0 and b˜ > 0, respectively. Put t ˜ τ )P˜ (τ, s) dτ, 0 ≤ s < t ≤ T . ˜ s) = Q(t, R(t, s

Then, R˜ belongs to C(; L(X)) and satisfies the estimate ˜ ˜ ν) ˜ ν −1 ˜ s) ≤ a˜ b(μ)(˜ , R(t, (t − s)μ+˜ (μ˜ + ν˜ )

0 ≤ s < t ≤ T.

(1.49)

Proof We observe in view of (1.38) that R˜ is well defined and satisfies (1.49). ˜ But this is not obvious because P˜ So, it suffices to verify the continuity of R. ˜ and Q have singularities. We here introduce a sequence of cutoff functions P˜k defined for all integers k > T1 such that P˜k ’s are defined for all 0 ≤ s ≤ t ≤ T and are continuous functions with values in L(X), that P˜k (t, s) = P˜ (t, s) for 0 ≤ s ≤ T − k1 , s + k1 ≤ t ≤ T , and that P˜k ’s satisfy the same estimate as (1.47). We also introduce a sequence Q˜ k satisfying (1.48), which is defined in a quite analogous way to P˜k . ˜ k ∈ C(; L(X)), it is easy to see that the function As P˜k , Q t 1 R˜ k (t, s) = Q˜ k (t, τ )P˜k (τ, s) dτ, 0 ≤ s ≤ t ≤ T , k > , T s is also in C(; L(X)). Moreover, for 0 ≤ s ≤ T − k1 , s + R(t, ˜ s) − R˜ k (t, s) ≤

≤t ≤T,

Q(t, ˜ τ ) − Q˜ k (t, τ )P˜ (τ, s) dτ

t

s

Q ˜ k (t, τ )P˜ (τ, s) − P˜k (τ, s) dτ

t

+ s

≤C

1 k

t t− k1

+

s+ k1

˜ (t − τ )μ−1 (τ − s)ν˜ −1 dτ

s

˜ 1 ν˜ −1 1 μ−1 −1 −μ˜ −1 −˜ν ≤ C μ˜ k + ν˜ k t −s− t −s− . k k ˜ s) in L(X) as k → ∞ This estimate indeed shows that R˜ k (t, s) converges to R(t, and, for arbitrarily fixed 0 < δ < T , the convergence is uniform for t and s such that

9 Integral Equations and Inequalities of Volterra Type

33

˜ s) for 0 ≤ s ≤ T − δ, s + δ ≤ t ≤ T . As a result, we observe the continuity of R(t, 0≤s 0, then u(t) ≤ e−δt u(0) +

t

e−δ(t−s) f (s) ds,

0 < t ≤ T.

(1.57)

0

In addition, if f (t) ≡ f > 0, then it follows that u(t) ≤ e−δt u(0) + f δ −1 ,

0 < t ≤ T.

(1.58)

We next consider the differential inequality du + δu ≤ f (t), dt

0 < t ≤ T,

(1.59)

with δ > 0 and f ∈ C([0, T ]; R). We assume that f (t) satisfies the condition t f (τ ) dτ ≤ α(t − s) + β, 0 ≤ s < t ≤ T , (1.60) s

with some constants α ≥ 0 and β ≥ 0. Then, as seen above, if u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfies (1.59), then u is estimated by (1.57). By integration by parts, t t t d −δ(t−s) −δ(t−s) − e f (s) ds = e f (τ ) dτ ds ds s 0 0 t t t = e−δt f (τ ) dτ + δ e−δ(t−s) f (τ ) dτ ds. 0

0

s

Therefore, (1.60) yields that t t e−δ(t−s) f (s) ds ≤ e−δt (αt + β) + δ e−δ(t−s) [α(t − s) + β] ds 0

0

= αδ

−1

(1 − e

−δt

) + β.

(1.61)

Hence, we have proved the following proposition. Proposition 1.4 Let δ > 0, and let f ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0. If u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfies the differential inequality (1.59), then u is estimated by u(t) ≤ e−δt u(0) + αδ −1 + β,

0 < t ≤ T.

(1.62)

On the basis of these results, we can prove some other propositions which will be often utilized in showing a dissipative condition for nonlinear diffusion equations. Proposition 1.5 Let δ > 0, γ > 0, and let f ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0. Assume that 0 ≤ u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) and 0 ≤ v ∈

10 Differential Inequalities

37

C([0, T ]; R) satisfy the differential inequality du + δu + γ v ≤ f (t), dt

0 < t ≤ T.

Then, it holds for v that t v(τ ) dτ ≤ αγ −1 (t − s) + [u(0) + αδ −1 + 2β]γ −1 ,

(1.63)

0 ≤ s < t ≤ T . (1.64)

s

Proof Since γ v ≥ 0, u satisfies (1.59), too. So, u is estimated by (1.62). Fix 0 ≤ s < t ≤ T . Integrate both hand sides of (1.63) for τ in the interval [s, t]. Then, t t t u(t) − u(s) + δ u(τ ) dτ + γ v(τ ) dτ ≤ f (τ ) dτ. s

Since u ≥ 0,

γ

t

s

s

v(τ ) dτ ≤ u(s) +

s

t

f (τ ) dτ. s

By (1.60) and (1.65), we see that t γ v(τ ) dτ ≤ e−δs u(0) + αδ −1 + β + α(t − s) + β. s

Hence, (1.64) is obtained.

Proposition 1.6 Let δ > 0, let f ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0, and let 0 ≤ p ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0, where δ > α ≥ 0. If 0 ≤ u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfies the differential inequality du + δu ≤ p(t)u + f (t), dt

0 < t ≤ T,

(1.65)

then u is estimated by u(t) ≤ eβ e−(δ−α)t u(0) + eβ α (δ − α)−1 + β ,

0 < t ≤ T.

(1.66)

Proof Obviously, u satisfies (1.55) with a(t) = δ − p(t). So, the assumption on p yields that t t t − 0 [δ−p(τ )]dτ u(t) ≤ e u(0) + e− s [δ−p(τ )]dτ f (s) ds ≤ eβ e−(δ−α)t u(0) + eβ

0 t

e−(δ−α)(t−s) f (s) ds.

0

Therefore, (1.66) is verified by (1.61).

38

1 Preliminaries

Proposition 1.7 Let δ > 0, γ > 0, and let f ∈ C([0, T ]; R) (resp. g ∈ C([0, T ]; R)) satisfy (1.60) with α ≥ 0 and β ≥ 0 (resp. α ≥ 0 and β ≥ 0). Let 0 ≤ p ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0. Assume that 0 ≤ u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) and 0 ≤ v ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfy the two differential inequalities du 0 < t ≤ T, dt + δu + γ v ≤ f (t), (1.67) dv + δv ≤ p(t)v + g(t), 0 < t ≤ T. dt Then, v is estimated by v(t) ≤ eβ e−δt v(0) + eβ αγ −1 [u(0) + α γ −1 + α δ −1 + 2β ] + eβ (α δ −1 + β ),

0 ≤ t ≤ T.

(1.68)

Proof Let ξ > 0 be a parameter. From the second inequality of (1.67) we have dv + [δ + ξ − p(t)]v ≤ ξ v(t) + g(t). dt Therefore, using (1.56) with a(t) = δ + ξ − p(t), we obtain that t t − 0t [δ+ξ −p(τ )]dτ v(t) ≤ e v(0) + e− s [δ+ξ −p(τ )]dτ [ξ v(s) + g(s)] ds. 0

The assumption on p then yields that v(t) ≤ eβ e−(δ+ξ −α)t v(0) + eβ

t

e−(δ+ξ −α)(t−s) [ξ v(s) + g(s)] ds.

0

We here fix the ξ such as ξ = α. Then, since v satisfies (1.64), we observe by (1.61) that t e−δ(t−s) αv(s) ds ≤ αγ −1 [u(0) + α γ −1 + α δ −1 + 2β ]. 0

Similarly, by (1.61),

t

e−δ(t−s) g(s) ds ≤ α δ −1 + β .

0

Hence, the desired estimate has been shown.

11 Sobolev–Lebesgue Spaces An open and connected subset Ω of Rn is called a domain in Rn . Throughout the book, unless otherwise notified, the functions in Ω are always complex valued.

11 Sobolev–Lebesgue Spaces

39

Let Ω be a domain in Rn . For 1 ≤ p ≤ ∞, the set of all Lp functions in Ω is denoted by Lp (Ω). With the usual Lp norm · Lp , Lp (Ω) is a Banach space. When Ω is a bounded domain, C(Ω) (resp. Ck (Ω), k = 1, 2, . . .) is the set of all continuous (resp. k-times continuously differentiable)functions on Ω. With the norm uC = maxx∈Ω |u(x)| (resp. uCk = uC + |α|≤k D α uC ), C(Ω) (resp. Ck (Ω)) becomes a Banach space. For 0 < σ < 1, Cσ (Ω) is the set of all Hölder continuous functions on Ω with exponent σ . Cσ (Ω) is a Banach space k+σ (Ω), k = with the norm uCσ = uC + supx,y∈Ω, x =y |u(x)−u(y)| |x−y|σ . Similarly, C 1, 2, . . . , are the sets of all Ck functions on Ω with kth-order derivatives in Cσ (Ω). Ck+σ (Ω)’s are also Banach spaces with the norms uCk+σ = uCk + α |α|=k D uCσ . Meanwhile, the space of Lipschitz continuous functions on Ω is denoted by C0,1 (Ω). C0,1 (Ω) is a Banach space with the norm uC0,1 = uC + k,1 supx,y∈Ω, x =y |u(x)−u(y)| |x−y| . In an analogous way, the spaces C (Ω), k = 1, 2, . . . , and their norms are defined. By D(Ω) we denote the space of all infinitely differentiable functions in Ω with compact supports.

11.1 Boundaries of Domains Following [Gri85], we review basic things concerning the boundary of domain in Rn . In this subsection, we are concerned with a bounded domain Ω in Rn . We say that Ω has a continuous (resp. Lipschitz, Cm class (m = 1, 2, 3, . . .)) boundary ∂Ω if for every x ∈ ∂Ω, there exists a neighborhood V of x in Rn and a new orthogonal coordinate system (y1 , . . . , yn ) such that (1) V is a cube in the new coordinates: V = {(y1 , . . . , yn ); −ai < yi < ai , i = 1, . . . , n}. (2) There exists a continuous (resp. Lipschitz, Cm class) function ϕ which is defined in V = {(y1 , . . . , yn−1 ); −ai < yi < ai , i = 1, . . . , n − 1} and satisfies |ϕ(y )| ≤ an /2 for every y = (y1 , . . . , yn−1 ) ∈ V , Ω ∩ V = {y = (y , yn ) ∈ V ; yn > ϕ(y )}, ∂Ω ∩ V = {y = (y , yn ) ∈ V ; yn = ϕ(y )}. (3) ϕC(V ) ≤ c (resp. ϕLip(V ) ≤ c, ϕCm (V ) ≤ c) with some constant c > 0. If Ω is an n-dimensional manifold of Cm class (m = 1, 2, . . .) with boundary, then its boundary ∂Ω is automatically of Cm class. However, the same is not necessarily true for a Lipschitz manifold. Even if Φ(y1 , . . . , yn ) is a Lipschitz continuous function for which the equation Φ(y1 , . . . , yn ) = 0 defines a hypersurface

40

1 Preliminaries

yn = ϕ(y1 , . . . , yn−1 ), the function ϕ may be non-Lipschitz. A counterexample is given in [Gri85, Lemma 1.2.1.4]. Let Ω be a bounded domain with Lipschitz boundary. Then, on the boundary ∂Ω, a measure dS can be defined in a similar manner as in the case of smooth boundary, and an integral with respect to dS is defined for the functions on ∂Ω. If a function f has a support in ∂Ω ∩ V , where V is one of the neighborhoods mentioned above, then its integral is given in the form f dS = f 1 + |∇ ϕ(y )|2 dy , V

∂Ω

where ∇ ϕ = (D1 ϕ, . . . , Dn−1 ϕ). Furthermore, for almost all boundary points x ∈ ∂Ω, the outer normal vectors ν(x) are defined. Indeed, if (y , yn ) ∈ ∂Ω ∩ V , then ν is given by ν(y , yn ) = (D1 ϕ(y ), . . . , Dn−1 ϕ(y ), −1) 1 + |∇ ϕ(y )|2 . Then, the formula of integration by parts Di u · v dx + u · Di v dx = [uv]νi (x) dS, Ω

Ω

i = 1, 2, . . . , n,

(1.69)

∂Ω

is valid for u, v ∈ C1 (Ω), where ν(x) = (ν1 (x), . . . , νn (x)), x ∈ ∂Ω.

11.2 Sobolev Spaces of Integral Order Let Ω be a domain in Rn . For 1 ≤ p ≤ ∞ and k = 0, 1, 2, . . . , the space Hpk (Ω) is defined as a set of all complex-valued Lp functions u whose derivatives D α u in the distribution sense up to order k belong to Lp (Ω), where α = (α1 , α2 , . . . , αn ) denotes a multiindex, and |α| = α1 + α2 + · · · + αn . The space Hpk (Ω) becomes a Banach space with the norm uHpk =

|α|≤k

1 D

α

p uLp

p

,

u ∈ Hpk (Ω).

When p = 2, H2k (Ω) is obviously a Hilbert space with the inner product (D α u, D α v)L2 , u, v ∈ H2k (Ω). (u, v)H k = 2

|α|≤k

Let Ω be the half-space Rn+ = {x = (x , xn ); x ∈ Rn−1 , xn > 0} or a bounded domain in Rn with Lipschitz boundary. According to [Ste70, Chap. VI, Theorems 5 and 5 ], we can construct an extension operator of functions in Ω to functions in Rn universally in 1 ≤ p ≤ ∞ and k = 0, 1, 2, . . . .

11 Sobolev–Lebesgue Spaces

41

Theorem 1.33 Let Ω be Rn+ or a bounded domain with Lipschitz boundary. There exists a linear operator E mapping functions in Ω to functions in Rn with the following properties: (1) (Eu)|Ω = u. (2) E is a continuous operator from Hpk (Ω) to Hpk (Rn ) for any 1 ≤ p ≤ ∞ and any k = 0, 1, 2, . . . , namely, EuHpk (Rn ) ≤ Ap,k uHpk (Ω) ,

u ∈ Hpk (Ω),

with some constant Ap,k > 0 depending on p and k.

11.3 Sobolev–Lebesgue Spaces in Rn When 1 < p < ∞, the Sobolev spaces Hpk (Ω) can be extended for nonintegral orders. In this subsection, we consider the case where Ω = Rn . Let s ≥ 0 be a nonnegative number. The space Hps (Rn ) is defined by s Hps (Rn ) = u ∈ S(Rn ) ; F−1 1 + |ξ |2 2 Fu ∈ Lp (Rn ) , where S(Rn ) denotes the set of tempered distributions in Rn , and F and F−1 denote the Fourier transform and the inverse Fourier transform on S(Rn ) , respectively. The space Hps (Rn ) is a Banach space with the norm s uHps = F−1 1 + |ξ |2 2 Fu L , p

u ∈ Hps (Rn ).

By the theory of Fourier multipliers, it is possible to verify that the two definitions of Hpk (Rn ) and Hps (Rn ) are equivalent for integral orders s = 0, 1, 2, . . . , that is, Hpk (Rn ) = Hps (Rn ) with norm equivalence for nonnegative integers s = k. When p = 2, the space H2s (Rn ) is seen to be a Hilbert space with the inner product (u, v)H2s =

1 + |ξ |2

s

2

s Fu, 1 + |ξ |2 2 Fv L , 2

u, v ∈ H2s (Rn ).

Furthermore, for s = k + σ , where k = [s] is an integer and 0 < σ < 1, the norm of H2s (Rn ) is equivalent to u H s 2

= uL2 +

|α|≤k

see [Tri78, 2.5.1, (15)].

Rn ×Rn

|D α u(x) − D α u(y)|2 dx dy |x − y|n+2σ

1 2

,

u ∈ H2s (Rn ),

42

1 Preliminaries

11.4 Sobolev–Lebesgue Spaces in Rn+ or Bounded Domains We intend to extend the definition of Hpk (Ω) for nonintegral orders. Let Ω denote the half-space Rn+ or a bounded domain in Rn with Lipschitz boundary. For 1 < p < ∞ and s ≥ 0, Hps (Ω) is defined as the set of all restrictions u of the functions in Hps (Rn ) to Ω. That is, a function u ∈ Lp (Ω) is in Hps (Ω) if and only if there exists a function U ∈ Hps (Rn ) such that U|Ω = u almost everywhere in Ω. For u ∈ Hps (Ω), its Hps norm is defined by uHps (Ω) =

inf

U ∈Hps (Rn ),U|Ω =u

U Hps (Rn ) .

With this norm, Hps (Ω) is a Banach space. Indeed, since the space K = {U ∈ Hps (Rn ); U = 0 in Ω} is a closed subspace of Hps (Rn ), Hps (Ω) is nothing more than the quotient space Hps (Rn ) K. By virtue of Theorem 1.33, we can easily verify that such a definition is consistent with the former one when s is an integer. Let p = 2, and let s = k + σ , where k = [s] is an integer and 0 < σ < 1. Then, the norm of H2s (Ω) is as before equivalent to u H s 2

= uL2 +

|α|≤k

Ω×Ω

|D α u(x) − D α u(y)|2 dx dy |x − y|n+2σ

1

u ∈ H2s (Ω),

2

, (1.70)

see [Tri78, Remark 4.4.2/2]. One sees that, for 0 < s0 < s1 < ∞, Hps1 (Ω) ⊂ Hps0 (Ω) ⊂ Lp (Ω) with continuous embeddings. Theorem 1.34 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. Let 1 < p < ∞ and s ≥ 0. Then, u ∈ Hps (Ω) if and only if Re u ∈ Hps (Ω) and Im u ∈ Hps (Ω). It holds that max{ Re uHps , Im uHps } ≤ uHps ≤ Re uHps + Im uHps ,

u ∈ Hps (Ω).

Proof We will prove that u ∈ Hps (Ω) implies u ∈ Hps (Ω) with uHps = uHps , where u(x) = u(x) is the complex conjugate of u(x). Clearly, all the assertions of the theorem are easily verified by this property. Consider first the case where Ω = Rn . By definition, u ∈ Hps (Rn ) is equivalent to s ei(y−x)·ξ 1 + |ξ |2 2 u(x) dx dξ ∈ Lp Rny . Rn ×Rn

11 Sobolev–Lebesgue Spaces

43

Taking the complex conjugate of this function and changing integral variables by ξ = −ξ , we have s n (−1) ei(y−x)·ξ 1 + |ξ |2 2 u(x) dx dξ ∈ Lp Rny . Rn ×Rn

This then means that u ∈ Hps (Rn ) with uHps = uHps . Consider next the general case of Ω. By definition, u ∈ Hps (Ω) implies that there exists a sequence of functions Uk ∈ Hps (Rn ) such that u = Uk|Ω and Uk Hps (Rn ) → uHps (Ω) as k → ∞. Since u = U k|Ω , we conclude that u ∈ Hps (Ω) with uHps ≤ infk U k Hps = infk Uk Hps = uHps . Similarly, uHps = uHps ≤ uHps . These spaces Hps (Ω) are called by different names. When the order is integral, Hpk (Ω) is called the Sobolev space. When p = 2, H2s (Ω) is also called the Sobolev space. In the meantime, when 1 < p < ∞, p = 2, Hps (Ω) is called the Lebesgue space. When p = 2, H2s (Ω) is often abbreviated by H s (Ω). The Sobolev–Lebesgue spaces consisting of real-valued functions are denoted by Hps (Ω; R), 1 < p < ∞, s ≥ 0.

11.5 Interpolation Property In this subsection, we present the interpolation property of the Sobolev–Lebesgue spaces. This property is extremely useful for deriving various properties for the Sobolev–Lebesgue spaces. According to [Tri78, Theorems 2.4.2/1, 2.10.1 and 4.3.1/2], we have the following theorem. Theorem 1.35 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. Let 1 < p < ∞ and 0 ≤ s0 < s1 < ∞. Then s (1.71) Hp0 (Ω), Hps1 (Ω) θ = Hps (Ω) with norm equivalence, where 0 ≤ θ ≤ 1 and s = (1 − θ )s0 + θ s1 . Let us use here this theorem to verify the boundedness of the extension operator E introduced above from Hps (Ω) to Hps (Rn ). Let s > 0 be nonintegral, and let Ω = Rn . Let k = [s] + 1 > s be an integer. Since E is a bounded operator from Lp (Ω) to Lp (Rn ) and from Hpk (Ω) to Hpk (Rn ) simultaneously, the interpolation theorem (Theorem 1.15) is applied to E to conclude that E is a bounded operator from Hps (Ω) to Hps (Rn ). In other words, if u ∈ Hps (Ω), then Eu ∈ Hps (Rn ) with the estimate EuHps (Rn ) ≤ Ap,s uHps (Ω) . We thus observe the following estimates: A−1 p,s EuHps (Rn ) ≤ uHps (Ω) ≤ EuHps (Rn ) ,

u ∈ Hps (Ω).

(1.72)

44

1 Preliminaries

As an immediate consequence, we get that a function u ∈ Lp (Ω) belongs to Hps (Ω) if and only if its extension Eu is in Hps (Rn ). Estimate (1.72) shows also that the norm of Hps (Ω) is equivalent to the graph norm E · Hps (Rn ) of E. When p = 2, since H2s (Rn ) is a Hilbert space, this fact yields that H s (Ω) = H2s (Ω) also enjoys a Hilbert structure with the inner product (u, v)H s (Ω) = (Eu, Ev)H s (Rn ) ,

u, v ∈ H s (Ω).

(1.73)

11.6 Embedding Theorems According to [Tri78, Theorem 2.8.1/Remark 2 and Theorem 4.6.1], we get the following embedding theorem. Theorem 1.36 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. Let 1 < p < ∞ and 0 ≤ s < ∞. (1) If 0 ≤ s < pn , then Hps (Ω) ⊂ Lr (Ω) where p ≤ r ≤ (2) If s = pn , then

with continuous embedding,

(1.74)

Hpp (Ω) ⊂ Lr (Ω) with continuous embedding,

(1.75)

pn n−ps .

n

where r is any finite number such that p ≤ r < ∞. (3) If s > pn , then Hps (Ω) ⊂

C(Rn ) (resp. C(Rn+ )) when Ω = Rn (resp. Rn+ ), when Ω is bounded. C(Ω)

(1.76)

When Ω is bounded, the embedding is continuous. The following estimates are known as Gagliardo–Nirenberg’s inequality. When Ω = Rn , the proof is given in [Tan97, Theorems 3.3, 3.4, and 3.5]. Thanks to the extension theorem, Theorem 1.33, the proofs for other cases are immediately reduced to this case. Theorem 1.37 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary, and let 1 ≤ q ≤ p ≤ ∞. (1) If 1 ≤ p < n, then Hp1 (Ω) ∩ Lq (Ω) ⊂ Lr (Ω)

11 Sobolev–Lebesgue Spaces

45

with the estimate uLr ≤ Cp,q,r uaH 1 u1−a Lq , p

where q ≤ r ≤

np n−p ,

u ∈ Hp1 (Ω) ∩ Lq (Ω),

and a is given by 1 1 1 1−a =a − + . r p n q

(1.77)

(2) If p = n, then Hn1 (Ω) ∩ Lq (Ω) ⊂ Lr (Ω) with the estimate uLr ≤ Cp,q,r uaH 1 u1−a Lq , n

u ∈ Hn1 (Ω) ∩ Lq (Ω),

where r is any finite number such that q ≤ r < ∞, and a is given by 1 1−a = . r q

(1.78)

(3) If n < p ≤ ∞, then Hp1 (Ω) ∩ Lq (Ω) ⊂ Lr (Ω) with the estimate uLr ≤ Cp,q,r uaH 1 u1−a Lq , p

u ∈ Hp1 (Ω) ∩ Lq (Ω),

where q ≤ r ≤ ∞, and a is given by 1 1−a 1 1 =a − + . r p n q

(1.79)

We finally present the compact embedding theorem. The proof is given in [Tri78, Theorem 4.10.1]. Theorem 1.38 Let Ω be a bounded domain in Rn with Lipschitz boundary. For 1 < p < ∞ and s > 0, the embedding i : Hps (Ω) → Lp (Ω) is compact.

11.7 Traces We are concerned with defining the values of f (x) on the boundary ∂Ω for the functions f (x) in Ω. Let us first consider the case where Ω = Rn+ . If 1 < p < ∞ and s > pn , we see from (1.76) that H s (Rn+ ) ⊂ C(Rn+ ). Therefore, the trace operator γ : f → f|∂Rn+ is

46

1 Preliminaries

defined from H s (Rn+ ) to C(∂Rn+ ), where ∂Rn+ = {x = (x , 0); x ∈ Rn−1 }. According to [Tri78, Theorem 2.9.3], if s > p1 , the operator γ can be extended as a bounded operator from Hps (Rn+ ) to Lp (∂Rn+ ). We want to extend this result to bounded domains with Lipschitz boundaries. In [Tri78], only the case where the boundary ∂Ω is sufficiently smooth is treated (see [Tri78, Theorem 4.7.1]). In [Gri85], the main interest is only in the Sobolev spaces Wps (Ω). So, it may be necessary to write up the full proof for Lebesgue spaces. However, we are able to carry out the proof in a quite analogous way to the case of Wps (Ω). So, we will omit the proof of the following theorem. It is the same for the proofs of Theorems 1.40, 1.41, and 1.42 below. Theorem 1.39 Let Ω be Rn+ or a bounded domain with Lipschitz boundary. Let 1 < p < ∞. If s > p1 , then the trace γ : f → f|∂Ω is a bounded operator from Hps (Ω) to Lp (∂Ω). On the contrary, the following two theorems show that the similar trace is no longer meaningful for s such that 0 ≤ s ≤ p1 . Theorem 1.40 Let Ω be as above, and let 1 < p < ∞. For 0 ≤ s ≤ p1 , the space D(Ω) is dense in Hps (Ω). Theorem 1.41 Let Ω be as above, and let 1 < p < ∞. For 0 ≤ s < p1 , the zero extension operator f → f ∼ , where f ∼ = f in Ω and f ∼ = 0 in Rn − Ω, is a bounded operator from Hps (Ω) to Hps (Rn ). For

1 p

< s ≤ 1, we can verify the following fact.

Theorem 1.42 Let Ω be as above, and let 1 < p < ∞. For p1 < s ≤ 1, a function u ∈ Hps (Ω) belongs to the closure of D(Ω) in Hps (Ω) if and only if u|∂Ω = 0.

11.8 Spaces H˚ ps (Ω) and Hp−s (Ω) For 1 < p < ∞ and s ≥ 0, the space H˚ ps (Ω) is defined as the closure of the set D(Ω) in the space Hps (Ω). Therefore, u ∈ H˚ ps (Ω) if and only if there exists a sequence un ∈ D(Ω) such that un → u in Hps (Ω). When Ω = Rn , H˚ ps (Rn ) = Hps (Rn ) for all 0 ≤ s < ∞. When Ω is Rn+ or a bounded domain with Lipschitz boundary, Theorem 1.40 implies that H˚ ps (Ω) = Hps (Ω)

for all 0 ≤ s ≤

1 , p

(1.80)

11 Sobolev–Lebesgue Spaces

47

but H˚ ps (Ω) = Hps (Ω)

for any

1 < s < ∞. p

In addition, Theorem 1.42 implies that 1 H˚ ps (Ω) = u ∈ Hps (Ω); u|∂Ω = 0 for < s ≤ 1. p

(1.81)

When p = 2, the space H˚ 2s (Ω) is abbreviated by H˚ s (Ω). For s ≥ 0, the space Hp−s (Ω) is defined as the dual space of H˚ ps (Ω), where 1 < p < ∞, 1 + 1 = 1. Therefore, {H˚ s (Ω), Hp−s (Ω)} is an adjoint pair with p

p

p

duality product ·, ·H˚ s

×Hp−s p

on H˚ ps (Ω) × Hp−s (Ω). Since D(Ω) ⊂ H˚ ps (Ω) with

dense embedding, we have Hp−s (Ω) = H˚ ps (Ω) ⊂ D(Ω) , namely, ϕ ∈ Hp−s (Ω) is characterized as a distribution in Ω which is continuous in D(Ω) with respect to the Hps (Ω) topology. As an immediate consequence, Lp (Ω) ⊂ Hp−s (Ω) with the relation u, f H˚ s

p

×Hp−s

= u, f Lp ×Lp ,

u ∈ H˚ ps (Ω), f ∈ Lp (Ω).

(1.82)

When p = p = 2, H2−s (Ω) is abbreviated as H −s (Ω). We easily verify that the three spaces H˚ s (Ω) ⊂ L2 (Ω) ⊂ H −s (Ω),

0 < s < ∞,

(1.83)

form a triplet. When Ω = Rn , Hp−s (Rn ) is given by − s Hp−s (Rn ) = f ∈ S(Rn ) ; F−1 1 + |ξ |2 2 Ff ∈ Lp (Rn ) . According to [Tri78, Theorem 2.6.1], for any −∞ < s < ∞, s n 1 1 −s n Hp (R ) = Hp (R ) =1 . + p p This in particular shows that, for any s ≥ 0 and 1 < p < ∞, Hps (Rn ) is a reflexive Banach space. Estimate (1.72) then yields that Hps (Ω) is also a reflexive Banach space. In fact, (1.72) shows that Hps (Ω) is isomorphic to the subspace E(Hps (Ω)) of H s (Rn ). But it is clear that the image E(Hps (Ω)) is a closed subspace of H s (Rn ). It is then known that any closed subspace of a reflexive Banach space is reflexive. Therefore, Hps (Ω) must be reflexive. Furthermore, by the similar reason, H˚ ps (Ω) is also reflexive. We remember that H s (Ω), s ≥ 0, is a Hilbert space with the inner product given by (1.73). It is the same for H˚ s (Ω). We can finally show the following fundamental fact. Since the proof is quite analogous to the case of the Sobolev spaces Wps (Ω) (cf. [Gri85, Theorem 1.4.4.6]), we will omit the proof.

48

1 Preliminaries

Theorem 1.43 Let Ω be Rn+ or a bounded domain with Lipschitz boundary. Let 1 < p < ∞. For any −∞ < s < ∞, s = p1 , the partial derivation Di (i = 1, . . . , n) is a bounded operator from Hps (Ω) to Hps−1 (Ω).

11.9 Product Spaces Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. For 1 ≤ p ≤ ∞, the product space Lp (Ω) is defined by ⎧ ⎫ ⎛ ⎞ f1 ⎪ ⎪ ⎨ ⎬ ⎜ .. ⎟ Lp (Ω) = F = ⎝ . ⎠ ; fj ∈ Lp (Ω) for j = 1, . . . , ⎪ ⎪ ⎩ ⎭ f

(1.84)

p 1 and is equipped with the product norm F Lp = ( j =1 fj Lp ) p if 1 ≤ p < ∞ and F L∞ = max{f1 L∞ , . . . , f L∞ } if p = ∞. Similarly, for 1 < p < ∞ and s ≥ 0, the product space Hsp (Ω) is defined by ⎧ ⎫ ⎛ ⎞ u1 ⎪ ⎪ ⎨ ⎬ ⎜ ⎟ Hsp (Ω) = U = ⎝ ... ⎠ ; uj ∈ Hps (Ω) for j = 1, . . . , ⎪ ⎪ ⎩ ⎭ v

(1.85)

1 p and is equipped with the product norm U Hsp = ( j =1 uj H s ) p . p All the results concerning Lp (Ω) and Hps (Ω) can be naturally generalized to Lp (Ω) and Hsp (Ω). For example, when p = 2, L2 (Ω) and Hs (Ω) = Hs2 (Ω) are Hilbert spaces. We shall also use the notation t (f1 , . . . , f ) (resp. t (u1 , . . . , u )), in order to denote the vectors F ∈ Lp (Ω) (resp. U ∈ Hsp (Ω)). Here, t (· · · ) means the transposition of matrices.

11.10 Miscellaneous Results Throughout this subsection, Ω denotes a bounded domain with Lipschitz boundary. (1) For 1 ≤ p, q, r ≤ ∞ such that

1 p

+

1 q

uvLr ≤ uLp vLq ,

= 1r , it holds that u ∈ Lp (Ω), v ∈ Lq (Ω).

(1.86)

Indeed, this estimate is immediately observed by using the usual Hölder inequality for p1 + q1 = 1, where p = pr and q = qr .

11 Sobolev–Lebesgue Spaces

49

(2) Let k be a positive integer. If a ∈ Ck (Ω), then the multiplication u → au by a is a bounded operator on Hps (Ω) for any 1 < p < ∞ and any 0 ≤ s ≤ k with the estimate s

1− s

auHps ≤ Ck aCk k aC k uHps ,

u ∈ Hps (Ω).

(1.87)

It is clear that auLp ≤ aC uLp and that auHpk ≤ Ck aCk uHpk . Then, the estimate is immediately verified by applying the interpolation theorem and property, namely, Theorems 1.15 and 1.35. (3) Let n < p < ∞. Then, u, v ∈ Hp1 (Ω) implies uv ∈ Hp1 (Ω) with the estimate uvHp1 ≤ Cp uHp1 vHp1 ,

u, v ∈ Hp1 (Ω).

(1.88)

Indeed, due to (1.76), Hp1 (Ω) ⊂ L∞ (Ω). Then, the estimate is verified directly from ∇(uv) = v∇u + u∇v. (4) Let s ≥ 1 and s > n2 . Then, similarly, u, v ∈ H s (Ω) implies uv ∈ H s (Ω) with the estimate uvH s ≤ Cs uH s vH s ,

u, v ∈ H s (Ω).

(1.89)

In order to show this result, let us prove first that for any a ∈ H s (Ω), the multiplication u → au by a is a bounded operator from H s−1 (Ω) into itself and the estimate auH s−1 ≤ Cs aH s uH s−1 ,

u ∈ H s−1 (Ω)

(1.90)

holds. Indeed, let s = k + σ with an integer k ≥ 1 and 0 ≤ σ < 1. Then, u → au is a bounded operator from H k−1 (Ω) into itself, for we have D α (av) =

Cα ,α (D α a)(D α u),

|α |+|α |=k−1

where α denotes any multiindex such that |α| = k − 1. Here, due to (1.74), (1.75), and (1.76), we observe that (D α a)(D α u) ∈ L2 (Ω); hence, au ∈ H k−1 (Ω) with auH k−1 ≤ Ck−1 aH s uH k−1 . In a similar way, we can observe that u → au is a bounded operator from H k (Ω) into itself with auH k ≤ Ck aH s uH k . Then, the interpolation theorem and property (Theorems 1.15 and 1.35) provide that u → au is a bounded operator from H k−1+σ (Ω) into itself with auH k−1+σ ≤ Ck−1+σ aH s uH k−1−σ , namely, (1.90) is proved. Now, the proof of (1.89) is immediate. Let u, v ∈ H s (Ω); obviously, ∇(uv) = v∇u + u∇v and ∇u, ∇v ∈ H s−1 (Ω); (1.90) then ensures ∇(uv) ∈ H s−1 (Ω); consequently, for any multiindex α such that |α| = k, it holds that D α (uv) ∈ H σ (Ω). Therefore, on account of (1.70), we conclude that uv ∈ H k+σ (Ω). At the same time, estimate (1.90) is also verified.

50

1 Preliminaries

(5) Let F : R → R be a C1 function. Let n < p < ∞. Then, the correspondence u → F (u) is an operator from Hp1 (Ω; R) into itself. Moreover, there is an increasing function p(·) determined from F such that F (u)Hp1 ≤ p(uHp1 ),

u ∈ Hp1 (Ω; R).

(1.91)

We have ∇F (u) = F (u)∇u. So, (1.91) is verified directly from (1.76). Let, more strongly, F : R → R be a C2 function, and n < p < ∞. Then, the operator u → F (u) from Hp1 (Ω; R) into itself satisfies the Lipschitz condition F (u) − F (v)Hp1 ≤ sup p1 (θ u + (1 − θ )vHp1 )u − vHp1 , 0≤θ≤1

u, v ∈ Hp1 (Ω; R),

(1.92)

where p1 (·) is an increasing function determined from F . Indeed, since 1 F (u) − F (v) = 0 F (θ u + (1 − θ )v) dθ (u − v), this estimate is verified by (1.88) and (1.91). (6) Let s ≥ 1 and s > n2 . Let F : R → R be a Cm function, where m is an integer such that m ≥ s. Then, the correspondence u → F (u) is an operator from H s (Ω; R) into itself. Moreover, there is an increasing function p(·) determined from F such that F (u)H s ≤ p(uH s ),

u ∈ H s (Ω; R).

(1.93)

Let s = k + σ , where 0 ≤ σ < 1, and k ≥ 1 is an integer. For any u ∈ H s (Ω; R), consider the linear operator Tu w = F (u)∇w. Due to (1.74), (1.75), and (1.76), we can observe that Tu is a bounded operator from H k (Ω) into H k−1 (Ω) with Tu wH k−1 ≤ p(uH s )wH k and is a bounded operator from H k+1 (Ω) into H k (Ω) with Tu wH k ≤ p(uH s )uH k+1 . The interpolation theorem and property (Theorems 1.15 and 1.35) then provide that Tu is a bounded operator from H k+σ (Ω) into H k−1+σ (Ω) with Tu wH k−1+σ ≤ p(uH s )wH k−1+σ . We now apply Tu to u itself to verify that ∇F (u) = F (u)∇u = Tu u ∈ H k−1+σ (Ω). Then, for any multiindex α such that |α| = k, we have D α F (u) ∈ H σ (Ω). Hence, on account of (1.70), we conclude that F (u) ∈ H k+σ (Ω). Estimate (1.93) is also verified. Let s ≥ 1 and s > n2 . Let, more strongly, F : R → R be a Cm+1 function with m ≥ s. Then, the operator u → F (u) from H s (Ω; R) into itself satisfies the Lipschitz condition F (u) − F (v)H s ≤ sup p1 (θ u + (1 − θ )vH s )u − vH s , 0≤θ≤1

u, v ∈ H s (Ω; R),

(1.94)

where p1 (·) is an increasing function determined from F . This estimate is verified in an analogous way for (1.92) by using (1.89) and (1.93).

11 Sobolev–Lebesgue Spaces

51

(7) Let F : R → R be a piecewise smooth function with F ∈ L∞ (R) and with F (0) = 0 (hence, |F (u)| ≤ F L∞ |u| for u ∈ R). Then, according to [GT83, Theorem 7.8], the correspondence u → F (u) is an operator from Hp1 (Ω; R) into itself for 1 ≤ p ≤ ∞ with the estimate F (u)Hp1 ≤ CF L∞ uHp1 ,

u ∈ Hp1 (Ω; R).

(1.95)

i = 1, . . . , n,

(1.96)

In addition, F (u) has the property / Fs , F (u)Di u if u(x) ∈ Di F (u) = 0 if u(x) ∈ Fs ,

where Fs denotes the set of singular points of F (u). (8) The correspondence f → ef is an operator from L∞ (Ω) into itself. The operator is Fréchet differentiable in L∞ (Ω). Its Fréchet differential at f ∈ L∞ (Ω) is the multiplication operator by the function ef , i.e., [Def ]h = ef h,

f, h ∈ L∞ (Ω).

(1.97)

This fact is easily verified by a direct calculation. (9) For 1 ≤ q < ∞, let us introduce the quantity Nq,log (u) = |u|q log(|u| + 1) dx. Ω

All the results announced in Theorem 1.37 can be modified in the following form. For any ζ > 0, estimates in (1), (2), and (3) can be rewritten as q uLr ≤ Cp,q,r ζ 1−a uaH 1 Nq,log (u)1−a + Cζ uLr q , p

u ∈ Hp1 (Ω; R) ∩ Lq (Ω),

(1.98)

with some constant Cζ > 0 dependent on ζ . The relations among p, q, and r np ; if p = n, then are the same as before, that is, if 1 ≤ p < n, then q ≤ r ≤ n−p q ≤ r < ∞; and if n < p ≤ ∞, then q ≤ r ≤ ∞. The exponent a is also given in each case as before, that is, by (1.77), (1.78), and (1.79), respectively. To prove the estimate, we use a cutoff function χR (u) such that χR (u) = −R for u ≤ −R, χR (u) = u for |u| ≤ R, and χR (u) = R for R ≤ u. Then, by Theorem 1.37 and (1.95), we observe in each case that u − χR (u)Lr ≤ Cp,q,r u − χR (u)aH 1 u − χR (u)1−a Lq p

1−a

≤

Cp,q,r uaH 1 p

|u| dx

≤

Cp,q,r uaH 1 Nq,log (u)1−a . p [log(R + 1)]1−a

q

|u|≥R

52

1 Preliminaries

Meanwhile, q

q

χR (u)Lr ≤ R 1− r uLr q . These two estimates readily yield the desired estimate (1.98). (10) Let H (u) be a C1,1 function defined by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. For any function u ∈ C1 ((a, b); L2 (Ω; R)), if we put ψ(t) = H (u(t)) dx, a < t < b, (1.99) Ω

then ψ

∈ C1 (a, b)

with the derivative H (u(t))u (t) dx, ψ (t) =

(1.100)

a < t < b.

Ω

In fact, we notice that, since H : R → R is Lipschitz continuous, the mapping u → H (u) is also Lipschitz continuous from L2 (Ω; R) into itself. From 1 H (θ u(t + h, x) + (1 − θ )u(t, x)) dθ H (u(t + h, x)) − H (u(t, x)) = 0

× [u(t + h, x) − u(t, x)],

a.e. x ∈ Ω,

we have ψ(t + h) − ψ(t) =

1

H (θ u(t + h) + (1 − θ )u(t)) dθ, u(t + h) − u(t) .

0

L2

Dividing this equality by h and letting h → 0, we obtain (1.100). (11) As a variation of (1.100), for any u ∈ C((a, b); H 1 (Ω; R)) ∩ C1 ((a, b); H 1 (Ω)∗ ), the function ψ(t) defined similarly by (1.99) is in C1 (a, b) with the derivative ψ (t) = H (u(t)), u (t)H 1 ×H 1∗ ,

(1.101)

a < t < b.

In fact, due to (1.95), H (u) ∈ H 1 (Ω; R) and H (u)H 1 ≤ uH 1 for any u ∈ H 1 (Ω; R). Furthermore, as v → 0 in H 1 (Ω; R), H (u + v)H 1 ≤ u + vH 1 and H (u + v) → H (u) in L2 (Ω; R); therefore, H (u + v) → H (u) in the weak topology of H 1 (Ω). Hence, u → H (u) is continuous in H 1 (Ω; R) from strong topology to weak topology. In the meantime, we have ψ(t + h) − ψ(t) 1 = H (θ u(t + h) + (1 − θ )u(t)) dθ, u(t + h) − u(t) (

0

= 0

) 1 H (θ u(t + h) + (1 − θ )u(t)) dθ, u(t + h) − u(t)

L2

. H 1 ×H 1∗

Notes and Further Researches

53

Hence, (1.101) is obtained. The continuity of ψ (t) is also verified by the continuity of the mapping u → H (u) from strong topology to weak topology. (12) Consider the Cauchy problem du dt + p(t)u = q(t), 0 < t ≤ T , (1.102) u(0) = u0 , in the space L∞ (Ω). Here, p and q are given functions such that p ∈ C((0, T ]; L∞ (Ω)) ∩ B((0, T ]; L∞ (Ω)), t 1−ν q ∈ B((0, T ]; L∞ (Ω)), q ∈ C((0, T ]; L∞ (Ω)), with some exponent 0 < ν ≤ 1. Then, for any initial value u0 ∈ L∞ (Ω), (1.102) has a unique solution u in the function space C([0, T ]; L∞ (Ω)) ∩ C1 ((0, T ]; L∞ (Ω)), which is given by the formula t t t e− s p(τ )dτ q(s) ds, u(t) = e− 0 p(s)ds u0 +

0 ≤ t ≤ T.

(1.103)

(1.104)

0

Indeed, by (1.97), the operator g → eg from L∞ (Ω) into itself is Fréchet differentiable with the derivative [Deg ]h = eg h. Therefore, we verify that the function u given by (1.104) certainly belongs to (1.103) and satisfies the equation in (1.102). Conversely, let u be any solution of (1.102). Then, for 0 < s ≤ t , we have t t d − t p(τ )dτ u(s) = e− s p(τ )dτ p(s)u(s) + e− s p(τ )dτ u (s) e s ds

= e−

t s

p(τ )dτ

q(s).

Integrating this equality in [0, t], we obtain (1.104).

Notes and Further Researches For the basic matters of the theory of functional analysis, we quote some of the classics, e.g., Hille–Phillips [HP57], Yosida [Yos80], and Brezis [Bre83]. For the basic matters of fractal dimension for the compact sets, see Eden–Foias– Nicolaenko–Temam [EFNT94]. Especially, for Mañé–Hölder-type projection, see [EFNT94, Chap. 10] and also Ben Artzi–Eden–Foias–Nicolaenko [BEFN93], Foias–Olson [FO96], and Hunt–Kaloshin [HY99]. Theorem 1.2 was obtained in [HY99]. For Dunford integrals, see Dunford–Schwartz [DS58, DS63]. Basics of multivalued linear operators are given in Favini–Yagi [FY99, Chap. I]. Basics of

54

1 Preliminaries

nonlinear operators are given in Deimling [Dei85] and Zeidler [Zei86]. Basics of interpolation spaces are given in Bergh–Löfström [BL76] and Tribel [Tri78, Chap. 1]. The triplets of spaces were presented and systematically used in Lions–Magenes [LM72], cf. also Dautray–Lions [DL88]. In these literatures, the triplet was defined by identifying X and its dual X , i.e., Z ⊂ X = X ⊂ Z . Such an identification is possible, since Z ⊂ X implies X ⊂ Z with dense and continuous embedding and since X and X are canonically isomorphic by the Riesz Theorem. By the uniqueness of triplet Z ⊂ X ⊂ Z ∗ , such a definition is of course essentially equivalent to ours presented in Sect. 7. For Sobolev–Lebesgue spaces, we quote Adams [Ada75], Grisvard [Gri85, Chap. 1], and Tribel [Tri78, Chap. 4]. The fundamental extension theorem, Theorem 1.33, was presented by Stein [Ste70].

Chapter 2

Sectorial Operators

Let X be a Banach space with norm · . Let A be a densely defined, closed linear operator in X. We assume that the spectrum of A is contained in an open sectorial domain such that σ (A) ⊂ Σω = {λ ∈ C; |arg λ| < ω},

0 < ω ≤ π,

(2.1)

and its resolvent satisfies the estimate (λ − A)−1 ≤ M , |λ|

λ∈ / Σω ,

(2.2)

with some constant M ≥ 1. We call such an operator A a sectorial operator of X. The condition (2.1) implicitly means that the origin is not in σ (A), namely, A has a bounded inverse A−1 on X. By (1.11) it then follows that λ ∈ ρ(A) if |λ| < A−1 −1 with the estimate (λ − A)−1 ≤

A−1 , 1 − A−1 |λ|

|λ| < A−1 −1 .

(2.3)

By the similar reason, for λ0 = r0 e±iω , r0 > 0, we see that r0 ⊂ ρ(A) λ ∈ C; |λ − λ0 | < M with the estimate (λ − A)−1 ≤

M , r0 − M|λ − λ0 |

|λ − λ0 |

δ > 0. Let A be the linear operator associated with the form, and let A|X and A|Z be its parts in the spaces X and Z, respectively. As already shown in Chap. 1, Sect. 8, A, A|X , and A|Z are densely defined, closed linear operators acting in Z ∗ , X, and Z, respectively. For each Re λ ≤ 0, consider the sesquilinear form a(U, V ) − λ(U, V ),

U, V ∈ Z.

Obviously this form is also continuous and coercive on Z. Therefore, Theorem 1.24 shows that the associated operator, which is clearly nothing more than A − λ, is an isomorphism from Z onto Z ∗ . In particular, λ ∈ ρ(A) if A is considered as an operator of Z ∗ . Let us next establish various estimates concerning the resolvent (λ − A)−1 for Re λ ≤ 0. Let U ∈ Z. Then, δU 2 ≤ Re a(U, U ) − Re λ|U |2 = Re (A − λ)U, U ≤ (A − λ)U ∗ U . Putting Φ = (λ − A)U , we obtain that (λ − A)−1 Φ ≤ δ −1 Φ∗ .

(2.8)

Since λ(λ − A)−1 Φ = A(λ − A)−1 Φ + Φ, the estimate then yields that |λ|(λ − A)−1 Φ ∗ ≤ AL(Z,Z ∗ ) (λ − A)−1 Φ + Φ∗ ≤ (Mδ −1 + 1)Φ∗ . Consequently, |λ|(λ − A)−1 Φ ∗ ≤ (Mδ −1 + 1)Φ∗ ,

Φ ∈ Z∗.

Let now U ∈ D(A|X ). For Re λ ≤ 0, δU 2 ≤ Re a(U, U ) − Re λ|U |2 = Re((A − λ)U, U ) ≤ |(A − λ)U ||U |. From λ|U |2 = a(U, U ) + ((λ − A)U, U ) it follows that |λ||U |2 ≤ MU 2 + |(λ − A)U | |U | ≤ (Mδ −1 + 1)|(λ − A)U | |U |. Putting F = (λ − A)U , we obtain that |λ|(λ − A)−1 F ≤ (Mδ −1 + 1)|F |,

F ∈ X.

58

2 Sectorial Operators

Let again U ∈ Z. From λ(λ − A)−1 U − (λ − A)−1 AU = U it follows that |λ|(λ − A)−1 U ≤ (λ − A)−1 L(Z ∗ ,Z) AL(Z,Z ∗ ) U + U . Therefore, by (2.8), |λ|(λ − A)−1 U ≤ (Mδ −1 + 1)U ,

U ∈ Z.

We have therefore shown for Re λ ≤ 0 the estimates |λ|(λ − A)−1 Φ ∗ ≤ (Mδ −1 + 1)Φ∗ , Φ ∈ Z ∗ , |λ|(λ − A)−1 F ≤ (Mδ −1 + 1)|F |, F ∈ X, |λ|(λ − A)−1 U ≤ (Mδ −1 + 1)U , U ∈ Z.

(2.9) (2.10) (2.11)

These estimates mean that all the operators A, A|X , and A|Z satisfy conditions (2.1) and (2.2) with angle ω = π2 and constant Mδ −1 + 1 in the spaces Z ∗ , X, and Z, respectively. In this way, we have proved that they are sectorial operators. Theorem 2.1 Let a(U, V ) be a sesquilinear form on Z satisfying (2.6) and (2.7), and let A, A|X , and A|Z be the linear operators of Z ∗ , X, and Z, respectively, which are determined from the form. Then, they satisfy (2.1) and (2.2) with angle ∗ ω = π2 and constant M+δ δ . Hence, they are sectorial operators of Z , X, and Z, π respectively, with angles < 2 . Remark 2.1 For λ ≤ 0 and U ∈ D(A|X ), we have δU 2 + |λ||U |2 ≤ Re a(U, U ) − λ|U |2 = Re((A − λ)U, U ) ≤ |(λ − A)U | |U |, (c−2 δ + |λ|)|U |2 ≤ |(λ − A)U ||U |, where c is the constant of embedding: |U | ≤ cU , U ∈ Z. Putting F = (λ − A)U , we obtain that (|λ| + c−2 δ)(λ − A)−1 F ≤ |F |. Hence, A|X satisfies the estimate (λ − A|X )−1 ≤ 1/(|λ| + c−2 δ),

λ ≤ 0.

(2.12)

1.2 Sectorial Operators in L2 Spaces Let us utilize Theorem 2.1 for constructing various sectorial operators in L2 spaces.

1 Sectorial Operators in Hilbert Spaces

59

Consider the sesquilinear form a(u, v) =

n n i,j =1 R

aij (x)Di uDj v dx +

Rn

c(x)uv dx,

u, v ∈ H 1 (Rn ), (2.13)

defined on H 1 (Rn ). Here, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Rn satisfying the conditions aij ∈ L∞ (Rn ), n

1 ≤ i, j ≤ n,

aij (x)ξi ξj ≥ δ|ξ |2 ,

(2.14)

ξ = (ξ1 , . . . , ξn ) ∈ Rn , a.e. x ∈ Rn ,

(2.15)

i,j =1

with some constant δ > 0, and c(x) is a real-valued function satisfying c ∈ L∞ (Rn )

and c(x) ≥ c0 > 0,

Clearly, a(u, v) fulfills (2.6) with M = max{ tion, (2.15) implies that Re a(u, u) =

n n i,j =1 R

i,j

a.e. x ∈ Rn .

(2.16)

ai,j L∞ , cL∞ }. In addi-

aij (x)[(Re Di u)(Re Dj u) + (Im Di u)(Im Dj u)] dx

+

Rn

c(x)|u|2 dx ≥ δ∇u2L2 + c0 u2L2 .

Therefore, a(u, v) fulfills (2.7), too. We consider a triplet H 1 (Rn ) ⊂ L2 (Rn ) ⊂ H 1 (Rn )∗ = H −1 (Rn ) and an operator A associated with (2.13). Since

n n aij (x)Di uDj v dx = − Dj [aij (x)Di u], v n i,j =1 R

i,j =1

H −1 ×H 1

in the distribution sense, it naturally holds that Au = −

n

Dj [aij (x)Di u] + c(x)u

in H −1 (Rn ).

i,j =1

Theorem 2.1 then yields the following results. Theorem 2.2 Let (2.14), (2.15), and (2.16) be satisfied. Then, the operator A associated with the form (2.13) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H −1 (Rn ), L2 (Rn ), and H 1 (Rn ), respectively, with angles < π2 .

60

2 Sectorial Operators

Let now Ω be any domain in Rn , and let Z be any closed subspace of H 1 (Ω) such that H˚ 1 (Ω) ⊂ Z ⊂ H 1 (Ω). We consider the sesquilinear form n

a(u, v) =

aij (x)Di uDj v dx +

i,j =1 Ω

c(x)uv dx,

u, v ∈ Z,

(2.17)

Ω

on Z. As before, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Ω satisfying the conditions aij ∈ L∞ (Ω), n

1 ≤ i, j ≤ n,

aij (x)ξi ξj ≥ δ|ξ |2 ,

(2.18)

ξ = (ξ1 , . . . , ξn ) ∈ Rn , a.e. x ∈ Ω,

(2.19)

i,j =1

with some constant δ > 0, and c(x) is a real-valued function in Ω satisfying c ∈ L∞ (Ω)

and c(x) ≥ c0 > 0,

a.e. x ∈ Ω.

(2.20)

By the same reason as above, these conditions readily imply that a(u, v) fulfills (2.6) and (2.7) on Z. So, if we consider a triplet Z ⊂ L2 (Ω) ⊂ Z ∗ (note that, due to D(Ω) ⊂ H˚ 1 (Ω) ⊂ Z, Z is always dense in L2 (Ω)), then the operator A associated with the form (2.17) and its parts are all sectorial operators of Z ∗ , L2 (Ω), and Z, respectively, with angles < π2 . Let us investigate below more specific cases. Case where Z = H˚ 1 (Ω). In this case, Z ∗ coincides with H −1 (Ω). Since the space D(Ω) is dense in H˚ 1 (Ω), H −1 (Ω) is contained in the space of distributions D(Ω) . Therefore, it is possible to represent Au as Au = −

n

Dj [aij (x)Di u] + c(x)u

in distribution sense.

(2.21)

i,j =1

This shows us that A and its parts are regarded as realizations of the differential operator (2.21) in H −1 (Ω), L2 (Ω), and H˚ 1 (Ω), respectively. We also know that, if Ω is a bounded domain with Lipschitz boundary, then D(A) = H˚ 1 (Ω) is given by (1.81), that is, u ∈ D(A) satisfies the Dirichlet boundary conditions γ u = 0 on ∂Ω.

(2.22)

Theorem 2.3 Let Ω be any domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied. Then, the operator A associated with the form (2.17) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H −1 (Ω), L2 (Ω), and H˚ 1 (Ω), respectively, with angles < π2 . In addition, if Ω is a bounded domain with Lipschitz boundary, then the functions in D(A) satisfy (2.22).

1 Sectorial Operators in Hilbert Spaces

61

Case where Z = H 1 (Ω). In this case, Z ∗ coincides no longer with any subspace of distributions. So, the associated operator A is not represented as a usual differential operator. But, let u ∈ D(A|L2 ), i.e., Au ∈ L2 (Ω), and assume that all are regular enough to justify the following arguments. Then, by Green’s formula, (Au, v)L2 = a(u, v) = −

+

n

Dj [aij (x)Di u]v dx

i,j =1 Ω

n

[νj (x)aij (x)Di u]v dS +

i,j =1 ∂Ω

c(x)uv dx,

v ∈ H 1 (Ω),

Ω

where ν(x) = (ν1 (x), . . . , νn (x)) denotes the outer normal vector at x ∈ ∂Ω. Since a(u, v) must be continuous in v with respect to the L2 (Ω) topology, the integral on ∂Ω must vanish. In other words, u must satisfy the boundary conditions n ∂u ≡ νj (x)aij (x)Di u = 0 ∂νA

on ∂Ω.

(2.23)

i,j =1

Such conditions are called the Neumann-type boundary conditions on ∂Ω. We then arrive at n Dj [aij (x)Di u] + c(x)u, v , v ∈ H 1 (Ω); (Au, v)L2 = − i,j =1

L2

therefore, Au = −

n

Dj [aij (x)Di u] + c(x)u

in L2 (Ω).

(2.24)

i,j =1

In this way, u satisfies the boundary conditions (2.23), and Au is given by (2.24). We also notice that, if Au = ϕ with u ∈ H 1 (Ω) and ϕ ∈ H 1 (Ω)∗ , then there exist sequences uk ∈ D(A|L2 ) and fk ∈ L2 (Ω) such that A|L2 uk = fk and that uk → u in H 1 (Ω) and fk → ϕ in H 1 (Ω)∗ . Indeed, it is sufficient to choose any sequence fk → ϕ in H 1 (Ω)∗ with fk ∈ L2 (Ω) and to put uk = (A|L2 )−1 fk . In this sense, A and its parts are regarded as realizations of the differential operator (2.21) under the Neumann-type boundary conditions (2.23) in H 1 (Ω)∗ , L2 (Ω), and H 1 (Ω), respectively. Theorem 2.4 Let Ω be any domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied. Then, the operator A associated with the form (2.17) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H 1 (Ω)∗ , L2 (Ω), and H 1 (Ω), respectively, with angles < π2 . Case where Z = H˚ D1 (Ω). Let Ω be a bounded domain with Lipschitz boundary. We now consider a splitting of the boundary ∂Ω into ΓD and ΓN , namely, ∂Ω =

62

2 Sectorial Operators

ΓD ∪ ΓN , ΓD ∩ ΓN = ∅, and ΓD is a nonempty open set of ∂Ω. We set Z = H˚ D1 (Ω) = {u ∈ H 1 (Ω); γ u = 0 on ΓD }.

(2.25)

The operator A associated with the form (2.17) and its parts are regarded as realizations of the differential operator (2.21) under the splitting boundary conditions γu=0

on ΓD

and

∂u = 0 on ΓN . ∂νA

(2.26)

Theorem 2.5 Let Ω be a bounded domain with Lipschitz boundary. Let (2.18), (2.19), and (2.20) be satisfied. Then, the operator A associated with the form (2.17) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H˚ D1 (Ω)∗ , L2 (Ω), and H˚ D1 (Ω), respectively, with angles < π2 . Remark 2.2 When aij (x) ≡ δij (Kronecker’s delta), the differential operator n ∂u 2 Δ n= i=1 Di is called the Laplace operator and the boundary conditions ∂ν ≡ i=1 νi (x)Di u = 0 are called the Neumann boundary conditions. Following the procedure described above, we can define a realization of the Laplace operator in Ω under the Dirichlet, Neumann, or splitting boundary conditions on ∂Ω. Remark 2.3 Let Ω be a bounded domain with Lipschitz boundary. When Z = H˚ 1 (Ω) or H˚ D1 (Ω), it is possible to show the coercivity (2.7) without using the positivity of the function c(x). In fact, according to the Poincaré inequality (e.g., see [BF02, 1.1.3.b]), uL2 ≤ C∇uL2 for all u ∈ H˚ D1 (Ω), a fortiori for all u ∈ H˚ 1 (Ω). Therefore, (2.19), together with the condition c(x) ≥ 0, can imply (2.7).

1.3 Shift Property in L2 When condition (2.19) is satisfied, the differential operator A(D)u = −

n

Dj [aij (x)Di u] + c(x)u

(2.27)

i,j =1

is said to be strongly elliptic. In general, strongly elliptic operators are expected to enjoy the regularity property that A(D)u ∈ L2 (Ω) implies u ∈ H 2 (Ω). Such a property is called the shift property. However, our operators introduced in the preceding subsection cannot enjoy this property, because aij (x) are merely measurable functions. In order to have the shift property, we assume that Ω is a bounded domain in Rn with C2 boundary and that aij (x) satisfy aij ∈ C1 (Ω),

1 ≤ i, j ≤ n.

(2.28)

1 Sectorial Operators in Hilbert Spaces

63

The shift property is verified by the widely developed theory of elliptic operators. Let A(D) be the elliptic operator in Ω given by (2.27). Let the conditions (2.19), (2.20), and (2.28) be satisfied. We will first consider the case where A(D) is equipped with the Dirichlet conditions. According to the theory of elliptic operators (see [Gri85, Theorem 2.2.2.3 and Corollary 2.2.2.4]), the mapping u → (A(D) + k)u is an isomorphism from the space {u ∈ H 2 (Ω); γ u = 0 on ∂Ω} onto L2 (Ω) if k is sufficiently large. In addition, it holds that uH 2 ≤ C[A(D) + k]uL2 ,

u ∈ H 2 (Ω), γ u = 0.

Let then A be the operator associated with the form (2.17) on H˚ 1 (Ω), and let A|L2 be its part in L2 (Ω). If A|L2 u = f ∈ L2 (Ω), then there exists a unique function u˜ ∈ ˜ v) = (A(D)u, ˜ v) = H 2 (Ω) such that [A(D) + k]u˜ = f + ku and γ u˜ = 0; since a(u, (k(u − u) ˜ + f, v), v ∈ H˚ 1 (Ω), it follows that A|L2 u˜ = k(u − u) ˜ + f ; therefore, (A|L2 +k)(u− ˜ u) = 0, i.e., u˜ = u. In this way, u ∈ D(A|L2 ) implies that u ∈ H 2 (Ω). We have therefore verified that

D(A|L2 ) = {u ∈ H 2 (Ω); γ u = 0 on ∂Ω}, (2.29) A|L2 u = A(D)u, as well as uH 2 ≤ C(A|L2 uL2 + uL2 ),

u ∈ D(A|L2 ).

(2.30)

In what follows, the domain will often be denoted by HD2 (Ω) = {u ∈ H 2 (Ω); γ u = 0 on ∂Ω}.

(2.31)

We next consider the case where A(D) is equipped with the Neumann-type boundary conditions. Similarly, according to the theory of elliptic operators (see [Gri85, Theorem 2.2.2.5 and Corollary 2.2.2.6]), the mapping u → (A(D) + k)u is ∂u an isomorphism from the space {u ∈ H 2 (Ω); ∂ν = 0 on ∂Ω} onto L2 (Ω) if k is A sufficiently large. In addition, it holds that uH 2 ≤ C[A(D) + k]uL2 ,

u ∈ H 2 (Ω),

∂u = 0. ∂νA

Let A be the operator associated with the form (2.17) on H 1 (Ω). Let A|L2 be its part in L2 (Ω). If A|L2 u = f ∈ L2 (Ω), then there exists a unique function u˜ ∈ H 2 (Ω) ∂ u˜ = 0 on ∂Ω. As before, it is easy to see such that [A(D) + k]u˜ = f + ku and ∂ν A that u˜ = u. Therefore, u ∈ D(A|L2 ) implies that u ∈ H 2 (Ω). We thus conclude that

∂u D(A|L2 ) = {u ∈ H 2 (Ω); ∂ν = 0 on ∂Ω}, A (2.32) A|L2 u = A(D)u as well as uH 2 ≤ C(A|L2 uL2 + uL2 ),

u ∈ D(A|L2 ).

(2.33)

64

2 Sectorial Operators

In what follows, the domain will often be denoted by ∂u HN2 (Ω) = u ∈ H 2 (Ω); = 0 on ∂Ω . ∂νA

(2.34)

The following theorem has therefore been verified. Theorem 2.6 Let Ω be a bounded domain with C2 boundary, and let (2.19), (2.20), and (2.28) be satisfied. Then, the parts A|L2 of the operators associated with the form (2.17) on H˚ 1 (Ω) and H 1 (Ω) are characterized by (2.29) and (2.32), respectively. In the case where Ω is convex and where aij (x) are symmetric in the sense that aij (x) = aj i (x),

1 ≤ i, j ≤ n,

(2.35)

we can weaken the C2 regularity of ∂Ω to Lipschitz. Indeed, according to Grisvard [Gri85, Theorems 3.1.3.1 and 3.1.3.3], the constants C appearing in the energy estimates (2.30) and (2.33) depend (roughly speaking) only on the negative part of the curvature of ∂Ω and are independent of the positive part. On the basis of this fact, the same shift properties are established for bounded convex domains. Note that any bounded convex domain has a Lipschitz boundary. Theorem 2.7 Let Ω be a bounded convex domain, and let (2.19), (2.20), (2.28), and (2.35) be satisfied. Then, the parts A|L2 of the operators associated with the form (2.17) on H˚ 1 (Ω) and H 1 (Ω) are again characterized by (2.29) and (2.32), respectively. Estimates (2.30) and (2.33) also hold true. For the proofs, see [Gri85, Theorems 3.2.1.2 and 3.2.1.3]. Let us now consider a very particular case where aij (x) = a(x)δij but a(x) is less regular than (2.28). Our elliptic operator is then written as A(D)u = −

n

Di [a(x)Di u] + c(x)u.

(2.36)

i=1

We assume (2.19) for a(x)δij , which reads as a(x) ≥ δ > 0 in Ω. We assume also that a(x) belongs to either a ∈ Hp1 (Ω; R)

with n < p < ∞

(2.37)

n with s ≥ 1 and s > . 2

(2.38)

or a ∈ H s (Ω; R)

Note that, due to (1.76), Hp1 (Ω; R) ⊂ C(Ω) and H s (Ω; R) ⊂ C(Ω).

1 Sectorial Operators in Hilbert Spaces

65

Theorem 2.8 Let Ω be a C2 or convex, bounded domain in Rn . Let (2.19) and (2.20) be satisfied, and let a(x) belong to either (2.37) or (2.38). Let A(D) be the elliptic operator given by (2.36). Then, realizations A|L2 of A(D) in L2 (Ω) under the boundary conditions (2.22) and (2.23) (cf. Remark 2.2) have their domains given by (2.31) and (2.34), respectively. Estimates (2.30) and (2.33) also hold true. Proof Let us describe here the proof only for the Dirichlet boundary conditions. For the other case, it is quite similar. Let u ∈ D(A|L2 ). Then, Au = f ∈ L2 (Ω), and −

n

Di [a(x)Di u] + c(x)u = f

in distribution sense.

(2.39)

i=1

We rewrite −a(x)Δu −

n [Di a(x)]Di u + c(x)u = f i=1

and (−Δ + 1)u −

n

a(x)−1 [Di a(x)]Di u + a(x)−1 c(x) − 1 u = a(x)−1 f.

i=1

In view of this formula, let us introduce a realization Λ of elliptic operator −Δ + 1 under (2.22) in L2 (Ω). We know by Theorems 2.6 and 2.7 that Λ is a positive definite self-adjoint operator of L2 (Ω) with domain (2.31). In addition, by Theorems 16.12 and 16.13 in Chap. 16, we know also characterization of the domains of fractional powers Λθ for 0 < θ < 1, in particular D(Λθ ) ⊂ H 2θ (Ω). Keeping in mind this fact, we consider a realization B of the differential operator B(D)v = −

n

a(x)−1 [Di a(x)]Di v + a(x)−1 c(x) − 1 v

i=1 1+ n

in L2 (Ω) whose domain is given by D(B) = H p (Ω) (resp. D(B) = n H 2−s+ 2 (Ω)) if (2.37) (resp. (2.38)) is assumed. Since 1 + pn < 2 (resp. 2 − s + n 2 < 2), the operator B is dominated by Λ in the sense of (3.8). As a consequence, if β > 0 is a sufficiently large number, then −β ∈ ρ(Λ + B), see Theorem 3.3. We recall that Au = f implies that (Λ + B)u = a(x)−1 f . Then, let u˜ be the solution of (Λ + B + β)u˜ = βu + a(x)−1 f . As shown above, u˜ ∈ HD2 (Ω) with the estimate u ˜ H 2 ≤ Cβu + a(x)−1 f L2 . We can follow the calculations above inversely to obtain the equality −

n i=1

Di [a(x)Di u˜ ] + c(x)u˜ + βa(x)u˜ = βa(x)u + f

in L2 (Ω).

66

2 Sectorial Operators

Subtract (2.39) from this. Then, −

n

Di [a(x)Di (u˜ − u)] + c(x)(u˜ − u) + βa(x)(u˜ − u) = 0.

i=1

Hence, u˜ = u. In addition, uH 2 ≤ Cβu + a(x)−1 f L2 ≤ Cδ −1 f L2 .

We often need a higher shift property of strongly elliptic operators. For this, we assume that aij ∈ Cm+1 (Ω),

1 ≤ i, j ≤ n; m = 1, 2, 3, . . . .

(2.40)

Theorem 2.9 Let Ω be a bounded domain with Cm+2 boundary, and let (2.19), (2.20), and (2.40) be satisfied. Then, the parts A|L2 of the operators associated with the form (2.17) on H˚ 1 (Ω) and H 1 (Ω), respectively, have the property that A|L2 u ∈ H m (Ω) implies u ∈ H m+2 (Ω). The estimate uH m+2 ≤ C(A|L2 uH m + uH m ) also holds true. For the proof, see [Gri85, Theorem 2.5.1.1]. Let us consider finally the splitting boundary conditions. Let Ω be a bounded domain of Rn with Lipschitz boundary. Consider a splitting of ∂Ω, ∂Ω = ΓD ∪ ΓN and ΓD ∩ ΓN = ∅, ΓD being a nonempty open subset. We assume the two conditions |B(x0 ; R) ∩ ΓD | ≥ γ R n−1 |B(x0 ; R) ∩ Ω| ≥ γ R

n

for any x0 ∈ ΓD ,

for any x0 ∈ ΓN , B(x0 ; R) ∩ ΓD = ∅

(2.41) (2.42)

with some constant γ > 0 (cf. [BF02, (1.22)]), where B(x0 ; R) denotes the open ball with center x0 ∈ ∂Ω and radius R > 0. Theorem 2.10 Let Ω be a bounded domain with Lipschitz boundary which is split into two parts ΓD and ΓN as above with conditions (2.41) and (2.42). Let (2.18), (2.19), and (2.20) be satisfied. Then, the part A|L2 of the operator associated with the form (2.17) on H˚ D1 (Ω) satisfies D(A|L2 ) ⊂ Wp1 (Ω) and uWp1 ≤ C(A|L2 uL2 + uL2 ),

u ∈ D(A|L2 ),

(2.43)

for a certain exponent p > 2. This property was proved by Bensoussan–Frehse [BF02, Theorem 2.2].

1.4 Sectorial Operators in Product Spaces of L2 Let Ω be a bounded domain in Rn . We introduce two product Hilbert spaces L2 (Ω) and H1 (Ω) with = 2 (see (1.84) and (1.85)). Consider the triplet of spaces

1 Sectorial Operators in Hilbert Spaces

67

H1 (Ω) ⊂ L2 (Ω) ⊂ H1 (Ω)∗ . By definition, it is easy to see that H1 (Ω)∗ is also given by the product space ϕ H1 (Ω)∗ = ; ϕ ∈ H 1 (Ω)∗ and ψ ∈ H 1 (Ω)∗ (2.44) ψ with duality product ϕ u , = ϕ, u + ψ, v , ψ v

ϕ u 1 ∗ ∈ H (Ω) , ∈ H1 (Ω). ψ v

Consider the sesquilinear form ˜ a(U, U ) = [a(x)∇u · ∇ u˜ + b(x)∇v · ∇ u˜ + c(x)∇u · ∇ v˜ Ω

˜ dx + k + d(x)∇v · ∇ v] u u˜ ˜ U= ,U= ∈ H1 (Ω), v v˜

(uu˜ + v v) ˜ dx, Ω

(2.45)

defined on H1 (Ω). Here, the functions a(x), b(x), c(x), d(x) are real-valued continuous functions on Ω. These functions are assumed to satisfy inf a(x) = a0 > 0,

x∈Ω

inf d(x) = d0 > 0,

x∈Ω

(2.46)

and to satisfy either b(x)c(x) ≤ 0

on Ω

(2.47)

or b(x)c(x) ≥ 0 on Ω

and

inf [a(x)d(x) − b(x)c(x)] = η0 > 0.

x∈Ω

(2.48)

The constant k is a positive constant which will be fixed below. Our goal is to show that the associated operator A and its parts in L2 (Ω) and H1 (Ω) are sectorial operators in H1 (Ω)∗ , L2 (Ω), and H1 (Ω), respectively. It seems, however, difficult to apply Theorem 2.1 to the form (2.45) directly. Our way to arrive at the goal will be indirect. We begin with verifying the following lemma. ϕ(x) 0 b(x) Lemma 2.1 Let P (x) = a(x) c(x) d(x) . There exist a diagonal matrix D(x) = 0 1 with ϕ ∈ C1 (Ω) and infx∈Ω ϕ(x) = ϕ0 > 0 and a constant δ > 0 for which it holds that D(x)P (x)D(x)−1 U, U ≥ δU 2 , x ∈ Ω, U ∈ R2 . (2.49) In addition, when (a, b, c, d) varies in a compact subset of C(Ω; R)4 satisfying (2.46) and either (2.47) or (2.48), the quantities ϕC1 , ϕ0 , and δ can be uniform.

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2 Sectorial Operators

Proof (I) Let us first consider a case where b(x) and c(x) are C1 functions on Ω. We have D(x)P (x)D(x)−1 U, U = a(x)u2 + b(x)ϕ(x) + c(x)ϕ(x)−1 uv + d(x)v 2 for U = t (u, v) ∈ R2 . Let condition (2.47) be satisfied. In this case, we put −b(x)c(x) + ε ϕε (x) = b(x)2 + ε 2 with parameter ε > 0. Then, b(x)ϕε (x) + c(x)ϕε (x)−1 =

|b(x)ε + c(x)ε 2 |

√ ≤ (1 + cC ) ε.

(b(x)2 + ε 2 )(−b(x)c(x) + ε) √ √ If ε > 0 is fixed in such a way that (1 + cC ) ε ≤ a0 d0 , then a(x)u2 + b(x)ϕε (x) + c(x)ϕε (x)−1 uv + d(x)v 2 ≥ a0 u2 −

1 a0 d0 |u||v| + d0 v 2 ≥ (a0 u2 + d0 v 2 ). 2

Let next condition (2.48) be satisfied. In this case, we put b(x)c(x) + ε ϕε (x) = , ε > 0. b(x)2 + ε 2 By (2.48), 2 c(x) + b(x)ε + c(x)ε 2 | b(x)ϕε (x) + c(x)ϕε (x)−1 = |2b(x) (b(x)2 + ε 2 )(b(x)c(x) + ε) √ ≤ 2 b(x)c(x) + (1 + cC ) ε √ ≤ 2 a(x)d(x) − η0 + (1 + cC ) ε.

Therefore, a(x)u2 + b(x)ϕε (x) + c(x)ϕε (x)−1 uv + d(x)v 2 ≥ [a(x) − μ]u2 − 2 a(x)d(x) − η √ + (1 + cC ) ε |u||v| + [d(x) − μ]v 2 + μ(u2 + v 2 ). Let μ =

η0 a(x)+d(x) .

Then, since

[a(x) − μ]u2 + [d(x) − μ]v 2 ≥ 2 (a(x) − μ)(d(x) − μ)|uv| > 2 a(x)d(x) − η0 |uv|,

1 Sectorial Operators in Hilbert Spaces

69

it follows that a(x)u2 + b(x)ϕε (x) + c(x)ϕε (x)−1 uv + d(x)v 2 √ η0 ≥ (u2 + v 2 ) − (1 + cC ) ε|uv|. a + dC Hence, if ε > 0 is sufficiently small, then the desired estimate (2.49) is satisfied. Since b(x) and c(x) are C1 functions, it is the same for ϕε (x). (II) Secondly, let us consider the general case. We notice that the C1 regularity of b(x) and c(x) is necessary only for ensuring that ϕε (x) is a C1 function and not for yielding estimate (2.49). So, if a new function ϕ(x), ˜ which is smooth on Ω, is taken sufficiently close to ϕε (x) in the topology of C(Ω), then (2.49) can be valid with ϕ(x). ˜ We also notice that, if (2.49) is satisfied for P (x) with some matrix D(x) = diag{ϕ(x), 1}, ϕ(x) being smooth, then it is satisfied for every P˜ (x), provided that ˜ ˜ a(x), ˜ b(x), c(x), ˜ and d(x) are all sufficiently close to a(x), b(x), c(x), and d(x), respectively, in the topology C(Ω). This shows that, when a(x), b(x), c(x), and d(x) vary in a compact subset of C(Ω), one can choose a finite number of matrices so that (2.49) is valid for every P (x) with at least one of these matrices. We notice that multiplication of D(x) is an isomorphism of H1 (Ω). In view of this fact, we here introduce the auxiliary sesquilinear form aD (U, V ) = a D(x)−1 U, D(x)V , U, V ∈ H1 (Ω), on H1 (Ω). On account of Lemma 2.1, aD (U, V ) is seen to satisfy the coercivity (2.7). In fact, we see that a D(x)−1 U, D(x)U = a(x)|∇u|2 + b(x)ϕ(x)∇v · ∇u Ω

+ c(x)ϕ(x)−1 ∇u · ∇v + d(x)|∇v|2 dx 2 +k |u| + |v|2 dx + b(U, U ), u U= ∈ H1 (Ω), v

Ω

where b(U, V ) denotes a sesquilinear form containing first derivatives of ϕ in its coefficients. Obviously, b(U, V ) satisfies |b(U, U )| ≤ Cϕ U |U |. Therefore, Re aD (U, U ) ≥ δ |∇u|2 + |∇v|2 + k |u|2 + |v|2 − Cϕ U |U | δ ≥ U 2 + (k − Cϕ )|U |2 . 2

70

2 Sectorial Operators

Fix k large enough to satisfy k ≥ Cϕ . Then, aD (U, V ) satisfies (2.7) and (2.6). Hence, Theorem 2.1 is available to conclude that the operator AD associated with the form aD (U, V ) and its parts satisfy (λ − AD )−1

H1∗

C , + (λ − AD|L2 )−1 L + (λ − AD|H1 )−1 H1 ≤ 2 |λ|

λ∈ / Σω ,

(2.50)

with some 0 ≤ ω < π2 . We now notice that AU, V = a(U, V ) = aD D(x)U, D(x)−1 V = AD D(x)U, D(x)−1 V . In addition, we observe by the lemma below that multiplications of D(x) and D(x)−1 are isomorphisms of H1 (Ω)∗ as well as of L2 (Ω) and H1 (Ω) and that the duality product has the property D(x)−1 Φ, V = Φ, D(x)−1 V , Φ ∈ H1 (Ω)∗ , V ∈ H1 (Ω). Then, it is deduced that A = D(x)−1 AD D(x) and the resolvent satisfies the relation (λ − A)−1 = D(x)(λ − AD )−1 D(x)−1 ,

λ∈ / Σω .

(2.51)

As a consequence of (2.50), (λ − A)−1

H1∗

C , + (λ − A|L2 )−1 L + (λ − A|H1 )−1 H1 ≤ 2 |λ|

λ∈ / Σω . (2.52)

Lemma 2.2 The operation of multiplication by D(x) can be extended as a bounded operator on H1 (Ω)∗ with the property D(x)Φ, V = Φ, D(x)V ,

Φ ∈ H1 (Ω)∗ , V ∈ H1 (Ω).

(2.53)

It is the same for D(x)−1 . Proof For F ∈ X, we have D(x)F ∗ = sup |(D(x)F, V )| = sup |(F, D(x)V )| ≤ CϕC1 F ∗ . V ≤1

V ≤1

Since L2 (Ω) is dense in H1 (Ω)∗ , this shows that multiplication by D(x) can be extended to H1 (Ω)∗ uniquely as a bounded operator. Since D(x)Φ, V = Φ, D(x)V holds for Φ ∈ L2 (Ω) and V ∈ H1 (Ω), (2.53) is also verified by the usual argument. We are now able to announce the following result.

2 Sectorial Operators in Banach Spaces

71

Theorem 2.11 Let Ω be any domain in Rn . Let a, b, c, d ∈ C(Ω; R) satisfy (2.46) and either (2.47) or (2.48). Then, for sufficiently large k, the operator A associated with the form (2.45) and its parts in L2 (Ω) and H1 (Ω) satisfy (2.1) and (2.2) in H1 (Ω)∗ , L2 (Ω), and H1 (Ω), respectively, with ω = π2 and some constant M. When (a, b, c, d) varies in a compact subset of C(Ω; R)4 satisfying (2.46) and either (2.47) or (2.48), the constants k and M can be uniform. We finally investigate a representation of A. Let us consider a case where a(x), b(x), c(x), and d(x) are all C1 functions. Let Λ be the operator associated with the sesquilinear form ∇u · ∇v dx + uv dx, u, v ∈ H 1 (Ω), a(u, v) = Ω

Ω

H 1 (Ω)∗

i.e., Λ is a realization of −Δ + 1 in under the Neumann boundary conditions. Then the first integral in the right-hand side of (2.45) is written as a(x)∇u · ∇ u˜ dx = ∇u · ∇(a u) ˜ dx − (∇u · ∇a)u˜ dx Ω

Ω

Ω

= a(x)Λu, u

˜ − ∇a · ∇u + a(x)u, u . ˜ It is the same for other integrals. Therefore, we have AU, V = a(U, V ) = a(x)Λu + b(x)Λv, u

˜ + c(x)Λu + d(x)Λv, v

˜ − ∇a · ∇u + ∇b · ∇v, u

˜ − ∇c · ∇u + ∇d · ∇v, v

˜ + (k − a(x))u − b(x)v, u

˜ + −c(x)u + (k − d(x))v, v . ˜ This then means that AU is given explicitly by the formula a(x) b(x) ∇a · ∇ ∇b · ∇ AU = (Λ − 1)U − U + kU, c(x) d(x) ∇c · ∇ ∇d · ∇

U ∈ H1 (Ω). (2.54)

Furthermore, for A|L2 , we can see that ∇ · (a∇·) ∇ · (b∇·) A|L2 U = − U + kU, ∇ · (c∇·) ∇ · (d∇·)

U ∈ H2N (Ω).

(2.55)

2 Sectorial Operators in Banach Spaces 2.1 Sectorial Operators in Lp (1 < p < ∞) Spaces Let Ω be a bounded domain in Rn . Consider the sesquilinear form a(u, v) =

n i,j =1 Ω

aij (x)Di uDj v dx +

c(x)uv dx, Ω

u, v ∈ Z,

(2.56)

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2 Sectorial Operators

on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Here, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Ω satisfying conditions (2.18) and (2.19), and c(x) is a real-valued function in Ω satisfying (2.20). We consider the triplet Z ⊂ L2 (Ω) ⊂ Z ∗ and the operator A associated with the form (2.56). We already know that its part A|L2 in L2 (Ω) is a sectorial operator of L2 (Ω). In this section, we shall study A in the space Lp (Ω), where 1 < p < ∞. It is shown for 2 < p < ∞ that its part in Lp (Ω) becomes a sectorial operator of Lp (Ω). On the other hand, it is shown for 1 < p < 2 that a natural extension of A in Lp (Ω) defines a sectorial operator of Lp (Ω). Consider first the case where 2 < p < ∞. We begin with showing the following lemma. Lemma 2.3 Let 2 < p < ∞. For u ∈ Z (= H˚ 1 (Ω) or H 1 (Ω)), put u∗ε = |u|2 p−2 2 u with a parameter ε > 0. Then, 1+ε|u|2 Re Au, u∗ε ≥ c0 u, u∗ε

(2.57)

and Im Au, u∗ ≤

Cnp Re Au, u∗ε √ 2δ p − 1

ε

(2.58)

with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20). Proof It is easily seen that u∗ε ∈ H 1 (Ω) with ∇u∗ε =

|u|2 1 + ε|u|2

|u|2 1 + ε|u|2

=

p−2 2

∇u +

p − 2 |u|p−4 u 2 p ∇|u| 2 (1 + ε|u|2 ) 2

∇u +

p − 2 |u|p−4 2 2 p |u| ∇u + u ∇u . 2 2 (1 + ε|u| ) 2

p−2 2

We therefore have Au, u∗ε =

Ω

|u|2 1 + ε|u|2

p−2 + 2 Ω

n

|u|p−4

Ω (1 + ε|u|2 )

(1 + ε|u|2 )

p−2 2

n

|u|p−2 (1 + ε|u|2 ) 2

p 2

aij (x)Di u u2 Dj u + |u|2 Dj u dx

i,j =1

dx

p

Ω

aij (x)Di uDj u dx

i,j =1

c(x)|u|p

+ =ε

p−2 n 2

i,j =1

aij (x)|u|2 Di uDj u dx + Ω

c(x)|u|p (1 + ε|u|2 )

p−2 2

dx

2 Sectorial Operators in Banach Spaces

1 + 2

73 n

|u|p−4

Ω (1 + ε|u|2 )

p 2

aij (x) p|u|2 Di uDj u

i,j =1

+ (p − 2)u2 Di uDj u dx.

Here, we put uDj u = αj + iβj , 1 ≤ j ≤ n. Then, it turns out that Au, u∗ε = ε

n

|u|p−2

aij (x)[(αi αj + βi βj ) + i(βi αj − αi βj )] dx

p

Ω

(1 + ε|u|2 ) 2

i,j =1 n

|u|p−4

+

aij (x){[(p − 1)αi αj + βi βj ]

p

Ω

(1 + ε|u|2 ) 2

i,j =1

c(x)|u|p

+ i[(p − 1)βi αj − αi βj ]} dx + Ω

(1 + ε|u|2 )

p−2 2

dx.

Then, Re(Au, u∗ε ) can be estimated from below by using (2.19) and (2.20). In particular, we verify the desired estimate (2.57). In the meantime, | Im(Au, u∗ε )| is estimated as follows. We have n

n

aij |βi αj − αi βj | ≤ 2C

i,j =1

|αi βj | ≤ C

i,j =1

= Cn

n 2 αi + βj2 i,j =1

n n 2 Cn αj + βj2 ≤ aij (αi αj + βi βj ). δ j =1

i,j =1

In a similar way, n

aij |(p − 1)βi αj − αi βj | ≤ Cp

i,j =1

≤√

≤

Noting that

√p 2 p−1

n

n Cp |αi βj | = √ p − 1|αi βj | p − 1 i,j =1 i,j =1

n Cp (p − 1)αi2 + βj2 p − 1 i,j =1

n Cnp aij [(p − 1)αi αj + βi βj ]. √ 2δ p − 1 i,j =1

≥ 1, we obtain from these the desired estimate (2.58).

Let Re λ ≤ 0. For the solution u to the equation (A − λ)u = f , where f ∈ Lp (Ω) ⊂ L2 (Ω), we verify the following fact.

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2 Sectorial Operators

Proposition 2.1 Let 2 < p < ∞, and let Re λ ≤ 0. If (A − λ)u = f with f ∈ Lp (Ω), then u belongs to Lp (Ω), and the estimates (| Re λ| + c0 )uLp ≤ f Lp , Cnp + 2 f Lp |λ|uLp ≤ √ 2δ p − 1

(2.59) (2.60)

hold with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20). Proof Taking the real part of the inner product of (A − λ)u = f and u∗ε in L2 (Ω), we have Re Au, u∗ε − Re λ u, u∗ε = Re f, u∗ε . Since Re(Au, u∗ε ) ≥ 0 and (u, u∗ε ) ≥ 0, it follows that Re Au, u∗ε + | Re λ|(u, u∗ ) = Re f, u∗ε ≤ f Lp u∗ε Lq , where

1 p

+

1 q

(2.61)

= 1. This, together with (2.57), yields that

(c0 + | Re λ|) Ω

|u|p (1 + ε|u|2 )

p−2 2

dx ≤ f Lp

Ω

(1 + ε|u|2 )

≤ f Lp

1

|u|(p−1)q

(1 + ε|u|2 )

dx 1

|u|p

Ω

q

(p−2)q 2

q

p−2 2

dx

.

Hence, (c0 + | Re λ|)

1

|u|p Ω

(1 + ε|u|2 )

p

p−2 2

dx

≤ f Lp .

Letting ε → 0, we deduce by Fatou’s lemma that u ∈ Lp (Ω) with (2.59). In order to prove (2.60), we notice that |λ| u, u∗ε ≤ Au, u∗ε + f, u∗ε ≤ Re Au, u∗ε + Im Au, u∗ε + f, u∗ε . Then, it follows from (2.58) and (2.61) that Cnp |λ| u, u∗ε ≤ + 2 f Lp u∗ε L . √ q 2δ p − 1 Letting ε → 0, we obtain that p |λ|uLp

≤

Cnp p−1 + 2 f Lp uLp . √ 2δ p − 1

Hence, estimate (2.60) is deduced.

2 Sectorial Operators in Banach Spaces

75

Let us next consider the case where 1 < p < 2. We verify the following analogous lemma. Lemma 2.4 Let 1 < p < 2. For u ∈ Z, put u∗ε = (|u|2 + ε) ε > 0. Then, Re Au, u∗ε ≥ c0 u, u∗ε

p−2 2

u with parameter (2.62)

and Im Au, u∗ ≤ ε

Cnp Re Au, u∗ε √ 2δ p − 1

(2.63)

with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20). Proof We can argue in an analogous way as before. It is observed that u∗ε ∈ H 1 (Ω) with p−2 p−4 p − 2 2 |u| + ε 2 |u|2 ∇u + u2 ∇u . ∇u∗ε = |u|2 + ε 2 ∇u + 2 From this we can write Au, u∗ε = ε

+

n 2 p−4 −2 2 |u| + ε |u| aij (x)|u|2 Di uDj u dx

Ω

1 2

i,j =1

Ω

n 2 p−4 |u| + ε 2 aij (x) p|u|2 Di uDj u i,j =1

+ (p − 2)u2 Di uDj u dx +

p−2 c(x) |u|2 + ε 2 |u|2 dx.

Ω

By putting uDj u = αj +iβj , 1 ≤ j ≤ n, we can repeat the same argument as before. Then, the desired estimates (2.62) and (2.63) are obtained. Let Re λ ≤ 0. For the solution u to the equation (A−λ)u = f , where f ∈ L2 (Ω), we verify the following proposition. Proposition 2.2 Let 1 < p < 2, and let Re λ ≤ 0. If (A − λ)u = f with f ∈ L2 (Ω), then u ∈ H 1 (Ω) satisfies the estimates (| Re λ| + c0 )uLp ≤ f Lp , Cnp + 2 f Lp |λ|uLp ≤ √ 2δ p − 1

(2.64) (2.65)

with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20).

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2 Sectorial Operators

Proof We consider the real part of the inner product (A − λ)u = f and u∗ε in L2 (Ω). Then, since Re(Au, u∗ε ) ≥ 0 and (u, u∗ε ) ≥ 0, it follows that Re Au, u∗ε + | Re λ| u, u∗ε ≤ f Lp u∗ε L . (2.66) q

By (2.62),

|u|2 + ε

(c0 + | Re λ|)

p−2 2

Ω

p−1

|u|2 dx ≤ f Lp uLp .

We here used the fact that u∗ε Lq ≤ uLp . Hence, letting ε → 0, we deduce by the Lebesgue dominate convergence theorem the desired estimate (2.64). Estimate (2.65) is obtained in a similar way from |λ| u, u∗ε ≤ Re Au, u∗ε + Im Au, u∗ε + f, u∗ε . p−1

Indeed, by (2.63) and (2.66), 2 p−2 2 2 |λ| |u| + ε |u| dx ≤ Ω

Cnp + 2 f Lp u∗ε L . √ q 2δ p − 1

Hence, letting ε → 0, (2.65) is also deduced.

We are now ready to define from A a linear operator Ap for each 1 < p < ∞. Let us first consider the case where 2 < p < ∞. In this case, the operator Ap is defined as the part of A in the space Lp (Ω). As Proposition 2.1 (λ = 0) shows that Au ∈ Lp (Ω) implies u ∈ Lp (Ω), we have in fact

D(Ap ) = {u ∈ Z; Au ∈ Lp (Ω)}, Ap u = Au. Proposition 2.1 shows also that, for every Re λ ≤ 0, the resolvent (λ − Ap )−1 exists and that the following estimates (λ − Ap )−1 ≤ 1/(| Re λ| + c0 ), Re λ ≤ 0, (2.67) Lp Cnp (λ − Ap )−1 ≤ + 2 |λ|, Re λ ≤ 0, λ = 0 (2.68) √ Lp 2δ p − 1 hold due to (2.59) and (2.60). Consider next the case where 1 < p < 2. Proposition 2.2 shows that, for every Re λ ≤ 0, the resolvent (λ − A)−1 acting on L2 (Ω) is continuous with respect to the Lp (Ω) norm. This means that the resolvent can be extended on Lp (Ω) as a bounded operator. Denote the extended operator by Rp (λ) for Re λ ≤ 0. Then the family of operators Rp (λ) defines a pseudo-resolvent in L(Lp (Ω)). According to Theorem 1.8, there exists a multivalued linear operator Ap of Lp (Ω) such that

2 Sectorial Operators in Banach Spaces

77

Rp (λ) = (λ − Ap )−1 for every Re λ ≤ 0. In this way, A−1 p is a continuous extension on L (Ω). Therefore, D(A ) ⊂ D(A ) and f = A2 u implies that f ∈ Ap u; of A−1 p 2 p 2 in particular, Ap is densely defined in Lp (Ω) (Indeed, D(A2 ) is dense in L2 (Ω), a fortiori in Lp (Ω)). Moreover, it is readily verified that u ∈ D(Ap ) and f ∈ Ap u if and only if there exist sequences un ∈ D(A2 ) and fn = A2 un such that un → u and fn → f in Lp (Ω). Finally, (2.64) and (2.65) yield that Ap also satisfies the same estimates as (2.67) and (2.68). To conclude for 1 < p < ∞ that Ap is a sectorial operator of Lp (Ω), it therefore remains to verify that Ap is densely defined in Lp (Ω) in the case where 2 < p < ∞ and that Ap is single valued in the case where 1 < p < 2. For these purposes, we notice the following proposition. Proposition 2.3 Let 1 < p < ∞ and f ∈ Lp (Ω). As λ → −∞, λ(λ − Ap )−1 f converges weakly to f in Lp (Ω). Proof When p = 2, the result is already known. Indeed, λ(λ − A2 )−1 f − f = A2 (λ − A2 )−1 f ; therefore, if f ∈ D(A2 ), then λ(λ − A2 )−1 f − f → 0 in L2 (Ω). By the denseness of D(A2 ) in L2 (Ω), λ(λ − A2 )−1 indeed converges strongly to the identity operator on L2 (Ω). Let 2 < p < ∞. Since the norms λ(λ − Ap )−1 f Lp are uniformly bounded for all λ ≤ 0, we can extract a sequence λn → −∞ such that λn (λn − Ap )−1 f → g weakly in Lp (Ω). However, since the sequence is strongly convergent to f in L2 (Ω), it must hold that g = f . Therefore, without extracting a sequence for λ, λ(λ − Ap )−1 f is weakly convergent to f in Lp (Ω). Let 1 < p < 2. If f ∈ L2 (Ω), then λ(λ − Ap )−1 f is strongly convergent to f in Lp (Ω). Then, by the uniform boundedness of λ(λ − Ap )−1 Lp for λ ≤ 0 and the denseness of L2 (Ω) in Lp (Ω), we conclude that λ(λ − Ap )−1 f → f in Lp (Ω) as λ → −∞. We can now state the final result and complete its proof. Theorem 2.12 Let Ω be a bounded domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied, and let A be an operator associated with the form (2.56) on Z = H˚ 1 (Ω) or H 1 (Ω). Then, for any 1 < p < ∞, the operator Ap defined from A in the way √ + 2, where specified above satisfies (2.1) and (2.2) with ω = π2 and constant 2δCnp p−1 C = maxi,j aij L∞ , δ, and c0 being the constants in (2.19) and (2.20). Therefore, Ap is a sectorial operator of Lp (Ω) with angle ωAp < π2 . Proof Let 2 < p < ∞. Since R((λ − Ap )−1 ) ⊂ D(Ap ), Proposition 2.3 shows that D(Ap ) is dense in the weak topology of Lp (Ω). But since D(Ap ) is a linear subspace, its denseness is valid even in the usual topology of Lp (Ω). Let 1 < p < 2. Assume that Ap 0 f ; then, (λ − Ap )0 f for any λ ≤ 0; that is, 0 = (λ − Ap )−1 f ; then Proposition 2.3 yields that f = 0. Therefore, Ap is a single-valued operator.

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2 Sectorial Operators

The operator Ap is called a realization of the elliptic operator (2.21) in Lp (Ω), 1 < p < ∞, under the Dirichlet- or Neumann-type boundary conditions ((2.22) or (2.23)) if Z = H˚ 1 (Ω) or H 1 (Ω), respectively. Remark 2.4 We notice that the resolvent of Ap satisfies the optimal estimate along the negative real axis (−∞, 0]. In fact, (2.67) shows that (λ − Ap )−1 ≤ 1/(|λ| + c0 ), λ ≤ 0, (2.69) L p

holds for every 1 < p < ∞.

2.2 Extreme Cases p = 1 and p = ∞ In this subsection, we consider the extreme cases p = 1 and p = ∞. It is possible to define analogously sectorial operators A1 and A∞ acting in the spaces L1 (Ω) and L∞ (Ω), respectively. However, (2.60) and (2.65) have no more meaning in these spaces. Let Ω be a bounded domain in Rn . Let A be a linear operator associated with the form (2.56) on Z = H˚ 1 (Ω) or H 1 (Ω). Conditions (2.18), (2.19), and (2.20) are assumed as before. First, let us consider the case p = ∞. For Re λ ≤ 0, consider the equation (A − λ)u = f , where f ∈ L∞ (Ω), and let u ∈ H 1 (Ω) be its solution. By 1

(2.59), we see that c0 uLp ≤ f Lp ≤ |Ω| p f L∞ for any finite p > 2. Using the fact that uL∞ = limp→∞ uLp , we can conclude that u ∈ L∞ (Ω). Furthermore, due to (2.59), (| Re λ| + c0 )uL∞ ≤ f L∞ .

(2.70)

The operator A∞ is then defined as the part of A in the space L∞ (Ω). That is, D(A∞ ) = {u ∈ Z; Au ∈ L∞ (Ω)} and A∞ u = Au. Due to (2.70), λ − A∞ has a bounded inverse on L∞ (Ω) with the estimates (λ − A∞ )−1 ≤ 1/(| Re λ| + c0 ), Re λ ≤ 0. (2.71) L ∞

In this way, we have observed that A∞ satisfies (2.1) and (2.2) with any ω > π2 . However, it is no longer possible to verify that D(A∞ ) is dense in L∞ (Ω), because Proposition 2.3 cannot be verified for p = ∞. We observe that λ(λ − A∞ )−1 f → f in L∞ (Ω) as λ → −∞ for f ∈ D(A∞ ) only. So, we have to consider again a part of A∞ in such a way that D(A˜ ∞ ) = {u ∈ D(A∞ ); A∞ u ∈ D(A∞ )} and A˜ ∞ u = A∞ u, where the closure is taken in L∞ (Ω). Then, the operator A˜ ∞ is densely defined in D(A∞ ). In fact, for any u ∈ D(A∞ ), uλ = λ(λ − A∞ )−1 u ∈ D(A˜ ∞ ) for λ < 0, and uλ → u in L∞ (Ω) as λ → −∞. Hence, D(A˜ ∞ ) is dense in D(A∞ ). Of course, A˜ ∞ also satisfies the same condition as (2.71). Secondly, let us consider the case p = 1. For Re λ ≤ 0, consider the equation (A − λ)u = f with f ∈ L2 (Ω). It is verified from (2.64) that (| Re λ| + c0 )uL1 ≤ f L1 .

(2.72)

2 Sectorial Operators in Banach Spaces

79

Here, we used the fact that uL1 = limp→1 uLp . This means that the resolvent (λ − A2 )−1 is continuous with respect to the L1 topology. As a consequence, each operator (λ − A2 )−1 is extended to a bounded operator R1 (λ) on L1 (Ω), and the family R1 (λ), Re λ ≤ 0, defines a pseudo-resolvent in L(L1 (Ω)). By a multivalued linear operator A1 of L1 (Ω), we have (λ − A1 )−1 = R1 (λ). Since D(A2 ) ⊂ D(A1 ), A1 is densely defined. Furthermore, as Proposition 2.3 is valid for p = 1, too, A1 is a single-valued linear operator. Finally, it is clear from (2.72) that (λ − A1 )−1

L1

≤ 1/(| Re λ| + c0 ),

Re λ ≤ 0.

(2.73)

In the cases where p = 1 and p = ∞, we have therefore obtained the following results. Theorem 2.13 Let Ω be a bounded domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied, and let A be an operator associated with the form (2.56) on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Then, the linear operators A1 and A˜ ∞ defined from A as above satisfy (2.1) and (2.2) in L1 (Ω) and D(A∞ ) ⊂ L∞ (Ω), respectively, with any angle ω > π2 and constant M = | cos1 ω| , and are sectorial operators of L1 (Ω) and D(A∞ ), respectively, with angles ωA1 ≤

π 2

and ωA˜ ∞ ≤ π2 .

2.3 Adjoint Operators We return to the study for the case where 1 < p < ∞. In Sect. 2.1 above, we defined the realization Ap of strongly elliptic operator (2.27) in Lp (Ω) under the Dirichletor Neumann type-boundary conditions. We are then concerned with the adjoint operator A∗p with respect to the adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1. Let Ω be a bounded domain in Rn . Under conditions (2.18), (2.19), and (2.20), consider the sesquilinear form (2.56) on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Let A be the operator associated with the from. Furthermore, let Ap be the sectorial operator of Lp (Ω) determined from A, where 1 < p < ∞. We consider an adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1, and denote by A∗p the adjoint of A∗p with respect to this adjoint pair. In order to characterize A∗p , we use the adjoint form whose notion was already presented in Chap. 1, Sect. 8.2. Actually, the adjoint form to (2.56) is given by a ∗ (u, v) = a(v, u) =

n i,j =1 Ω

aj i (x)Di uDj v dx +

c(x)uv dx,

u, v ∈ Z,

Ω

(2.74) on Z. Let A∗ be the operator associated with this form. It is of course possible to determine a sectorial operator Bp of Lp (Ω) from A∗ , where 1 < p < ∞. Our goal is to show that A∗p = Bp if p1 + p1 = 1.

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Theorem 2.14 Let Ω be a bounded domain in Rn , and let (2.18), (2.19), and (2.20) be satisfied. For 1 < p, p < ∞, let Ap and Bp be sectorial operators of Lp (Ω) and Lp (Ω) determined from the operators associated with the form (2.56) and the adjoint form (2.74), respectively, on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Then, A∗p = Bp if p1 + p1 = 1. Proof When p = 2, L2 (Ω) is a Hilbert space. So, A∗2 = B2 is observed directly by applying Theorem 1.25 to the form (2.56) and its adjoint form (2.74). Let 2 < p < ∞ and p1 + p1 = 1. Let u ∈ D(Ap ), and let v ∈ D(Bp ). Since D(Ap ) ⊂ D(A2 ), we have Ap u = A2 u. In the meantime, we recall that Bp v = g if and only if there exist sequences vn ∈ D(B2 ) and gn = B2 vn such that vn → v and gn → g in Lp (Ω). Then, since Ap u, vn Lp ×Lp = (A2 u, vn )L2 = (u, B2 vn )L2 = u, Bp vn Lp ×Lp , it follows that Ap u, v Lp ×Lp = u, Bp v Lp ×Lp ,

u ∈ D(Ap ), v ∈ D(Bp ).

This then shows that, if v ∈ D(Bp ), then v ∈ D(A∗p ) with A∗p v = Bp v. On the other hand, we know that both Bp and A∗p have their bounded inverses on Lp (Ω). Hence, it must hold that D(A∗p ) = D(Bp ) and A∗p = Bp . Let next 1 < p < 2 and p1 + p1 = 1. Since 2 < p < ∞, we can apply the result just obtained to Bp to conclude that Bp∗ = Ap . By virtue of Theorem 1.19, it then follows that A∗p = Bp∗∗ = Bp . As a consequence of Theorem 2.14, we know that, if aij (x) are symmetric in the sense of (2.35), then A∗2 = B2 = A2 . Therefore, A2 = A is a positive definite selfadjoint operator of L2 (Ω). Furthermore, A has a domain D(A) ⊂ Z (= H˚ 1 (Ω) or H 1 (Ω)). By Theorem 1.38, A−1 is a compact operator of L2 (Ω). Then, the theory of spectral resolutions (see [Yos80, Sect. XI.9]) provides that A possesses an infinite number of positive eigenvalues 0 < λ1 < λ2 < · · · and is written by

∞

A= 0

λ dE(λ) =

∞

λk Pk ,

k=1

where Pk = E(λk ) − E(λk − 0) denotes the orthogonal projection to the eigenspace Nk corresponding to the eigenvalue λk . Each Nk is a finite-dimensional subspace of L2 (Ω), and L2 (Ω) is the direct sum of Nk . Since ai,j (x) and c(x) are real-valued functions, we see, for each k, that u ∈ Nk implies u ∈ Nk , that is, u ∈ Nk implies Re u ∈ Nk and Im u ∈ Nk . This means that Nk can be generated by real functions alone. By the usual manner, we can then construct an orthonormal basis of Nk which is composed by real-valued functions alone. Corollary 2.1 Let Ω be a bounded domain in Rn . Let (2.18), (2.19), (2.20), and (2.35) be satisfied. Then, A2 is a positive definite self-adjoint operator of L2 (Ω).

2 Sectorial Operators in Banach Spaces

81

There exists an orthonormal basis of L2 (Ω) which is composed by real-valued eigenfunctions of A2 .

2.4 Shift Property in Lp Spaces For 1 < p < ∞, let Ap be the realization of strongly elliptic operator (2.27) in Lp (Ω) under the Dirichlet- or Neumann-type boundary conditions. In the case where p = 2, we already know that regularity of ∂Ω and aij (x) imply the shift property D(A2 ) ⊂ H 2 (Ω). This subsection is devoted to remarking that a similar property is valid for the general Ap , 1 < p < ∞, too. Let Ω be a bounded domain in Rn with C2 boundary. Let (2.19), (2.20), and (2.28) be satisfied. Let A(D) be the elliptic operator given by (2.27). Let us first consider the realization Ap which is equipped with the Dirichlet boundary conditions. According to Grisvard [Gri85, Theorem 2.4.1.3], the mapping u → (A(D) + k)u is an isomorphism from the space {u ∈ Hp2 (Ω); γ u = 0} onto Lp (Ω) if k is sufficiently large. For u ∈ D(Ap ), put Ap u = f ∈ Lp (Ω); then, there exists a function u˜ ∈ Hp2 (Ω) satisfying γ u˜ = 0 such that (A(D) + k)u˜ = f + ku, i.e., (Ap + k)u˜ = f + ku. Therefore, u˜ = u. In this way, we verify that

D(Ap ) = {u ∈ Hp2 (Ω); γ u = 0 on ∂Ω}, (2.75) Ap u = A(D)u, with uHp2 ≤ Cp (Ap uLp + uLp ),

u ∈ D(Ap ).

(2.76)

It is quite similar for the realization Ap equipped with the Neumann-type boundary conditions. According to Grisvard [Gri85, Theorem 2.4.1.3], the mapping u → ∂u = 0} onto Lp (Ω) (A(D) + k)u is an isomorphism from the space {u ∈ Hp2 (Ω); ∂ν A if k is sufficiently large. Then, by the same reason, we verify that

∂u = 0 on ∂Ω}, D(Ap ) = {u ∈ Hp2 (Ω); ∂ν A (2.77) Ap u = A(D)u, with uHp2 ≤ Cp (Ap uLp + uLp ),

u ∈ D(Ap ).

(2.78)

Theorem 2.15 Let Ω be a bounded domain with C2 boundary. Let (2.19), (2.20), and (2.28) be satisfied. Then, for each 1 < p < ∞, the realizations Ap of A(D) in Lp (Ω) under the Dirichlet- and Neumann-type boundary conditions are characterized by (2.75) and (2.77), respectively. Remark 2.5 A similar shift property is verified even in the extreme cases p = 1 and p = ∞. But the optimal shift property (i.e., D(A1 ) ⊂ H12 (Ω) and D(A∞ ) ⊂ 2 (Ω)) fails except the one-dimensional case. For characterization of these operH∞ ators, we refer the reader to Tanabe [Tan97, Chap. 5].

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2 Sectorial Operators

3 Sectorial Operators in Product Spaces Let X1 and X2 be Banach spaces with norms · 1 and · 2 , respectively, and let X be the product Banach space of X1 and X2 with norm F = f1 1 + f2 2 , F = t (f , f ) ∈ X. 1 2 Let A1 and A2 be sectorial operators of X1 and X2 , respectively, with angles 0 ≤ ω1 < π and 0 ≤ ω2 < π . We consider a matrix operator of the form A1 B , 0 A2 u1 ; u1 ∈ D(A1 ), u2 ∈ D(A2 ) ∩ D(B) , D(A) = u2

A=

(2.79)

where B is a third linear operator from D(B) ⊂ X2 into X1 . If B is a bounded operator from D(A2 ) into X1 , then A is a sectorial operator of X. Theorem 2.16 Let Ak be two sectorial operators of Xk with angles 0 ≤ ωk < π , respectively, where k = 1, 2. Let B ∈ L(D(A2 ), X1 ). Then, the matrix operator A defined by (2.79) satisfies (2.1) and (2.2) for any ω such that ωA < ω ≤ π with constant Mω = max{M1,ω , M2,ω + M1,ω (M2,ω + 1)BA−1 2 }, where ωA = max{ω1 , ω2 }. Therefore, A is a sectorial operator of X with angle ≤ ωA . Proof The equality f u (λ − A) 1 = 1 u2 f2 is obviously equivalent to

(λ − A1 )u1 − Bu2 = f1 , (λ − A2 )u2 = f2 .

/ Σω , then λ ∈ ρ(A), and the resolvent is written by Let ωA < ω < π . If λ ∈ −1

(λ − A)

(λ − A1 )−1 = 0

−1 (λ − A1 )−1 BA−1 2 A2 (λ − A2 ) . (λ − A2 )−1

The resolvent is estimated by (λ − A)−1 ≤ max M1,ω , M2,ω + M1,ω (M2,ω + 1)BA−1 |λ|−1 , 2

This shows that the assertion of theorem is true.

λ∈ / Σω .

4 Yosida Approximation

83

4 Yosida Approximation 4.1 Uniform Estimates Let A be a sectorial operator of X with angle ωA . We define the sequence An of bounded linear operators on X by An = A(1 + n−1 A)−1 = n − n2 (n + A)−1 ,

n = 1, 2, 3, . . . .

(2.80)

This sequence {An }n=1,2,3,... is called the Yosida approximation for A. It is observed by some calculations that nλ λ − An = n2 − (n − λ)(n + A) (n + A)−1 = (n − λ) − A (n + A)−1 . n−λ Therefore, if

nλ n−λ

∈ ρ(A), then λ ∈ ρ(An ), and the following representation is valid: −1

(λ − An )

−1 nλ 1 (n + A) −A = n−λ n−λ 2 −1 n nλ 1 + −A = . λ−n n−λ n−λ

(2.81) (2.82)

Let ωA < ω < π . By definition, C − Σω ⊂ ρ(A). Note that λ ∈ C − Σω implies ∈ C − Σω . Indeed, consider the transform

nλ n−λ

λ →

nλ = n−λ

1 1 λ

−

1 n

(2.83)

in C which is composed with two elementary transforms λ → λ−1 and λ → λ − n1 . nλ These two transforms map C − Σω into itself. Consequently, the transform λ → n−λ also maps C − Σω into itself. In view of (2.81), we have C − Σω ⊂ ρ(An ), i.e., σ (An ) ⊂ Σω ,

n = 1, 2, 3, . . . .

(2.84)

Let us estimate next (λ − An )−1 . By the cosine theorem, we have |λ − n|2 = |λ|2 + n2 − 2|λ|n cos(arg λ) ≥ |λ|2 + n2 − 2|λ|n cos ω

(1 − cos ω)(|λ|2 + n2 ) if 0 < ω ≤ π2 , λ ∈ / Σω , ≥ π 2 2 if 2 < ω < π, λ ∈ / Σω . |λ| + n

(2.85)

We then obtain from (2.82) that ˜ (λ − An )−1 ≤ Mω , |λ|

λ∈ / Σω , n = 1, 2, 3, . . . ,

with some constant M˜ ω ≥ Mω independent of n.

(2.86)

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2 Sectorial Operators

We have thus verified that the Yosida approximation An satisfies (2.1) and (2.2) for any ω such that ωA < ω < π with some constant M˜ ω independent of n. In particular, An are sectorial operators of X with angles ωAn ≤ ωA .

4.2 Convergence As remarked in Chap. 1, Sect. 3.4, (λ − A)−1 is an L(X)-valued analytic function for λ ∈ ρ(A). It then follows from (2.82) that, for each λ ∈ / Σω , as n → ∞, (λ − An )−1 → (λ − A)−1

in L(X).

(2.87)

Let us see the convergence of An itself. To this end, we introduce the sequence of bounded operators Jn = (1 + n−1 A)−1 = 1 − A(n + A)−1 ,

n = 1, 2, 3, . . . .

(2.88)

It is clear that Jn ≤ M. Meanwhile, (1 − Jn )u = (n + A)−1 Au → 0 for u ∈ D(A). The denseness of D(A) in X then yields that Jn converges to 1 strongly on X

(2.89)

(see Chap. 1, Sect. 3). We therefore observe for U ∈ D(A) that, as n → ∞, An U = Jn AU → AU

in X.

(2.90)

Let k be any positive integer. Then, as n → ∞, Jnk also converges to 1 strongly on X. On the other hand, R(Jnk ) is contained in D(Ak ). This then shows that D(Ak ) is dense in X for any integer k = 1, 2, 3, . . . .

(2.91)

5 Exponential Functions In this section, we consider a sectorial operator A of a Banach space X with angle 0 ≤ ωA < π2 . By ω, we denote an angle such that ωA < ω < π2 . By definition, σ (A) ⊂ Σω = {λ ∈ C; | arg λ| < ω},

ωA < ω

0 such that e−zA ≤ Cφ e−δφ |z| ,

π 2

− ω, there exist a positive exponent

z ∈ Σφ − {0}.

(2.97)

Proof First, let us consider variables such that 0 < |z| ≤ 1 and | arg z| ≤ φ. We shift the integral contour in (2.95) to C1 ∪ Γ1 , where C1 : λ = |z|−1 e−iϕ ,

ω ≤ ϕ ≤ 2π − ω,

and

Γ1 : λ = ρe±iω ,

|z|−1 ≤ ρ < ∞.

5 Exponential Functions

87

Then, since |e−zλ | ≤ e for λ ∈ C1 , −zλ −1 e (λ − A) dλ ≤ C1

2π−ω

ω

eMω −1 |z| dϕ ≤ 2πeMω . |z|−1

Since |e−zλ | ≤ e−|z||λ| cos(φ+ω) for λ ∈ Γ1 , ∞ ∞ dρ −zλ −1 −ρ|z| cos(φ+ω) dρ ≤ Mω = M . e (λ−A) dλ e e−ρ cos(φ+ω) ω −1 ρ ρ |z| 1 Γ1 Therefore, e−zA ≤ Cφ if 0 < |z| ≤ 1 and | arg z| ≤ φ. Next, let us consider variables such that |z| ≥ 1 and | arg z| ≤ φ. Let 0 < δ < A−1 −1 . On account of (2.3), we can shift the integral contour to C2 ∪ Γ2 , where C2 : λ = δeiϕ ,

and Γ2 : λ = ρe±iω ,

−ω ≤ ϕ ≤ ω,

δ ≤ ρ < ∞.

Then, since |e−zλ | ≤ e−δ|z| cos(φ+ω) for λ ∈ C2 , ω −zλ −1 ≤ Mω e (λ − A) dλ e−δ|z| cos(φ+ω) dϕ ≤ 2ωMω e−δ|z| cos(φ+ω) . C2

−ω

In addition, −zλ −1 e (λ − A) dλ ≤ Mω Γ2

∞

e−ρ|z| cos(φ+ω) dρ =

δ

Mω e−δ|z| cos(φ+ω) . |z| cos(φ + ω)

Therefore, e−zA ≤ Cφ e−δ|z| cos(φ+ω) if |z| ≥ 1 and | arg z| ≤ φ. We have thus obtained the desired estimate (2.97).

We are now concerned with the limit of e−zA as z → 0. Proposition 2.6 For any φ such that 0 < φ < on X as z → 0 with z ∈ Σφ − {0}.

π 2

− ω, e−zA converges to 1 strongly

Proof First, let U ∈ D(A). Then, by the same argument as in the proof of Proposition 2.4, 1 −zA e U= e−zλ (λ − A)−1 (1 + A)−1 dλ(1 + A)U 2πi Γ 1 e−zλ (λ − A)−1 + (1 + A)−1 dλ(1 + A)U = 2πi Γ λ + 1 1 e−zλ = (λ − A)−1 dλ(1 + A)U. 2πi Γ λ + 1 Then, by the Lebesgue dominated convergence theorem, we see that

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2 Sectorial Operators

e−zA U →

1 2πi

1 =− 2πi

Γ

1 (λ − A)−1 dλ(1 + A)U λ+1

|λ+1|=1

1 (λ − A)−1 dλ(1 + A)U = U λ+1

in X.

Next, let F ∈ X be a general vector. The denseness of D(A) in X and uniform boundedness of the norms e−zA verified by (2.97) readily yield that e−zA F → F in X. In view of this convergence, we are naturally led to define the value of e−zA at z = 0 by e−0A = 1 (the identity mapping on X). We finally show the differentiability of e−zA U at z = 0 for U ∈ D(A). Proposition 2.7 Let U ∈ D(A). For any φ such that 0 < φ < z−1 [e−zA − 1]U converges to −AU in X as z → 0 with z ∈ Σφ − {0}.

π 2

− ω,

Proof Let z ∈ Σφ − {0} and ε > 0. Then, [e−zA − e−εA ]U = (ε − z)

1

e−[θz+(1−θ)ε]A AU dθ.

0

Letting ε → 0, we have [e−zA − 1]U = −z z−1 [e−zA − 1]U → −AU .

1 0

e−θzA AU dθ . Therefore, as z → 0,

In this subsection, we have proved for a sectorial operator A satisfying (2.92) and (2.93) that its exponential function e−zA is an analytic function in Σ π2 −ω enjoying the properties described in Propositions 2.4, 2.5, 2.6, and 2.7. It is also verified that e−zA satisfies the semigroup property

e−zA e−z A = e−(z+z )A ,

z, z ∈ Σ π2 −ω .

(2.98)

Such an analytic function is called an analytic semigroup on X generated by −A. The precise definitions of analytic semigroups and their generators will be presented in Chap. 3.

5.2 Contraction Semigroups Let A be a sectorial operator of X satisfying (2.92) and (2.93). We assume that A satisfies the optimal estimate on the negative real axis, i.e., (λ − A)−1 ≤ 1/|λ|, λ < 0. (2.99) This condition implies that e−tA ≤ 1 for all t ≥ 0. Such a semigroup is called a contraction semigroup.

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89

Theorem 2.17 Let A be a sectorial operator of X satisfying (2.92), (2.93), and (2.99). Then, it holds that e−tA ≤ 1 for any 0 ≤ t < ∞. Proof Let U ∈ D(A). By Proposition 2.7, limh0 h−1 [e−hA − 1]U = −AU . In the meantime, by (1.12), h−1 [(1+hA)−1 −1]U = −(1+hA)−1 AU → −AU as h 0. Therefore, we observe that limh0 h−1 [e−hA − (1 + hA)−1 ]U = 0. Furthermore, we verify from (2.99) that e−hA U − U [e−hA − (1 + hA)−1 ]U ≤ , h h

h > 0.

Hence, lim suph0 h−1 [e−hA U − U ] ≤ 0. By the same argument, we verify that D + e−tA U ≤ 0 for every t ≥ 0, where D + denotes Dini’s derivative. We then have e−tA U ≤ U . Indeed, suppose the contrary; then there exists some τ > 0 such that e−τ A U > U . Let α = (e−τ A U − U )/τ , and let the continuous function e−tA U − αt on [0, τ ] hit its minimum at t0 . Since e−τ A U − ατ = U , it is possible to assume that 0 ≤ t0 < τ . Then, e−tA U − e−t0 A U ≥ α(t − t0 ) for any t0 < t ≤ τ , and we conclude that (D + e−tA U )|t=t0 ≥ α > 0, which is a contradiction. Estimate (2.99) thus implies that, for U ∈ D(A), e−tA U ≤ U for every t ≥ 0. But, since D(A) is dense in X, the same estimate is valid for general F ∈ X, too.

5.3 Adjoint Semigroups Let X be a reflexive Banach space, and let X ∗ be an adjoint space of X. Let {X, X ∗ } be an adjoint pair with duality product ·, · . Let A be a sectorial operator of X satisfying (2.92) and (2.93). So, −A generates an analytic semigroup e−zA on X which is given by (2.95). Let A∗ be the adjoint operator of A with respect to {X, X ∗ }. By Theorem 1.19, ∗ A is a sectorial operator of X ∗ with the same angle ωA as A. Therefore, −A∗ also generates an analytic semigroup. We observe that, for any F ∈ X and any G ∈ X ∗ , −zλ 1 e (λ − A)−1 F, G dλ e−zA F, G = 2πi Γ 1 F, e−zλ (λ − A∗ )−1 G dλ. = 2πi Γ Here, let us choose an integral contour as Γω : λ = ρe±iω , 0 ≤ ρ < ∞, where ωA < ω < π2 . If the integral variable λ varies on Γω counterclockwise, namely, from ∞eiω through 0 to ∞e−iω , then the conjugate λ varies on Γω in the opposite direction. Therefore, 1 ∗ e−zA F, G = F, − e−zλ (λ − A∗ )−1 G dλ = F, e−zA G , 2πi −Γω

90

2 Sectorial Operators

i.e., ∗

e−zA = [e−zA ]∗ .

(2.100)

∗

We call the exponential function e−zA the adjoint semigroup of e−zA .

5.4 Approximate Semigroups Let A be a sectorial operator of X satisfying (2.92) and (2.93). Let An be the Yosida approximation of A defined by (2.80). As verified by (2.84) and (2.86), An are also sectorial operators of X with angles ≤ ωA . So, each −An generates an analytic semigroup e−zAn given by the integral (2.95). By (2.97), for any 0 < φ < π2 − ω, e−zAn ≤ Cφ e−δφ |z| ,

z ∈ Σφ ,

(2.101)

with some exponent δφ > 0 and constant Cφ which are independent of n. The sequence of analytic semigroups e−zAn is then called approximate semigroups of e−zA . As shown by (2.87), as n → ∞, (λ − An )−1 converges to (λ − A)−1 in L(X). Therefore, by the Lebesgue dominate convergence theorem, we verify for any z ∈ Σ π2 −ωA that e−zAn → e−zA

in L(X).

(2.102)

6 Generation of Analytic Semigroups 6.1 Generation Results in Hilbert Spaces Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Consider a continuous and coercive sesquilinear form a(U, V ) defined on Z, that is, (2.6) and (2.7) are satisfied. Let A be a linear operator associated with a(U, V ), and let A|X and A|Z be its parts in X and Z, respectively. By Theorem 2.1, we already know that A, A|X , and A|Z are all sectorial operators with angles < π2 . Therefore, they are negative generators of analytic semigroups on Z ∗ , X, and Z, respectively. Summarizing known results for A and its parts, we can state the following generation theorem. Theorem 2.18 Let a(U, V ) satisfy (2.6) and (2.7). Let A be the associated operator, and let A|X and A|Z be its parts. Then, A, A|X , and A|Z satisfy (2.92) and (2.93) δ < ω < π2 with a in Z ∗ , X, and Z, respectively, for any ω such that π2 − sin−1 M+δ (M+δ) sin ω constant δ−(M+δ) cos ω , where M and δ are the constants in (2.6) and (2.7).

6 Generation of Analytic Semigroups

91

Proof By (2.8), (2.9), (2.10), and (2.11), A, A|X , and A|Z satisfy (2.1) and (2.2) in Z ∗ , X, and Z, respectively, with angle π2 and constant M+δ δ . Then the results are obtained by applying (2.4) and (2.5). We know by (2.12) that the part A|X in X satisfies (2.99). Consequently, Theorem 2.17 yields that |e−tA|X | ≤ 1,

0 ≤ t < ∞.

We also remark that the operator A given by (2.54) and its parts are negative generators of analytic semigroups on the product Hilbert spaces. See Theorem 2.11.

6.2 Generation Results in Lp Spaces Let Ω be a bounded domain in Rn . For 1 < p < ∞, let Ap be the realization of the elliptic operator A(D) given by (2.27) in Lp (Ω) under the Dirichlet- or Neumanntype boundary conditions. By Theorem 2.12, we already know that Ap satisfies (2.1) and (2.2) with angle π2 and constant Mp given by (2.68). If we apply (2.4) and (2.5), then Ap is seen to satisfy (2.92) and (2.93). On the other hand, due to (2.69), Ap also satisfies (2.99). We thus obtain the following generation theorem. Theorem 2.19 Let Ω be a bounded domain in Rn . For 1 < p < ∞, let Ap be the realization of A(D) given by (2.27) in Lp (Ω) under the Dirichlet- and Neumanntype boundary conditions. Then, Ap satisfies (2.92) and (2.93) for any ω such M sin ω that π2 − sin−1 M1p < ω < π2 with constant 1−Mp p cos ω , where Mp is the constant in (2.68). The analytic semigroup e−tAp generated by −Ap satisfies the estimate e−tAp L(Lp ) ≤ 1 for 0 ≤ t < ∞.

For p = 1 and p = ∞, let A1 and A˜ ∞ be the realizations of A(D) in L1 (Ω) and L∞ (Ω), respectively. As seen by Theorem 2.13, there is no guarantee which ensures generation of analytic semigroup for A1 and A˜ ∞ . But (2.71) and (2.73) mean that they satisfy (2.99). As will be mentioned in Sect. 7.8, such a sectorial operator can generate a continuous semigroup. The following result is an immediate consequence of Theorem 2.28 announced below. Theorem 2.20 The realizations A1 and A˜ ∞ of A(D) are negative generators of continuous and contraction semigroups on L1 (Ω) and D(A∞ ), respectively. Remark 2.6 It is known in the regular case (see Sect. 2.4) that −A1 and −A˜ ∞ can generate analytic semigroups. Especially, let AC be a realization of A in the continuous function space C(Ω), then −AC generates an analytic semigroup in C(Ω). For details, see Tanabe [Tan97, Chap. 5] and Taira [Tai03].

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2 Sectorial Operators

7 Fractional Powers of Linear Operators Let X be a Banach space with norm · . Let A be a sectorial operator of X with angle 0 ≤ ωA < π . We know that, for any integer n ∈ Z, the operator An is defined; indeed, when n > 0, An is a densely defined, closed operator of X (cf. (2.91)); when n < 0, An = (A−1 )−n = (A−n )−1 is a bounded operator of X; and, when n = 0, A0 = 1 (the identity on X). We are then concerned with extending the definition for all real exponents x ∈ R in a natural manner. By ω we denote an angle such that ωA < ω < π . By definition, σ (A) ⊂ Σω = {λ ∈ C; | arg λ| < ω},

ωA < ω < π,

(2.103)

and (λ − A)−1 ≤ Mω , |λ|

λ∈ / Σω , ωA < ω < π,

(2.104)

with some constant Mω ≥ 1. As noticed in the beginning of the chapter, (2.103) implicitly means that 0 ∈ ρ(A). We may remark again that {λ ∈ C; |λ| ≤ δ} ⊂ ρ(A),

(2.105)

provided that 0 < δ < A−1 −1 .

7.1 Integral Formula We define, for each complex number z such that Re z > 0, the bounded linear operator 1 A−z = λ−z (λ − A)−1 dλ, (2.106) 2πi Γ using the Dunford integral in L(X). As a branch of the analytic function λ−z , we take the principal branch {λ ∈ C; | arg λ| < π} obtained by excluding the negative real axis from the complex plane. So, Γ is an integral contour surrounding σ (A) counterclockwise in C − (∞, 0] ∩ ρ(A). Here, we take Γ = Γ− ∪ Γ0 ∪ Γ+ such that Γ± : λ = ρe±iω ,

δ ≤ ρ < ∞,

and Γ0 : λ = δeiϕ ,

−ω ≤ ϕ ≤ ω,

(2.107)

where ω and δ are as in (2.103), (2.104), and (2.105). In addition, Γ is oriented from ∞eiω to δeiω , from δeiω to δe−iω , and from δe−iω to ∞e−iω . Since |λ−z | = |e−z log λ | = |e−z(log ρ±iω) | = e±(Im z)ω ρ − Re z ,

λ ∈ Γ± ,

we observe that the integral (2.106) is convergent in L(X). When z = n is a positive integer, we can shift the contour from Γ to a circle Cδ : |λ| = δ. Consequently, by

7 Fractional Powers of Linear Operators

93

the Cauchy integral formula, 1 1 λ−n (λ − A)−1 dλ = − λ−n (λ − A)−1 dλ 2πi Γ 2πi Cδ n−1 d 1 =− (λ − A)−1 (n − 1)! dλn−1 λ=0 n −n −n = (−1) (λ − A) λ=0 = A . This formula shows that the new definition (2.106) is consistent with the former one for any positive integer. By a direct calculation, we can verify that A−z is an analytic function for Re z > 0 with values in L(X). Indeed, we have 1 d −z A =− (log λ)λ−z (λ − A)−1 dλ. dz 2πi Γ However, since the proof is direct, we may omit it. We are now concerned with convergence of A−z as z → 0. Theorem 2.21 For any 0 < φ < to 1 strongly on X.

π 2,

as z → 0 with z ∈ Σφ − {0}, A−z converges

Proof We shall first verify the uniform boundedness of A−z when z varies in the set {z ∈ C; | arg z| ≤ φ, 0 < |z| < 1}. We notice that for such a z, the contour can be shifted from Γ to the reduplicated half lines (∞eiπ , 0] ∪ [0, ∞e−iπ ) by letting ±ω → ±π and δ → 0. From (2.107) it indeed follows that ∞ πi −z 1 A−z = (ρe ) − (ρe−πi )−z (ρ + A)−1 dρ 2πi 0 e−πiz − eπiz ∞ −z = ρ (ρ + A)−1 dρ 2πi 0 sin(πz) ∞ −z =− ρ (ρ + A)−1 dρ. (2.108) π 0 By (2.104) and (2.105), there exists some constant C ≥ 1 such that (ρ + A)−1 ≤

C , ρ +1

0 ≤ ρ < ∞.

Therefore, C| sin(πz)| A ≤ π −z

0

∞

C| sin(πz)| ρ − Re z dρ = . ρ +1 sin(π Re z)

Noting that (cos φ)|z| ≤ Re z ≤ |z|, we conclude that sup

| arg z|≤φ, 0 0, ∞ 1 1 A−z = λ−x (λ − A)−1 dλ = λ−z (λ − μ)−1 dE(μ) dλ 2πi Γ 2πi Γ δ ∞ ∞ 1 −z −1 λ (λ − μ) dλ dE(μ) = μ−z dE(μ). (2.110) = 2πi Γ δ δ

7.2 Law of Exponent We shall show that the fractional power A−z of sectorial operator satisfies the law of exponent, i.e.,

A−z A−z = A−(z+z ) ,

Re z > 0, Re z > 0.

(2.111)

By (1.12) we have −z

−z

A A

= =

1 2πi 1 2πi

2 Γ

2

Γ

Γ

Γ

λ−z (λ )−z (λ − A)−1 (λ − A)−1 dλ dλ λ−z (λ )−z (λ − A)−1 − (λ − A)−1 dλ dλ, λ −λ

where Γ is the integral contour given by (2.107), and Γ is another integral contour obtained by substituting ω and δ for ω and δ in such a way that ω < ω < π and 0 < δ < δ, respectively. Then, by the Cauchy integral formula, 1 2πi

Γ

(λ )−z 1 dλ = λ −λ 2πi

|λ −λ|= 2δ

(λ )−z dλ = λ−z , λ −λ

7 Fractional Powers of Linear Operators

1 2πi

Γ

95

λ−z dλ = 0. λ − λ

Therefore, we obtain the desired law 1 A−z A−z = λ−(z+z ) (λ − A)−1 dλ = A−(z+z ) . 2πi Γ We have thus verified the following fact. Theorem 2.22 The L(X)-valued function A−z is an analytic semigroup defined in the half-plane {z ∈ C; Re z > 0}. Using this fact, let us next observe that A−z is invertible for every Re z > 0. Indeed, if A−z0 F = 0 for some z0 , then, for any x > 0, A−(z0 +x) F = A−x A−z0 F = 0; since A−z F is analytic, A−z F must vanish identically. Then, by Theorem 2.21, F is necessarily equal to 0. Hence, for Re z > 0, A−z is one-to-one, namely, its inverse Az is a single-valued linear operator of X. The following definition is meaningful: Az = (A−z )−1

for Re z > 0.

If 0 < Re z1 < Re z2 , then A−z2 = A−z1 A−(z2 −z1 ) ; therefore, R(A−z2 ) ⊂ R(A−z1 ); this then means that D(Az2 ) ⊂ D(Az1 ). In particular, if 0 < Re z < k, where k is an integer, then D(Ak ) ⊂ D(Az ); therefore, D(Az ) is dense in X due to (2.91). Hence, Az is a densely defined, closed linear operator of X. In view of Theorem 2.21 it is natural to define A0 = 1. Then, for every real number −∞ < x < ∞, the fractional power Ax of A has been defined. The following properties are already established: (1) Ax are bounded operators on X for −∞ < x < 0, A0 = 1, and Ax are densely defined, closed linear operators of X for 0 < x < ∞. (2) D(Ax2 ) ⊂ D(Ax1 ) for 0 ≤ x1 < x2 < ∞. (3) Ax Ax = Ax Ax = Ax+x for any −∞ < x, x < ∞.

7.3 Law of Exponent, II Let α be a fixed exponent such that 0 < α < 1. We shall show that the fractional power Aα of a sectorial operator A is a sectorial operator with angle ≤ αωA and satisfies the second law of exponent, i.e., (Aα )z = Aαz ,

z ∈ C, Re z = 0.

We consider the family of bounded operators defined by 1 1 Rα (λ) = (μ − A)−1 dμ, 2πi Γ λ − μα

(2.112)

(2.113)

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2 Sectorial Operators

where Γ is the same integral contour given in (2.107). Let ω be another angle satisfying ω < ω < π . Then, if λ ∈ / Σαω , then λ = μα for any μ ∈ Γ . So, Rα (λ) is defined for λ ∈ / Σαω . By a direct calculation, Rα (λ) is seen to be an analytic function with values in L(X). Furthermore, by similar techniques as used to verify (2.111), we can verify that 1 1 / Σαω . (μ − A)−1 dμ, λ, λ ∈ Rα (λ)Rα (λ ) = 2πi Γ (λ − μα )(λ − μα ) 1 1 1 1 Since (λ−μα )(λ −μα ) = − λ−λ λ−μα − λ −μα , it follows that Rα (λ)Rα (λ ) = −

1 [Rα (λ) − Rα (λ )], λ − λ

λ, λ ∈ / Σαω ,

that is, Rα (λ) define a pseudo-resolvent. Since 0 − R(0)−1 = Aα , we see by Theorem 1.8 that this pseudo-resolvent is generated by Aα , i.e., Rα (λ) = (λ − Aα )−1 . Hence, we conclude that σ (Aα ) ⊂ Σαω . In order to verify (2.2), however, we need more careful calculations in which the integral contour in (2.113) must be shifted suitably depending on φ = | arg λ|, π−ωA (φ − αωA ) + ωA . Clearly, where αω ≤ φ ≤ π . We introduce the angle ω˜ = π−αω A ω˜ satisfies

π(1−α) (φ − αωA ), π > φ − α ω˜ = π−αω A (2.114) π(1−α) 2π(1 − α) > α(ω˜ − 2π) − (φ − 2π) > π−αω (φ − αωA ). A As a branch of the function μα , we retake the domain {μ; ω˜ − 2π < arg μ < ω}. ˜ Then, in view of the relation {arg μα ; ω˜ − 2π < arg μ < ω} ˜ ⊂ [α(ω˜ − 2π), α ω] ˜ ⊂ (φ − 2π, φ) (due to (2.114)), the equation λ − μα = 0 (λ being fixed) has no solution in this domain. Noting this fact, we then shift the integral contour Γ to the replicated ˜ ). After some calculations, it turns out that half-lines (∞ei ω˜ , 0] ∪ [0, ∞ei(ω−2π) ˜ ω−π)} ˜ sin πα ei{ω+α( π ∞ ρα × (ρei ω˜ − A)−1 dρ. ˜ (λ − ρ α eiα ω˜ )(λ − ρ α eiα(ω−2π) ) 0

Rα (λ) = −

From this representation we have

∞

Mω˜ ρ α−1 dρ ˜ |λ − ρ α eiα ω˜ ||λ − ρ α eiα(ω−2π) | 0 ∞ α−1 Mω˜ sin πα ρ dρ ≤ iφ α iα ω ˜ i(φ−2π) ˜ π|λ| |e − ρ e ||e − ρ α eiα(ω−2π) | 0 ∞ α−1 ρ Mω˜ sin πα = 2α α π|λ| 0 ρ − 2ρ cos(φ − α ω) ˜ +1

Rα (λ) ≤

sin πα π

7 Fractional Powers of Linear Operators

×

97

dρ ρ 2α

− 2ρ α cos[α(ω˜ − 2π) − (φ

− 2π)] + 1

.

Due to (2.114), cos(φ − α ω) ˜ < 1 and cos[α(ω˜ − 2π) − (φ − 2π)] < 1. Therefore, C for λ ∈ / Σαω . we conclude that Rα (λ) ≤ |λ| Theorem 2.23 For 0 < α < 1, Aα is a sectorial operator of X with angle ≤ αωA . Proof To complete the proof, it suffices to notice that ω , together with ω, can be taken arbitrarily close to ωA . We now know that Aα is a sectorial operator and its resolvent is given by (2.113). Then, verification of (2.112) is immediate. The proof may be left to the reader. Remark 2.8 For λ < 0, we can obtain more precise estimates. Indeed, if φ = ω˜ = π , then ∞ ρ α−1 dρ (λ − Aα )−1 ≤ Mπ sin πα π|λ| ρ 2α − 2ρ α cos π(1 − α) + 1 0 ∞ dρ Mπ Mπ sin πα = = , λ < 0, 2 2 πα|λ| |λ| (ρ + cos πα) + sin πα 0 (2.115) where Mπ is the constant appearing in (2.2) with ω = π .

7.4 Moment Inequality This subsection is devoted to showing the moment inequality of the fractional powers. For 0 < θ < 1, let Aθ be the fractional powers of a sectorial operator A. Let Mπ π be the constant in (2.2) with ω = π , i.e., (λ − A)−1 ≤ M |λ| for λ < 0. Applying θ−1 (2.108) to A , we have for U ∈ D(A), sin(πθ ) ∞ θ−1 Aθ U = Aθ−1 AU = − ρ (ρ + A)−1 AU dρ. (2.116) π 0 Here,

0

R

ρ θ−1 A(ρ + A)−1 U dρ ≤

R

ρ θ−1 A(ρ + A)−1 U dρ

0

≤ (Mπ + 1) 0

R

ρ θ−1 dρ U =

Mπ + 1 θ R U , θ

98

2 Sectorial Operators

where 0 < R < ∞ is an arbitrary number. Similarly, ∞ ∞ θ−1 −1 ρ (ρ + A) AU dρ ρ θ−1 (ρ + A)−1 AU dρ ≤ R

R

≤ Mπ

∞

Mπ θ−1 R AU . 1−θ

ρ θ−2 dρAU =

R

Therefore,

sin(πθ ) U θ AU θ−1 A U ≤ , (Mπ + 1) R + R π θ 1−θ θ

0 < R < ∞.

The right-hand side hits the minimum at R = AU /U . Hence, Aθ U ≤

sin(πθ ) (Mπ + 1)AU θ U 1−θ , πθ(1 − θ )

U ∈ D(A).

(2.117)

This inequality is called a moment inequality for Aθ . If we put U = A−1 F , then we have another moment inequality of the form Aθ−1 F ≤

sin(πθ ) (Mπ + 1)F θ A−1 F 1−θ , πθ(1 − θ )

F ∈ X.

(2.118)

Let 0 ≤ α < β < γ ≤ 1. We know that Aγ −α is also a sectorial operator. So, applying the moment inequality to Aγ −α with θ = γβ−α −α , we obtain that Aβ−α V ≤

2 (sin π(β−α) γ −α )(γ − α)

π(γ − β)(β − α)

β−α

γ −β

(Mπ + 1)Aγ −α V γ −α V γ −α .

Putting V = Aα U , we conclude that Aβ U ≤

2 (sin π(β−α) γ −α )(γ − α)

π(γ − β)(β − α) β−α

(Mπ + 1) γ −β

× Aγ U γ −α Aα U γ −α , Remark 2.9 We remark that the function θ=

1 2

and hence

sin(πθ) πθ(1−θ)

≤

sin(πθ) πθ(1−θ)

U ∈ D(Aγ ).

(2.119)

for 0 < θ < 1 hits its maximum at

4 π.

Remark 2.10 The generalized moment inequality (2.119) can be shown for any triplet of exponents 0 ≤ α < β < γ < ∞.

7.5 Comparison of Domains of Fractional Powers Let A be a sectorial operator of a Banach space X. Let Aθ , 0 < θ < 1, be the fractional powers of A. We are concerned with comparing D(Aθ ) with some other spaces.

7 Fractional Powers of Linear Operators

99

Let us introduce the spaces " ! Dθ (A) = U ∈ X; sup ρ θ A(ρ + A)−1 U < ∞ ,

0 ≤ θ ≤ 1.

(2.120)

0 θ ), we observe that ∞ θ −1 −ν ν−θ −1 ρ (ρ + B) [A − B]A A (ρ + A) U dρ 1 ∞ ρ θ−1 ρ max{ν−θ−1,−1} dρU . ≤ (Mπ + 1)2 (1 + N ) 1

Therefore, [Aθ − B θ ]A−θ U ≤ C(A−θ U + U ),

U ∈ D(A).

The denseness of D(A) in X and closedness of B θ yield that this estimate holds true for any F ∈ X. It then follows that D(B θ ) ⊃ R(A−θ ) = D(Aθ ).

7.6 Fractional Powers of Adjoint Operators Let X be a reflexive Banach space, and let X ∗ be the adjoint space of X. Let {X, X ∗ } be an adjoint pair. Let A be a sectorial operator of X, and let A∗ be its adjoint operator. By Theorem 1.19, A∗ is seen to be a sectorial operator of X ∗ . So, their fractional powers A−z and A∗(−z) are defined for Re z > 0 by the integral formula (2.106). We can then easily prove that A∗(−z) = A(−z)∗ for Re z > 0 (by the same argument as for (2.100)). Considering these inverses, we prove that A∗z = Az∗ for Re z > 0. In particular, for real powers, we see that A∗x = Ax∗ ,

−∞ < x < ∞.

(2.126)

7.7 Fractional Powers and Analytic Semigroups Let A be a sectorial operator of X satisfying (2.92) and (2.93). As verified in Sect. 5, −A generates an analytic semigroup e−zA on X. By Proposition 2.4, R(e−tA ) ⊂ D(Aθ ) for any t > 0 and any 0 < θ < ∞. In this subsection, we shall investigate various properties concerning Aθ e−tA . We first estimate the norm of Aθ e−tA . Consider the operator defined by 1 θ E (t) = λθ e−tλ (λ − A)−1 dλ, 0 < t < ∞, 0 < θ < ∞, 2πi Γ

102

2 Sectorial Operators

where Γ is the integral contour given by (2.107) but with ωA < ω < π2 . Then, by arguing in a similar way as for (2.111), it is easily verified that A−θ E θ (t) = E θ (t)A−θ = e−tA . Hence, E θ (t) = Aθ e−tA = e−tA Aθ , i.e., 1 λθ e−tλ (λ − A)−1 dλ, 0 < t < ∞, 0 < θ < ∞. (2.127) Aθ e−tA = 2πi Γ We here shift Γ to Γω : λ = ρe±iω , 0 ≤ ρ < ∞ (i.e., δ → 0). Then, we obtain that ∞ |λ|θ−1 e−t Re λ |dλ| = Mω ρ θ−1 e−tρ cos ω dρ Aθ e−tA ≤ Mω 0

Γw

= Mω (cos ω)−θ (θ )t −θ ,

0 < t < ∞, 0 < θ < ∞,

where (·) denotes the gamma function. Let now 0 < θ ≤ 1. We have t t 1−θ −τ A ≤ Cθ [e−tA − 1]A−θ ≤ A e dτ τ θ−1 dτ ≤ Cθ t θ , 0

0

(2.128)

0 < t < ∞. (2.129)

Meanwhile, we can verify for each 0 < θ ≤ 1 that, as t → 0, t θ Aθ e−tA converges to 0 strongly on X.

(2.130)

Indeed, if U ∈ D(Aθ ), it is clear that t θ e−tA Aθ U → 0 in X as t → 0. Then, for general F ∈ X, the convergence is verified by the uniform boundedness of t θ Aθ e−tA due to (2.128) and the denseness of D(Aθ ) in X. Similarly, for each 0 ≤ θ < 1, as t → 0, t −θ [e−tA − 1]A−θ converges to 0 strongly on X.

(2.131)

Indeed, if U ∈ D(A1−θ ), then t −θ [e−tA − 1]A−θ U = t −θ [e−tA − 1]A−1 A1−θ U → 0 in X as t → 0 due to (2.129). Then, for general F ∈ X, the convergence is verified by the uniform boundedness of t −θ [e−tA − 1]A−θ due to (2.129) and the denseness of D(A1−θ ) in X. Using the properties obtained above, we will show the following result. Theorem 2.27 Let 0 < β ≤ 1 and U0 ∈ D(Aβ ). Then, for any σ such that 0 < σ < β ≤ 1 and any 0 < T < ∞, Ae−tA U0 ∈ Fβ,σ ((0, T ]; X) with the estimate Ae−tA U0 Fβ,σ ((0,T ];X) ≤ CT Aβ U0 X . Proof In fact, since t 1−β Ae−tA U0 = t 1−β A1−β e−tA Aβ U0 ,

(2.132)

7 Fractional Powers of Linear Operators

103

(2.130) yields that lim t 1−β Ae−tA U0 = 0 when 0 < β < 1,

t→0

lim t 1−β Ae−tA U0 = AU0

t→0

when β = 1.

Therefore, t 1−β Ae−tA U0 has a limit as t → 0. By (2.128) and (2.129), s 1−β+σ A[e−tA − e−sA ]U0 [e−(t−s)A − 1]A−σ ≤ (t − s)σ (t − s)σ × s 1−β+σ A1−β+σ e−sA (Aβ U0 ) ≤ CT Aβ U0 ,

0 < s < t ≤ T.

Therefore, (1.6) is fulfilled. In a similar way, s 1−β+σ A[e−tA − e−sA ]U0 ≤ CT s 1−β+σ A1−β+σ e−sA (Aβ U0 ). (t − s)σ Hence, by (2.130), lim sup

t→0 0<s 0. Hence, by (1.10), e−tAU = U (t).

1.3 Perturbation for Generators of Analytic Semigroups We shall consider some perturbation of generators of analytic semigroup. Let A be a sectorial operator of a Banach space X with angle ωA < π2 . Let B a linear operator of X. For some exponent 0 < ν ≤ 1, we assume that B is dominated by A1−ν , where A1−ν is the fractional power of A, in the sense that D(A1−ν ) ⊂ D(B), and the estimate BAν−1 ≤ D

(3.8)

holds with some constant D > 0. Let ω > ωA . Since D(A) ⊂ D(B), we have

λ − (A + B) = 1 − B(λ − A)−1 (λ − A)

for λ ∈ / Σω .

Here, B(λ − A)−1 ≤ BAν−1 A1−ν (λ − A)−1 . In addition, by the moment inequality (2.117) (with θ = 1 − ν), 1−ν A (λ − A)−1 ≤ C A(λ − A)−1 1−ν (λ − A)−1 ν ≤ C|λ|−ν . Therefore, if λ ∈ / Σω and |λ| ≥ R and if R > 0 is fixed sufficiently large, then λ ∈ ρ(A + B), and the resolvent is given by

−1 (λ − (A + B))−1 = (λ − A)−1 1 − B(λ − A)−1 . This then means that A + B satisfies (3.2) and (3.3) with the ω fixed above and β = −(sin ω)−1 R. We thus obtain the following theorem. Theorem 3.3 Let A be a sectorial operator with angle ωA < π2 . If a linear operator B satisfies (3.8), then −(A + B) satisfies (3.2) and (3.3) with any angle ω > ωA . So, A + B generates an analytic semigroup e−z(A+B) .

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3 Linear Evolution Equations

2 Cauchy Problems 2.1 Construction of Solutions Let us consider the Cauchy problem for a linear evolution equation dU 0 < t ≤ T, dt + AU = F (t), U (0) = U0 ,

(3.9)

in a Banach space X, where 0 < T < ∞ is fixed time. Here, A is a sectorial operator of X with angle ωA < π2 , that is, A satisfies (2.92) and (2.93). Throughout this section, the notation C will stand for a universal constant which is determined in each occurrence by the exponents and constants in (2.92)–(2.93) and by A−1 in a specific way. The function F is a given external force function defined in (0, T ] with values in X. We assume that F belongs to F ∈ Fβ,σ ((0, T ]; X),

0 0. So, it suffices only to show that Aβ U (t) is continuous at t = 0. We use the expression A [U (t) − U0 ] = [e−tA − 1]Aβ U0 +

β

t

Aβ e−(t−τ )A [F (τ ) − F (t)] dτ

0

+ [1 − e

−tA

]A

β−1

F (t)

= J1 (t) + J2 (t) + J3 (t).

2 Cauchy Problems

127

Clearly, limt→0 J1 (t) = 0 in X. In addition, it is seen that

t

J2 (t)X ≤ C

(t − τ )σ −β τ β−σ −1 dτ sup

0

0 0 as an L(X)-valued function. Hence, the proof of proposition will be completed if we verify the following lemma.

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3 Linear Evolution Equations

Lemma 3.2 For 0 ≤ θ ≤ 2, A(t)θ e−τ A(t) − A(s)θ e−τ A(s) ≤ Cτ ν−θ−1 |t − s|μ ,

τ > 0, 0 ≤ s, t ≤ T .

Proof Notice the formula (λ − A(t))−1 − (λ − A(s))−1

= −A(t)(λ − A(t))−1 A(t)−1 − A(s)−1 A(s)(λ − A(s))−1 . Then, by the same techniques as for (3.32), we observe that (λ − A(t))−1 − (λ − A(s))−1 ≤ C|λ|−ν |t − s|μ , λ ∈ / Σω , 0 ≤ s, t ≤ T . (3.33) We use (2.127) and write A(t)θ e−τ A(t) − A(s)θ e−τ A(s) =

1 2πi

λθ e−τ λ (λ − A(t))−1 − (λ − A(s))−1 dλ, Γω

where Γω : λ = ρe±iω , 0 ≤ ρ < ∞, is an integral contour. Then, using (3.33), we obtain that A(t)θ e−τ A(t) − A(s)θ e−τ A(s) ≤ C |λ|θ−ν e−τ Re λ |dλ||t − s|μ Γω

≤ Cτ ν−θ−1 |t − s|μ . When θ = 0 and ν = 1, this integral may not be convergent in a neighborhood λ = 0. But in this case, we are able to shift the integral contour Γω in the same way as for proving (2.128). Then, the assertion of lemma is proved without any change. When θ = 0, the continuity of the semigroup is extended at τ = 0. In fact, we verify the following proposition. Proposition 3.2 e−τ A(t) is strongly continuous on X for τ ≥ 0 and 0 ≤ t ≤ T . Proof Obviously, it suffices to prove the result at τ = 0 only. We already know that −τ A(t) [e − 1]A(t)−ν ≤ Cτ ν , τ ≥ 0, 0 ≤ t ≤ T . Let U ∈ D(A(0)). Then, [e−τ A(t) − 1]U ≤ [e−τ A(t) − 1]A(t)−ν A(t)ν A(0)−1 A(0)U ≤ CA(0)U τ ν ,

τ ≥ 0, 0 ≤ t ≤ T .

Furthermore, the denseness of D(A(0)) in X, together with the uniform boundedness of the norms e−τ A(t) − 1, yields that e−τ A(t) F → F in X as τ → 0 for every F ∈ X, the convergence being uniform in 0 ≤ t ≤ T .

4 Nonautonomous Abstract Evolution Equations

137

4.3 Structural Assumptions and Yosida Approximation Let a family of sectorial operators A(t) satisfy (3.27), (3.28), (3.29), (3.30), and (3.31). For each n = 1, 2, 3, . . . , let An (t), 0 ≤ t ≤ T , be a family of Yosida approximations of A(t). The purpose of this subsection is to show that An (t), 0 ≤ t ≤ T , also satisfy these conditions uniformly in n; especially ω, μ, and ν can be the same as for A(t). As already verified by (2.84) and (2.86), σ (An (t)) ⊂ Σω , and the estimate (λ − An (t))−1 ≤ M/|λ|, ˜ (3.34) λ∈ / Σω , 0 ≤ t ≤ T , holds with some constant M˜ ≥ 1 determined by M and ω only. In the meantime, it is possible to verify from (3.30) that

An (t)ν An (t)−1 − An (s)−1 ≤ N˜ |t − s|μ , 0 ≤ s, t ≤ T .

(3.35)

In fact, since An (t)−1 = n−1 + A(t)−1 due to (2.81), we have An (t)−1 − An (s)−1 = A(t)−1 − A(s)−1 . Then, (3.35) follows as a consequence of the two propositions presented below. Proposition 3.3 Let 0 < θ ≤ 1. For n = 1, 2, 3, . . . , it holds that An (t)θ A(t)−θ = (1 + n−1 A(t))−θ ,

0 ≤ t ≤ T.

(3.36)

Proof When θ = 1, the assertion is clear. So, we consider the case where 0 < θ < 1 only. Let us prove that A(t)−θ = An (t)−θ (1 + n−1 A(t))−θ . From (λ − (1 + n−1 A(t)))−1 = n(n(λ − 1) − A(t))−1 it is seen that σ (1 + −1 n A(t)) ⊂ Σω and the estimate (λ − (1 + n−1 A(t)))−1 ≤

M M ≤ , |λ − 1| |λ| sin ω

λ∈ / Σω ,

(3.37)

holds. This means that 1 + n−1 A(t) are also sectorial operators with angles < ω. Their fractional powers are then given by 1 (1 + n−1 A(t))−θ = λ−θ (λ − (1 + n−1 A(t)))−1 dλ 2πi Γλ 1 λ−θ n(n(λ − 1) − A(t))−1 dλ. = 2πi Γλ Since 0 < θ < 1, we can take an integral contour which passes the origin. Here, we choose Γλ : λ = re±iω , 0 ≤ r < ∞. Then, change of variable η = n(λ − 1) yields that 1 (1 + n−1 η)−θ (η − A(t))−1 dη, (1 + n−1 A(t))−θ = 2πi Γη where Γη = Γλ − n, and the branch of the function (1 + n−1 η)−θ is C − (−∞, −n].

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3 Linear Evolution Equations

In the meantime, we observe by (2.81) that −1 λ−θ 1 nλ dλ, An (t)−θ = (n + A(t)) − A(t) 2πi Γλ n − λ n−λ where Γλ is the same contour as above. We here use the integral formula −1 −1 nλ 1 nλ = (η − A(t))−1 dη , λ ∈ Γλ , − A(t) −η n−λ 2πi Γη n − λ

choosing a contour Γη = Γω ∪ Γδ such that Γω : η = re±iω , δ ≤ r < ∞, and Γδ : η = δeiϕ , −ω ≤ ϕ ≤ ω , where ω is an angle such that ωA(t) < ω < ω, and δ is a radius such that 0 < δ < A(t)−1 −1 . Then, on D(A(t)), we obtain that −1 λ−θ 1 nλ (η − A(t))−1 dλ dη (n + A(t)) − η An (t)−θ = (2πi)2 Γλ Γη n − λ n − λ 1 1 nη −θ = (η − A(t))−1 dη (n + A(t)). 2πi Γη n + η n + η Noting that

nη n+η

1 2πi

∈ Σω for η ∈ Γη , we used the formula Γλ

λ−θ dλ = λ − nη (n + η )−1

nη n + η

−θ ,

η ∈ Γη .

nη −θ is given by C − [−n, 0]. We easily notice that the branch of the function n+η We now consider the product 1 nη −θ 1 (η − A(t))−1 dη 2πi Γη n + η n + η 1 × (1 + n−1 η)−θ (η − A(t))−1 dη. 2πi Γη Then, it is deduced by the same techniques as for (2.98) and (2.111) (cf. also (16.4)) that this product is equal to (1 + n−1 λ)−θ nλ −θ 1 (λ − A(t))−1 dλ. 2πi Γλ n+λ n+λ nλ −θ n −θ −θ nλ Furthermore, we observe that n+λ = = n+λ λ for λ ∈ Γλ , since arg n+λ arg(nλ) − arg(n + λ) and −π < arg(nλ) − arg(n + λ) < π . We therefore arrive at (1 + n−1 λ)−θ nλ −θ 1 (λ − A(t))−1 dλ = A(t)−θ (n + A(t))−1 . 2πi Γλ n+λ n+λ Hence, we proved that A(t)−θ = An (t)−θ (1 + n−1 A(t))−θ .

5 Evolution Operators

139

Proposition 3.4 Let 0 < θ ≤ 1. For n = 1, 2, 3, . . . , it holds that An (t)θ A(t)−θ ≤ 2 M + 1 M θ , 0 ≤ t ≤ T . sin ω

(3.38)

In addition, for each t , An (t)θ A(t)−θ converges to 1 strongly on X as n → ∞. Proof Applying the moment inequality (2.118) to 1 + n−1 A(t), we have (1 + n−1 A(t))−θ ≤ 2 M + 1 (1 + n−1 A(t))−1 θ ≤ 2 M + 1 M θ sin ω sin ω due to (3.37). Hence, (3.38) is an immediate consequence of (3.36). The strong convergence of An (t)θ A(t)−θ to 1 is first verified on the domain D(A(t)1−θ ). Indeed, it is seen that An (t)θ A(t)−θ U = An (t)θ−1 (1 + n−1 A(t))−1 A(t)1−θ U → U, U ∈ D A(t)1−θ . Then, the convergence on X is verified by the uniform boundedness of An (t)θ A(t)−θ and the denseness of D(A(t)1−θ ) in X.

5 Evolution Operators In this section, we shall construct under (3.27), (3.28), (3.29), (3.30), and (3.31) the evolution operator U (t, s) for the family of sectorial operators A(t). For each n = 1, 2, 3, . . . , let Un (t, s), 0 ≤ s ≤ t ≤ T , be the evolution operator for the family of Yosida approximations An (t), 0 ≤ t ≤ T , of A(t). In fact, Un (t, s) is obtained as a unique solution to the Cauchy problem of the abstract differential equation dU n dt + An (t)Un = 0, s ≤ t ≤ T , Un (s) = 1, in the space L(X); note that An (·) ∈ C([0, T ]; L(X)) (due to (3.32) and (3.33)). Obviously, the problem is equivalent to the integral equation t An (τ )Un (τ ) dτ, s ≤ t ≤ T . Un (t) = 1 + s

Then, the solution is constructed by the recurrence method as in the proof of Theorems 1.29 and 1.30. The evolution operator Un (t, s) is a continuous function defined for 0 ≤ s ≤ t ≤ T with values in L(X). By the definition, Un (t, s) enjoys the generalized semigroup property Un (t, r)Un (r, s) = Un (t, s), 0 ≤ s ≤ r ≤ t ≤ T , (3.39) 0 ≤ s ≤ T. Un (s, s) = 1,

140

3 Linear Evolution Equations

In addition, Un (t, s) is differentiable both in t and s with the derivatives ∂ Un (t, s) = −An (t)Un (t, s), ∂t ∂ Un (t, s) = Un (t, s)An (s). ∂s

(3.40) (3.41)

Verification of (3.41) is not direct. We have to consider another family of operators U˜ n (t, s), 0 ≤ s ≤ t ≤ T , where U˜ n (t, s) is a solution to the backward Cauchy problem ˜ d Un ˜ ds − Un An (s) = 0, 0 ≤ s ≤ t, U˜ n (t) = 1, for 0 < t ≤ T . Clearly, U˜ n (t, s) satisfies (3.41). Moreover, Un (t, s) = U˜ n (t, s) for every 0 ≤ s ≤ t ≤ T . Indeed, since ∂ ˜ Un (t, τ )Un (τ, s) = 0 for all τ in (s, t), ∂τ it is deduced that U˜ n (t, τ )Un (τ, s) is constant in τ ; hence, taking τ = s and t, we conclude that Un (t, s) = U˜ n (t, s) for every 0 ≤ s ≤ t ≤ T .

5.1 Integral Equations We first derive two integral equations for Un (t, s). By using (3.41) we have t ∂ [Un (t, τ )e−(τ −s)An (s) ] dτ Un (t, s) − e−(t−s)An (s) = − ∂τ s t

Un (t, τ )An (τ ) An (τ )−1 − An (s)−1 = s

× An (s)e−(τ −s)An (s) dτ. Operating An (s)1−ν to this equality from the right-hand side and putting Vn (t, s) = Un (t, s)An (s)1−ν , 0 ≤ s ≤ t ≤ T ,

Dn (t, s) = An (t)ν An (t)−1 − An (s)−1 , 0 ≤ s ≤ t ≤ T , we obtain the first integral equation Vn (t, s) = An (s)1−ν e−(t−s)An (s) t + Vn (t.τ )Dn (τ, s)An (s)2−ν e−(τ −s)An (s) dτ. s

(3.42)

5 Evolution Operators

141

To obtain the other one, we use (3.40). Then, Un (t, s) − e−(t−s)An (t) t

=− An (t)e−(t−τ )An (t) An (t)−1 − An (τ )−1 An (τ )Un (τ, s) dτ. s

Operating An (t) to this equality from the left-hand side and putting Wn (t, s) = An (t)Un (t, s) − An (t)e−(t−s)An (t) ,

0 ≤ s ≤ t ≤ T,

we obtain the integral equation t Wn (t, s) = Rn (t, s) − An (t)2−ν e−(t−τ )An (t) Dn (t, τ )Wn (τ, s) dτ,

(3.43)

s

where

Rn (t, s) = −

t

An (t)2 e−(t−τ )An (t)

s

× [An (t)−1 − An (τ )−1 ]An (τ )e−(τ −s)An (τ ) dτ.

(3.44)

Secondly, we shall derive some norm estimates for Un (t, s) which hold uniformly in n. From (2.133) and (3.35) it is readily seen that the integral kernels of (3.42) satisfy An (s)1−ν e−(t−s)An (s) ≤ C(t − s)ν−1 , Dn (t, s)An (s)2−ν e−(t−s)An (s) ≤ C(t − s)μ+ν−2 , respectively. Then, we can apply Theorem 1.30 to (3.42). As a direct consequence, we have the following proposition. Proposition 3.5 The operator Un (t, s)An (s)1−ν satisfies Un (t, s)An (s)1−ν ≤ C(t − s)ν−1 , 0 ≤ s < t ≤ T ,

(3.45)

C > 0 being independent of n. Concerning the integral kernels of (3.43), we readily verify from (2.133) and (3.35) that An (t)2−ν e−(t−s)An (t) Dn (t, s) ≤ C(t − s)μ+ν−2 . (3.46) As for Rn (t, s), however, we need more careful calculations. Lemma 3.3 Rn (t, s) satisfies the estimates Rn (t, s) ≤ C(t − s)μ+ν−2 , C > 0 being independent of n.

0 ≤ s < t ≤ T,

(3.47)

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3 Linear Evolution Equations

Proof Because of

An (t)2 e−(t−τ )An (t) An (t)−1 − An (τ )−1 An (τ )e−(τ −s)An (τ ) ≤ C(t − τ )μ+ν−2 (τ − s)−1 , we cannot obtain the desired estimate by direct estimation of the integral (3.44). We have to decompose the integral suitably into several terms which admit uniform estimates. As a matter of fact, we write Rn (t, s) in the form t

An (t)2 e−(t−τ )An (t) An (τ )−1 − An (t)−1 An (τ )e−(τ −s)An (τ ) dτ Rn (t, s) = r

r

+

An (t)2 e−(t−τ )An (t) [e−(τ −s)An (τ ) − e−(τ −s)An (s) ] dτ

s r

+

An (t)e−(t−τ )An (t) [An (s)e−(τ −s)An (s) − An (τ )e−(τ −s)An (τ ) ] dτ

s

+ An (t)e−(t−r)An (t) [e−(r−s)An (s) − e−(r−s)An (t) ], where r =

t+s 2 .

(3.48)

Here, we used the formula r An (t)e−(t−τ )An (t) [An (t) − An (s)]e−(τ −s)An (s) dτ s

= An (t)e−(t−r)An (t) [e−(r−s)An (s) − e−(r−s)An (t) ]. As the family of Yosida approximations An (t) also satisfies the same structural assumptions, we can apply Lemma 3.2 to An (t). In other words, for 0 ≤ θ ≤ 2, it holds that An (t)θ e−τ An (t) − An (s)θ e−τ An (s) ≤ Cτ ν−θ−1 |t − s|μ , τ > 0, 0 ≤ s, t ≤ T . (3.49) This estimate is then available to estimate the decomposed terms in the right-hand side of (3.48). Indeed, by (2.133), (3.35), and (3.49) (with θ = 0, 1), we can obtain estimate (3.47). The details of the proof are however left to the reader. It is now ready to apply Theorem 1.29 to (3.43). In fact, we obtain the following estimate. Proposition 3.6 The operator An (t)Un (t, s) satisfies An (t)Un (t, s) ≤ C(t − s)−1 ,

0 ≤ s < t ≤ T,

(3.50)

C > 0 being independent of n. Proof Applying Theorem 1.29 to (3.43), we immediately obtain that Wn (t, s) ≤ C(t − s)μ+ν−2 .

(3.51)

5 Evolution Operators

143

Then, since An (t)e−(t−s)An (t) ≤ C(t − s)−1 , we conclude the estimate of proposition.

5.2 Convergence of Un (t, s) In this subsection, we intend to see convergence of Un (t, s) as n → ∞. To this end, we will utilize the dominated convergence theorem, Theorem 1.31, to the integral equation (3.43). From (2.102) and Proposition 3.4, we observe that, as n → ∞, An (t)2−ν e−(t−s)An (t) Dn (t, s) → A(t)2−ν e−(t−s)A(t) D(t, s) strongly on X, where

D(t, s) = A(t)ν A(t)−1 − A(s)−1 , As

0 ≤ s < t ≤ T.

A(t)2−ν e−(t−s)A(t) D(t, s) = A(t)2 e−(t−s)A(t) A(t)−1 − A(s)−1 ,

the limit function A(t)2−ν e−(t−s)A(t) D(t, s) is continuous for 0 ≤ s < t ≤ T in L(X). For verifying convergence of Rn (t, s), we return to the decomposition (3.48). Then, by (2.102), Rn (t, s) is seen to converge strongly to the bounded linear operator t

R(t, s) = A(t)2 e−(t−τ )A(t) A(τ )−1 − A(t)−1 A(τ )e−(τ −s)A(τ ) dτ r

r

+

A(t)2 e−(t−τ )A(t) [e−(τ −s)A(τ ) − e−(τ −s)A(s) ] dτ

s r

+

A(t)e−(t−τ )A(t) [A(s)e−(τ −s)A(s) − A(τ )e−(τ −s)A(τ ) ] dτ

s

+ A(t)e−(t−r)A(t) [e−(r−s)A(s) − e−(r−s)A(t) ], where r = t+s 2 , for all 0 ≤ s < t ≤ T . Furthermore, by the same techniques as in the proof of Lemma 1.3, R(t, s) is verified to be continuous in L(X) for 0 ≤ s < t ≤ T . It is now ready to apply Theorem 1.31 to (3.43). Indeed, in view of (3.46) and (3.47), Theorem 1.31 immediately provides the following convergence result. Proposition 3.7 As n → ∞, Wn (t, s) converges to a bounded linear operator W (t, s)

(3.52)

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3 Linear Evolution Equations

strongly on X for all 0 ≤ s < t ≤ T . The limit W (t, s) is characterized as a solution to the integral equation t W (t, s) = R(t, s) − A(t)2−ν e−(t−τ )A(t) D(t, τ )W (τ, s) dτ. (3.53) s

Since Un (t, s) = e−(t−s)An (t) + An (t)−1 Wn (t, s), as a consequence of the proposition it follows that Un (t, s) is also strongly convergent. We then define the operator U (t, s) as the limit of Un (t, s), namely, as n → ∞, Un (t, s) → U (t, s) = e−(t−s)A(t) + A(t)−1 W (t, s)

(3.54)

strongly on X for all 0 ≤ s < t ≤ T . We see from relation (3.54) that R(U (t, s)) ⊂ D(A(t)) and have A(t)U (t, s) = A(t)e−(t−s)A(t) + W (t, s). As a consequence, An (t)Un (t, s) → A(t)U (t, s) = A(t)e−(t−s)A(t) + W (t, s)

(3.55)

strongly on X for all 0 ≤ s < t ≤ T . If U ∈ D(A(s)1−ν ), then Un (t, s)An (s)1−ν U → U (t, s)A(s)1−ν U . But, since the norms Un (t, s)An (s)1−ν are uniformly bounded in n (due to (3.45)), we verify that U (t, s)A(s)1−ν has a bounded extension on X. Moreover, as n → ∞, Un (t, s)An (s)1−ν → U (t, s)A(s)1−ν

(3.56)

strongly on X for all 0 ≤ s < t ≤ T .

5.3 Basic Properties of U (t, s) For 0 ≤ s ≤ T , we define U (s, s) = 1. Then, by (3.39), U (t, s) also enjoys the generalized semigroup property U (t, s) = U (t, r)U (r, s), 0 ≤ s ≤ r ≤ t ≤ T , (3.57) U (s, s) = 1, 0 ≤ s ≤ T . The operator U (t, s) has the following properties. Proposition 3.8 A(t)U (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T satisfying the estimate A(t)U (t, s) ≤ C(t − s)−1 , with some constant C > 0.

0 ≤ s < t ≤ T,

(3.58)

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145

Proof As W (t, s) has been obtained as a solution to the integral equation (3.53), W (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T . This, together with Proposition 3.1 (θ = 1), then yields the continuity of A(t)U (t, s) in the uniform topology. Estimate (3.58) is an immediate consequence of (3.50) and (3.55). Proposition 3.9 U (t, s) is an L(X)-valued strongly continuous function for 0 ≤ s ≤ t ≤ T with the estimate U (t, s) ≤ C,

0 ≤ s ≤ t ≤ T,

(3.59)

with some constant C > 0. Proof From (3.42) we have Un (t, s) = e

−(t−s)An (s)

+

t

Vn (t, τ )Dn (τ, s)An (s)e−(τ −s)An (s) dτ.

s

Therefore, by (2.102), (3.35), and (3.45), Un (t, s) is estimated by t

Un (t, s) ≤ C + C (t − τ )ν−1 (τ − s)μ−1 dτ ≤ C 1 + (t − s)μ+ν−1 ≤ C. s

This, together with (3.54), then yields estimate (3.59). As U (t, s) = A(t)−1 A(t)U (t, s), Proposition 3.8, together with (3.32), yields the continuity of U (t, s) in L(X) for 0 ≤ s < t ≤ T . So, it suffices to verify that U (t, s) → 1 strongly as (t, s) → (r, r) with s ≤ t , where 0 ≤ r ≤ T . By the same reason as above, we have Un (t, s) − e−(t−s)An (s) ≤ C(t − s)μ+ν−1 . So, letting n → ∞, U (t, s) − e−(t−s)A(s) ≤ C(t − s)μ+ν−1 . Therefore, as (t, s) → (r, r), U (t, s) − e−(t−s)A(s) → 0. Then, the desired strong convergence of U (t, s) is deduced from Proposition 3.2. We finally show the differentiability of U (t, s) in t . Let 0 ≤ s < t ≤ T . In view of (3.50), let n → ∞ in the formula t+h −1 −1 h [Un (t + h, s) − Un (t, s)] = −h An (τ )Un (τ, s) dτ. t

By (3.55) we obtain that −1

h

−1

t+h

[U (t + h, s) − U (t, s)] = −h

A(τ )U (τ, s) dτ, t

146

3 Linear Evolution Equations

which shows that U (t, s) is differentiable in t in the uniform topology with the derivative ∂ U (t, s) = −A(t)U (t, s), 0 ≤ s < t ≤ T . (3.60) ∂t We here sum up the obtained results concerning the evolution operator. Theorem 3.8 Let A(t) be a family of sectorial operators of X satisfying the structural assumptions (3.27), (3.28), (3.29), (3.30), and (3.31). Then, there exists a unique evolution operator U (t, s) satisfying (3.57) and having the following properties. A(t)U (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T with estimate (3.58), and U (t, s) is an L(X)-valued strongly continuous function for 0 ≤ s ≤ t ≤ T with estimate (3.59). Furthermore, for t > s, U (t, s) is differentiable in t with the derivative (3.60). Proof It remains only to show the uniqueness of the evolution operator. Let U˜ (t, s) be any other evolution operator for the family A(t). For s < τ ≤ t , we have ∂ Un (t, τ )U˜ (τ, s) = Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ, s). ∂τ Integrating this equality in τ on the interval [s + ε, t], where ε > 0 is a small positive number, we obtain that t U˜ (t, s) − Un (t, s + ε)U˜ (s + ε, s) = Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ, s) dτ. s+ε

Letting n → ∞, U˜ (t, s) − U (t, s + ε)U˜ (s + ε, s) = 0 (due to (2.89)). Finally, letting ε → 0, we conclude that U˜ (t, s) = U (t, s). Hence, the evolution operator is uniquely determined from A(t).

6 Cauchy Problems We consider the Cauchy problem dU dt + A(t)U = F (t),

0 < t ≤ T,

U (0) = U0 .

(3.61)

Here, A(t), 0 ≤ t ≤ T , is a family of sectorial operators of X which satisfies all the structural assumptions (3.27), (3.28), (3.29), (3.30), and (3.31). The function F is a given function on (0, T ] which belongs to the function space: F ∈ Fβ,σ ((0, T ]; X), The initial value U0 is taken in X.

0 < σ < β ≤ 1.

(3.62)

6 Cauchy Problems

147

Let U (t, s) be the evolution operator for the family of the operators A(t) constructed in the preceding section. We show the following existence and uniqueness result. Theorem 3.9 Under (3.27), (3.28), (3.29), (3.30), and (3.31), for any F belonging to (3.62) and any U0 ∈ X, there exists a unique solution U to (3.61) in the function space: U ∈ C([0, T ]; X) ∩ C1 ((0, T ]; X),

AU ∈ C((0, T ]; X),

with the estimate dU U (t) + t (t) + tA(t)U (t) ≤ C(U0 + F Fβ,σ ), dt Moreover, U is necessarily given by the formula t U (t, τ )F (τ ) dτ, U (t) = U (t, 0)U0 +

(3.63)

0 < t ≤ T . (3.64)

0 ≤ t ≤ T.

(3.65)

0

Proof In order to prove this theorem, we will go back to the stage of constructing the evolution operator U (t, s). Let An (t) be the Yosida approximations for A(t) for each n = 1, 2, 3, . . . , and let Un (t, s) be the evolution operator for An (t). Using the operator Un (t, s), we define an X-valued function of the form t Un (t) = Un (t, 0)U0 + Un (t, τ )F (τ ) dτ, 0 ≤ t ≤ T . 0

In view of (3.54), as n → ∞, this function converges to the function U given by (3.65) in X pointwise. It is easy to see that U is an X-valued continuous function on [0, T ]. By a direct calculation, we verify from (3.40) that Un is differentiable for t > 0 and the derivative is given by dUn (t) = −An (t)Un (t) + F (t), dt

0 < t ≤ T.

Therefore, for t > 0, −1

h

−1

[Un (t + h) − Un (t)] = h

t+h

[F (τ ) − An (τ )Un (τ )] dτ.

(3.66)

t

In the meantime, we consider the function An (t)Un (t). This function can be decomposed in the form t An (t)Un (t) =An (t)Un (t, 0)U0 + An (t)Un (t, τ )[F (τ ) − F (t)] dτ + 0

0 t

Wn (t, τ ) dτ F (t) + [1 − e−tAn (t) ]F (t).

148

3 Linear Evolution Equations

Here, we used the formula

t

An (t)e−(t−τ )An (t) dτ = 1 − e−tAn (t)

0

(which follows from (2.130)). By (1.8), (3.51), and (3.50), we then obtain the estimate An (t)Un (t) ≤ CF,U0 t −1 ,

0 < t ≤ T.

(3.67)

The constant CF,U0 > 0 depends on F and U0 . Furthermore, by (3.52) and (3.55), the function is seen to converge to the function W (t) = A(t)U (t, 0)U0 +

t

A(t)U (t, τ )[F (τ ) − F (t)] dτ

0

t

+

W (t, τ ) dτ F (t) + [1 − e−tA(t) ]F (t)

0

in X pointwise. The limit function is also verified to be an X-valued continuous function for 0 < t ≤ T by arguing in a similar way as in the proof of Lemma 1.3. Since Un (t) = An (t)−1 An (t)Un (t), it follows that U (t) = A(t)−1 W (t); therefore, U (t) ∈ D(A(t)) and A(t)U (t) = W (t) for each 0 < t ≤ T . In view of (3.67), let n → ∞ in (3.66). Then, h−1 [U (t + h) − U (t)] = h−1

t+h

[F (τ ) − A(τ )U (τ )] dτ,

0 < t ≤ T.

t

Hence, dU (t) = −A(t)U (t) + F (t), dt

0 < t ≤ T,

which shows that U is a C1 function on (0, T ] and satisfies the equation of (3.61). It is easy to verify that U satisfies (3.64). It now remains only to show the uniqueness of solution. Let U˜ be any other solution in the space (3.63). Let 0 < ε < t ≤ T . By (3.41), ∂ Un (t, τ )U˜ (τ ) = Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ ) + Un (t, τ )F (τ ). ∂τ So, U˜ (t) = Un (t, ε)U˜ (ε) +

t

Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ ) dτ

ε

t

+

Un (t, τ )F (τ ) dτ. ε

7 Conditions (3.29) and (3.30)

149

Letting n → ∞, we obtain that U˜ (t) = U (t, ε)U˜ (ε) +

t

U (t, τ )F (τ ) dτ. ε

Finally, letting ε → 0, we conclude that U˜ (t) = U (t) for every 0 ≤ t ≤ T .

Remark 3.3 We needed the derivative of Un (t, s) in s in establishing the uniqueness of solution to problem (3.61). On the other hand, we used the derivative of Un (t, s) in t in constructing the solution.

7 Conditions (3.29) and (3.30) The objective of this section is to see how one can verify conditions (3.29) and (3.30) in concrete examples. The case where ν = 1 is the most favorable one. In this case, since

(3.68) A(t) A(t)−1 − A(s)−1 = −[A(t) − A(s)]A(s)−1 , we can calculate directly the difference A(t) − A(s) in order to verify (3.30). On the other hand, when ν < 1, there is no convenient representation for the term A(t)ν [A(t)−1 − A(s)−1 ], for we have to consider the fractional power A(t)ν .

7.1 Case of Hilbert Spaces Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. We consider a family of sesquilinear forms a(t; U, V ), 0 ≤ t ≤ T , which are all defined on Z and satisfy (2.6) and (2.7) with some uniform constants M > 0 and δ > 0, respectively. In addition, we assume the Hölder condition |a(t; U, V ) − a(s; U, V )| ≤ N |t − s|μ U V ,

0 ≤ s, t ≤ T ; U, V ∈ Z, (3.69)

with some fixed exponent 0 < μ ≤ 1 and some constant N > 0. We already know (see Chap. 1) that, from the sesquilinear forms, sectorial operators A(t), 0 ≤ t ≤ T , of Z ∗ and equally sectorial operators A(t), 0 ≤ t ≤ T , of X are defined. Consider first the family of A(t)’s. By Theorem 1.24, D(A(t)) coincides with Z, namely, (3.29) is fulfilled with ν = 1. In addition, using the formula (3.68) for A(t), we observe that

A(t) A(t)−1 − A(s)−1 Φ, U

= − a t; A(s)−1 Φ, U − a s; A(s)−1 Φ, U , Φ ∈ Z ∗ , U ∈ Z. (3.70)

150

3 Linear Evolution Equations

So, (3.69) yields directly that

A(t) A(t)−1 − A(s)−1 Φ ≤ N δ −1 |t − s|μ Φ∗ , ∗

Φ ∈ Z∗.

Thus, under (3.69), A(t)’s satisfy (3.29) and (3.30) with ν = 1 and the μ.

(3.71)

Secondly, consider the family of A(t)’s. According to Theorem 2.32, we have D(A(t)∗θ ) ⊂ Z for any 12 < θ ≤ 1. Therefore, for any ν such that 0 < ν < 12 , we observe from (3.70) that

A(t)ν A(t)−1 − A(s)−1 F, G = − a t; A(s)−1 F, A(t)∗(ν−1) G

(3.72) − a s; A(s)−1 F, A(t)∗(ν−1) G , F, G ∈ X. Hence, (3.69) implies that

A(t)ν A(t)−1 − A(s)−1 F, G ≤ C|t − s|μ |F ||G|, namely,

A(t)ν A(t)−1 − A(s)−1 F ≤ N |t − s|μ |F |,

F ∈ X, G ∈ X,

F ∈ X.

Thus, under (3.69), A(t) satisfy (3.29) and (3.30) with any 0 < ν

0

7 Conditions (3.29) and (3.30)

151

uniformly with respect to the variable t, that c(x, t) satisfies condition (2.20), and that aij ∈ C([0, T ]; C1 (Ω)) ∩ Cμ ([0, T ]; Lp (Ω)), (3.74) c ∈ C([0, T ]; L∞ (Ω)) ∩ Cμ ([0, T ]; Lp (Ω)), with some exponents 0 < μ ≤ 1 and 1 < p < ∞. We already know that one can define, from the forms a(t; ·, ·), realizations

Ap (t) of the differential operators − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in the space Lp (Ω), 1 < p < ∞, under the Neumann-type boundary conditions ∂ν∂u = 0 on A(t) ∂Ω. We are concerned with the family of sectorial operators Ap (t), 0 ≤ t ≤ T , in the Banach space Lp (Ω). It is known by (2.77) and (2.78) that D(Ap (t)) = {u ∈ Hp2 (Ω); ∂ν∂u = 0 on ∂Ω} and A(t) uHp2 ≤ CAp (t)uLp ,

u ∈ D(Ap (t)).

(3.75)

Here, the constant C > 0 can be uniform with respect to t , because aij (·, t) vary only in some relatively compact subset of C1 (Ω). We consider also the adjoint operators Bp (t) = Ap (t)∗ (see Theorem 2.14) of Ap (t) with respect to the adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1, Bp (t) being determined by the adjoint forms a ∗ (t; ·, ·). As will be shown by Theorems 16.7 and 16.10 in Chap. 16, we have, in favorable cases, that D(Bp (t)θ ) ⊂ Hp2θ (Ω) for any 0 < θ ≤ 1, and uH 2θ ≤ C Bp (t)θ uL , p

p

u ∈ D Bp (t)θ ,

(3.76)

C > 0 being uniform for t . In the general case, according to (16.36), it is at least true that D(Bp (t)θ ) ⊂ Hp2θ (Ω) for any 0 < θ < θ ≤ 1 with the estimate uH 2θ ≤ Cθ ,θ Bp (t)θ uL , p

p

u ∈ D Bp (t)θ ,

(3.77)

Cθ ,θ > 0 being uniform for t . In what follows, let us consider the optimal case (3.76). For the general case, we shall have some weaker result, see Remark 3.4. ν Let 12 ≤ ν < p+1 2p . Then, by (16.36) again, we see that D(A(s)) ⊂ D(A(t) ) for all 0 ≤ s, t ≤ T ; hence, (3.29) is fulfilled. When p = 2, the formula (3.72) reads as

A2 (t)ν A2 (t)−1 − A2 (s)−1 f, g n [ai,j (t, x) − aij (s, x)]Di A2 (s)−1 f · Dj A2 (t)∗(ν−1) g dx =− i,j =1 Ω

−

[c(x, t) − c(s, x)]A2 (s)−1 f · A2 (t)∗(ν−1) g dx,

Ω

f ∈ L2 (Ω), g ∈ D(A2 (t)∗ ).

152

3 Linear Evolution Equations

For the general 1 < p < ∞, this formula yields that

Ap (t)ν Ap (t)−1 − Ap (s)−1 f, g L

p ×Lp

n

=−

[aij (t, x) − aij (s, x)]Di Ap (s)−1 f · Dj Bp (t)ν−1 g dx

i,j =1 Ω

[c(x, t) − c(s, x)]Ap (s)−1 f · Bp (t)ν−1 g dx,

− Ω

f ∈ Lp (Ω), g ∈ D(Bp (t)). Since 0 ≤ 2ν − 1 < p1 , we observe by (1.80) and (1.82) that [aij (t, x) − aij (s, x)]Di Ap (s)−1 f · Dj Bp (t)ν−1 g dx Ω ≤ [aij (t) − aij (s)]Di Ap (s)−1 f H˚ 2ν−1 Dj Bp (t)ν−1 g H 1−2ν p

p

≤ aij (t) − aij (s)L(H 1 ,H 2ν−1 ) p

p

Di Ap (s)−1 f

ν−1 g 1 Dj Bp (t)

Hp

Hp1−2ν

.

(3.78) In addition, by (3.75), Di Ap (s)−1 f Hp1 ≤ Cf Lp . By the boundedness of dif2(1−ν)

ferentiation Dj ∈ L(Hp (Ω), Hp1−2ν (Ω)) (due to Theorem 1.43) and estimate ν−1 (3.76), we have Dj Bp (t) gH 1−2ν ≤ Cν gLp . Finally, Lemma 3.4 below, top

gether with assumption (3.74), provides that ⎧ 2pμ(1−ν)/n ⎪ ⎨C|t − s| aij (t) − aij (s)L(H 1 ,Hp2ν−1 ) ≤ Cμ |t − s|2μ (1−ν) p ⎪ ⎩ C|t − s|2μ(1−ν)

if 1 < p < n, if p = n, if n < p < ∞,

where μ can be taken arbitrarily in such a way that 0 < μ < μ. In this way, under (3.74) and (3.76), we have shown that, for any condition (3.30) is fulfilled as ⎧ 2pμ(1−ν)/n ⎪ ⎨Cν |t − s|

ν −1 −1 Ap (t) Ap (t) − Ap (s) ≤ Cν,μ |t − s|2μ (1−ν) L(Lp ) ⎪ ⎩ Cν |t − s|2μ(1−ν) μ being an arbitrary exponent such that 0 < μ < μ.

1 2

≤ν

0. Let 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1. We write A(t)θ [e−τ A(t) − e−τ A(s) ]A(s)−ϕ 1 = e−τ λ A(t)θ−ν+1 (λ − A(t))−1 D(t, s)A(s)1−ϕ (λ − A(s))−1 dλ. 2πi Γ By assumption, it holds that −2 ≤ θ − ϕ − ν ≤ 0. In the case where −1 < θ − ϕ − ν, A(t)θ [e−τ A(t) − e−τ A(s) ]A(s)−ϕ ≤C |λ|θ−ϕ−ν e−τ Re λ |dλ| |t − s|μ ≤ Cτ ϕ−θ+ν−1 |t − s|μ . Γω

In the case where −2 ≤ θ − ϕ − ν ≤ −1, we can always use the shifting techniques to obtain (3.91). For 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1, we write

A(t)θ e−τ A(t) A(t)−ϕ − e−τ A(s) A(s)−ϕ 1 λ−ϕ e−τ λ A(t)θ−ν+1 (λ − A(t))−1 D(t, s)A(s)(λ − A(s))−1 dλ. = 2πi Γ When −1 < θ − ϕ − ν, (3.92) is verified by the same arguments as before. In the critical case where θ − ϕ − ν = −1, the shifting of integral contour is not available except the special case θ = ν and ϕ = 1. When θ − ϕ − ν < −1, we have |λ|θ−ϕ−ν e−τ Re λ |dλ| ≤ C(ϕ + ν − θ − 1)−1 . Γδ,ω Let 0 < ϕ ≤ 1. To see (3.83), we notice that A(t)U (t, s)A(s)−ϕ is a solution to the integral equation A(t)U (t, s)A(s)−ϕ = A(t)e−(t−s)A(t) A(s)−ϕ t − A(t)2−ν e−(t−τ )A(t) D(t, τ )A(τ )U (τ, s)A(s)−ϕ dτ. s

(3.93) Moreover, by (3.89), A(t)e−(t−s)A(t) A(s)−ϕ L(X) −(t−s)A(t) ≤ [A(t)e − A(s)e−(t−s)A(s) ]A(s)−ϕ L(X) + A(s)1−ϕ e−(t−s)A(s) L(X) ≤ C(t − s)ϕ−1 .

158

3 Linear Evolution Equations

Hence, (3.83) is verified for 0 < ϕ ≤ 1 and θ = 1. When ϕ = 0, (3.83) is already known. Let 0 ≤ θ < μ + ν and 0 < ϕ ≤ 1. From (3.93) it also follows that A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ t =− A(t)θ−ν+1 e−(t−τ )A(t) D(t, τ )A(τ )U (τ, s)A(s)−ϕ dτ. s

Therefore, A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ

L(X)

t

≤C

(t − τ )−θ+μ+ν−1 (τ − s)ϕ−1 dτ ≤ C(t − s)ϕ−θ+μ+ν−1 ,

s

and hence (3.84) is verified for 0 < ϕ ≤ 1, too. Let 0 ≤ ϕ ≤ 1 and 0 ≤ ϕ ≤ θ < μ + ν. The estimate just established, together with Lemma 4.1, yields that A(t)θ U (t, s)A(s)−ϕ − A(s)θ−ϕ e−(t−s)A(s) L(X) ≤ A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ L(X)

+ A(t)θ e−(t−s)A(t) − A(s)θ e−(t−s)A(s) A(s)−ϕ ≤ Cθ log (t − s)−1 + 1 (t − s)ϕ−θ+μ+ν−1 .

L(X)

(3.94)

Therefore, (3.83) holds. Let θ = ϕ = 0 in (3.84). Then, U (t, s) − e−(t−s)A(t) L(X) ≤ C(t − s)μ+ν−1 . This, together with (3.91) (θ = ϕ = 0), therefore yields (3.85) in the case k = 0. Similarly, let θ = ϕ = 1 in (3.84); then (3.85) of the case k = 1 is verified by (3.89) (θ = ϕ = 1). Equation (3.86) is also verified in the same manner. Indeed, for 0 < θ = ϕ < 1, A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−θ ≤ C(t − s)μ+ν−1 . L(X) This, together with (3.90) (θ = ϕ), then yields (3.86). For 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1, (3.84), together with (3.91), yields (3.87). We have thus established all the properties of U (t, s) to be shown. Remark 3.5 In some problems we may also need the property (3.82) for θ ≥ μ. In order to extend (3.82) to such an exponent θ , say θ = 1, we have to assume a dual condition of (3.30). That is, for some exponent 0 < ν ≤ 1, the operators

8 Maximal Regularity

159

[A(t)−1 − A(s)−1 ]A(s)ν have bounded extensions on X and satisfy the Hölder condition

A(t)−1 − A(s)−1 A(s)ν ≤ N |t − s|μ , 0 ≤ s, t ≤ T , (3.95) L(X) with another exponent 0 < μ ≤ 1 with the relation 1 < μ + ν .

(3.96)

Furthermore, we assume that X is a reflexive Banach space. In the adjoint space X ∗ , let us consider the family of sectorial operators A∗ (t) ≡ A(T − t)∗ , 0 ≤ t ≤ T . By Theorem 2.32, the family of A∗ (t)’s is easily seen to satisfy (3.27) and (3.28). In addition, by (2.126), (3.95), together with (3.96), implies that A∗ (t) satisfies (3.29), (3.30), and (3.31) with μ = μ and ν = ν . Consequently, there exists an evolution operator U ∗ (t, s), 0 ≤ s ≤ t ≤ T , for A∗ (t). Then, it is easily verified that U ∗ (t, s) = U (T − s, T − t)∗ for any 0 ≤ s ≤ t ≤ T . Indeed, for τ such that s ≤ τ ≤ t, we have ∂ [U (T − τ, T − t)∗ U ∗ (τ, s)] = [−A(T − τ )U (T − τ, T − t)]∗ U ∗ (τ, s) ∂τ + U (T − τ, T − t)∗ A∗ (τ )U ∗ (τ, s) = 0. Since A∗ (t)U ∗ (t, s) = A(T − t)∗ U (T − s, T − t)∗ = [U (T − s, T − t)A(T − t)]∗ , estimate (3.58) for A∗ (t) shows that U (T − s, T − t)A(T − t)θ L(X) ≤ C(t − s)−θ for 0 < θ < μ + ν . We have thus seen that, when X is reflexive, (3.95), together with (3.96), yields property (3.82) for μ ≤ θ < μ + ν , too.

8.2 Maximal Regularity We return to problem (3.61). Utilizing the refined properties of U (t, s), we shall show the maximal regularity of solutions. Theorem 3.10 Under (3.27), (3.28), (3.29), (3.30), and (3.31), let F ∈ Fβ,σ ((0, T ]; X), and let

0 < σ < min{β, μ + ν − 1},

U0 ∈ D A(0)β ,

0 < β ≤ 1.

(3.97)

(3.98)

Then, the solution U to (3.61) given by (3.65) possesses the following regularity:

160

3 Linear Evolution Equations

A(t)β U ∈ C([0, T ]; X), dU , dt

(3.99)

A(t)U ∈ Fβ,σ ((0, T ]; X),

with the estimates

A(t)β U ≤ C A(0)β U0 + F Fβ,σ , C dU

β dt β,σ + A(t)U Fβ,σ ≤ C A(0) U0 + F Fβ,σ . F

(3.100)

(3.101) (3.102)

Proof Let us first verify (3.99). For 0 < t ≤ T , we notice that A(t)β U (t) = A(t)β−1 A(t)U (t). Since A(·)β−1 ∈ C([0, T ]; L(X)), (3.63) yields that A(t)β U (t) is continuous for t > 0. So, it suffices to show that A(t)β U (t) is continuous at t = 0. We use the expression

A(t)β U (t) = A(0)β U0 + A(t)β U (t, 0)A(0)−β − 1 A(0)β U0 t + A(t)β U (t, τ )[F (τ ) − F (t)] dτ 0

+

t

A(t)β [U (t, τ ) − e−(t−τ )A(t) ] dτ F (t)

0

+t

β−1

[1 − e−tA(t) ]A(t)β−1 t 1−β F (t).

Then, by the same arguments as in the proof of Theorem 3.5, we directly see that A(t)β U (t) → A(0)β U0 as t → 0. Hence, (3.99) is proved. Estimate (3.101) is also verified in the same way. Let us now prove (3.100). We use the expression A(t)U (t) = A(0)e−tA(0) U0

+ A(t)U (t, 0)A(0)−β − A(0)1−β e−tA(0) A(0)β U0 t t + A(t)U (t, τ )[F (τ ) − F (t)] dτ + W (t, τ ) dτ F (t) 0

0

+ [1 − e−tA(t) ]F (t) = W1 (t) + W2 (t) + W3 (t) + W4 (t) + W5 (t), where W (t, s) = A(t)[U (t, s) − e−(t−s)A(t) ]. By (2.132), it is already known that W1 ∈ Fβ,σ ((0, T ]; X). So, it remains to verify Wi ∈ Fβ,σ ((0, T ]; X) for 2 ≤ i ≤ 5. Proof for W2 We use (3.94) to observe that t 1−β W2 (t)X ≤ Ct μ+ν−1 log(t −1 + 1) → 0 as t → 0. In the meantime, W2 (t) − W2 (s) is seen to be decomposed as

8 Maximal Regularity

161

W2 (t) − W2 (s) = A(t)U (t, s)A(s)−1 − e−(t−s)A(s) A(s)U (s, 0)A(0)−β A(0)β U0 + [e−(t−s)A(s) − 1]A(s)−σ A(s)1+σ U (s, 0)A(0)−β A(0)β U0 − [e−(t−s)A(0) − 1]A(0)−σ A(0)1−β+σ e−sA(0) A(0)β U0 . Then, by (2.129), (3.83), and (3.85),

W2 (t) − W2 (s)X ≤ C s β−1 (t − s)μ+ν−1 + s β−σ −1 (t − s)σ A(0)β U0 X . Hence, since σ < μ + ν − 1, W2 satisfies (1.6). It is also readily verified by (2.130) and (3.94) that W2 satisfies (1.7). Proof for W3 We shall argue as in the proof of Theorem 3.5. It is easy to see that limt→0 t 1−β W3 (t) = 0. In addition, W3 (t) − W3 (s) is decomposed as W3 (t) − W3 (s) = s

t

A(t)U (t, τ )[F (τ ) − F (t)] dτ

s

+

W (t, τ ) dτ + [e−(t−s)A(t) − e−tA(t) ] [F (s) − F (t)]

0

+ A(t)U (t, s)A(s)−1 − e−(t−s)A(s) s × A(s)U (s, τ )[F (τ ) − F (s)] dτ 0

+ [e−(t−s)A(s) − 1]A(s)−σ + [e−(t−s)A(s) − 1]

s

A(s)σ W (s, τ )[F (τ ) − F (s)] dτ

0 s

A(s)e−(s−τ )A(s) [F (τ ) − F (s)] dτ.

0

Then, using (2.129), (3.81), (3.84), and (3.85) and especially using (3.21) for the last term, we can verify that

W3 (t) − W3 (s)X ≤ C s 1−β (t − s)μ+ν−1 + s 1−β+σ (t − s)σ F Fβ,σ .

Hence, W3 satisfies (1.6). Similarly, (1.7) is also verified. Proof for W4 It is clear that limt→0 t 1−β W4 (t) = 0. We use the decomposition

t

W4 (t) − W4 (s) =

s

W (t, τ ) dτ F (t) +

W (t, τ ) dτ [F (t) − F (s)]

0

s

+ A(t)U (t, s)A(s)−1 − e−(t−s)A(s) + [e

−(t−s)A(s)

−σ

− 1]A(s)

0

t

W (s, τ ) dτ F (s) s

s

A(s)σ W (s, τ ) dτ F (s)

162

3 Linear Evolution Equations

+

s

[A(s)e−(t−τ )A(s) − A(t)e−(t−τ )A(t) ] dτ F (s)

0

+ A(t)U (t, s)A(s)−1 − e−(t−s)A(s) [1 − e−sA(s) ]F (s). Then, by (2.129), (3.49), (3.84), and (3.85),

W4 (t) − W4 (s)X ≤ C s β−1 (t − s)μ+ν−1 + t μ+ν−1 s β−σ −1 (t − s)σ F Fβ,σ .

Hence, W4 satisfies (1.6) and (1.7). Proof for W5 We see that W5 is decomposed as W5 (t) − W5 (s) = [1 − e−tA(t) ][F (t) − F (s)] − [e−tA(t) − e−tA(s) ]F (s) − [e−(t−s)A(s) − 1]A(s)−σ A(s)σ e−sA(s) F (s). Then, by (2.129) and (3.49), W5 (t) − W5 (s)X ≤ Cs β−σ −1 (t − s)σ F Fβ,σ .

It is also verified that W5 satisfies (1.7). We have thus verified that AU belongs to Fβ,σ ((0, T ]; X) and satisfies estimate (3.102). Since dU dt = −A(t)U + F (t), we have accomplished the proof of (3.100) and (3.102). Remark 3.6 When 0 < β < 1, t 1−β A(t)U (t) tends to 0 as t → 0. In the meantime, when β = 1, A(t)U (t) tends to A(0)U0 as t → 0.

8.3 Spatial Regularity We remark that, if the external force function F ∈ Fβ,σ ((0, T ]; X) has the spatial regularity β−1−ρ

A(t)ρ F ∈ B{0}

((0, T ]; X),

0 < ρ < σ,

(3.103)

the solution U to (3.61) has the same regularity. Theorem 3.11 Let U0 ∈ D(A(0)β ), 0 < β ≤ 1, and let F ∈ Fβ,σ ((0, T ]; X), 0 < 1+ρ U also belong to σ < min{β, μ + ν − 1}, satisfy (3.103). Then, A(t)ρ dU dt , A(t) β−1−ρ

B{0}

((0, T ]; X) with the estimate A(t)ρ dU dt

β−1−ρ

B{0}

+ A(t)1+ρ U Bβ−1−ρ {0}

≤ C A(0)β U0 + F Fβ,σ + A(t)ρ F Bβ−1−ρ . {0}

9 Variational Methods

163

Proof The result is easily verified from the representation A(t)1+ρ U (t) = A(t)1+ρ U (t, 0)A(0)−β A(0)β U0 t + A(t)1+ρ U (t, τ )[F (τ ) − F (t)] dτ 0

+

t

A(t)1+ρ [U (t, τ ) − e−(t−τ )A(t) ] dτ F (t)

0

+ [1 − e−tA(t) ]A(t)ρ F (t). Note that 1 + ρ < 1 + σ < μ + ν.

9 Variational Methods Let Z and X be two separable Hilbert spaces such that Z ⊂ X densely and continuously with norms · and | · |, respectively. Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces, the norm of Z ∗ being denoted by · ∗ . Consider a family of sesquilinear forms a(t; U, V ), 0 ≤ t ≤ T , defined on Z and assume that the following conditions are satisfied: a(t; ·, ·) is a measurable function with respect to t,

(3.104)

|a(t; U, V )| ≤ MU V ,

(3.105)

Re a(t; U, U ) ≥ δU , 2

U, V ∈ Z,

U ∈ Z,

(3.106)

with some constant M > 0 and positive constant δ > 0 independent of t . Let A(t) be a family of sectorial operators of Z ∗ which are associated with the forms a(t; U, V ). We then consider the Cauchy problem for an abstract evolution equation dU dt + A(t)U = F (t), 0 < t ≤ T , (3.107) U (0) = U0 , in the space Z ∗ . The following theorem is known as the fundamental existence and uniqueness result. For the proof, see Theorems 1, 2, and 3 of Dautray–Lions [DL88, Chap. XVIII]. Theorem 3.12 Let A(t) be a family of sectorial operators of Z ∗ associated with sesquilinear forms satisfying (3.104), (3.105), and (3.106). For any F ∈ L2 (0, T ; Z ∗ ) and any U0 ∈ X, there exists a unique solution U to (3.107) in the function space: U ∈ L2 (0, T ; Z) ∩ C([0, T ]; X) ∩ H 1 (0, T ; Z ∗ ). In addition, U satisfies the estimates

164

3 Linear Evolution Equations

U L2 (0,T ;Z) + U C([0,T ];X) + U H 1 (0,T ;Z ∗ ) ≤ C(|U0 | + F L2 (0,T ;Z ∗ ) ), where C > 0 is a constant determined by M, δ, and T only.

10 Nonautonomous Parabolic Equations Let us consider a nonautonomous parabolic equation of the form n ∂u Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ) = ∂t i,j =1

in a domain Ω ⊂ Rn , where 0 < T < ∞ is fixed time. It is possible to treat a number of different initial-boundary-value problems concerning this equation. In this section, however, we will describe only a few examples. As for the concrete characterization of the domains of fractional powers of strongly elliptic operators, see Chap. 16.

10.1 Problems in Hilbert Spaces Case where Ω = Rn . Consider the initial-value problem ∂u n n i,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in R × (0, T ), ∂t = u(x, 0) = u0 (x) Here, aij and c are such that aij ∈ Cμ ([0, T ]; L∞ (Rn ; R)), c ∈ Cμ ([0, T ]; L∞ (Rn ; R)),

in Rn .

1 ≤ i, j ≤ n,

(3.108)

(3.109)

with some exponent 0 < μ ≤ 1. We assume further that, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.15) with some uniform constant δ > 0, and c(x, t) satisfies (2.16) with some uniform constant c0 > 0. Let us handle (3.108) in the space H −1 (Rn ) by formulating (3.108) as an abstract problem of the form (3.61) in which the coefficient linear operator is a realization A(t) of the differential operator − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in H −1 (Rn ). By Theorem 2.2, A(t) is a sectorial operator of H −1 (Rn ) with uniform domain D(A(t)) ≡ H 1 (Rn ) and with angle ωA(t) ≤ ωA < π2 , ωA being independent of t . Therefore, the family of operators A(t) fulfills conditions (3.27), (3.28), and (3.29) with ν = 1. In addition, it is clear that (3.109) implies (3.69); therefore, by virtue of (3.71), condition (3.30) is also fulfilled. Since ν = 1 in the present case, (3.31) is trivially fulfilled.

10 Nonautonomous Parabolic Equations

165

It is now ready to apply Theorem 3.10 to (3.108) to conclude the following result. For any f ∈ Fβ,σ ((0, T ]; H −1 (Rn )), where 0 < σ < β ≤ 1, and any u0 ∈ D(A(0)β ), (3.108) possesses a unique solution in the function space: u ∈ C((0, T ]; H 1 (Rn )) ∩ C1 ((0, T ]; H −1 (Rn )), β,σ ((0, T ]; H −1 (Rn )), A(t)β u ∈ C([0, T ]; H −1 (Rn )); du dt , A(t)u ∈ F with the estimates

β A(t)β u + du dt β,σ + A(t)uFβ,σ ≤ C A(0) u0 H −1 + f Fβ,σ . C F

Dirichlet conditions. Consider the initial-boundary-value problem ⎧ ∂u n ⎪ ⎨ ∂t = i,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ), u=0 on ∂Ω × (0, T ), (3.110) ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a domain Ω ⊂ Rn . Here, aij and c are such that aij ∈ Cμ ([0, T ]; L∞ (Ω; R)), 1 ≤ i, j ≤ n, c ∈ Cμ ([0, T ]; L∞ (Ω; R)),

(3.111)

with some exponent 0 < μ ≤ 1. In addition, it is assumed that, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.19) with some uniform constant δ > 0, and c(x, t) satisfies (2.20) with some uniform constant c0 > 0. Let us handle (3.110) again in H −1 (Ω) = [H˚ 1 (Ω)]∗ . Indeed, (3.110) is formulated as an abstract problem of the form (3.61) in which the coefficient linear operator is a realization A(t) of the differential operator − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in H −1 (Ω) under the Dirichlet boundary conditions. By Theorem 2.3, A(t) is a sectorial operator of H −1 (Ω) with uniform domain D(A(t)) ≡ H˚ 1 (Ω) and with angles ωA(t) ≤ ωA < π2 , ωA being independent of t . Therefore, the family of operators A(t) fulfills conditions (3.27), (3.28), and (3.29) with ν = 1. In addition, (3.111) implies (3.69); therefore, by (3.71), condition (3.30) is also fulfilled. Clearly, (3.31) is also fulfilled. Theorem 3.5 readily provides that, for any f ∈ Fβ,σ ((0, T ]; H −1 (Ω)), where 0 < σ < β ≤ 1, and any u0 ∈ D(A(0)β ), (3.110) possesses a unique solution in the function space: u ∈ C((0, T ]; H˚ 1 (Ω)) ∩ C1 ((0, T ]; H −1 (Ω)), β,σ ((0, T ]; H −1 (Ω)), A(t)β u ∈ C([0, T ]; H −1 (Ω)); du dt , A(t)u ∈ F with the estimates

β A(t)β u + du dt β,σ + A(t)uFβ,σ ≤ C A(0) u0 H −1 + f Fβ,σ . C F

166

3 Linear Evolution Equations

Neumann-type conditions. Consider the initial-boundary-value problem ⎧

n ∂u ⎪ ⎪ i,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ), ∂t = ⎨ n on ∂Ω × (0, T ), (3.112) i,j =1 νj (x)aij (x, t)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a domain Ω ⊂ Rn . Here, aij and c are as in (3.111). In addition, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.19) with some uniform constant δ > 0, and c(x, t) satisfies (2.20) with some uniform constant c0 > 0. We handle this problem in H 1 (Ω)∗ . Indeed, (3.112) is written as (3.61) in which

the coefficient linear operator is a realization A(t) of the differential operator − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in H 1 (Ω)∗ under the Neumann-type boundary conditions. By Theorem 2.4, A(t) is a sectorial operator of H 1 (Ω)∗ with uniform domain D(A(t)) ≡ H 1 (Ω) and with angle ωA(t) ≤ ωA < π2 , ωA being independent of t . Therefore, the family of operators A(t) fulfills conditions (3.27), (3.28), and (3.29) with ν = 1. In addition, (3.111) implies (3.69); therefore, by (3.71), condition (3.30) is also fulfilled. Clearly, (3.31) is also fulfilled. Theorem 3.5 readily provides that, for any f ∈ Fβ,σ ((0, T ]; H 1 (Ω)∗ ), where 0 < σ < β ≤ 1, and any u0 ∈ D(A(0)β ), (3.112) possesses a unique solution in the function space: u ∈ C((0, T ]; H 1 (Ω)) ∩ C1 ((0, T ]; H 1 (Ω)∗ ), β,σ ((0, T ]; H 1 (Ω)∗ ), A(t)β u ∈ C([0, T ]; H 1 (Ω)∗ ); du dt , A(t)u ∈ F with the estimate

β A(t)β u + du + A(t)u β,σ ≤ C A(0) u0 1∗ + f F β,σ . F C H dt Fβ,σ

10.2 Problems in Lp Spaces Consider the initial-boundary-value problem ⎧

∂u ⎪ = ni,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ), ⎪ ∂t ⎨ n on ∂Ω × (0, T ), (3.113) i,j =1 νj (x)aij (x, t)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω. Here, aij and c are as in (3.74). In addition, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.19) with some uniform constant δ > 0, and c(x, t) satisfies (2.20) with some uniform constant c0 > 0. We now want to handle (3.113) in the space Lp (Ω), where n < p < ∞. Let Ap (t) be a realization of the differential operator − i,j Dj [aij (x, t)Di ] + c(x, t)

10 Nonautonomous Parabolic Equations

167

in Lp (Ω) under the Neumann-type boundary conditions ∂ν∂u = 0 on ∂Ω. Using A(t) Ap (t), we formulate (3.113) as an abstract problem of the form (3.61). Fix an exponent ν in such a way such that 12 ≤ ν < p+1 2p . As discussed in Sect. 7.2, a family of the operators Ap (t) can fulfill condition (3.29) with the ν. Similarly, by Remark 3.4, we know that the family can fulfill condition (3.30) with the Hölder exponent 2μ(1 − ν ), where ν is an arbitrarily fixed exponent such that ν < ν < p+1 2p . Therefore, if μ in (3.74) is such that μ > 12 (and if ν is taken sufficiently close to ν), then relation (3.31) can be fulfilled. In this way, we have verified that all the structural assumptions are fulfilled in Lp (Ω), too. Theorem 3.10 readily provides the following result. For any f ∈ Fβ,σ ((0, T ]; Lp (Ω)), where 0 < σ < β ≤ 1, and any u0 ∈ D(Ap (0)β ), (3.113) possesses a unique solution in the function space:

u ∈ C((0, T ]; Hp2 (Ω)) ∩ C1 ((0, T ]; Lp (Ω)), β,σ ((0, T ]; L (Ω)), Ap (t)β u ∈ C([0, T ]; Lp (Ω)); du p dt , Ap (t)u ∈ F

with the estimate β Ap (t)β u + du dt β,σ + Ap (t)uFβ,σ ≤ C Ap (0) u0 Lp + f Fβ,σ . C F

10.3 Some Example of the Critical Case ν = 0 Let us consider a very particular example such that A(t) is a multiplicative operator of the form A(t)f (x) = (x − t)−2 f (x) in Lp (0, 1) (1 < p < ∞) for 0 ≤ t ≤ 1 = T . It is easy to see that A(t) is a sectorial operator of Lp (Ω). The domain D(A(t)) of A(t) consists of the functions (x − t)2 f (x), where f ∈ Lp (0, 1). It is also verified that, for 0 < θ < 1, A(t)θ f (x) = |x − t|−2θ f (x) with D(A(t)θ ) = {|x − t|2θ f ; f ∈ Lp (0, 1)}. Therefore, condition (3.29) is not the case for any ν > 0. Nevertheless, there exists an evolution operator for A(t) which is given by

U (t, s) = e−

t s

(x−τ )−2 dτ

⎧ [(t−x)−1 +(x−s)−1 ] , ⎪ ⎨e = 0, ⎪ ⎩ [(t−x)−1 +(x−s)−1 ] e ,

t < x ≤ 1, s ≤ x ≤ t, 0 ≤ x < s,

for 0 ≤ s < t ≤ 1 and by U (s, s) = 1 for 0 ≤ s ≤ 1. We can in fact extend our abstract theory built in this chapter to the critical case where ν = 0. Let X be a Banach space. Consider a family of sectorial operators A(t), 0 ≤ t ≤ T , of X satisfying (3.27) and (3.28). We assume that A(·)−1 is strongly continuously differentiable on X for 0 ≤ t ≤ T

(3.114)

168

3 Linear Evolution Equations

d and that there exists some fixed exponent 0 < ν ≤ 1 such that R( dt A(t)−1 ) ⊂ ν D(A(t) ) for all t with the estimates A(t)ν d A(t)−1 ≤ N , 0 ≤ t ≤ T , (3.115) dt

N > 0 being a constant. Under (3.27), (3.28), (3.114), and (3.115), we can construct a unique evolution operator U (t, s) for A(t) and can furthermore show similar results as Theorems 3.8, 3.9, and 3.10, see Tanabe [Tan64] (cf. also [Yag76] and [Wat77]).

11 Perturbed Problems We are concerned with the perturbed problem dU dt + A(t)U + B(t)U = F (t),

0 < t ≤ T,

U (0) = U0 ,

(3.116)

in a Banach space X. Here, A(t), 0 ≤ t ≤ T , is a family of sectorial operators of X satisfying (3.27), (3.28), (3.29), (3.30), and (3.31). Meanwhile, B(t), 0 < t ≤ T , is a family of linear operators of X with singularity at t = 0. We assume that D(A(t)1−˜ν ) ⊂ D(B(t)) for all 0 < t ≤ T with the estimates ˜ B(t)A(t)ν˜ −1 ≤ Dt ˜ μ−1 , 0 < t ≤ T, (3.117) D˜ > 0 being a constant. Here, μ˜ > 0 and ν˜ > 0 are some fixed exponents such that 0 < μ˜ ≤ 1,

0 < ν˜ ≤ 1 and 1 < μ˜ + ν˜ .

(3.118)

In addition, B(t)A(t)ν˜ −1 is assumed to satisfy a Hölder condition of the form ˜ σ˜ −1 B(t)A(t)ν˜ −1 − B(s)A(s)ν˜ −1 ≤ Ls ˜ μ− (t − s)σ˜ , 0 < s ≤ t ≤ T , (3.119) with some fixed exponent 0 < σ˜ < min{μ + ν − 1, μ˜ + ν˜ − 1},

(3.120)

L˜ > 0 being a constant. The function F is a given external force function, and U0 is an initial value in X.

11.1 Evolution Operator The purpose of this subsection is to construct an evolution operator U˜ (t, s), 0 ≤ s ≤ t ≤ T , for the family of operators A(t) + B(t).

11 Perturbed Problems

169

We begin with introducing the approximation A˜ n (t) = An (t) + Bn (t),

n = 1, 2, 3, . . . , 0 < t ≤ T ,

where An (t) is the Yosida approximation of A(t), and Bn (t) = B(t)Jn (t)1−˜ν = B(t)A(t)ν˜ −1 An (t)1−˜ν ,

(3.121)

see Proposition 3.3. For any 0 < ε < T , A˜ n (t) is an L(X)-valued continuous function on [ε, T ]. So, we have an evolution operator U˜ n (t, s), ε ≤ s ≤ t ≤ T , for A˜ n (t) such that ∂ ˜ Un (t, s) = −A˜ n (t)U˜ n (t, s), ε ≤ s ≤ t ≤ T , ∂t ∂ ˜ Un (t, s) = U˜ n (t, s)A˜ n (s), ε ≤ s ≤ t ≤ T . ∂s Of course, U˜ n (t, s) does not depend on the choice of ε > 0. Therefore, the evolution operator is defined for any (t, s) such that 0 < s ≤ t ≤ T . The evolution operator U˜ n (t, s) satisfies integral equations. Due to (3.40), t ∂ ˜ U˜ n (t, s) − Un (t, s) = − Un (t, τ )Un (τ, s) dτ s ∂τ t =− U˜ n (t, τ )Bn (τ )Un (τ, s) dτ, s

where Un (t, s), 0 ≤ s ≤ t ≤ T , is the evolution operator for An (t). Therefore, U˜ n (t, s) is connected with Un (t, s) by t ˜ Un (t, s) = Un (t, s) − U˜ n (t, τ )Bn (τ )Un (τ, s) dτ, 0 < s ≤ t ≤ T . (3.122) s

Operating An (t) to this equation, we obtain also an integral equation for W˜ n (t, s) = An (t)U˜ n (t, s) − An (t)Un (t, s) such that t W˜ n (t, τ )Bn (τ )Un (τ, s) dτ, 0 < s ≤ t ≤ T , (3.123) W˜ n (t, s) = R˜ n (t, s) + s

where R˜ n (t, s) = −

t

An (t)Un (t, τ )Bn (τ )Un (τ, s) dτ. s

Let us now show the convergence of U˜ n (t, s) as n → ∞. From (3.121) we have Bn (τ )Un (τ, s) = B(τ )A(τ )ν˜ −1 An (τ )1−˜ν Un (τ, s). Therefore, by (3.81) and (3.117), ˜ ˜ ν −2 (τ − s)ν˜ −1 ≤ C(τ − s)μ+˜ , Bn (τ )Un (τ, s) ≤ Cτ μ−1

0 < s < τ ≤ T,

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3 Linear Evolution Equations

with some constant C > 0 independent of n. In addition, as n → ∞, Bn (τ )Un (τ, s) → B(τ )A(τ )ν˜ −1 A(τ )1−˜ν U (τ, s)

strongly on X.

We can then apply the dominated convergence theorems, Theorems 1.30 and 1.32, to (3.122) to conclude that U˜ n (t, s) satisfies the uniform estimate U˜ n (t, s) ≤ C0 and is strongly convergent on X to a bounded operator, which will be denoted by U˜ (t, s) for all 0 < s ≤ t ≤ T . Furthermore, U˜ (t, s) is characterized as a solution to the integral equation t ˜ U (t, s) = U (t, s) − U˜ (t, τ )B(τ )U (τ, s) dτ, 0 < s ≤ t ≤ T . s

In the meantime, we notice for 0 < s < s < T that U˜ (t, s ) − U˜ (t, s) = U (t, s ) − U (t, s) + +

s

U˜ (t, τ )B(τ )U (τ, s) dτ

s t s

U˜ (t, τ )B(τ )U (τ, s ) dτ [1 − U (s , s)].

This implies that U˜ (t, s) is uniformly continuous in s ∈ (0, t] in the strong topology of L(X) (t > 0 being fixed). Therefore, as s → 0, U˜ (t, s) is strongly convergent on X, and the strong convergence is uniform in t . In this way, we conclude that U˜ (t, s) is defined even at s = 0, namely, for all 0 ≤ s ≤ t ≤ T , and is strongly continuous on X for 0 ≤ s ≤ t ≤ T . Let us next consider (3.123). We have the following proposition. Proposition 3.10 For any n = 1, 2, 3, . . . , R˜ n (t, s) satisfies ˜ ν −2 , R˜ n (t, s) ≤ C(t − s)μ+˜

0 ≤ s < t ≤ T,

with some uniform constant C > 0. As n → ∞, Rn (t, s) converges strongly to a ˜ s) on X. R(t, ˜ s) satisfies the same estimate and is bounded linear operator R(t, strongly continuous on X for 0 ≤ s < t ≤ T . Proof We can decompose R˜ n (t, s) in the form t ˜ Rn (t, s) = − [An (t)Un (t, τ ) − An (t)e−(t−τ )An (t) ]Bn (τ )Un (τ, s) dτ s

−

t

An (t)e−(t−τ )An (t) B(τ )A(τ )ν˜ −1 − B(t)A(t)ν˜ −1 An (τ )1−˜ν Un (τ, s) dτ

t

An (t)e−(t−τ )An (t) B(t)A(t)ν˜ −1 1 − An (t)1−˜ν Un (t, τ )An (τ )ν˜ −1

s

− s

× An (τ )1−˜ν Un (τ, s) dτ + [e−(t−s)An (t) − 1]B(t)A(t)ν˜ −1 An (t)1−˜ν Un (t, s).

11 Perturbed Problems

171

We used the formula −

t

An (t)e−(t−τ )An (t) dτ = e−(t−s)An (t) − 1.

s

We notice by (3.81), (3.85), and (3.120) that

1 − An (t)1−˜ν Un (t, τ )An (τ )ν˜ −1 An (τ )1−˜ν Un (τ, s)

≤ 1 − An (t)1−˜ν Un (t, τ )An (τ )ν˜ −1 An (τ )−σ˜ An (τ )1−˜ν +σ˜ Un (τ, s) ≤ C(t − τ )σ˜ (τ − s)ν˜ −σ˜ −1 ,

0 ≤ s < τ < t ≤ T.

Using this together with (3.81), (3.84), (3.117), and (3.119), we deduce that

˜ ν −3 ˜ ν −2 . + (t − s)μ+˜ R˜ n (t, s) ≤ C (t − s)μ+ν+μ+˜ Hence, the desired estimate is proved. The strong convergence of R˜ n (t, s) is also verified by using the decomposition presented above, since each term is easily seen to be strongly convergent on X as n → ∞ by (3.55) and Proposition 3.4. It is now ready to apply Theorem 1.32 to (3.123) to deduce that W˜ n (t, s) is strongly convergent to a bounded linear operator W˜ (t, s) for each 0 ≤ s < t ≤ T , which is characterized as a solution to the integral equation ˜ s) + W˜ (t, s) = R(t,

t

W˜ (t, τ )B(τ )U (τ, s) dτ,

0 < s ≤ t ≤ T.

s

Consequently, An (t)U˜ n (t, s) = An (t)Un (t, s) + W˜ n (t, s) is strongly convergent to A(t)U (t, s) + W˜ (t, s). Since U˜ (t, s) = U (t, s) + A(t)−1 W˜ (t, s), it follows that R(U˜ (t, s)) ⊂ D(A(t)) and A(t)U˜ (t, s) = A(t)U (t, s) + W˜ (t, s). We can repeat the same arguments as in Sect. 5.3 for proving the analogous results to (3.57), (3.58), (3.59), and (3.60) for A(t) + B(t) and U˜ (t, s). In particular, since U˜ (t, s) ≤ C0 and A(t)U˜ (t, s) ≤ C1 (t − s)−1 hold, the moment inequality yields that for any 0 ≤ θ ≤ 1, A(t)θ U˜ (t, s) ≤ Cθ (t − s)−θ , 0 ≤ s < t ≤ T . (3.124) It is also seen that for 0 ≤ θ ≤ μ, ˜ A(t)U˜ (t, s)A(s)−θ ≤ Cθ (t − s)θ−1 ,

0 ≤ s < t ≤ T.

(3.125)

In fact, in the proof of Proposition 3.10, it is possible to utilize (3.83) with θ = 1 − ν˜ to obtain that for 0 ≤ θ ≤ 1 − ν˜ , ˜ ν +θ−2 R˜ n (t, s)An (s)−θ ≤ C(t − s)μ+˜ , 0 ≤ s < t ≤ T.

172

3 Linear Evolution Equations

˜ s)A(s)−θ . Then, for 0 ≤ θ ≤ 1 − ν˜ , It is the same for R(t, ˜ ν +θ−2 W˜ (t, s)A(s)−θ ≤ C(t − s)μ+˜ , 0 ≤ s < t ≤ T. This means that (3.125) is valid for 0 ≤ θ ≤ 1 − ν˜ . For 1 − ν˜ < θ ≤ μ, ˜ it is observed that ˜ W˜ (t, s)A(s)−θ ≤ C W˜ (t, s)A(s)ν˜ −1 ≤ C(t − s)μ−1 , 0 ≤ s < t ≤ T. This then means that (3.125) is extensively valid for 1 − ν˜ < θ ≤ μ, ˜ too. We sum up the results obtained for the evolution operator U˜ (t, s). Theorem 3.13 Let (3.27), (3.28), (3.29), (3.30), and (3.31) be satisfied for A(t), and let (3.117), (3.118), (3.119), and (3.120) be satisfied for B(t). Then, there exists a unique evolution operator U˜ (t, s) for A(t) + B(t) satisfying (3.57), having the regularity that A(t)U˜ (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T , and U˜ (t, s) is strongly continuous on X for 0 ≤ s ≤ t ≤ T , satisfies estimates (3.124) and (3.125), and is differentiable in t for t > s with ∂t∂ U˜ (t, s) = −[A(t) + B(t)]U˜ (t, s).

11.2 Cauchy Problem Let us consider problem (3.116) under conditions (3.27), (3.28), (3.29), (3.30), and (3.31) and (3.117), (3.118), (3.119), and (3.120). Let U˜ (t, s) be an evolution operator for A(t) + B(t) constructed above. As a matter of fact, we can repeat the same arguments as in Sect. 2 to prove the following theorem. So, the proof is left to the reader. Theorem 3.14 Let (3.27), (3.28), (3.29), (3.30), and (3.31) be satisfied, and let (3.117), (3.118), (3.119), and (3.120) be also satisfied. Then, for any function F satisfying (3.62) and any initial value U0 ∈ X, (3.116) possesses a unique solution U in the function space: U ∈ C([0, T ]; X) ∩ C1 ((0, T ]; X),

A(t)U ∈ C((0, T ]; X).

Moreover, U is given by the formula t U˜ (t, s)F (s) ds, U (t) = U˜ (t, 0)U0 +

0 ≤ t ≤ T.

(3.126)

(3.127)

0

Concerning the regularity of B(t)U (t), we have the following result. Corollary 3.1 Let U be the solution obtained in Theorem 3.14. Then, B(t)U (t) satisfies the following estimate: ˜ ν −2 B(t)U (t) ≤ C U0 + F Bβ−1 t μ+˜ , 0 < t ≤ T, (3.128) {0}

11 Perturbed Problems

173

˜ ν −σ˜ −2 B(t)U (t) − B(s)U (s) ≤ C U0 + F Bβ−1 (t − s)σ˜ s μ+˜ , {0}

0 < s < t ≤ T.

(3.129)

Proof The first estimate is verified directly from (3.127). Indeed, t ν˜ −1 1−˜ν ˜ ˜ U (t, s)F (s) ds B(t)U (t) ≤ B(t)A(t) U (t, 0)U0 + A(t) 0

˜ ν −2 ≤ C U0 + F Bβ−1 t μ+˜ , {0}

due to (3.124). To prove the second one, we shall use rather a formula of the form t U (t, s)[F (s) − B(s)U (s)] ds, 0 ≤ t ≤ T , U (t) = U (t, 0)U0 + 0

where U (t, s) denotes the evolution operator for A(t). Then, from

B(t)U (t, 0) − B(s)U (s, 0) = B(t)A(t)ν˜ −1 − B(s)A(s)ν˜ −1 A(s)1−˜ν U (s, 0)

+ B(t)A(t)ν˜ −1 A(t)1−˜ν U (t, s)A(s)ν˜ −1 − 1 A(s)−σ˜ × A(s)1−˜ν +σ˜ U (s, 0) we observe that ˜ ν −σ˜ −2 [B(t)U (t, 0) − B(s)U (s, 0)]U0 ≤ CU0 (t − s)σ˜ s μ+˜ .

Similarly, from t s B(t) U (t, τ )F (τ ) dτ − B(s) U (s, τ )F (τ ) dτ 0

= B(t)A(t)ν˜ −1

0 t

A(t)1−˜ν U (t, τ )F (τ ) dτ

s

+ B(t)A(t)ν˜ −1 A(t)1−˜ν U (t, s)A(s)ν˜ −1 − 1 A(s)−σ˜ s

× A(s)1−˜ν +σ˜ U (s, τ )F (τ ) dτ + B(t)A(t)ν˜ −1 − B(s)A(s)ν˜ −1 0

×

s

A(s)1−˜ν U (s, τ )F (τ ) dτ

0

we obtain that t s B(t) U (t, τ )F (τ ) dτ − B(s) U (s, τ )F (τ ) dτ 0

0

˜ ν −σ˜ −2 ≤ CF Bβ−1 (t − s)σ˜ s μ+˜ . {0}

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3 Linear Evolution Equations

It is the same for B(t) μ+˜ ˜ ν −2 ((0, T ]; X) B{0}

t 0

U (t, τ )B(τ )U (τ ) dτ , since (3.128) means that B(·)U (·) ∈

with B(t)U Bμ+˜ ˜ ν −2 ≤ C U0 + F β−1 . B {0}

{0}

˜

We notice from (3.128) and (3.129) that B(·)U (·) belongs to Fβ,σ˜ ((0, T ]; X) ˜ with any β˜ such that σ˜ < β˜ < μ+ ˜ ν˜ − 1. Therefore, if F ∈ Fβ,σ˜ ((0, T ]; X) and U0 ∈ ˜ D(A(0)β ), then Theorem 3.10 provides the maximal regularity that dU dt , A(t)U ∈ ˜

Fβ,σ˜ ((0, T ]; X) with the estimate dU β˜ dt β,˜ σ˜ + A(t)U Fβ,˜ σ˜ ≤ C A(0) U0 + F Fβ,˜ σ˜ . F

(3.130)

Remark 3.7 Theorems 3.13 and 3.14 are applicable even for the autonomous case. Namely, A(t) ≡ A is a sectorial operator satisfying (2.92) and (2.93), and B(t) ≡ B is a linear operator satisfying (3.8). Then, it is easily verified that the evolution operator for A + B obtained by Theorem 3.13 is nothing more than the analytic semigroup e−t (A+B) given by Theorem 3.3. Therefore, in this case, we can read all the results in Theorem 3.14 by substituting e−(t−s)(A+B) for U˜ (t, s).

Notes and Further Researches There are three main approaches for linear abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. The semigroup methods have originated in the birth of the notion of analytic semigroups (see Notes of Chap. 2). The analytic semigroup was first used to study autonomous abstract parabolic evolution equations by Solomyak [Sol58], and those abstract results were then applied for solving concrete autonomous parabolic differential equations, see Hille [HP57], Yosida [Yos80], Krein [Kre67], and so on. Afterward, Sobolevskii [Sob61a] and Tanabe [Tan60] independently began to study nonautonomous abstract parabolic evolution equations. On the basis of analytic semigroups, they could construct an evolution operator for the Cauchy problem of the form (3.33). Sobolevskii and Tanabe considered first the case where the domains D(A(t)) of coefficient linear operators are independent of the variable t . Sobolevskii [Sob61b] and Kato [Kat61c] then considered the case where the domains D(A(t)) are variable with t but, for some exponent 0 < θ < 1, the domains D(A(t)θ ) of their fractional powers A(t)θ are independent of t . Tanabe [Tan64] then considered the case where the domains D(A(t)) vary completely with t in the sense that, for any exponent, D(A(t)θ ) are possibly variant (see also Kato–Tanabe [KT62]). The semigroup methods are the most constructive ones and provide us the solution formulas (3.13) and (3.51) which furthermore give us much information on the solutions of equations. For the details, see the books by Friedman

Notes and Further Researches

175

[Fri69] and Tanabe [Tan75, Tan97]. In particular, we can get the maximal regularity with respect to the initial values and the external force functions, i.e., Theorems 3.5 and 3.10. The variational methods have originated with Lions [Lio61]. Abstract equations are considered in Hilbert spaces. For L2 external force functions with values in a Hilbert space, unique H 1 solutions are immediately constructed together with the L2 maximal regularity, Theorem 3.12. For details, see Lions–Magenes [LM72] and Dautray–Lions [DL88]. Da Prato–Grisvard [DaPG75] (see also [DaPG84]) devised a method for solving operational equations of the form AU + BU = F given by the sum of two linear operators A and B. Acquistapace–Terreni [AcT86, AcT87] used this method for constructing C1 solutions for abstract parabolic evolution equations in Banach spaces. Lunardi [Lun95] studied the maximal regularity of C 1 solutions. In the meantime, Dore–Venni [DV87] used the method for constructing Wp1 (1 < p < ∞) solutions for Lp external force functions with values in a Banach space enjoying HT property (namely, a Banach space in which the Hilbert transform is bounded) under the condition that the coefficient linear operators have bounded purely imaginary powers. This result gives the Lp (1 < p < ∞) maximal regularity (see also Yamamoto [Yamm93]). Afterward, many authors have studied extensions of this result in the context of H∞ functional calculus in various directions. We refer the reader to the monograph due to Prüss [Pru02] and references therein. There is of course a great deal of important literature which deals with the parabolic differential equations directly by the functional analytical methods. However, we will here quote only some monographs by Ladyzhenskaya–Solonnikov– Uraltseva [LSU68], Amann [Ama95], and so forth. The main Theorems 3.8 and 3.9 were proved when D(A(t)) ≡ D(A(0)) (i.e., ν = 1) in [Sob61a] and [Tan60]. When the domain condition (3.29) fails for any ν > 0 (i.e., ν = 0, see Sect. 10.3), as shown by [Tan64], we have to assume condition (3.115). So, (3.29), (3.30), and (3.31) can be regarded as those of interpolating these conditions of the two limit cases that ν = 1 (with μ > 0) and ν = 0 (with μ > 1). In [Sob61b] and [Kat61c], it was assumed that D(A(t)θ ) are independent of t for some suitable exponent 0 < θ < 1, but we need not to assume such an independence condition. Theorems 3.8 and 3.9 were essentially presented by the paper [Yag88]. So far, maximal regularity was mainly investigated for the correspondence F → dU dt (from the external force function to the derivative of solution), and the initial value U0 was usually assumed to be such that U0 ∈ D(A(0)). Theorem 3.5 or 3.10 newly provides the maximal regularity for both initial value U0 ∈ D(A(0)β ) and external force function F ∈ Fβ,σ ((0, T ]; X). The results obtained in this chapter can be extended in various directions. For example, we are able to get rid of the density of domains D(A(t)) (cf. [Yag88]). Conditions (3.27) and (3.28) can be relaxed to σ (A(t)) ⊂ Σα = λ ∈ C; c|Im λ|α < Re λ , (3.131) (λ − A(t))−1 ≤ M , |λ|β

λ∈ / Σα ,

(3.132)

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3 Linear Evolution Equations

respectively, with some exponents 0 < β ≤ α ≤ 1, see [Yag89, Yag90]. We are able to generalize the results to the abstract parabolic evolution equations of the form dU + A(t)U F (t), dt

0 < t ≤ T,

where A(t) is a family of sectorial multivalued linear operators (Chap. 1, Sect. 3.4), see Favini–Yagi [FY99]. Muramatsu–Tojima [MT01] have relaxed (3.30) to some integrability condition.

Chapter 4

Semilinear Evolution Equations

This chapter is devoted to the Cauchy problem for a semilinear evolution equation of the form (4.1) in a Banach space X. We want to present existence and uniqueness results in a sophisticated way enough to apply them to various models considered in the subsequent chapters (Chaps. 8, 9, 10, 11, 12, and 13). The equation in (4.1) contains two operators, a linear operator A and a nonlinear operator F . The linear operator A is a sectorial operator of X with angle ωA < π2 , and hence −A generates an analytic semigroup on X. Meanwhile, the nonlinear operator F is dominated by the fractional power Aη of A with some exponent 0 ≤ η < 1, and F satisfies a Lipschitz condition described by two fractional powers Aη and Aβ , where β is another exponent such that 0 ≤ β ≤ η < 1. In applications, we intend to take the β as small as possible in view of the fact that η is allowed to be arbitrarily close to 1 by the following two reasons. First, the initial values of (4.1) can be taken in D(Aβ ). Secondly, as Corollaries 4.1 and 4.3 show, the a priori estimates for local solutions of (4.1) with respect to the Aβ graph norm ensure extension of local solutions without limit so as to construct the global solutions. So, the case β = 0 is the optimal one.

1 Semilinear Abstract Evolution Equations 1.1 Cauchy Problems Let X be a Banach space with norm · . We consider the Cauchy problem for a semilinear abstract evolution equation dU 0 < t ≤ T, dt + AU = F (U ) + G(t), (4.1) U (0) = U0 , in X. Here, A is a sectorial operator of X satisfying (2.92) and (2.93). Meanwhile, F is a nonlinear operator from D(Aη ) into X, where 0 < η < 1. F is assumed to A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_4, © Springer-Verlag Berlin Heidelberg 2010

177

178

4 Semilinear Evolution Equations

satisfy a Lipschitz condition of the form F (U ) − F (V ) ≤ ϕ(Aβ U + Aβ V )[Aη (U − V ) + (Aη U + Aη V )Aβ (U − V )],

U, V ∈ D(Aη ), (4.2)

where β is a second exponent such that 0 < β ≤ η < 1,

(4.3)

ϕ(·) being some increasing continuous function. In particular, (4.2) implies the estimate F (U ) ≤ ψ(Aβ U )(Aη U + 1),

U ∈ D(Aη ),

(4.4)

where ψ(ξ ) = F (0) + ϕ(ξ )(ξ + 1). The function G(t) is an external force function given in the space Fβ,σ ((0, T ]; X), 0 < σ < β. The initial value U0 is taken in D(Aβ ). We shall show the following result. Theorem 4.1 Let (2.92), (2.93), (4.2), and (4.3) be satisfied. Then, for any G ∈ Fβ,σ ((0, T ]; X), where 0 < σ < 1 − η, and any U0 ∈ D(Aβ ), (4.1) possesses a unique local solution U in the function space: U ∈ C((0, TG,U0 ]; D(A)) ∩ C([0, TG,U0 ]; D(Aβ )) ∩ C1 ((0, TG,U0 ]; X), (4.5) dU AU ∈ Fβ,σ ((0, TG,U0 ]; X), dt , TG,U0 > 0 depending only on the norms GFβ,σ and Aβ U0 . In addition, U satisfies the estimates dU (4.6) Aβ U C + dt β,σ + AU Fβ,σ ≤ CG,U0 F with some constant CG,U0 > 0 depending on the norms GFβ,σ and Aβ U0 .

1.2 Proof of Theorem 4.1 For each S ∈ (0, T ], we set the following Banach space which consists of D(Aη ) valued continuous functions on (0, S]: X(S) = U ∈ C((0, S]; D(Aη )) ∩ C([0, S]; D(Aβ )); sup t η−β Aη U (t) < ∞ , 0 A0 Aβ U0 + Aβ B(1 − β, β)Gβ,σ , and if S is sufficiently small, then Φ(U ) satisfies (4.7) and (4.8). For 0 < s < t ≤ S, we see by the semigroup property that t −(t−s)A −sA {ΦU }(t) = e e U0 + e−(t−τ )A [F (U (τ )) + G(τ )] dτ + e−(t−s)A

s s

e−(s−τ )A [F (U (τ ) + G(τ )] dτ.

0

From this it is verified that {ΦU }(t) − {ΦU }(s) = [e−(t−s)A − 1]{ΦU }(s) t + e−(t−τ )A [F (U (τ )) + G(τ )] dτ. s

Then, by (4.10) and (4.11) (θ = η + σ < 1), we obtain that Aη [{ΦU }(t) − {ΦU }(s)] ≤ [e−(t−s)A − 1]A−σ Aη+σ {ΦU }(s) t + Aη e−(t−τ )A [F (U (τ )) + G(τ )] dτ s

≤ CG,U0 A1−σ (t − s)σ s β−σ −η t + CG,U0 Aη (t − τ )−η (τ β−1 + 1) dτ. s

Here, we also used property (2.129). Writing β − 1 as β − 1 = (η + σ − 1) + (β − σ − η), we have t t (t − τ )−η τ β−1 dτ ≤ (t − τ )−η (τ − s)η+σ −1 s β−σ −η dτ s

s

= B(1 − η, η + σ )(t − s)σ s β−σ −η . Hence, we have obtained that Aη [{ΦU }(t) − {ΦU }(s)] ≤ CG,U0 (t − s)σ s β−σ −η ,

0 < s < t ≤ S.

(4.12)

This in particular shows that ΦU ∈ C((0, S]; D(Aη )). It is also verified, by a similar calculation, that σ −σ

A [{ΦU }(t) − {ΦU }(s)] ≤ CG,U0 (t − s) s β

+ CG,U0

s

t

(t − τ )−β τ β−1 dτ.

1 Semilinear Abstract Evolution Equations

181

Using now the decomposition β − 1 = (β + σ − 1) − σ , we have t t (t − τ )−β τ β−1 dτ ≤ (t − τ )−β (τ − s)β+σ −1 s −σ dτ s

s

= B(1 − β, β + σ )(t − s)σ s −σ . Therefore, Aβ [{ΦU }(t) − {ΦU }(s)] ≤ CG,U0 (t − s)σ s −σ ,

0 < s < t ≤ S.

(4.13)

t We already know by Theorem 3.5 that e−tA U0 + 0 e−(t−s)A G(s)ds is continuous at t = 0 in the graph norm of D(Aβ ). In addition, it follows from (4.10) that, as t → 0, t t β −(t−s)A A ≤ Aβ e F (U (s)) ds (t − s)−β (s β−η + 1) ds 0

0

≤ CAβ (t 1−η + t 1−β ) → 0. Therefore, ΦU ∈ C([0, S]; D(Aβ )). The mapping Φ has been shown to map K(S) into itself, provided that S > 0 is sufficiently small. Step 2. We next verify that Φ is a contraction mapping of X(S). Let U, V ∈ K(S) be any two functions. For θ such that β ≤ θ < 1, we can write t θ A [{ΦU }(t) − {ΦV }(t)] = Aθ e−(t−s)A [F (U (s)) − F (V (s))] ds. 0

From (4.2), (4.7), and (4.8) it is seen that t

θ−β

A [{ΦU }(t) − {ΦV }(t)] ≤ Aθ ϕ(2C2 )t θ

θ−β

t

(t − s)−θ

0

× {A [U (s) − V (s)] + (2C1 s + 1)A [U (s) − V (s)]} ds t ≤ CAθ (C1 + 1)ϕ(2C2 )t θ−β (t − s)−θ (s β−η + 1) dsU − V X(S) . η

β−η

β

0

Applying these estimates with θ = η and θ = β, we easily conclude that ΦU − ΦV X(S) ≤ C(C1 + 1)ϕ(2C2 )S 1−η U − V X(S) , which shows that Φ is a contraction in X(S), provided that S > 0 is sufficiently small. Step 3. Let S > 0 be sufficiently small in such a way that Φ maps K(S) into itself and is a contraction with respect to the norm X(S). By the arguments of Steps 1 and 2, S = TG,U0 > 0 can be determined by the two norms Gβ,σ and Aβ U0

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4 Semilinear Evolution Equations

only. Thanks to the fixed point theorem for contraction mappings, Theorem 1.11, there exists a unique function U ∈ K(TG,U0 ) such that U = ΦU . From now on, U stands for this function. Clearly, t e−(t−s)A [F (U (s)) + G(s)] ds, 0 ≤ t ≤ TG,U0 . (4.14) U (t) = e−tA U0 + 0

We next verify that F (U ) ∈ Fβ,σ ((0, TG,U0 ]; X).

(4.15)

In fact, by(4.10), as t → 0, t 1−β F (U (t)) ≤ CG,U0 (t 1−η + t 1−β ) → 0. Furthermore, from (4.2), (4.12), and (4.13) it follows that F (U (t)) − F (U (s)) = F ({ΦU }(t)) − F ({ΦU }(s)) ≤ CG,U0 (t − s)σ s β−σ −η + (s β−η + 1)(t − s)σ s −σ ≤ CG,U0 (t − s)σ s β−σ −η . Consequently, s 1−β+σ F (U (t)) − F (U (s)) ≤ CG,U0 s 1−η , (t − s)σ

0 < s < t ≤ TG,U0 .

This shows that (1.6) and (1.7) are valid for F (U (t)). Hence, (4.15) is verified. We now know that F (U ) + G ∈ Fβ,σ ((0, T ]; X). In view of (4.14), we can apply Theorem 3.5 to conclude that U belongs to the space (4.5) with estimates (4.6) and is a solution to problem (4.1). Step 4. Let us finally verify the uniqueness of solutions. Let U˜ be any other solution to (4.1) on the interval [0, TG,U0 ] which belongs to the space (4.5). By the definition, t 1−β AU˜ (t) + Aβ U˜ (t) ≤ CU˜ ,

0 < t ≤ TG,U0 .

Moreover, we can verify the intermediate estimate t η−β Aη U˜ (t) ≤ CU˜ ,

0 < t ≤ TG,U0 .

Indeed, by the moment inequality (2.117) for the fractional power A1−β applied with θ = η−β 1−β , we have η−β

1−η

Aη−β V ≤ CA1−β V 1−β V 1−β ,

V ∈ D(A1−β ).

Then, taking V = Aβ U˜ (t), we obtain the desired estimate.

1 Semilinear Abstract Evolution Equations

183

The formula U˜ (t) = e−tA U0 +

t

e−(t−s)A [F (U˜ (s)) + G(s)] ds,

0

0 ≤ t ≤ TG,U0 ,

together with (4.14), yields that U (t) − U˜ (t) =

t

e−(t−s)A [F (U (s)) − F (U˜ (s))] ds,

0

0 ≤ t ≤ TG,U0 .

We can then repeat the same arguments as in Step 2 to deduce that U − U˜ X(S) ≤ CU,U˜ S 1−η U − U˜ X(S) ,

0 < S ≤ TG,U0 .

This means that, if S > 0 is sufficiently small, then U (t) = U˜ (t) for every 0 ≤ t ≤ S. Let S˜ = sup{S; U (t) = U˜ (t) for all 0 ≤ t ≤ S}, and suppose that S˜ < TG,U0 . Then, ˜ = U˜ (S). ˜ We repeat the same procedure with the initial time S˜ and we have U (S) ˜ ˜ to derive that U (S˜ + t) = U˜ (S˜ + t) for every sufficiently initial value U (S) = U˜ (S) small t > 0, which is a contradiction. Hence, U (t) = U˜ (t) for every 0 < t ≤ TG,U0 .

1.3 Regularity for More Regular Initial Data For an external force function G ∈ Fγ ,σ ((0, T ]; X), β < γ ≤ 1, and an initial value U0 ∈ D(Aγ ), β < γ ≤ 1, we can verify a stronger regularity than (4.5) for the local solution of (4.1). Theorem 4.2 Let (2.92), (2.93), (4.2), and (4.3) be satisfied. Then, for any G ∈ Fγ ,σ ((0, T ]; X), 0 < σ < 1 − η, β < γ ≤ 1, and any U0 ∈ D(Aγ ), β < γ ≤ 1, the local solution U of (4.1) constructed in Theorem 4.1 has the following regularity: U ∈ C((0, TG,U0 ]; D(A)) ∩ C([0, TG,U0 ]; D(Aγ )) ∩ C1 ((0, TG,U0 ]; X), (4.16) dU AU ∈ Fγ ,σ ((0, TG,U0 ]; X). dt , In addition, U satisfies the estimates dU Aγ U C + dt γ ,σ + AU Fγ ,σ ≤ CG,U0 F

(4.17)

with some constant CG,U0 > 0 depending on the norms GFγ ,σ and Aγ U0 . Proof From (4.14) we have A U (t) = A η

η−γ −tA

e

A U0 + γ

0

t

Aη e−(t−s)A [F (U (s)) + G(s)] ds.

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4 Semilinear Evolution Equations

Therefore, in view of (4.4), Aη U (t) ≤ Aη−γ e−tA Aγ U0 t +C (t − s)−η [Aη U (s) + GFγ ,σ s γ −1 + 1] ds. 0

t When η ≤ γ ≤ 1, we see that Aη U (t) ≤ CG,U0 + C 0 (t − s)−η Aη U (s) ds. Theorem 1.27 then provides that Aη U (t) ≤ CG,U0 . When β < γ < η, however, we see that t η γ −η +C (t − s)−η Aη U (s) ds. A U (t) ≤ CG,U0 t 0

In this case, the function

p(t) = t η−γ Aη U (t)

p(t) ≤ CG,U0 + C

t

satisfies

t η−γ (t − s)−η s γ −η p(s) ds.

0

Let us solve the integral equation as follows. Let ε > 0 denote a small parameter. For 0 ≤ t ≤ ε, t p(t) ≤ CG,U0 + C t η−γ (t − s)−η s γ −η ds sup p(s) 0

0≤s≤ε

≤ CG,U0 + Ct 1−η sup p(s) ≤ CG,U0 + Cε 1−η sup p(s). 0≤s≤ε

0≤s≤ε

Hence, (1 − Cε 1−η ) sup0≤s≤ε p(s) ≤ CG,U0 . If ε is taken sufficiently small so that Cε1−η ≤ 12 , then we conclude that p(t) ≤ CG,U0 , 0 ≤ t ≤ ε. Meanwhile, for ε < t ≤ TG,U0 , we have t ε p(t) ≤ CG,U0 + C t η−γ (t − s)−η s γ −η ds + CT η−γ ε γ −η (t − s)−η p(s) ds 0

≤ CG,U0 + C

ε

t

(t − s)−η p(s) ds.

ε

Then, Theorem 1.27 again provides that p(t) ≤ CU0 ,G , ε ≤ t ≤ TG,U0 . We have thus shown that Aη U (t) ≤ CG,U0 (t γ −η + 1),

0 < t ≤ TG,U0 .

(4.18)

Using (4.18) (instead of (4.7)) and (4.8), we will repeat the same argument as in Step 1 of the proof of Theorem 4.1. Then, it is concluded that Aη [U (t) − U (s)] ≤ CG,U0 (t − s)σ (s γ −η−σ + 1),

0 < s ≤ t ≤ TG,U0 ,

Aβ [U (t) − U (s)] ≤ CG,U0 (t − s)σ (s γ −β−σ + 1),

0 < s ≤ t ≤ TG,U0 .

and

1 Semilinear Abstract Evolution Equations

185

Furthermore, these estimates immediately imply that F (U ) ∈ Fγ ,σ ((0, TG,U0 ]; X) with the estimate F (U )Fγ ,σ ≤ CG,U0 . The desired regularity (4.16) is then an immediate consequence of Theorem 3.5.

1.4 Global Existence We will return to the case where G ∈ Fβ,σ ((0, T ]; X) and U0 ∈ D(Aβ ). We recall that, according to Theorem 4.1, the interval [0, TG,U0 ] on which the local solution was constructed is determined by the norms GFβ,σ and Aβ U0 only. This fact immediately provides the following global existence of solutions which is utilized very often in applications. Corollary 4.1 Under the assumptions of Theorem 4.1, let G ∈ Fβ,σ ((0, T ]; X), 0 < σ < 1 − η, and U0 ∈ D(Aβ ). Assume that any local solution U of (4.1) in the function space C((0, TU ]; D(A)) ∩ C([0, TU ]; D(Aβ )) ∩ C1 ((0, TU ]; X) satisfies the estimate Aβ U (t) ≤ CG,U0 ,

0 ≤ t ≤ TU ,

(4.19)

with some constant CG,U0 > 0 independent of TU . Then, (4.1) possesses a unique global solution on the interval [0, T ]. Proof We extend the function G to a function G defined on the whole interval [0, ∞) by putting G(t) ≡ G(T ) for T < t < ∞. Clearly, we have GFβ,σ ((a,b];X) ≤ GFβ,σ ((0,T ];X) for any interval (a, b]. Let U1 be any vector in D(Aβ ) satisfying Aβ U1 ≤ CG,U0 . We here consider the Cauchy problem dV dt + AV = F (V ) + G(t), t1 < t < ∞, (4.20) V (t1 ) = U1 , with initial time t1 such that 0 ≤ t1 < ∞. Theorem 4.1 then provides that there exists a length τ > 0 determined by GFβ,σ and CG,U0 only such that (4.20) always possesses a local solution on an interval [t1 , t1 + τ ]. On account of this fact, we can prove the assertion of the corollary. Let us start with the local solution to (4.1) on [0, TG,U0 ] obtained in Theorem 4.1. Clearly, TG,U0 ≥ τ . Setting t1 = TG,U0 − τ2 and U1 = U (t1 ), we consider problem (4.20) to obtain a local solution V on an interval [t1 , t1 + τ ]. By the uniqueness of solution, we have U (t) = V (t) for t1 ≤ t ≤ TG,U0 ; this means that we have constructed

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4 Semilinear Evolution Equations

a local solution to (4.1) on the interval [0, TG,U0 + τ2 ]. The a priori estimate (4.19) allows us to continue this procedure unlimitedly. Each time, the local solution is extended over the fixed length τ2 of interval. So, by finite times, the extended interval can cover the given interval [0, T ].

1.5 Lipschitz Continuity of Solutions in Initial Data Let G be a bounded ball G = {G ∈ Fβ,σ ((0, T ]; X); Gβ,σ ≤ R1 },

0 < R1 < ∞,

of the space Fβ,σ ((0, T ]; X), and let B be a bounded ball B = {U ∈ D(Aβ ); Aβ U ≤ R2 },

0 < R2 < ∞,

of D(Aβ ). Then, for all G ∈ G and U0 ∈ B, there exists a local solution of (4.1) on some interval [0, TG,U0 ]. There is then time TG,B > 0 such that [0, TG,B ] ⊂ [0, TG,U0 ] for all pairs of (G, U0 ) ∈ G × B. We shall show the continuous dependence of solutions on (G, U0 ). Theorem 4.3 Under (2.92), (2.93), (4.2), and (4.3), let U and V be the solutions of (4.1) for the data (G, U0 ) and (H, V0 ) in G × B, respectively. Then, t η Aη [U (t) − V (t)] + t β Aβ [U (t) − V (t)] + U (t) − V (t) ≤ CG,B [U0 − V0 + t β G − H β,σ ],

0 < t ≤ TG,B .

Proof This theorem is proved by analogous arguments as in the proof of Theorem 4.1. Indeed, for 0 ≤ θ < 1, we see that t Aθ [U (t) − V (t)] = Aθ e−tA (U0 − V0 ) + Aθ e−(t−s)A 0

× {[F (U (s)) − F (V (s))] + [G(s) − H (s)]} ds, 0 < t ≤ TB,G . As a consequence, t θ Aθ [U (t) − V (t)] ≤ Aθ U0 − V0 + Aθ t θ + CG,B Aθ t θ

t

(t − s)−θ G(s) − H (s) ds

0 t

(t − s)−θ {Aη [U (s) − V (s)]

0

+ (s β−η + 1)Aβ [U (s) − V (s)]} ds

1 Semilinear Abstract Evolution Equations

187

≤ Aθ U0 − V0 + Aθ G − H β,σ t θ + CG,B Aθ t

θ

t

(t − s)−θ s β−1 ds

0 t

(t − s)−θ s −η p(s) ds,

0

where p(t) = t η Aη [U (t) − V (t)] + t β Aβ [U (t) − V (t)],

0 < t ≤ TG,B .

Using these estimates with θ = β and η, we obtain the integral inequality p(t) ≤ C[U0 − V0 + t β G − H β,σ ] t η + CG,B t (t − s)−η + t β (t − s)−β s −η p(s) ds,

0 < t ≤ TG,B .

0

To solve this inequality, we use the same techniques as in the proof of Theorem 4.2. Arguing first in a small interval [0, ε] and then in the other interval [ε, TG,B ], we in fact conclude that p(t) ≤ CG,B [U0 − V0 + t β G − H β,σ ],

0 < t ≤ TG,B .

Finally, taking θ = 0, we conclude also that U (t) − V (t) ≤ CG,B [U0 − V0 + t β G − H β,σ ],

0 ≤ t ≤ TG,B .

Corollary 4.2 Under the conditions of Theorem 4.3, t η−β Aη [U (t) − V (t)] + Aβ [U (t) − V (t)] ≤ CG,B [Aβ (U0 − V0 ) + G − H β,σ ],

0 < t ≤ TG,B .

Proof The techniques used in the proof of Theorem 4.3 are available for verifying the assertion. It indeed suffices to substitute the estimate Aθ e−tA (U0 − V0 ) ≤ Aθ−β t β−θ Aβ (U0 − V0 ) with θ = β and η for Aθ e−tA (U0 − V0 ) ≤ Aθ t −θ U0 − V0 .

1.6 Case β = 0 It is possible to generalize some results of Theorem 4.1 to the critical case where β = 0. Let us assume that (4.2) is valid with β = 0, i.e., F (U ) − F (V ) ≤ ϕ(U + V )[Aη (U − V ) + (Aη U + Aη V )U − V ],

U, V ∈ D(Aη ),

(4.21)

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4 Semilinear Evolution Equations

with 0 ≤ η < 1.

(4.22)

We can then verify the following result. Theorem 4.4 Let (2.92), (2.93), (4.21), and (4.22) be satisfied. Then, for any G ∈ Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1 − η, and any U0 ∈ X, (4.1) possesses a unique local solution U in the function space: U ∈ C([0, TG,U0 ]; X) ∩ C1 ((0, TG,U0 ]; X), AU ∈ C((0, TG,U0 ]; X),

(4.23)

TG,U0 > 0 depending only on the norms GFβ,σ and U0 . In addition, U satisfies the estimates dU (4.24) (t) U (t) + t + tAU (t) ≤ CG,U0 , 0 < t ≤ TG,U0 , dt with some constant CG,U0 > 0 depending on the norms GFβ,σ and U0 . Proof The method of proving this theorem is quite analogous to that of Theorem 4.1. We set for 0 < S ≤ T , the basic Banach space X(S) = U ∈ C((0, S]; D(Aη )) ∩ C([0, S]; X); sup t η Aη U (t) < ∞ , 0 0 (i = 1, 2) are chosen appropriately and S > 0 is sufficiently small, then Φ maps the set K(S) into itself and is a contraction with respect to the norm X(S). Consequently, Φ possesses a unique fixed point U ∈ K(S). Since F (U ) = F (ΦU ) ∈ F β,σ ((0, S]; X),

(4.27)

we conclude that U is a unique solution of (4.1) in the function space (4.23). Estimate (4.24) is verified by Theorem 3.4.

2 Variational Methods

189

By the same argument as in Corollary 4.1, we can similarly construct a global solution to (4.1). Corollary 4.3 Under the assumptions of Theorem 4.4, let G ∈ Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1 − η, and U0 ∈ X. Assume that any local solution U of (4.1) in the function space C((0, TU ]; D(A)) ∩ C([0, TU ]; X) ∩ C1 ((0, TU ]; X)

(4.28)

satisfies the estimate U (t) ≤ CG,U0 ,

0 ≤ t ≤ TU ,

(4.29)

with some constant CG,U0 > 0 independent of TU . Then, (4.1) possesses a unique global solution in the function space: U ∈ C((0, T ]; D(A)) ∩ C([0, T ]; X) ∩ C1 ((0, T ]; X). Let us finally verify the Lipschitz continuity of solutions in initial data. Let G be a bounded ball G = {G ∈ Fβ,σ ((0, T ]; X); Gβ,σ ≤ R1 },

0 < R1 < ∞,

of the space Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1 − η, and let B be a bounded ball B = {U ∈ X; U ≤ R2 },

0 < R2 < ∞,

of X. Then, there is an interval [0, TG,B ] on which (4.1) has a unique local solution for any pair (G, U0 ) ∈ G × B. Then, we have the following result. Theorem 4.5 Under (2.92), (2.93), (4.21), and (4.22), let U and V be the solutions of (4.1) for the data (G, U0 ) and (H, V0 ) in G × B, respectively. Then, t η Aη [U (t) − V (t)] + U (t) − V (t) ≤ CG,B [U0 − V0 + t β G − H β,σ ],

0 < t ≤ TG,B .

As the proof of this theorem is quite analogous to that of Theorem 4.3, we may omit it.

2 Variational Methods We here present existence and uniqueness results for problem (4.1) by the variational methods. Let Z and X be two separable Hilbert spaces with dense and compact embedding Z ⊂ X. Let · and | · | be the norms of Z and X, respectively. Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Let · ∗ be the norm of Z ∗ . Let a(U, V ) be a sesquilinear

190

4 Semilinear Evolution Equations

form defined on Z satisfying (2.6) and (2.7), and let A be a sectorial operator of Z ∗ associated with the form. We are concerned with the Cauchy problem for a semilinear evolution equation dU dt + AU = F (U ) + G(t), 0 < t ≤ T , (4.30) U (0) = U0 , in Z ∗ . Here, F is a nonlinear operator from Z to Z ∗ and is assumed to satisfy the following conditions. For any positive number ζ > 0, there exist continuous increasing functions φζ (·) ≥ 0 and ψζ (·) ≥ 0 such that the following estimates hold: F (U )∗ ≤ ζ U + φζ (|U |),

U ∈ Z;

(4.31)

F (U ) − F (V )∗ ≤ ζ U − V + (U + V + 1)ψζ (|U | + |V |)|U − V |, U, V ∈ Z.

(4.32)

The external force function G is a Z ∗ -valued function belonging to L2 (0, T ; Z ∗ ), and the initial value U0 is taken in X. We then obtain the following theorem. For the proof, see Lions [Lio69] (see also [RY01]). Theorem 4.6 Let (4.31) and (4.32) be satisfied. Then, for any G ∈ L2 (0, T ; Z ∗ ) and any U0 ∈ X, there exists a unique local solution U to (4.30) in the function space: U ∈ L2 (0, TU0 ,G ; Z) ∩ C([0, TU0 ,G ]; X) ∩ H 1 (0, TU0 ,G ; Z ∗ ), where TU0 ,G > 0 is determined by the norms |U0 | and GL2 (0,T ;Z ∗ ) . In addition, U satisfies the estimates U L2 (0,TG,U0 ;Z) + U C([0,TG,U0 ];X) + U H 1 (0,TG,U

0

;Z ∗ )

≤ CG,U0 ,

where CG,U0 > 0 is a constant determined by the norms GL2 (0,T ;Z ∗ ) and |U0 |.

3 Semilinear Parabolic Equations We consider the initial-boundary-value problem for a semilinear parabolic equation ⎧ n ∂u ⎪ ⎪ i,j =1 Dj [aij (x)Di u] + f (x, u, ∇u) + g(x, t) in Ω × (0, T ), ∂t = ⎨ n on ∂Ω × (0, T ), (4.33) i,j =1 νj (x)aij (x)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω, where 0 < T < ∞ is fixed time. We assume that aij (x) satisfy (2.19) and (2.28). The function f (x, u, ζ ) is

4 Competition System with Diffusions

191

a complex-valued function for x ∈ Ω, u ∈ R + iR, and ζ ∈ (R + iR)n , which is a smooth function with respect to the real variables x ∈ Ω, Re u, Im u, Re ζ , and Im ζ . The function g(x, t) is an external force function, and u0 (x) is an initial function. We want to handle (4.33) in a Banach space X = Lp (Ω), where n < p < ∞. Let Ap be a realization of the differential operator − i,j Dj [aij (x)Di ] + 1 in Lp (Ω) ∂u under the Neumann-type boundary conditions ∂ν = 0 on ∂Ω. It is known that the A domain D(Ap ) is characterized by (2.77). Using this operator, we obtain an abstract formulation of (4.33) of the form (4.1). Here, the nonlinear operator F (u) is given by F (u) = u + f (x, u, ∇u),

u ∈ Hps (Ω),

where the exponent s is taken as 1 + pn < s < 2. We notice by (1.76) that Hps (Ω) ⊂ C1 (Ω) with continuous embedding. We choose exponents β and η as 2s < β = η < 1. Then, according to (16.36) in η Chap. 16, it is observed that D(Ap ) ⊂ Hps (Ω). In addition, by a direct calculation, F (u) is shown to fulfill condition (4.2). So, the abstract results obtained above are available to (4.33). By Theorem 4.1, for any g ∈ Fβ,σ ((0, T ]; Lp (Ω)), 0 < σ < β 1 − η, and any u0 ∈ D(Ap ), there exists a unique local solution u to (4.33) in the function space: β u ∈ C((0, Tg,u0 ]; D(Ap )) ∩ C([0, Tg,u0 ]; D(Ap )) ∩ C1 ((0, Tg,u0 ]; Lp (Ω)), du Ap u ∈ Fβ,σ ((0, Tg,u0 ]; Lp (Ω)), dt , β

Tg,u0 > 0 depending only on the norms gFβ,σ and Ap u0 . Furthermore, u satisfies the estimates β A U + du p dt β,σ + Ap uFβ,σ ≤ Cg,u0 . C F

4 Competition System with Diffusions Consider the initial-boundary-value problem for the competition system ⎧ ∂u ⎪ = aΔu + cu − γ11 u2 − γ12 uv in Ω × (0, ∞), ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂v = bΔv + dv − γ21 uv − γ22 v 2 in Ω × (0, ∞), ∂t ∂u ∂v ⎪ on ∂Ω × (0, ∞), ⎪ ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x) in Ω, 0

(4.34)

0

in a C2 or convex, bounded domain Ω ⊂ R3 . We consider two competing species of biological individuals in Ω. Their densities are denoted by u(x, t) and v(x, t), respectively. We consider diffusions of individuals, and the diffusion rates of species are denoted by a and b, respectively.

192

4 Semilinear Evolution Equations

We assume that a, b, c, d, and γij (i, j = 1, 2) are positive constants (> 0). Initial functions u0 , v0 are assumed to satisfy 0 ≤ u0 , v0 ∈ L2 (Ω).

4.1 Construction of Local Solutions We want to handle (4.34) in the product space X = L2 (Ω) (see (1.84)). The problem is then formulated as the Cauchy problem for an abstract equation dU 0 < t < ∞, dt + AU = F (U ), (4.35) U (0) = U0 , in X. Here, A is a diagonal matrix operator of X given by A = diag {A1 , A2 } with domain D(A) = H2N (Ω), where A1 and A2 are realizations of operators −aΔ + 1 and −bΔ + 1, respectively, in L2 (Ω) under the Neumann boundary conditions on ∂Ω. It is known by (2.34) that A1 and A2 are positive definite self-adjoint operators of L2 (Ω) with domains D(A1 ) = D(A2 ) = HN2 (Ω). Furthermore, according to Theorems 16.7 and 16.9 in Chap. 16, D(Aθ1 ) = D(Aθ2 ) = H 2θ (Ω) if 0 ≤ θ < 34 and D(Aθ1 ) = D(Aθ2 ) = HN2θ (Ω) if 34 < θ ≤ 1. As a consequence, A is a positive definite self-adjoint operator of X, and the domains of fractional powers are given 3 by D(Aθ ) = H2θ (Ω) if 0 ≤ θ < 34 and D(Aθ ) = H2θ N (Ω) if 4 < θ ≤ 1. The nonlinear operator F : L4 (Ω) → L2 (Ω) is defined by u (c + 1)u − γ11 u2 − γ12 uv , U= ∈ L4 (Ω). F (U ) = (d + 1)v − γ21 uv − γ22 v 2 v The space of initial functions is given by u K = U0 = 0 ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ v0 ∈ L2 (Ω) . v0 It is immediate to verify that the nonlinear operator F fulfills the condition (4.21) 2η with 34 < η < 1. Indeed, since D(Aη ) = HN (Ω) ⊂ L∞ (Ω), we have F (U ) − F (V )L2 ≤ C(U L∞ + V L∞ + 1)U − V L2 ≤ C(Aη U L2 + Aη V L2 + 1)U − V L2 ,

U, V ∈ D(Aη ).

Theorem 4.4 then provides the existence of local solutions. Indeed, for any U0 ∈ L2 (Ω), (4.35) possesses a unique local solution U in the function space: U ∈ C (0, TU0 ]; H2N (Ω) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)). (4.36) Here, TU0 > 0 is determined by the norm U0 L2 only. Furthermore, the solution satisfies the estimates U (t)L2 + tAU (t)L2 ≤ CU0 .

(4.37)

4 Competition System with Diffusions

193

4.2 Nonnegativity of Local Solutions We shall show that the local solution constructed above is nonnegative. For U0 ∈ K, let U = t(u, v) be the local solution in the function space (4.36) with estimate (4.37), and let us prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . We first notice that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (4.35) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) must be real valued. We will employ the truncation method. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞ 0 such that, for any local solution U in (4.38) with U0 ∈ K, it holds that U (t)L2 ≤ C(e−t U0 L2 + 1),

0 ≤ t ≤ TU .

(4.39)

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4 Semilinear Evolution Equations

Proof Multiply the first equation of (4.34) by u and integrate the product in Ω. Then, 1 d 2 dt

u2 dx + a Ω

|∇u|2 dx =

Ω

(cu2 − γ11 u3 − γ u2 v) dx

Ω

≤

(cu2 − γ11 u3 ) dx. Ω

Since 1 (2c + 1)3 cu2 − γ11 u3 ≤ − u2 + 2 2 27γ11

for all u ≥ 0,

it follows that 1 d 2 dt

1 u dx ≤ − 2 Ω

u2 dx +

2

Ω

|Ω|(2c + 1)3 . 2 27γ11

Solving this differential inequality (cf. (1.58)), we obtain that 2|Ω|(2c + 1)3 , 2 27γ11

u(t)2L2 ≤ e−t u0 2L2 +

0 ≤ t ≤ TU .

We verify by the similar arguments that v(t)2L2 ≤ e−t v0 2L2 +

2|Ω|(2d + 1)3 , 2 27γ22

0 ≤ t ≤ TU .

This, together with the above, yields the desired estimate (4.39).

The global existence is now obtained immediately by Corollary 4.3. Indeed, (4.39) shows that estimate (4.29) holds for any nonnegative local solution with initial value U0 ∈ K. We can then repeat the same arguments as in Corollary 4.3 to conclude the global existence of a solution. We have thus deduced that, for any U0 ∈ K, there exists a unique global solution of (4.34) in the function space: 0 ≤ u, v ∈ C (0, ∞); HN2 (Ω) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)).

5 Some Model in Immunology We are concerned with the initial-boundary-value problem for some mathematical model describing the immune system. The problem is given by

5 Some Model in Immunology

195

⎧ ∂u ⎪ ⎪ ∂t = c − du − αuρ ⎪ ⎪ ⎪ ∂v ⎪ ⎪ ⎪ ∂t = αuρ − ev − βvw ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ⎨ ∂t = aΔw − f w + γ (x) ∂ρ ∂t = bΔρ − gρ ⎪ ⎪ ⎪ ∂ρ ∂w ⎪ ⎪ = ∂n =0 ⎪ ⎪ ∂n

+ hv

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞),

(4.40)

on ∂Ω × (0, ∞), ⎪ ⎪ ⎪ ⎪ v(x, 0) = v0 (x), u(x, 0) = u0 (x), ⎪ ⎪ ⎩ ρ(x, 0) = ρ0 (x) in Ω, w(x, 0) = w0 (x), in a C2 or convex, bounded domain Ω ⊂ R3 . Here, Ω denotes an organism or a body. The function u(x, t) denotes the density of uninfected target cells in Ω at time t, and v(x, t) denotes the density of infected cells by viruses. By w(x, t) the density of killer cells is denoted. The killer cells are produced at a rate γ (x) in Ω by the effect of the immune system and kill the infected cells. By ρ(x, t) the density of viruses is denoted. Viruses are produced by the infected cells and diffuses in Ω with diffusion rate b. We assume that 0 ≤ γ ∈ L2 (Ω)

(4.41)

and that a, b, c, d, e, f , and g are positive constants (>0) and α, β are also positive constants (>0). We will handle this problem in the product space X = L2 (Ω) = t(u, v, w, ρ); u, v, w and ρ ∈ L2 (Ω) . The space of initial functions is given by K = t(u0 , v0 , w0 , ρ0 ); 0 ≤ u0 , v0 , w0 and ρ0 ∈ L2 (Ω) .

(4.42)

5.1 Construction of Local Solutions Problem (4.40) is formulated in X as the Cauchy problem of the form (4.35). Here, the linear operator A is a diagonal matrix operator of X given by A = diag{d, e, A1 , A2 }, where A1 and A2 are realizations of differential operators −aΔ + f and −bΔ + g, respectively, in L2 (Ω) under the Neumann boundary conditions on ∂Ω. So, D(A) is given by D(A) = t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ HN2 (Ω) (see (2.34)). Clearly, A is a positive definite self-adjoint operator of X. For 0 < θ < 1, its fractional powers Aθ are also diagonal operators diag{d θ , eθ , Aθ1 , Aθ2 } and, thanks to Theorems 16.7 and 16.9, their domains are characterized by D(Aθ ) = {t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ H 2θ (Ω)}, 0 ≤ θ < 34 , D(Aθ ) = {t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ HN2θ (Ω)}, 34 < θ ≤ 1.

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4 Semilinear Evolution Equations

The nonlinear operator F is given by F (U ) = t (c − αuρ, αuρ − βvw, γ (x), hv) with domain D(F ) = t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ L∞ (Ω) . Fix an exponent η such that 34 < η < 1. Then, by virtue of (1.76), we have D(Aη ) ⊂ D(F ). Furthermore, it is immediate to see that F satisfies condition (4.21). Thanks to Theorem 4.4, we can state that, for any U0 = t(u0 , v0 , w0 , ρ0 ) ∈ K, there exists a unique local solution U = t(u, v, w, ρ) to (4.40) in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; X) ∩ C1 ((0, TU0 ]; X),

(4.43)

where TU0 > 0 is determined by the norm U0 X only. In addition, the local solution satisfies the estimates dU U (t)L2 + t (4.44) dt + tAU (t)L2 ≤ CU0 , 0 < t ≤ TU0 , L2 CU0 > 0 being determined by U0 L2 .

5.2 Nonnegativity of Local Solutions For U0 ∈ K, let U be the local solution of (4.40) constructed above. We want to show that u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, and ρ(t) ≥ 0 for every 0 < t ≤ TU0 . For this purpose, however, we have to introduce the modified nonlinear operator F˜ (U ) = t (c − αuρ, αuρ − βvw, γ (x), hχ(Re v)),

U ∈ D(F˜ ) = D(F ),

where χ(v) denotes a cutoff function such that χ(v) ≡ 0 for −∞ < v < 0 and χ(v) = v for 0 ≤ v < ∞. We have to consider the auxiliary problem

d U˜ dt

+ AU˜ = F˜ (U ),

U˜ (0) = U0 ,

0 < t < ∞,

(4.45)

in X. It is clear that the new nonlinear operator F˜ also satisfies (4.21) with the same ˜ L2 for v, v˜ ∈ L2 (Ω). So, exponent η because of χ(Re v) − χ(Re v) ˜ L2 ≤ v − v ˜ v, ˜ w, ˜ ρ) ˜ on an interval [0, T˜U0 ] in (4.45) possesses a unique local solution U˜ = t(u, the same function space (4.43). Then, it is seen that u(t) ˜ ≥ 0, v(t) ˜ ≥ 0, w(t) ˜ ≥ 0, ρ(t) ˜ ≥ 0 for every 0 < t ≤ T˜0 . In fact, let us first verify that U˜ (t) is real valued. It is clear that the complex conjugate U˜ (t) of U˜ (t) is also a local solution to (4.45) with the same initial value; so, by the uniqueness of solution, U˜ (t) = U˜ (t); hence, U˜ (t) is real valued.

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197

Let us next use a cutoff function H (ρ) ˜ such that H (ρ) ˜ = 12 ρ˜ 2 for −∞ < ρ˜ < 0 and H (ρ) ˜ ≡ 0 for 0 ≤ ρ˜ < ∞. Since ρ˜ ∈ C([0, T˜U0 ];L2 (Ω)) ∩ C1 ((0, T˜U0 ]; L2 (Ω)) due to (4.43), we observe by (1.100) that ψ(t) = Ω H (ρ(t)) ˜ dx is continuously differentiable with the derivative

ψ (t) = b H (ρ)Δ ˜ ρ˜ dx − g H (ρ) ˜ ρ˜ dx + h H (ρ)χ( ˜ v) ˜ dx. Ω

Ω

Ω

By property (1.96), we have H (ρ)Δ ˜ ρ˜ dx = − ∇H (ρ) ˜ · ∇ ρ˜ dx = − |∇H (ρ)| ˜ 2 dx ≤ 0. Ω

Ω

Ω

˜ ≤ 0 and H (ρ) ˜ ρ˜ ≥ 0, it follows that ψ (t) ≤ 0; conseIn addition, since H (ρ) quently, ψ(t) ≤ ψ(0) for 0 < t ≤ T˜U0 . Thus, ψ(0) = 0 implies ψ(t) ≡ 0, namely, ρ(t) ˜ ≥ 0 for 0 ≤ t ≤ T˜U0 . We can repeat the same argument for u, ˜ v, ˜ w˜ also. In fact, knowing that ρ(t) ˜ ≥ 0, we are able to prove that u(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . Since ˜ ρ(t) ˜ ≥ 0 and w(t) ˜ ≥ 0, we γ (x) ≥ 0, w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . Finally, since u(t) conclude that v(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . We now notice that v(t) ˜ ≥ 0 implies that χ(v(t)) ˜ = v(t); ˜ in other words, U˜ is a local solution of the original problem (4.40), too. The uniqueness of solution then implies that U (t) ≡ U˜ (t). Hence, u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, ρ(t) ≥ 0 for 0 < t ≤ T˜U0 . If T˜U0 ≥ TU0 , we finish the proof. If not, we put T0 = sup{0 < T ≤ TU0 ; u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, ρ(t) ≥ 0 for every 0 < t ≤ T }. Since Ω H (u(T0 )) dx = limtT0 Ω H (u(t)) dx = 0, we see that u(T0 ) ≥ 0. By similar reasons, we have v(T0 ) ≥ 0, w(T0 ) ≥ 0, ρ(T0 ) ≥ 0. So, if T0 = TU0 , we finish the proof. If T0 < TU0 , we will consider again problem (4.45) but with the initial time T0 and the initial value U (T0 ). Repeating the same arguments as above, we conclude that there is τ > 0 such that u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, ρ(t) ≥ 0 for every t such that TU0 ≤ t ≤ TU0 + τ . This is a contradiction; hence, T0 = TU0 .

5.3 Global Solutions In order to construct a global solution, let us obtain a priori estimates for local solutions. For U0 ∈ K, let U = t(u, v, w, ρ) be a local solution of (4.40) on [0, TU ] in the function space: 0 ≤ U ∈ C((0, TU ]; D(A)) ∩ C([0, TU ]; X) ∩ C1 ((0, TU ]; X). Integrate the first equation of (4.40) in Ω. Then, d u dx = (c − du − αuρ) dx. dt Ω Ω

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Similarly, by the second equation of (4.40), d v dx = (αuρ − ev − βvw) dx. dt Ω Ω d Therefore, dt Ω (u + v) dx ≤ c|Ω| − δ Ω (u + v) dx, where δ = min{d, e}. Consequently (cf. (1.58)), it follows that u(t) + v(t)L1 ≤ e−δt (u0 + v0 L1 ) + cδ −1 |Ω|,

0 ≤ t ≤ TU .

(4.46)

Multiply the first equation by u and integrate the product in Ω. Then, 1 d u2 dx = (cu − du2 − αu2 ρ) dx. 2 dt Ω Ω Meanwhile, multiplying the first equation by v and integrating the product in Ω, we have du v dx = (cv − duv − αuvρ) dx. Ω dt Ω In a similar way, we can obtain from the second equation on v the equalities dv u dx = (αu2 ρ − euv − βuvw) dx Ω dt Ω and 1 d 2 dt

v 2 dx = Ω

(αuvρ − ev 2 − βv 2 w) dx. Ω

Therefore, summing up these four equalities, we have 1 d c(u + v) − δ(u + v)2 dx (u + v)2 dx ≤ 2 dt Ω Ω ≤ C(u0 + v0 L1 + 1) − δ (u + v)2 dx Ω

(due to (4.46)). Hence, solving this differential inequality, we conclude that u(t) + v(t)2L2 ≤ e−2δt u0 + v0 2L2 + C(u0 + v0 L1 + 1),

0 ≤ t ≤ TU .

Since u(t) ≥ 0 and v(t) ≥ 0, it then follows that u(t)2L2 + v(t)2L2 ≤ e−2δt u0 + v0 2L2 + C(u0 + v0 L1 + 1), 0 ≤ t ≤ TU .

(4.47)

Multiply the fourth equation on ρ by ρ and integrate the product in Ω. Then, after some calculations, 1 d g h2 2 2 2 ρ dx + b |∇ρ| dx + g ρ dx ≤ ρ 2 dx + v(t)2L2 . 2 dt Ω 2 Ω 2g Ω Ω

6 Nonautonomous Semilinear Abstract Evolution Equations

199

Using (4.47), we obtain that g 1 d ρ 2 dx + ρ 2 dx ≤ C u0 + v0 2L2 + 1 . 2 dt Ω 2 Ω Hence, ρ(t)2L2 ≤ e−gt ρ0 2L2 + C u0 + v0 2L2 + 1 ,

0 ≤ t ≤ TU .

(4.48)

In view of (4.41), we can use the similar techniques to conclude that w(t)2L2 ≤ e−f w0 2L2 + C, 0 ≤ t ≤ TU . Hence, this, together with (4.47) and (4.48), yields that U (t)L2 ≤ C(U0 L2 + 1),

0 ≤ t ≤ TU .

(4.49)

Now, when we have the a priori estimates (4.29), the global existence is concluded by applying Corollary 4.3. Indeed, for any U0 ∈ K, (4.40) possesses a unique global solution in the function space: 0 ≤ U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); X) ∩ C1 ((0, ∞); X).

6 Nonautonomous Semilinear Abstract Evolution Equations In this section, we will present some device for treating nonautonomous semilinear abstract evolution equations. Consider the Cauchy problem for a nonautonomous evolution equation dU dt

+ AU = F (t, U ) + G(t),

0 < t ≤ T,

U (0) = U0 ,

(4.50)

in a Banach space X. Here, A is a sectorial operator of X satisfying (2.92) and (2.93). For each 0 ≤ t ≤ T , F (t, ·) is a nonlinear operator from D(Aη ) into X, where η is some fixed exponent. We assume the following Lipschitz condition: F (t, U ) − F (s, V ) ≤ ϕ(Aβ U + Aβ V ) × Aη (U − V ) + (Aη U + Aη V + 1)[|t − s| + Aβ (U − V )] , (t, U ), (s, V ) ∈ [0, T ] × D(Aη ),

(4.51)

where β is a second fixed exponent satisfying the relation 0 ≤ β ≤ η < 1. The function G is a given external force function in Fβ,σ ((0, T ]; X). The initial value U0 is taken in D(Aβ ).

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We consider the product Banach space X = C × X and rewrite problem (4.50) in the form d τ 1 0 τ 0 τ +1 dt U + 0 A U = F (χ(Re τ ),U ) + G(t) , 0 < t ≤ T , (4.52) τ 0 U (0) = U0 , where χ(t) is a C0,1 cutoff function for −∞ < t < ∞ such that χ(t) ≡ 0 for −∞ < t < 0, χ(t) = t for 0 ≤ t ≤ T , and χ(t) ≡ T for T < t < ∞. We can consider (4.52) as an autonomous abstract evolution equation in X for the unknown function t(τ, U ). Since τ (t) = t, problem (4.52) is certainly equivalent to (4.50). It is easy to see that the operator A = diag{1, A} is a sectorial operator of X satisfying (2.92) and (2.93). Furthermore, for 0 < θ < 1, Aθ = diag{1, Aθ } with domain D(Aθ ) = C × D(Aθ ). Therefore, condition (4.51) implies (4.2) if β > 0 and (4.21) if β = 0 for the nonlinear F(τ, U ) = t(τ + 1, F (χ(Re τ ), U )), which is an operator from D(Aη ) into X. In addition, U0 ∈ D(Aβ ) implies that t(0, U0 ) ∈ D(Aβ ). Hence, Theorems 4.1 if β > 0 and 4.4 if β = 0 are available to (4.52).

Notes and Further Researches The theory of semilinear abstract parabolic evolution equations has been developed in the study of the Navier–Stokes equations by the functional analytical methods. As pioneering works, we quote Fujita–Kato [FK64], Giga [Gig81, Gig85, Gig86], and Giga–Miyakawa [GM85]. For the variational methods, we quote Lions [Lio69] and Temam [Tem84]. When the Lipschitz condition (4.2) (or (4.21)) is valid with α = η, Theorem 4.1 (or 4.4) is quite standard (e.g., [Zei90, Theorem 19.I]). The generalization for the case where α < η was first done by von Wahl [vWa85] and Hoshino–Yamada [HY91]. The proof of Theorem 4.1 was presented by Osaki–Yagi [OY02]. Theorem 4.4 can be obtained by a slight modification of Theorem 4.1. In this chapter, we have described applications of the abstract theory only for the Lotka–Volterra competition model and some model in immunology. It is actually possible to apply our abstract results to enormous reaction–diffusion models presented from the real world. (We will indeed treat some of these in Chaps. 8, 9, 10, 11, 12, and 13). A historical study on the Lotka–Volterra model is seen in Israel– Gasca [IG02]. Stationary problem of (4.34) were studied by Dancer–Du [DD95a, DD95b], Kan-on [Kan02, Kan03, Kan06, Kan07], and others. Nakashima–Wakasa [NW07] and Dancer–Zhang [DZ02] studied the asymptotic behavior of solutions of the evolutional problem of (4.34). In addition, there is a great deal of literature which treats related problems to (4.34) including the homogeneous Dirichlet boundary conditions, variable coefficients, nonautonomous equations, and so forth. For mathematical models describing the immune system against the threat of invasion by bacteria and viruses, we refer the reader to Murray [Mur02, Chap. 10]. The reaction–diffusion model in (4.40) was obtained by incorporating spreading of killer cells and viruses to the ordinary differential equation model presented by Britton [Bri03, Sect. 6.5].

Chapter 5

Quasilinear Evolution Equations

This chapter is devoted to the Cauchy problem for quasilinear evolution equations of the forms (5.1) and (5.43) in Banach spaces. The linear operators in the equations are sectorial operators with angles < π2 but are allowed to depend on the unknown functions. Besides an underlying space X in which the problem is given, we shall introduce two more Banach spaces Y and Z such that Z ⊂ Y ⊂ X. The sectorial operators A(U ) are defined for U ∈ Z. Like (3.29), we make the assumption that D(A(V )) ⊂ D(A(U )ν ) for any U, V ∈ Z with some fixed exponent 0 < ν ≤ 1. In addition, instead of (3.30), A(U )ν [A(U )−1 − A(V )−1 ] is estimated by the Y norm of U − V , i.e., U − V Y . We shall use two more exponents α and β as graded scales to measure the spaces Y and Z, respectively. That means that we assume the inclusions D(A(U )α ) ⊂ Y and D(A(U )β ⊂ Z for all U ∈ Z with 0 ≤ α < β < 1. The exponents are also assumed to satisfy the relation 1 + α < β + ν. The optimal one is of course the case where ν = 1, which means that the domains D(A(U )) are independent of U . When ν ≤ β, we have to add the assumption of reflexibility of Z. The initial value U0 is taken in Z; more precisely, U0 must satisfy a compatibility condition of the form U0 ∈ D(A(U0 )β ) (⊂ Z). In applying the abstract results of this chapter, we have to set the three spaces Z ⊂ Y ⊂ X in a suitable way. The space X is a function space in which we want to handle a quasilinear equation to be considered. The space Z is chosen as large as possible with the condition that, for all U ∈ Z, A(U ) are well defined as sectorial operators of X. Recall that the initial values can be taken only in Z. The space Y is also chosen as close as possible to X in order that the exponent condition 1 + α < β + ν is valid. The optimal one is the case where Y = X, i.e., α = 0. The norm which we have to estimate a priori for obtaining global solutions is the Z norm of the local solutions, more precisely, the norm A(U (t))β U (t)X . The abstract results are actually applicable to various quasilinear problems. In Chaps. 14 and 15, we shall apply them to physical and biological models. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_5, © Springer-Verlag Berlin Heidelberg 2010

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1 Quasilinear Abstract Evolution Equations 1.1 Structural Assumptions We consider the Cauchy problem of an abstract quasilinear evolution equation dU dt

+ A(U )U = F (t),

0 < t ≤ T,

(5.1)

U (0) = U0 ,

in a Banach space X. Let Z be a second Banach space continuously embedded in X, and let K be the open ball of Z, K = B Z (0; R) = {U ∈ Z; U Z < R},

0 < R < ∞.

For U ∈ K, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 and with domain D(A(U )) which is allowed to depend on U . The function F is a given external force function on (0, T ]. The initial value U0 is taken in K(⊂ Z). We make the following assumptions. The spectrum σ (A(U )) is contained in a fixed open sectorial domain, i.e., σ (A(U )) ⊂ Σω = {λ ∈ C; |arg λ| < ω},

U ∈ K,

(5.2)

λ∈ / Σω , U ∈ K,

(5.3)

with some angle 0 < ω < π2 , and the resolvent satisfies (λ − A(U ))−1 ≤ M/|λ|, L(X)

with a constant M ≥ 1. The domain D(A(U )) may vary with U ∈ K, but there is an exponent 0 < ν ≤ 1 such that D(A(V )) ⊂ D A(U )ν for any pair of U, V ∈ K. (5.4) In addition, A(U ) satisfies a Lipschitz condition of the form A(U )ν A(U )−1 − A(V )−1 ≤ N U − V Y , U, V ∈ K, L(X)

(5.5)

with a constant N > 0. Here, Y is a third Banach space such that Z ⊂ Y ⊂ X with continuous embeddings. There are two exponents 0 ≤ α < β < 1 such that, for any U ∈ K, D(A(U )α ) ⊂ Y and D(A(U )β ) ⊂ Z with the estimates U˜ Y ≤ D1 A(U )α U˜ X , U˜ ∈ D(A(U )α ), U ∈ K, (5.6) U˜ ∈ D(A(U )β ), U ∈ K, U˜ Z ≤ D2 A(U )β U˜ X , Di > 0 (i = 1, 2) being some constants. The exponents satisfy the relations 0≤α 0 is determined by F and U0 . In addition, U satisfies the estimates A(U )β U + dU (5.11) dt β,σ + A(U )U Fβ,σ ≤ CF,U0 , C F where CF,U0 > 0 is a constant depending on F and U0 . Remark 5.1 It is very important to investigate the dependence of TF,U0 on the initial datum. We must notice by the proof of the theorem described below (see Steps 3 and 4) that TF,U0 depends not only on the norms F Fβ,σ and A(U0 )β U0 X but also on the following moduli of continuity: ωF (t) = sup 0≤s 0 be small enough to make the mapping Φ : K(S) → K(S) a contraction. The fixed point theorem, Theorem 1.11, ensures the existence of a unique fixed point U ∈ K(S). Since U satisfies the formula

t UU (t, s)F (s) ds, 0 ≤ t ≤ TF,U0 , (5.28) U (t) = UU (t, 0)U0 + 0

U is shown to be a local solution to (5.1) on the interval [0, TF,U0 ]. Since 0 < σ < μ + ν − 1, U must satisfy all the conditions of (5.10), except that U ∈ Cβ−α ([0, TF,U0 ]; Y ) and U ∈ C([0, TF,U0 ]; Z); see Theorem 3.10. The Hölder condition was however verified by (5.22). Similarly, by (5.19) it is already seen that U (t) is continuous at t = 0 in the Z norm. Meanwhile, for 0 < t ≤ TU0 ,F , we have AU (t + h)β [U (t + h) − U (t)] = AU (t + h)β U (t + h) − AU (t)β U (t) + AU (t + h)β AU (t + h)−1 − AU (t)−1 AU (t)U (t) − AU (t + h)β−1 − AU (t)β−1 AU (t)U (t). Because of AU (·)β−1 ∈ C([0, TF,U0 ]; L(X)), we verify that U (t + h) − U (t)Z → 0

as h → 0.

Therefore, U (t) is also continuous in the topology of Z for t > 0. We already know that the family of operators AU (t), 0 ≤ t ≤ TF,U0 , satisfies (3.27)–(3.31) and that U (t) is represented by (5.28). Therefore, (5.11) is an immediate consequence of Theorem 3.10. Step 6. Let us finally verify the uniqueness of solutions of (5.1) in the space (5.10). Under (5.8) and (5.9), let U˜ be any other local solution belonging to the function space (5.10) on an interval [0, TU˜ ]. We assume that TU˜ ≤ TU0 ,F and prove that U˜ (t) = U (t) for every 0 ≤ t ≤ TU˜ . Since there exists an evolution operator UU˜ (t, s) for the family of operators AU˜ (t) = A(U˜ (t)), 0 ≤ t ≤ TU˜ , U˜ is written by the formula

t ˜ UU˜ (t, s)F (s) ds, 0 ≤ t ≤ TU˜ . U (t) = UU˜ (t, 0)U0 + 0

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211

Then, we can repeat a quite similar argument as in Step 4 to obtain the estimate U˜ − U Z(S) ≤ CU˜ S μ+ν−1 U˜ − U Z(S) ,

0 < S ≤ TU˜ .

This then shows that U˜ (t) = U (t) for every t ∈ [0, S] if S > 0 is sufficiently small. That is, the uniqueness is true in a neighborhood of the initial time. We here consider the interval I = {S ∈ (0, TU˜ ]; U˜ (t) = U (t) for every t ∈ [0, S]}. By the discussion above, I is nonempty. In addition, I is open in (0, TU˜ ]. Indeed, if S ∈ I and S < TU˜ , then VS = U˜ (S) = U (S) ∈ K satisfies the condition VS ∈ D(A(VS )). It is clear that F ∈ F1,σ ([S, TU˜ ]). Then, we can construct a local solution to the problem dV dt + A(V )V = F (t), S < t ≤ T , V (S) = VS . As the solution is unique in a neighborhood of the initial time S, we conclude that V (t) = U˜ (t) = U (t) for every t > S which is sufficiently close to S. Therefore, I is open. In the meantime, it is clear that I is closed in (0, TU˜ ]. Hence, I = (0, TU˜ ]. We have thus accomplished the proof of Theorem 5.1.

1.3 Stronger Results for More Regular Data When F and U0 in problem (5.1) satisfy stronger conditions than (5.8) and (5.9), respectively, we can strengthen the results of Theorem 5.1. Let β < γ ≤ 1. Let F belong to F ∈ Fγ ,σ ((0, T ]; X),

0 < σ < β − α + ν − 1.

Let U0 satisfy the stronger compatibility condition U0 ∈ D A(U0 )γ , β < γ ≤ 1.

(5.29)

(5.30)

Theorem 5.2 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied. Let F and U0 ∈ K satisfy (5.29) and (5.30), respectively. Then, there exists a unique local solution to (5.1) in the function space: U ∈ C([0, TF,U0 ]; Z) ∩ Cγ −α ([0, TF,U0 ]; Y ) ∩ C1 ((0, TF,U0 ]; X), (5.31) γ ,σ ((0, T A(U )γ U ∈ C([0, TF,U0 ]; X); dU F,U0 ]; X), dt , A(U )U ∈ F where TF,U0 > 0 is determined by the norms F Fγ ,σ ((0,T ];X)) and A(U0 )γ U0 X only. Furthermore, U satisfies the estimates A(U )γ U + dU (5.32) dt γ ,σ + A(U )U Fγ ,σ ≤ CF,U0 , C F

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5 Quasilinear Evolution Equations

where CF,U0 > 0 is a constant determined by the norms F Fγ ,σ and A(U0 )γ U0 X . Proof By the definition (5.13), we have ωU0 (t) ≤ sup [e−sA(U0 ) − 1]A(U0 )β−γ A(U0 )γ U0 X ≤ Ct γ −β A(U0 )γ U0 X . 0≤s≤t

Similarly, by the definitions (5.12) and (5.14), ωF (t) + ωU0 ,F (t) ≤ Ct γ −β F Fγ ,σ . These estimates mean that (5.20) is valid if S γ −β (F Fγ ,σ + A(U0 )γ U0 X ) is sufficiently small. Hence, TF,U0 depends only on these quantities. By the same arguments as in Step 3 of the proof of Theorem 5.1, (5.29) and (5.30) yield that, for γ ≤ θ ≤ 1, A(U (t))θ U (t) ≤ CU ,F t γ −θ , 0 < t ≤ TF,U . 0 0 X Then, the other properties of U can also be verified by the same techniques as in Steps 3 and 5 of the proof of Theorem 5.1. As an immediate consequence, we are able to state the global existence of solutions. Corollary 5.1 Under the assumptions of Theorem 5.2, let F and U0 ∈ K satisfy (5.29) and (5.30), respectively. Assume that any local solution U of (5.1) in the function space (5.31), with TF,U0 replaced by TU , satisfies the estimates U (t)Z ≤ R1 < R, 0 ≤ t ≤ TU , (5.33) A(U (t))γ U (t)X ≤ CF,U0 , 0 ≤ t ≤ TU , with some constants R1 and CF,U0 independent of TU . Then, (5.1) possesses a unique global solution on the interval [0, T ]. Proof We extend the function F to a function F defined on the whole interval [0, ∞) by putting F (t) ≡ F (T ) for T < t < ∞. Taking an initial value U1 which satisfies U1 Z ≤ R1 and A(U1 )γ U1 X ≤ CF,U0 and setting the initial time 0 ≤ T1 < ∞ arbitrarily, we consider the following problem dV dt + A(V )V = F (t), T1 < t < ∞, (5.34) V (T1 ) = U1 . Theorem 5.2 then ensures the existence of a unique local solution on an interval [T1 , T1 + τ ], where τ > 0 is uniform with respect to U1 and T1 . Note that F Fγ ,σ ((a,b];X) ≤ F Fγ ,σ ((0,T ];X) for any finite interval (a, b].

1 Quasilinear Abstract Evolution Equations

213

This fact proves the assertion of the corollary. Consider first the local solution U constructed in Theorem 5.2 on the interval [0, TF,U0 ], where TF,U0 ≥ τ . Set T1 = TF,U0 − τ2 and U1 = U (T1 ). Then, since U1 satisfies condition (5.33), there exists a local solution V to (5.34) on the interval [T1 , T1 +τ ]. By the uniqueness of solutions, U (t) = V (t) for every T1 ≤ t ≤ TF,U0 . This then means that U has been extended to the interval [0, TF,U0 + τ2 ] as a local solution. Due to (5.33), we can repeat this procedure unlimitedly. Each time, the local solution is extended over the length τ2 of interval. Hence, we can arrive at the full interval [0, T ] by finite times.

1.4 Case ν ≤ β We assumed the relation β < ν in (5.7) as one of the structural conditions of our abstract theory. But, this is not an indispensable one. We shall show that the lack of this relation can be recovered by the reflexivity of Z. Instead of (5.7), let the relations 0 ≤ α < ν ≤ β < 1 and 1 + α < β + ν

(5.35)

be satisfied. We assume the following condition: Z is a reflexive Banach space.

(5.36)

Theorem 5.3 Let (5.3), (5.4), (5.5), (5.6), (5.35), and (5.36) be satisfied. Let F be as in (5.8), and let U0 satisfy condition (5.9) with A(U0 )β U0 < R/D2 . X Then, there exists a unique local solution to (5.1) on an interval [0, TF,U0 ] in the function space: U ∈ B([0, TF,U0 ]; Z) ∩ Cβ−α ([0, TF,U0 ]; Y ) ∩ C1 ((0, TF,U0 ]; X), (5.37) β,σ ((0, T A(U )β U ∈ C([0, TF,U0 ]; X); dU F,U0 ]; X). dt , A(U )U ∈ F Here, TF,U0 > 0 is determined by F and U0 . Proof In the present case, Proposition 5.1 is not applicable with θ = β in showing (5.26). This fact makes us change the underlying Banach spaces from Z(S) to the space μ

Y(S) = C{0} ([0, S]; Y )

with 1 − ν + σ < μ < β − α.

By the new condition (5.36), the closed ball {U ∈ Z; U Z ≤ R1 }, where R1 is a fixed number such that D2 A(U0 )β U0 X < R1 < R,

214

5 Quasilinear Evolution Equations

is weakly compact. As a consequence, this ball is a closed subset of Y . (In fact, let Un Z ≤ R1 and Un → U in Y ; then, we can assume that Un → V weakly in Z with V Z ≤ R1 . Since Z ⊂ Y with continuous embedding, Φ ∈ Y implies Φ|Z ∈ Z ; therefore, Φ(Un ) = Φ|Z (Un ) → Φ(V ), i.e., Un → V weakly in Y , too. Hence, U = V . Therefore, the set K(S) introduced in the proof of Theorem 5.1 is a closed subset of Y(S). In showing that Φ maps K(S) into itself, we will utilize the following formula. For U = Φ(V ), by (5.18) it holds that AV (t)β U (t) = A(U0 )β U0 + [e−tA(U0 ) − 1]A(U0 )β U0 + AV (t)β UV (t, 0)AV (0)−β − e−tAV (0) A(U0 )β U0

t + AV (t)β UV (t, s)[F (s) − F (t)] ds 0

+

t

AV (t)β [UV (t, s) − e−(t−s)AV (t) ] ds F (t)

0

− [e−tAV (t) − 1]AV (t)β−1 F (t). Then, it is verified by (2.129), (3.80), (3.84), and (3.86) that AV (t)β U (t) ≤ A(U0 )β U0 + C[ωU (t) + ωF (t)] 0 X X μ+ν−1 β + Ct A(U0 ) U0 X + F Fβ,σ + ω˜ F,U0 (t), where

ω˜ F,U0 (t) = t β−1 [e−tAV (t) − 1]AV (t)β−1 t 1−β F (t)X .

Therefore, by (5.6), U (t)Z ≤ D2 A(U0 )β U0 X + C[ωU0 (t) + ωF (t)] + Ct μ+ν−1 A(U0 )β U0 X + F Fβ,σ + ω˜ F,U0 (t).

(5.38)

As will be seen by the lemma below, we have limt→0 ω˜ F,U0 (t) = 0. Hence, if S > 0 is sufficiently small, then sup U (t)Z ≤ R1 . 0≤t≤S

Lemma 5.1 As t → 0, ω˜ F,U0 (t) → 0. Proof We have ω˜ F,U0 (t) ≤ t β−1 [e−tAV (t) − 1]AV (t)β−1 [t 1−β F (t) − F0 ]X + t β−1 [e−tAV (t) − 1]AV (t)β−1 F0 X ,

2 Lipschitz Continuity of Solutions in Initial Data

215

where F0 = limt→0 t 1−β F (t). The first term in the right-hand side is seen to converge to 0. For the second term, we use the similar argument as for (2.131). When F0 ∈ D(A(0)), it follows that t β−1 [e−tAV (t) − 1]AV (t)β−1 F0 ≤ t β−1 [e−tAV (t) − 1]AV (t)β−ν−1 X

× AV (t)ν A(0)−1

L(X)

L(X)

A(0)F0 X → 0.

For general F0 ∈ X, the desired convergence is deduced by the uniform boundedness of t β−1 [e−tAV (t) − 1]AV (t)β−1 L(X) and the density of D(A(0)) in X. The same Lipschitz condition as (5.22) is verified. Hence, we conclude that Φ is a mapping from K(S) into itself if S > 0 is sufficiently small. As estimate (5.27) was proved without using the relation β < ν, Φ is a contraction with respect to the Y(S) norm if S > 0 sufficiently small. We can apply again the fixed point theorem to Φ : K(S) → K(S) to obtain the desired solution U . By analogous arguments as before, U is shown to belong to the space (5.37). In the present case, U may not be a continuous function with values in Z. We know only that A(U )β U is an X-valued continuous function on [0, TF,U0 ] by Theorem 3.10.

2 Lipschitz Continuity of Solutions in Initial Data Under (5.2)–(5.7), we shall show the Lipschitz continuity of the local solutions with respect to initial data (F, U0 ). We introduce the set of functions F by E(T ) = {F ∈ Fγ ,σ (T ); F Fγ ,σ ((0,T ];X) ≤ L1 },

0 < σ < ν + β − α − 1,

with some constant L1 > 0, and also introduce the set of initial values

B = U0 ∈ Z; U0 Z ≤ R1 and A(U0 )γ U0 X ≤ R2 , β < γ ≤ 1, with some constants 0 < R1 < R and 0 < R2 < ∞. By Theorem 5.1, there exists a unique local solution of (5.1) for each pair (F, U0 ) ∈ E(T ) × B on some interval [0, TF,U0 ]. Let TE,B = inf(F,U0 )∈E(T )×B TF,U0 ; then TE,B is positive. Our goal is to prove the following theorem. Theorem 5.4 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied. Let U (resp. V ) be the local solution of (5.1) with initial data (F, U0 ) ∈ E(T ) × B (resp. (G, V0 ) ∈ E(T ) × B). Then, there exist some constants CE,B > 0 depending only on E(T ) and B such that t β U (t) − V (t)Z + t α U (t) − V (t)Y + U (t) − V (t)X ≤ CB,E (U0 − V0 X + t β F − GFβ,σ ),

0 < t ≤ TE,B .

216

5 Quasilinear Evolution Equations

Proof Let UU (t, s) (resp. UV (t, s)) denote the evolution operator for a family of AU (t) = A(U (t)) (resp. AV (t) = A(V (t))). It follows from (5.28) that U (t) − V (t) = UU (t, 0)(U0 − V0 ) + [UU (t, 0) − UV (t, 0)]V0

t + [UU (t, s) − UV (t, s)]G(s) ds 0

t

+

UU (t, s)[F (s) − G(s)] ds.

0

Then, by (3.80) and (5.6) we have UU (t, 0)(U0 − V0 )Y ≤ D1 AU (t)α UU (t, 0)(U0 − V0 )X ≤ Ct −α U0 − V0 X . Similarly, t t α UU (t, s)[F (s) − G(s)] ds ≤ D1 AU (t) UU (t, s)[F (s) − G(s)] ds 0 0 Y X

t ≤ CF − GFβ,σ (t − s)−α s β−1 ds 0

≤ Ct

β−α

F − GFβ,σ .

Repeating the same arguments as in the proof of Proposition 5.1, we can verify for 0 ≤ θ < ν that AU (t)θ [UU (t, 0) − UV (t, 0)]V0 X

t ≤ CV0 β (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds

(5.39)

0

and

t AU (t)θ [UU (t, s) − UV (t, s)]G(s) ds 0 X

t ≤ CGFβ,σ (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds. 0

Therefore, applying these estimates with θ = α, we observe that t [UU (t, 0) − UV (t, 0)]V0 Y + [UU (t, s) − UV (t, s)]G(s) ds 0 Y

t ≤ CE,B (t − s)ν−α−1 s β−1 U (s) − V (s)Y ds. 0

(5.40)

2 Lipschitz Continuity of Solutions in Initial Data

217

In this way, we have obtained the integral inequality

t ϕ(t) ≤ CE(t) + CE,B t α (t − s)ν−α−1 s β−α−1 ϕ(s) ds

(5.41)

0

for the function ϕ(t) which is defined by ϕ(t) = t α U (t) − V (t)Y ,

0 ≤ t ≤ TE,B ,

where E(t) = U0 − V0 X + t β F − GFβ,σ ,

0 ≤ t ≤ TE,B .

To solve this integral inequality, we first consider small variable t . For any s such that 0 ≤ s ≤ t , we have

s ϕ(s) ≤ CE(s) + CE,B s α (s − σ )ν−α−1 σ β−α−1 dσ sup ϕ(σ ) 0≤σ ≤s

0

≤ CE(t) + CE,B t ν+β−α−1 sup ϕ(s). 0≤s≤t

Therefore, it follows that 1 − CE,B t ν+β−α−1 sup ϕ(s) ≤ CE(t). 0≤s≤t

Fix an ε > 0 small enough to satisfy CE,B ε ν+β−α−1 < 1. Then, for 0 ≤ t ≤ ε, ϕ(t) ≤ sup ϕ(s) ≤ CE(t),

0 ≤ t ≤ ε.

0≤s≤t

Secondly, let us consider variable t such that t ≥ ε. We observe that

ε (t − s)ν−α−1 s β−α−1 ds ϕ(t) ≤ CE(t) + CE,B t α E(ε) 0

+ CE,B t α ε β−α−1

≤ CE,B E(t) +

t

t

(t − s)ν−α−1 ϕ(s) ds

ε

(t − s)

ν−α−1

ϕ(s) ds .

ε

Applying Theorem 1.27, we conclude that ϕ(t) ≤ CE,B E(t),

ε ≤ t ≤ TE,B .

Thus, for all 0 ≤ t ≤ TE,B , ϕ(t) ≤ CE,B E(t),

i.e.,

t α U (t) − V (t)Y ≤ CE,B E(t),

0 ≤ t ≤ TE,B . (5.42)

218

5 Quasilinear Evolution Equations

Estimation for Z and X norms of U (t) − V (t) is now immediate. By (3.80) and (5.6), UU (t, 0)(U0 − V0 )Z ≤ Ct −β U0 − V0 X . Similarly, t UU (t, s)[F (s) − G(s)] ds 0 Z t β ≤ D2 AU (t) UU (t, s)[F (s) − G(s)] ds 0

≤ CF − GFβ,σ

0

t

X

(t − s)−β s β−1 ds ≤ CF − GFβ,σ .

In addition, applying (5.39) and (5.40) with θ = β, we have t [UU (t, 0) − UV (t, 0)]V0 Z + [UU (t, s) − UV (t, s)]G(s) ds

≤ CE,B

0

t

Z

(t − s)ν−β−1 s β−1 U (s) − V (s)Y ds

0

and, therefore, by (5.42),

t ≤ CE,B U0 − V0 X (t − s)ν−β−1 s β−α−1 ds ≤ CE,B t ν−α−1 U0 − V0 X . 0

Summing up these estimates, we conclude that t β U (t) − V (t)Z ≤ CE,B E(t),

0 ≤ t ≤ TE,B .

It is similar for the estimation of U (t) − V (t)X . It suffices to apply (5.39) and (5.40) with θ = 0.

3 Equations Including Semilinear Terms In this section, we consider the Cauchy problem for a quasilinear equation of the form dU dt + A(U )U = F (U ) + G(t), 0 < t ≤ T , (5.43) U (0) = U0 , in a Banach space X. For U ∈ K, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 , where K is an open ball K = {U ∈ Z; U Z < R}, 0 < R < ∞, of a second Banach space Z continuously embedded in X. The operator F is a nonlinear operator from another Banach space W into X, W being continuously embedded in Z. The function G is a given external force function. The initial value U0 is taken in K.

3 Equations Including Semilinear Terms

219

3.1 Existence Theorem For the linear operators A(U ), we make the same structural assumptions as in Sect. 1.1. That is, we use the three Banach spaces Z ⊂ Y ⊂ X. The operator A(U ), U ∈ K, is a sectorial operator of X satisfying conditions (5.2) and (5.3) uniformly. The domain D(A(U )) is allowed to vary with U ∈ K, but A(U ) is assumed to satisfy (5.4) and the Lipschitz condition (5.5). Furthermore, the domains of fractional powers D(A(U )α ) and D(A(U )β ) are contained in Y and Z with (5.6), respectively, with the exponents α and β satisfying relations (5.7). For the nonlinear operator F , we assume the Lipschitz condition F (U ) − F (V )X ≤ ϕ(U Z + V Z )[U − V W + (U W + V W ) × U − V Z ],

U, V ∈ W,

(5.44)

where ϕ(·) is some continuous increasing function. This condition obviously implies that F (U )X ≤ F (0)X + ϕ(U Z )(1 + U Z )U W ,

U ∈ W.

(5.45)

For the space W , we assume an analogous condition to that for Y and Z. There is an exponent 0 < η < 1 such that D(A(U )η ) ⊂ W with the estimate U˜ W ≤ D3 A(U )η U˜ X , U˜ ∈ D A(U )η for U ∈ K, (5.46) D3 > 0 being some constant. Finally, the exponents satisfy the relation β < η < ν.

(5.47)

As in the preceding section, the notation C will stand for a universal constant which is determined by the angle, the exponents, the norm A(0)−1 , and the constants appearing in (5.2)–(5.7). In this section, C may depend also on the function ϕ(·), the norm F (0)X , the exponent η, and the constant D3 . We shall prove the following theorem. Theorem 5.5 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let G belong to (5.8) with 0 < σ < min{β − α + ν − 1, 1 − η}. Let U0 ∈ K satisfy (5.9). Then, there exists a unique local solution U to (5.43) on an interval [0, TG,U0 ] in the function space: ⎧ β−η ⎪ ⎨U ∈ B{0} ((0, TG,U0 ]; W ) ∩ C([0, TG,U0 ]; Z) ∩Cβ−α ([0, TG,U0 ]; Y ) ∩ C1 ((0, TG,U0 ]; X), ⎪ ⎩ β,σ ((0, T A(U )β U ∈ C([0, TG,U0 ]; X); dU G,U0 ]; X); dt , A(U )U ∈ F (5.48) here TG,U0 > 0 is determined by G and U0 . In addition, U satisfies the estimates

220

5 Quasilinear Evolution Equations

A(U )β U + dU dt C

+ A(U )U Fβ,σ ≤ CG,U0 ,

F β,σ

(5.49)

where CG,U0 > 0 is a constant depending on G and U0 . Remark 5.2 The dependence of TG,U0 on G and U0 is quite similar to that in Theorem 5.1. As will be seen in the proof below (see Steps 3 and 4), TG,U0 is determined by GFβ,σ and A(U0 )β U0 X and by the moduli of continuity ωG (t), ωU0 (t), and ωG,U0 (t) as t → 0, which are already defined by (5.12), (5.13), and (5.14), respectively.

3.2 Proof of Theorem 5.5 The proof consists of several steps. Step 1. For each 0 < S ≤ T , we set the underlying Banach space −ρ

μ

W(S) = B{0} ((0, S]; W ) ∩ B([0, S]; Z) ∩ C{0} ([0, S]; Y ) with some fixed exponents μ and ρ such that 1 − ν + σ < μ < β − α (cf. the proof of Theorem 5.1) and 1 − σ − β < ρ < 1 − β. Therefore, we have the relations η − β < 1 − σ − β < ρ < 1 − β < 1 − μ − α. −ρ

(5.50)

μ

For the function spaces B{0} ((0, S]; W ) and C{0} ([0, S]; Y ), see Chap 1, Sect. 2. We also set a subset F(S) of W(S) as follows: F(S) = U ∈ W(S); U (0) = U0 , sup U (t)Z ≤ R1 , sup t ρ U (t)W ≤ 1, 0≤t≤S

sup 0≤s 0 is sufficiently small, then sup

sup {Ψ (V )}(t)Z ≤ R1 .

V ∈F(S) 0≤t≤S

We next apply the moment inequality (2.119) for estimating AU (t)θ U (t)X . Indeed, due to (3.101) and (3.102), for β ≤ θ ≤ 1, AU (t)θ U (t) ≤ Ct β−θ (FV + GFβ,σ + U0 β ), 0 ≤ t ≤ S. (5.57) X Therefore, in view of (5.46), t ρ U (t)W ≤ D3 t ρ AU (t)η U (t)X ≤ Ct ρ+β−η (U0 β + FV + GFβ,σ ) ≤ 1,

0 ≤ t ≤ S,

if S > 0 is sufficiently small. For proving the Hölder condition of U , we write

t

U (t) − U (s) =[UU (t, s) − 1]U (s) +

UU (t, τ )[FV (τ ) + G(τ )] dτ

s

={[UU (t, s) − e−(t−s)AU (s) ] + [e−(t−s)AU (s) − 1]}AU (s)−(η+σ )

t η+σ ×AU (s) U (s) + UU (t, τ )[FV (τ ) + G(τ )] dτ, s

0 ≤ s < t ≤ S. Then, by (2.129) and (3.87) (note that η < ν), AU (t)η [UU (t, s) − e−(t−s)AU (s) ]AU (s)−(η+σ ) ≤ C(t − s)σ +μ+ν−1 , L(X) AU (s)η [e−(t−s)AU (s) − 1]AU (s)−(η+σ ) ≤ C(t − s)σ . L(X) By (3.81),

3 Equations Including Semilinear Terms

AU (t)η UU (t, τ )

t

s

≤ C(t − s)σ

t

223

τ L(X)

β−1

dτ ≤ C

t

(t − τ )−η τ β−1 dτ

s

(t − τ )−η−σ τ β−1 dτ ≤ C(t − s)σ

t

(t − τ )−η−σ τ β−1 dτ

0

s

≤ C(t − s) t

σ β−η−σ

.

In view of (5.46) and (5.57) (θ = η + σ ), we therefore obtain that U (t) − U (s)W ≤ C(FV + GFβ,σ + U0 β )(t − s)σ s β−η−σ .

(5.58)

Hence, if S > 0 is sufficiently small, then sup 0<s 0 is sufficiently small, then sup 0<s 0 is sufficiently small. Step 4. We shall prove that Ψ : F(S) → W(S) is a contraction with respect to the W(S) norm if S > 0 is sufficiently small. In fact, let Vi ∈ F(S) and Ui = Ψ (Vi ), i = 1, 2. From (5.56) it is seen that U1 (t) − U2 (t) = [UU1 (t, 0) − UU2 (t, 0)]U0

t + [UU1 (t, s) − UU2 (t, s)][FV1 (s) + G(s)] ds 0

t

+ 0

UU2 (t, s)[FV1 (s) − FV2 (s)] ds.

For 0 ≤ θ < ν, using (5.23) of Proposition 5.1, we have AU (t)θ [UU (t, 0) − UU (t, 0)]U0 1

1

≤ Cθ t

2

β−θ+μ+ν−1

X

U0 β U1 − U2 Cμ

{0} ([0,S];Y )

Similarly, using (5.24),

t AU (t)θ [UU1 (t, s) − UU2 (t, s)][FV1 (s) + G(s)] ds 1 0

≤ Cθ (GFβ,σ + 1)t

β−θ+μ+ν−1

U1 − U2 Cμ

(5.59)

.

X

{0} ([0,S];Y )

.

(5.60)

224

5 Quasilinear Evolution Equations

By (5.44), we have

t AU (t)θ UU2 (t, s)[FV1 (s) − FV2 (s)] ds 2

≤C

0

t

X

(t − s)−θ s −ρ [s ρ V1 (s) − V2 (s)W + V1 (s) − V2 (s)Z ] ds

0

≤ Cθ t −θ−ρ+1 (V1 − V2 B−ρ ((0,S];W ) + V1 − V2 B([0,S];Z) ).

(5.61)

{0}

Applying these estimates with θ = α, we obtain that t −μ U1 (t) − U2 (t)Y ≤ C(GFβ,σ + U0 β + 1)S β−α+ν−1 U1 − U2 Cμ

{0} ([0,S];Y )

+ CS 1−μ−α−ρ (V1 − V2 B−ρ ((0,S];W ) + V1 − V2 B([0,S];Z) ), {0}

0 < t ≤ S.

(5.62)

Similarly, applying the estimates with θ = β, U1 (t) − U2 (t)Z ≤ C(GFβ,σ + U0 β + 1)S μ+ν−1 U1 − U2 Cμ

{0} ([0,S];Y )

+ CS 1−β−ρ (V1 − V2 B−ρ ((0,S];W ) {0}

+ V1 − V2 B([0,S];Z) ),

0 < t ≤ S.

(5.63)

Finally, applying the estimates with θ = η, we have t ρ U1 (t) − U2 (t)W ≤ C(GFβ,σ + U0 β + 1)S ρ+β−η+μ+ν−1 × U1 − U2 Cμ

{0} ([0,S];Y )

+ V1 − V2 B([0,S];Z) ),

+ CS 1−η (V1 − V2 B−ρ ((0,S];W ) {0}

0 < t ≤ S.

(5.64)

Therefore, summing up (5.62), (5.63), and (5.64), we have U1 − U2 W(S) ≤ C(GFβ,σ + U0 β + 1)S μ+ν−1 U1 − U2 Cμ

{0} ([0,S];Y )

+ CS 1−β−ρ (V1 − V2 B−ρ ((0,S];W ) + V1 − V2 B([0,S];Z) ) {0}

(due to (5.50)). In this way, we conclude that Ψ is a contraction with respect to the W(S) norm, provided that S > 0 is sufficiently small. Step 5. Let TG,U0 = S > 0 be small enough to make Ψ : F(S) → F(S) to be a contraction. Then, there exists a unique fixed point U ∈ F(S) of Ψ which satisfies the formula

t U (t) = UU (t, 0)U0 + UU (t, s)[FU (s) + G(s)] ds, 0 ≤ t ≤ TG,U0 . (5.65) 0

3 Equations Including Semilinear Terms

225

Then, U is a local solution to (5.43) on [0, TG,U0 ] belonging to the space (5.10). In addition, applying (5.57) with θ = η, we verify by (5.46) that U (t)W ≤ Ct β−η (FU + GFβ,σ + U0 β ). Hence, U belongs to the space (5.48). In view of (5.55), the family of operators AU (t), 0 ≤ t ≤ TG,U0 , satisfies (3.27)–(3.31). Therefore, (5.49) is readily verified by Theorem 3.10 and (5.54). The uniqueness of a local solution is also verified in an analogous way as in the proof of Theorem 5.1. We have thus accomplished the proof of Theorem 5.5.

3.3 Stronger Results for More Regular Data Let G belong to the function space (5.29). Let U0 satisfy the compatibility condition (5.30), which is stronger than (5.8). Then, Theorem 5.5 can be strengthened as follows. Theorem 5.6 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let G and U0 ∈ K satisfy (5.29) and (5.30), respectively, with 0 < σ < min{β − α + ν − 1, 1 − η}. Then, there exists a unique local solution U to (5.43) on an interval [0, TG,U0 ] in the function space (5.31). Here, TG,U0 > 0 is determined by the norms GFγ ,σ and A(U0 )γ U0 X alone. Furthermore, U satisfies the estimates A(U )γ U + dU (5.66) dt γ ,σ + A(U )U Fγ ,σ ≤ CG,U0 C F with a constant CG,U0 > 0 determined by the norms GFγ ,σ and A(U0 )γ U0 X . As noticed in Remark 5.2, the time TG,U0 > 0 is determined by ωG (t), ωU0 , and ωG,U0 (t) in addition to GFβ,σ and A(U0 )β U0 X . Then, the proof of this theorem is quite analogous to that of Theorem 5.2, and we may leave it to the reader. By the same argument as in Corollary 5.1, we verify the following global existence. The proof may now be obvious. Corollary 5.2 Under the assumptions of Theorem 5.6, let G and U0 ∈ K satisfy (5.29) and (5.30), respectively. Assume that any local solution U of (5.43) in the function space (5.31), with TG,U0 replaced by TU , satisfies the estimates 0 ≤ t ≤ TU , U (t)Z ≤ R1 < R, (5.67) A(U (t))γ U (t)X ≤ CG,U0 , 0 ≤ t ≤ TU , with some constants R1 and CG,U0 independent of TU . Then, (5.43) possesses a unique global solution on the interval [0, T ].

3.4 Case ν ≤ β The condition β < ν assumed above is not an indispensable one for the local existence. As in Sect. 1.4, the lack of this condition can be recovered by reflexivity of Z.

226

5 Quasilinear Evolution Equations

Instead of (5.44), we assume that F satisfies F (U ) − F (V )X ≤ ϕ(U Z + V Z )U − V W ,

U, V ∈ Z,

(5.68)

with some continuous increasing function ϕ(·), and that the space W satisfies the relation Z ⊂ W (⊂ Y ⊂ X) with continuous embedding. We also assume relations (5.35) and, instead of (5.47), (5.69)

α < η < ν.

Theorem 5.7 Let (5.2), (5.3), (5.4), (5.5), (5.6), (5.35), and (5.36) be satisfied, and let (5.46), (5.68), and (5.69) be satisfied. Let G belong to (5.8) with 0 < σ < min{β − α + ν − 1, 1 − η}. Let U0 satisfy condition (5.9) with A(U0 )β U0 < R D2 . X Then, there exists a unique local solution to (5.43) on an interval [0, TG,U0 ] in the function space: ⎧ β−η ([0, T ⎪ G,U0 ]; W ) ⎨U ∈ B([0, TG,U0 ]; Z) ∩ C ∩ Cβ−α ([0, TG,U0 ]; Y ) ∩ C1 ((0, TG,U0 ]; X), ⎪ ⎩ β,σ ((0, T A(U )β U ∈ C([0, TG,U0 ]; X); dU G,U0 ]; X). dt , A(U )U ∈ F (5.70) Here, TG,U0 > 0 is determined by G and U0 . Proof In the present case, (5.63) is no longer available. So, we have to change the underlying spaces from W(S) to ζ μ ˜ W(S) = C{0} ((0, S]; W ) ∩ C{0} ([0, S]; Y ),

0 < S ≤ T,

with some fixed exponents μ and ζ such that 1 − ν + σ < μ < β − α and 0 < ζ < ˜ β − η, respectively. Accordingly, we set the subset of W(S) by ˜ ˜ F(S) = U ∈ W(S); U (0) = U0 , sup U (t)Z ≤ R1 0≤t≤S

and

sup 0≤s 0. ˜

3 Equations Including Semilinear Terms

227

˜ Hence, by formula (5.56), we define a mapping U = Ψ V for V ∈ F(S) (0 < S ≤ ˜ TG,U0 ). ˜ If S > 0 is sufficiently small, then Ψ maps F(S) into itself. In fact, by (5.38), {Ψ V }(t)Z ≤ D2 A(U0 )β U0 X + C[ωU0 (t) + ωFV +G (t)] + Ct μ+ν−1 FV + GFβ,σ + A(U0 )β U0 X + ω˜ FV +G,U0 (t). This shows that, if S > 0 is sufficiently small, then sup

sup (Ψ V )(t)Z ≤ R1 .

0≤t≤S ˜ V ∈F(S)

For verifying the Hölder condition, we can use the same argument as for (5.58). In view of the relations η < ν and σ < 1 − η, estimate (5.58) is verified without any change. Therefore, if S > 0 is sufficiently small, then sup

sup

0≤s 0 is sufficiently small. Let us verify that, if S > 0 is sufficiently small, then Ψ is a contraction in the ˜ W(S) norm. This is also verified in the same way as before. In fact, let Ui = Ψ Vi ˜ with Vi ∈ F(S) (i = 1, 2). For 0 ≤ θ < ν, estimates (5.59) and (5.60) are valid for these Ui and Vi . Instead of (5.61), we verify that

t AU (t)θ U (t, s)[F (s) − F (s)] ds U2 V1 V2 2 0 X

t ≤C (t − s)−θ s ζ s −ζ V1 (s) − V2 (s)W ds 0

≤ Cθ t ζ −θ+1 V1 − V2 Cζ

{0} ((0,S];W )

.

Then, using these estimates with θ = η and α, we easily obtain that t −ζ U1 (t) − U2 (t)W + t −α U1 (t) − U2 (t)Y ≤ CG,U0 S μ+ν−1 U1 − U2 W(S) + C(S 1−η + S ζ +1−α−μ )V1 − V2 W(S) , ˜ ˜ 0 ≤ t ≤ S. ˜ This shows that Ψ is a contraction with respect to the W(S) norm, provided that S > 0 is sufficiently small (note that 1 − α − μ > 1 − β > 0). ˜ We thus conclude that Ψ has a unique fixed point U ∈ F(S) which satisfies formula (5.65). Then, U is the solution to (5.43) in the space (5.70) which enjoys all the desired properties.

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5 Quasilinear Evolution Equations

4 Lipschitz Continuity of Solutions in Initial Data We are concerned with the continuous dependence of the local solutions of (5.43) with respect to the initial data (G, U0 ). Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied, namely, we return to the case where β < ν. We consider the set of functions G given by E(T ) = {G ∈ Fγ ,σ ((0, T ]; X); GFγ ,σ ≤ L1 }, 0 < σ < min{β − α + ν − 1, 1 − η} with some constant L1 > 0. We consider also the set of initial values

B = U0 ∈ Z; U0 Z ≤ R1 and A(U0 )γ U0 X ≤ R2 , β < γ ≤ 1, with some constants 0 < R1 < R and 0 < R2 < ∞. By Theorem 5.6, there exists a unique local solution to (5.43) for each pair (G, U0 ) ∈ E(T ) × B on some interval [0, TG,U0 ]. Let TE,B = inf(G,U0 )∈E(T )×B TG,U0 > 0. Before proving the Lipschitz continuity, we shall show estimates of solutions by the initial data. Theorem 5.8 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let U be the local solution of (5.43) with (G, U0 ) ∈ E(T ) × B. Then, there exist some constants CE,B > 0 depending on E(T ) and B only such that t η U (t)W + t β U (t)Z + t α U (t)Y + U (t)X ≤ CE,B (U0 X + tF (0)X + t β GFβ,σ ),

0 ≤ t ≤ TE,B .

Proof Let FU (t) = F (U (t)), and let UU (t, s) be the evolution operator for the family of AU (t) = A(U (t)). Operating A(U (t))θ (0 ≤ θ < 1) to the solution formula (5.65), we have AU (t)θ U (t) = AU (t)θ UU (t, 0)U0

t + AU (t)θ UU (t, s)[FU (s) + G(s)] ds,

0 < t ≤ TB,E .

0

Here, by (5.45), FU (t)X ≤ F (0)X + D3 ϕ(R)(1 + R)AU (t)η U (t)X . Therefore,

t AU (t)θ U (t) ≤Cθ t −θ U0 X + (t − s)−θ F (0)X + AU (s)η U (s)X X 0

+ s β−1 GFβ,σ ds .

4 Lipschitz Continuity of Solutions in Initial Data

229

Consequently, t θ AU (t)θ U (t)X ≤ Cθ U0 + tF (0)X + t β GFβ,σ

t

+ 0

t θ (t − s)−θ s −η s η AU (s)η U (s)X ds .

If we put θ = η, then an integral inequality with respect to AU (s)η U (s)X is obtained, and the inequality can be easily solved by the same techniques used for (5.41). As a result, we conclude that t η AU (t)η U (t)X ≤ CE,B [U0 X + tF (0)X + t β GFβ,σ ], 0 < t ≤ TE,B . Hence, for 0 ≤ θ < 1, we have

t t θ AU (t)θ U (t)X ≤ Cθ U0 + tF (0)X + t β GFβ,σ + t θ (t − s)−θ s −η 0

× [U0 X + sF (0)X + s β GFβ,σ ] ds . Putting θ = β, α, and 0, we can obtain the desired estimates.

Corollary 5.3 Under the conditions of Theorem 5.8, we have t η−β U (t)W + U (t)Z ≤ CE,B A(U0 )β U0 X + t 1−β F (0)X + GFβ,σ ,

0 ≤ t ≤ TE,B .

Proof For β ≤ θ < 1, we use the estimate AU (t)θ UU (t, 0)U0 ≤ CE,B t β−θ U0 β . X Then, it follows that

t AU (t)θ U (t) ≤ Cθ CB,E t β−θ U0 β + (t − s)−θ F (0)X X 0

η β−1 + AU (s) U (s) X + s Gβ,σ ds . Putting θ = η, we have t

AU (t) U (t) X ≤ CB,E U0 β + t 1−β F (0)X + GFβ,σ

η−β

η

t

+ 0

t η−β (t − s)−η s β−η s η−β AU (s)η U (s)X ds .

230

5 Quasilinear Evolution Equations

So, solving this integral inequality, we conclude that t η−β AU (t)η U (t)X ≤ CB,E [U0 β + t 1−β F (0)X + Gβ,σ ], By putting θ = β, we also verify that AU (t)β U (t) ≤ CE,B [U0 β + t 1−β F (0)X + Gβ,σ ], X

0 < t ≤ TE,B .

0 < t ≤ TE,B .

Theorem 5.9 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let U (resp. V ) be the local solution of (5.43) with (G1 , U0 ) ∈ E(T ) × B (resp. (G2 , V0 ) ∈ E(T ) × B). Then, there exist some constants CE,B > 0 depending on E(T ) and B only such that t η U (t) − V (t)W + t β U (t) − V (t)Z + t α U (t) − V (t)Y + U (t) − V (t)X ≤ CE,B (U0 − V0 X + t β G1 − G2 Fβ,σ ),

0 ≤ t ≤ TE,B .

Proof Let FU (t) = F (U (t)) (resp. FV (t) = F (V (t))), and let UU (t, s) (resp. UV (t, s)) be the evolution operator for the family of AU (t) = A(U (t)) (resp. AV (t) = A(V (t))). By (5.65) we can write U − V as U (t) − V (t) = UU (t, 0)(U0 − V0 ) + [UU (t, 0) − UV (t, 0)]V0

t + [UU (t, s) − UV (t, s)][FV (s) + G2 (s)] ds

0 t

+

UU (t, s)[FU (s) − FV (s)] ds

0 t

+

UU (t, s)[G1 (s) − G2 (s)] ds.

(5.71)

0

For 0 ≤ θ < ν, operating AU (t)θ , we estimate the X norm of the right-hand side. By (5.44),

t AU (t)θ UU (t, s)[FU (s) − FV (s)] ds

≤C

0

t

X

(t − s)−θ [U (s) − V (s)W + s −ρ U (s) − V (s)Z ] ds.

0

The norms of other terms are estimated in the same way as in the proof of Theorem 5.4. Indeed, by (3.80), AU (t)θ UU (t, 0)(U0 − V0 ) ≤ Ct −θ U0 − V0 X (5.72) X and

t AU (t)θ UU (t, s)[G1 (s) − G2 (s)] ds ≤ Ct β−θ G1 − G2 Fβ,σ . 0

X

4 Lipschitz Continuity of Solutions in Initial Data

231

By the same arguments as for (5.39) and (5.40), we obtain that AU (t)θ [UU (t, 0) − UV (t, 0)]V0 X

t ≤ C A(V0 )β V0 X (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds 0

and

t AU (t)θ [U (t, s) − U (t, s)][F (s) + G (s)] ds U V V 2 0 X

t ≤ CFV + G2 Fβ,σ (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds. 0

Applying these estimates with θ = η, β, and α and using (5.6) and (5.46), we immediately observe that the function ϕ(t) = t η U (t) − V (t)W + t β U (t) − V (t)Z + t α U (t) − V (t)Y , 0 < t ≤ TE,B , satisfies the integral inequality ϕ(t) ≤ CE,B (U0 − V0 X + t β G1 − G2 Fβ,σ )

t θ −θ −η −ρ−β ν−1 β−α−1 ϕ(s) ds. + CE,B t (t − s) s + s + (t − s) s 0

θ=η,β,α

We can then solve this integral inequality by the same technique as for (5.41) (see (5.50)) and can in fact conclude that ϕ(t) ≤ CE,B (U0 − V0 X + t β G1 − G2 Fβ,σ ),

0 ≤ t ≤ TE,B .

As in the proof of Theorem 5.4, we can estimate the norm U (t) − V (t)X , too, directly from (5.71). Hence, we have accomplished the proof. Corollary 5.4 Under the conditions of Theorem 5.9, it holds that t η−α U (t) − V (t)W + t β−α U (t) − V (t)Z + U (t) − V (t)Y ≤ CB,E A(U0 )α (U0 − V0 )X + t β−α G1 − G2 Fβ,σ , 0 ≤ t ≤ TB,E . Proof For α ≤ θ ≤ 1, it holds that AU (t)θ UU (t, 0)(U0 − V0 ) X θ −α ≤ AU (t) UU (t, 0)AU (0) L(X) AU (0)α (U0 − V0 )X ≤ CE,B t α−θ AU (0)α (U0 − V0 )X , 0 < t ≤ TE,B .

232

5 Quasilinear Evolution Equations

We will use this instead of (5.72). Then, by a similar argument as in the proof Theorem 5.9, it is deduced that ϕ(t) ≤ CE,B t α A(U0 )α (U0 − V0 )X + t β G1 − G2 Fβ,σ , 0 ≤ t ≤ TE,B .

Hence, the desired estimate is obtained.

5 Variational Methods In this section, we will remark on the variational methods for studying problem (5.1). Let X and Z be two Hilbert spaces such that Z ⊂ X with dense and compact embedding. Let | · | (resp. · ) be the norm of X (resp. Z). Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Let · ∗ be the norm of Z ∗ . We consider the Cauchy problem for a quasilinear parabolic evolution equation dU dt + A(U )U = F (U ) + G(t), 0 < t ≤ T , (5.73) U (0) = U0 , in Z ∗ . Here, the sectorial operators A(U ) are given as follows. Let K be an open ball of X, K = {E ∈ X; |E| < R},

0 < R < ∞.

For each E ∈ K, let a(E; ·, ·) be a sectorial operator on Z and assume the following conditions: |a(E; U, V )| ≤ MU V , Re a(E; U, U ) ≥ δU , 2

E ∈ K, U, V ∈ Z,

E ∈ K, U ∈ Z,

(5.74) (5.75)

|a(E1 ; U, V ) − a(E2 ; U, V )| ≤ N |E1 − E2 |U V , Ei ∈ K (i = 1, 2), U, V ∈ Z,

(5.76)

with constants δ > 0 and M, N independent of E ∈ K. By Theorem 2.1, we know that the associated operator A(E) is defined and is a sectorial operator of Z ∗ with angle ωA(E) < π2 . The operator F : K ∩ Z → Z ∗ is a nonlinear operator satisfying the following conditions. For any ζ > 0, there exists a constant Lζ such that F (U )∗ ≤ ζ U + Lζ ,

U ∈ K ∩ Z,

(5.77)

F (U ) − F (V )∗ ≤ ζ U − V + Lζ (U + V + 1)|U − V |, U, V ∈ K ∩ Z.

(5.78)

The function G(t) is a given external force function belonging to L2 (0, T ; Z ∗ ). The initial value U0 is taken in K.

6 Quasilinear Parabolic Equations

233

We can then prove the following results in a quite analogous way to the proof of Theorem 4.6. Theorem 5.10 Let (5.74), (5.75), (5.76), (5.77), and (5.78) be satisfied. Then, for any G ∈ L2 (0, T ; Z ∗ ) and any U0 ∈ K, there exists a unique local solution U to (5.73) in the function space: U ∈ L2 ((0, TG,U0 ); Z) ∩ C([0, TG,U0 ]; X) ∩ H 1 ((0, TG,U0 ); Z ∗ ), where TG,U0 > 0 is determined by the norms GL2 (0,T ;Z ∗ ) and |U0 | only. In addition, U satisfies the estimates U L2 ((0,TG,U0 );Z) + U C([0,TG,U0 ];X) + U H 1 ((0,TG,U

0

);Z ∗ )

≤ CG,U0 ,

where CG,U0 > 0 depends on the norms |U0 | and GL2 ((0,T );Z ∗ ) .

6 Quasilinear Parabolic Equations We consider the initial-boundary-value problem for a quasilinear parabolic equation ⎧ ∂u n ⎪ ⎨ ∂t = i,j =1 Dj [aij (x, u)Di u] + f (x, u, ∇u) + g(x, t) n νj (x)aij (x, u)Di u = 0 ⎪ ⎩ i,j =1 u(x, 0) = u0 (x)

in Ω × (0, T ),

on ∂Ω × (0, T ), in Ω, (5.79) in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω, where 0 < T < ∞ is fixed time. Here, aij (x, u) are real-valued functions for (x, u) ∈ Ω × (R + iR) which are smooth functions with respect to the real variables x ∈ Ω, Re u and Im u. We assume that aij (x, u) satisfy conditions (2.19) and (2.28) for every u ∈ R + iR with some uniform constant δ > 0. The function f (x, u, ζ ) is a complex-valued function for (x, u, ζ ) ∈ Ω × (R + iR) × (R + iR)n which is a smooth function with respect to the real variables x ∈ Ω, Re u, Im u, Re ζ , and Im ζ . The function g(x, t) is an external force function, and u0 (x) is an initial function.

6.1 Abstract Formulation Let us formulate problem (5.79) as the Cauchy problem of the form (5.43) in X = Lp (Ω), where n < p < ∞. We set the Sobolev space Hps (Ω) as the space Z where s s 1 s is taken as p+n 2p < 2 < 1, Hp (Ω) being embedded in C (Ω) continuously due to (1.76). For each u ∈ K = {u ∈ Hps (Ω); uHps < R}, 0 < R < ∞, we consider a real ization Ap (u) of the differential operator − i,j Dj [aij (x, u(x))Di ·] + 1 in Lp (Ω)

234

5 Quasilinear Evolution Equations

u˜ under the Neumann-type boundary conditions ∂ν∂A(u) = 0 on ∂Ω. The domain of Ap (u) is given by (2.77) and depends on u. The nonlinear operator F (u) is given by

F (u) = u + f (x, u, ∇u),

u ∈ Hps (Ω).

We set Hp1 (Ω) as the space W . In this way, we rewrite (5.79) in the form (5.43).

6.2 Construction of Local Solutions We already know by Theorem 2.12 that Ap (u) satisfy (5.2) and (5.3). We fix an exponent ν as 12 < ν < p+1 2p . Then, according to the fact (16.36) presented in Chap. 16,

condition (5.4) is fulfilled. As for the third space Y , we set Hp2ν −1 (Ω) = Y , where

the exponent ν is fixed as ν < ν

0 being independent of u. Lemma 5.3 Let a(x, u) be a function for (x, u) ∈ Ω × (R + iR) which is a smooth function with respect to the real variables Re u and Im u. Then, for 0 ≤ s ≤ 1, a(x, u) − a(x, v)Hps ≤ ϕ(uC1 + vC1 )u − vHps ,

u, v ∈ C1 (Ω),

where ϕ(·) is some continuous increasing function determined by a(x, u). Proof Indeed,

1

a(x, u) − a(x, v) =

+ 0

0 1

∂a (x, θ Re u + (1 − θ ) Re v + i Im v) dθ · Re(u − v) ∂ Re u

∂a (x, Re v + iθ Im u + i(1 − θ ) Im v) dθ · i Im(u − v). ∂ Im u

It then suffices to apply (1.87) to the function in the right-hand side with m = 1. Let us now fix the exponents α and β so that (5.6) is verified. By (16.36), if we 1 take these exponents in such a way that 2ν 2−1 < α < 2p and 2s < β < 1, then the two space conditions in (5.6) are verified. In addition, it holds that 1+α 0).

7.1 Abstract Formulation We want to treat (5.80) in the product space ⎧⎛ ⎞ ⎫ u ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎟ ⎬ v 1 1 1 ⎟ ; u ∈ H (Ω), v ∈ H (Ω), w ∈ H (Ω), ρ ∈ L2 (Ω) , X= ⎜ p p p ⎝w ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ρ p being a fixed number such that 2 < p ≤ 4. Let 0 < ε < 1 be a fixed exponent. The space K of initial values is defined as follows. A vector U0 = t (u0 , v0 , w0 , ρ0 ) ∈ X belongs to K if and only if u0 , v0 , w0 ∈ Hp1 (Ω) and ρ0 ∈ H ε (Ω) with the conditions 0 ≤ u0 ≤ 1,

0 ≤ v0 ≤ 1,

0 ≤ w0 ≤ 1,

and 0 ≤ ρ0 , ρ0 L1 = 1.

(5.81)

7 Some Model for Honeybee Colonies

237

We then formulate (5.80) as the Cauchy problem for an abstract equation dU dt

+ A(U )U = F (U ),

0 < t < ∞,

(5.82)

U (0) = U0 ,

in X. Here, for U = t (u, v, w, ρ) ∈ X, A(U ) is a linear operator given by A(U ) = diag{1, 1, 1, A(u, v)}, where A(u, v) is a realization of the differential operator # $ k2 −d∇ · ∇· + 1 k2 + χ2 (Re u) + χ2 (Re v) in the space L2 (Ω) under the Neumann boundary conditions on ∂Ω, χ2 (u) being a smooth cutoff function for −∞ < u < ∞ such that χ2 (u) ≡ − k22 for −∞ < u ≤ − k22 , χ2 (u) = u for 0 ≤ u ≤ 1, and χ2 (u) ≡ 1 + k22 for 1 + k22 ≤ u < ∞. Actually, A(u, v) is a positive definite self-adjoint operator of L2 (Ω). Note that Theorem 2.8 is available to verify that the domain D(A(u, v)) is independent of (u, v) and is given by D(A(u, v)) = HN2 (Ω) (see (2.34)). Consequently, we observe that D(A(U )) ≡ D = Hp1 (Ω) × Hp1 (Ω) × Hp1 (Ω) × HN2 (Ω),

U ∈ X.

(5.83)

In particular, D(A(U )) is independent of U ∈ X. In addition, for 0 ≤ θ ≤ 1, we have A(U )θ = diag{1, 1, 1, A(u, v)θ }. Since A(u, v) is a self-adjoint operator, it follows that D(A(u, v)θ ) coincides with the interpolation space [L2 (Ω), HN2 (Ω)]θ characterized by Theorems 16.7 and 16.9. Therefore, for every U ∈ X,

D A(U )

θ

≡ Dθ =

[Hp1 (Ω)]3 × H 2θ (Ω) for 0 ≤ θ < 34 , [Hp1 (Ω)]3 × HN2θ (Ω) for

3 4

< θ ≤ 1.

(5.84)

The nonlinear operator F is defined by ⎛ ⎞ 1 (Re u) [1 + c(Re ˜ w)] u + a1 s˜ (Re u)θ˜1 (Re v)θ˜2 (Re w) − a2 k1χ+χ 1 (Re u) ⎜ ⎟ ⎜v + a3 s˜ (Re v)θ˜1 (Re u)θ˜2 (Re w) − a4 χ1 (Re v) [1 + c(Re ⎟ ˜ w)] ⎜ ⎟. k +χ (Re v) 1 1 F (U ) = ⎜ ⎟ ⎝ ⎠ ˜ ˜ w + a5 θ1 (Re u)θ1 (Re v)(1 − w)ρ ρ Here, s˜ (u), θ˜i (u) (i = 1, 2) denote C1,1 extensions of s(u), θi (u), respectively, to the whole real line (−∞, ∞) in such a way that s˜ (u) = s(u) for 0 ≤ u ≤ 1, s˜ (u) ≡ 1 for −∞ < u ≤ 0, and s˜ (u) ≡ 0 for 1 ≤ u < ∞. It is the same for θ˜i (u) (i = 1, 2). Similarly, c(w) ˜ is a C1,1 extension of c(w) for −∞ < w < ∞ such that c(w) ˜ = c(w) for 0 ≤ w ≤ 1 and − 12 ≤ c(w) ˜ ≤ c(1) + 12 for −∞ < w < ∞. While, χ1 (u) is a smooth cutoff function for −∞ < u < ∞ such that χ1 (u) ≡ − k21 for −∞ < u ≤ − k21 , χ1 (u) = u for 0 ≤ u ≤ 1, and χ1 (u) ≡ 1 + k21 for 1 + k21 ≤ u < ∞. Let us take 1 2θ 1 the exponent θ = p−1 p (> 2 ). Then, H (Ω) ⊂ Hp (Ω). Note also property (1.95)

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5 Quasilinear Evolution Equations

and the fact that Hp1 (Ω) is a Banach algebra (see (1.88)). As a result, we conclude that F : W ≡ Hp1 (Ω) × Hp1 (Ω) × Hp1 (Ω) × H 2θ (Ω) → X.

(5.85)

The initial values U0 are taken in K. In this way, problem (5.80) has been formulated in the form (5.82).

7.2 Construction of Local Solutions It is immediate to apply Theorem 5.6 to (5.82) for constructing local solutions. By the definition, it is clear that A(U ) is a sectorial operator of X with angle ωA(U ) ≤ ω < π2 , namely, (5.2) and (5.3) are fulfilled with some angle ω and constant M ≥ 1. In view of (5.83), (5.4) is also fulfilled with ν = 1. For verifying (5.5), we will use (3.68). Then, it is sufficient to verify that [A(U ) − A(V )]U˜ X ≤ CU − V Y U˜ D ,

U˜ ∈ D.

But, since ˜ H2, ∇ · [χ∇ u] ˜ L2 ≤ CχHp1 u

χ ∈ Hp1 (Ω), u˜ ∈ HN2 (Ω),

the estimate is observed directly by setting Y = X as the space Y . In addition, since X = Y = Z, (5.6) is also fulfilled with α = 0 and an arbitrarily fixed β such that 0 < β < 2ε , as well as (5.7). As verified by (5.85), F is an operator from W into X. Furthermore, F is seen to satisfy (5.44). Indeed, ˜s (u1 ) − s˜ (u2 )Hp1 ≤ C[∇[˜s (u1 ) − s˜ (u2 )]Lp + ˜s (u1 ) − s˜ (u2 )Lp ] ≤ C[˜s (u1 )∇u1 − s˜ (u2 )∇u2 Lp + u1 − u2 Lp ] ≤ C(u1 Hp1 + u2 Hp1 + 1)u1 − u2 Hp1 ,

u1 , u2 ∈ Hp1 (Ω).

It is the same for other functions contained in the components of F . On account of (5.84), (5.46) is fulfilled with η = θ = p−1 p , as well as (5.47) because of 0 < β < < 12 < p−1 p = η < 1. We finally verify that U0 satisfies (5.30) if γ is fixed as γ = 2ε . Thus, for any U0 ∈ K, (5.82) possesses a unique local solution in the function space: ε 2

U ∈ C((0, TU0 ]; D) ∩ C([0, TU0 ]; Dγ ) ∩ C1 ((0, TU0 ]; X), where TU0 > 0 is determined by the norm U0 D ε . By (5.66), 2 belong to dU , dt

with any 0 < σ < min 2ε , p1 .

ε

A(U )U ∈ F 2 ,σ ((0, TU0 ]; X)

dU dt

(5.86) and A(U )U

(5.87)

7 Some Model for Honeybee Colonies

239

7.3 Nonnegativity of Local Solutions For U0 ∈ K, let U (t) be the local solution of (5.82) constructed above in (5.86) and (5.87). We shall prove that the solution inherits all the properties described in (5.81). Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (5.82) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function given by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative $ #

χ1 (u) ˜ ˜ H (u) a1 s˜ (u)θ1 (v)θ2 (w) − a2 [1 + c(w)] ˜ dx. ψ (t) = k1 + χ1 (u) Ω ˜ and χ1 (u), we observe that By the definitions of s˜ (u), θ˜i (v) (i = 1, 2), c(w), # $

χ1 (u) H (u) [1 + c(w)] ˜ dx ψ (t) ≤ −a2 k1 + χ1 (u) Ω

≤C |H (u)χ1 (u)| dx ≤ C H (u) dx. Ω

Ω

Since ψ (t) ≤ Cψ(t), it follows that ψ(t) ≤ eCt ψ(0) = 0, i.e., ψ(t) ≡ 0. This shows that u(t) ≥ 0 for every 0 ≤ t ≤ TU0 . We next consider the function ψ1 (t) = Ω H (1 − u(t)) dx. By similar arguments, we observe that ψ1 (t) ≤ 0. Hence, ψ1 (t) ≤ ψ1 (0) = 0, i.e., u(t) ≤ 1 for every 0 ≤ t ≤ T U0 . We can repeat quite similar arguments as those above to prove that 0 ≤ v(t) ≤ 1 for 0 ≤ t ≤ TU0 . In the meantime, the nonnegativity of ρ(t) is easily verified by the same techniques as in Chap. 4, Sect. 4.2. From the differential equation of ρ, we d ρ(t) dx ≡ 0. Therefore, ρ(t)L1 = Ω ρ(t) dx ≡ Ω ρ0 dx = 1. Now, have dt Ω since ρ(t) ≥ 0, we can carry out the proof of 0 ≤ w(t) ≤ 1. We will in fact repeat the same arguments as for u and v in view of the known estimate 1

1−ε

1

ρ(t)L∞ ≤ Cρ(t)H 1+ε ≤ Cρ(t)H2−ε2 ρ(t)H2−εε ≤ CU t − 2 ,

0 < t ≤ TU 0 .

7.4 A Priori Estimates In this subsection, let U0 ∈ K satisfy a supplement condition that ρ0 ∈ L∞ (Ω) ∩ H 1 (Ω). Let U denote a local solution of (5.82) on [0, TU ] with U (t) ∈ K, 0 ≤ t ≤ TU , which belongs to the function space: U ∈ C((0, TU ]; D) ∩ C([0, TU ]; Dγ ) ∩ C1 ((0, TU ]; X). We shall establish a priori estimates for local solutions.

(5.88)

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5 Quasilinear Evolution Equations

Proposition 5.2 There exists a continuous increasing function p(·) such that the estimate U (t)Dγ ≤ p(U0 X + ρ0 L∞ + ρ0 H 1 + t),

0 ≤ t ≤ TU ,

(5.89)

holds for any local solution of (5.82) in the function space (5.88). Proof The proof will consist of a several steps. Step 1. Let 2 < q < ∞ be any number. Multiply the equation of ρ by qρ q−1 and integrate the product in Ω. Then,

d k2 ρ q−2 q |∇ρ|2 dx ≤ 0. ρ dx = −dq(q − 1) dt Ω Ω k2 + u + v Therefore, ρ(t)Lq ≤ ρ0 Lq , 0 ≤ t ≤ TU . Letting q → ∞, it follows that ρ(t)L∞ ≤ ρ0 L∞ ,

0 ≤ t ≤ TU .

(5.90)

k2 Step 2. Multiply now the equation of ρ by ∇ · [ k2 +u+v ∇ρ] and integrate the product in Ω. Then, 2

k2 k2 ∂ρ ∇· ∇ρ dx = ∇ · ∇ρ dx. k2 + u + v k2 + u + v Ω ∂t Ω

By a direct calculation, the left-hand side is equal to

k2 d ∂ρ k2 ∇· ∇ρ dx = − |∇ρ|2 dx k2 + u + v dt Ω 2(k2 + u + v) Ω ∂t

k2 (ut + vt ) + |∇ρ|2 dx. 2 2(k + u + v) 2 Ω Here, from the equations of u and v we have ut ≤ a1 and vt ≤ a3 . Therefore, 2

k2 k2 d 2 |∇ρ| dx + ∇ · ∇ρ dx dt Ω 2(k2 + u + v) k + u + v 2 Ω

k2 a1 + a3 |∇ρ|2 dx. ≤ k2 Ω 2(k2 + u + v) Consequently,

k2 2 Ω 2(k2 +u+v) |∇ρ| dx

∇ρ(t)2L2 ≤

≤ 12 e

a1 +a3 k2 t

ρ0 2H 1 , i.e.,

k2 + 2 a1k+a3 t e 2 ρ0 2H 1 , k2

0 ≤ t ≤ TU .

As well, we obtain that 2

t a1 +a3 k2 dx ≤ k2 + 2 e k2 t − 1 ρ0 2 1 , ∇ · ∇ρ H k2 + u + v 2k2 0 Ω

(5.91)

0 ≤ t ≤ TU . (5.92)

7 Some Model for Honeybee Colonies

241

Step 3. We shall use the notation p(ρ0 ) = p(ρ0 L∞ + ρ0 H 1 ), where p(·) in the right-hand side denotes a suitable continuous increasing function. Differentiating the equation of u with respect to the variable xi (i = 1, 2), we observe that ∂ Di u = f1 (u, v, w)Di u + f2 (u, v, w)Di v + f3 (u, v, w)Di w, ∂t where fj (u, v, w) (j = 1, 2, 3) are real smooth functions defined for 0 ≤ u ≤ 1, 0 ≤ v ≤ 1, 0 ≤ w ≤ 1. Multiply this equation by p|Di u|p−2 Di u and integrate the product in Ω. After some calculations, we have

d p p p |Di u|p dx ≤ C Di uLp + Di vLp + Di wLp . dt Ω d Note that du |u|p = p|u|p−2 u for −∞ < u < ∞. By using the equations of v and w (in view of (5.90)), we similarly obtain that

d p p p |Di v|p dx ≤ C Di uLp + Di vLp + Di wLp , dt Ω

d p p p p |Di w|p dx ≤ Cp(ρ0 ) Di uLp + Di vLp + Di wLp + Di ρLp . dt Ω

Therefore,

d |Di u|p + |Di v|p + |Di w|p dx dt Ω p p p p ≤ Cp(ρ0 ) Di u(t)Lp + Di v(t)Lp + Di w(t)Lp + Di ρ(t)Lp . Here, by (5.90) and (5.91), p

p−2

p−2

Di ρLp ≤ CDi ρH 1 Di ρ2L2 ≤ Cp(ρ0 )eμt ρ(t)H 2 , where μ =

a1 +a3 k2 .

In addition, from

$ k2 k2 k2 ∇ρ = Δρ − (∇u + ∇v) · ∇ρ ∇· k2 + u + v k2 + u + v (k2 + u + v)2 #

we see that ΔρL2

# $ k2 k2 + 2 1 ∇· ∇ρ ≤ + (∇uLp + ∇vLp )∇ρL 2p k2 k2 + u + v k2 p−2 L2 # $ k2 k2 + 2 ∇ · ∇ρ ≤ k2 k2 + u + v L2

242

5 Quasilinear Evolution Equations 2

p−2

+ C(∇uLp + ∇vLp )ρHp 2 ρHp1 # $ k2 k2 + 2 ∇· ≤ ∇ρ + ζ ρH 2 k2 k2 + u + v L2 p p & % μ p−2 + Cζ p(ρ0 )e 2 t ∇uLp−2 + ∇v Lp p with any number ζ > 0 and some constant Cζ > 0 depending on ζ . Since ρH 2 ≤ C(ΔρL2 + ρL2 ) for all ρ ∈ HN2 (Ω) (see (2.33)), it follows that # $ p p & ' μ % ( k2 p−2 p−2 2 t ∇u + Cp(ρ ) e + ∇v ∇ρ + 1 ρH 2 ≤ C ∇ · 0 Lp Lp k2 + u + v L2 and that p−2 ρH 2

# $p−2 k2 ≤ C ∇ · ∇ρ k2 + u + v L2 μ(p−2) t p p ∇uLp + ∇vLp + 1 . + Cp(ρ0 ) e 2

In this way, we arrive at the differential inequality

d |∇u|p + |∇v|p + |∇w|p dx dt Ω pμ p p p t 2 ∇uLp + ∇vLp + ∇wLp ≤ Cp(ρ0 )e # $p−2 k2 + ∇ · +1 . ∇ρ k2 + u + v L2 By virtue of (5.92) (recall that p was taken as 2 < p ≤ 4), we can solve this differential inequality to conclude that ∇u(t)Lp + ∇v(t)Lp + ∇w(t)Lp ≤ p(U0 X + ρ0 L∞ + ρ0 H 1 + t),

0 ≤ t ≤ TU .

This, together with (5.91), then yields the desired estimate (5.89).

7.5 Global Solutions Utilizing the a priori estimates (5.89), we shall construct a global solution to (5.82). For U0 ∈ K, we know that there exists a local solution having the properties (5.81) at least on an interval [0, TU0 ]. Let 0 < t1 < TU0 . Then, U1 = U (t1 ) ∈ K

8 Some Chemotaxis Model

243

and ρ1 = ρ(t1 ) ∈ HN2 (Ω). We next consider problem (5.82) with the initial value U1 on an interval [t1 , T ], where the end time T > 0 is any finite time. Proposition 5.2 ensures estimate (5.89) for any local solution V , i.e., V (t)Dγ ≤ p(U1 X + ρ1 L∞ + ρ1 H 1 + T ), t1 ≤ t ≤ TV . Then, the local solution V can always be extended over an interval [t1 , TV + τ ] as local solution, τ > 0 being dependent only on p(U1 X + ρ1 L∞ + ρ1 H 1 + T ) and hence being independent of the extreme time TV (cf. Corollary 5.2). This means that our Cauchy problem possesses a global solution on the interval [t1 , T ]. This argument is meaningful for any finite time T > 0. So, we conclude the global existence of solution. For any initial value U0 ∈ K, there exists a unique global solution to (5.82) with U (t) ∈ K, 0 ≤ t < ∞, in the function space: U ∈ C((0, ∞); D) ∩ C([0, ∞); Dγ ) ∩ C1 ((0, ∞); X).

8 Some Chemotaxis Model Let us consider the initial-boundary-value problem for a chemotaxis model in biology which is given by ⎧ ∂u 2 in Ω × (0, ∞), ⎪ ∂t = Δ(au + αu ) − μ∇ · (u∇ρ) ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu in Ω × (0, ∞), ∂t (5.93) ∂ρ ⎪ ∂u ⎪ = = 0 on ∂Ω × (0, ∞), ⎪ ∂n ∂n ⎪ ⎩ ρ(x, 0) = ρ0 (x) in Ω, u(x, 0) = u0 (x), in a C2 or convex, bounded domain Ω ⊂ R2 . In the domain Ω, biological individuals (bacteria) exist. The density of bacteria at time t is denoted by u(x, t). The function ρ(x, t) denotes the concentration of chemical substance at time t . Bacteria diffuse not only by random walking with diffusion rate a but also by self-diffusion with rate α. They also have a directed movement which is called the chemotaxis in the sense that they have a tendency to move toward higher concentration of the chemical substance. The flow by this effect is given by μu∇ρ with some mobility constant μ. Bacteria produce the chemical substance by themselves at a rate ν. The chemical substance diffuses with diffusion rate b. For the details, see Chap. 12. We assume that a, b, d, α, μ, and ν are positive constants (>0). Let 0 < ε < 12 be arbitrarily fixed. The initial functions are taken as 0 ≤ u0 ∈ H 1+ε (Ω) and 0 ≤ ρ0 ∈ H 1+ε (Ω).

(5.94)

8.1 Abstract Formulation Let us set X = L2 (Ω) (see (1.84)) as the underlying space X. We want to formulate (5.93) as the Cauchy problem for an abstract equation

244

5 Quasilinear Evolution Equations

+ A(U )U = F (U ), U (0) = U0 , dU dt

0 < t < ∞,

(5.95)

in X. Here, A(U ) is a linear operator of X defined for U ∈ Z = H1+ε1 (Ω), where 0 < ε1 < ε, which is given by # $ $ # u A1 (u) B(u) ∈ Z. with D(A(U )) = H2N (Ω), U = A(U ) = ρ 0 A2 The operator A1 (u) is a realization of the differential operator −{∇ · [a + 2αχ(Re u)]∇·} + 1 in L2 (Ω) under the Neumann boundary conditions on ∂Ω, where χ(u) is a smooth cutoff function such that χ(u) = u for u ≥ 0 and χ(u) ≡ −δ for u ≤ −δ, δ > 0 being some small positive constant such that 4αδ ≤ a. By Theorem 2.4, A1 (u) is a positive definite self-adjoint operator of L2 (Ω), and by Theorem 2.8, their domain is given by D(A1 (u)) = HN2 (Ω) (see (2.34)). The operator A2 is a realization of the operator −bΔ + d in L2 (Ω) under the same boundary conditions on ∂Ω and is a positive definite self-adjoint operator of L2 (Ω) with domain D(A2 ) = HN2 (Ω); and the operator B(u) is given by μ∇ · (u∇·). Since ∇ · (u∇v) = ∇u · ∇v + uΔu, we observe from (1.74), (1.75), and (1.76) that ∇ · (u∇v)L2 ≤ CuH 1+ε1 vH 2 , This means that D(B(u)) ⊃ H 2 (Ω). The operator F : X → X is given by # $ u F (U ) = , νu

u ∈ H 1+ε1 (Ω), v ∈ H 2 (Ω).

U=

# $ u ∈ X. ρ

The initial value U0 is given by U0 = t (u0 , ρ0 ), u0 and ρ0 satisfying (5.94).

8.2 Basic Properties of Operators A(U ) Let 0 < R < ∞, and let KR = {U ∈ Z; U Z < R} be an open ball of Z. Proposition 5.3 For any U ∈ Z, we have D(A(U )) = H2N (Ω). For each 0 < R < ∞, there is a constant CR > 0 such that CR−1 U˜ H2 ≤ A(U )U˜ X ≤ CR U˜ H2 , U˜ ∈ D(A(U )), holds for every U ∈ KR . Proof This result is easily verified by (2.33) which is ensured by Theorem 2.8. Proposition 5.4 For U ∈ KR , A(U ) is a sectorial operator of X with angle ωA(U ) ≤ ωR < π2 and satisfies (5.3) with MR ≥ 1 uniformly. Proof As A1 (u) and A2 are positive definite self-adjoint operators of L2 (Ω), this result is obtained by applying Theorem 1.8 to each operator A(U ).

8 Some Chemotaxis Model

245

For s > 32 , we put HsN (Ω) =

∂U U ∈ H (Ω); = 0 on ∂Ω . ∂n s

Proposition 5.5 For all U ∈ Z, we have D A(U )θ = H2θ (Ω), D A(U )θ = H2θ N (Ω),

3 0≤θ < , 4 3 < θ ≤ 1. 4

Let 0 ≤ θ ≤ 1, θ = 34 . For 0 < R < ∞, there is a constant CR,θ > 0 such that −1 ˜ U H2θ ≤ A(U )θ U˜ X ≤ CR,θ U˜ H2θ , CR,θ

U˜ ∈ D A(U )θ ; U ∈ KR .

Proof In order to prove the assertion of theorem, we will appeal to Theorem 16.4 presented in Chap. 16. For applying Theorem 16.4, it is in fact sufficient to verify that for each u, the adjoint operator B(u)∗ : D(B(u)∗ ) ⊂ L2 (Ω) → L2 (Ω) of B(u) has a domain such that D(B(u)∗ ) ⊃ HN2 (Ω) and is a bounded operator from HN2 (Ω) into L2 (Ω). However, this is easily verified. It is clear that HN2 (Ω) ⊂ D(B(u)∗ ) and B(u)∗ v = μ∇ · (u∇v) for v ∈ HN2 (Ω). Hence, B(u)∗ is bounded from HN2 (Ω) into L2 (Ω). We thus know that D(A(U )θ ) = [X, H2N (Ω)]θ for 0 ≤ θ ≤ 1. Then, the desired characterization of D(A(U )θ ) is verified by Theorem 16.7.

8.3 Construction of Local Solutions We are ready to apply Theorem 5.6 to construct local solutions to (5.95). Let U0 be any initial value satisfying (5.94). We can then choose a radius R > 0 in such a way that U0 ∈ KR . It is already known that, for U ∈ KR , A(U ) fulfills (5.2) and (5.3). Since D(A(U )) = H2N (Ω) for every U ∈ KR , (5.4) is also fulfilled with ν = 1. In view of (3.68), (5.5) is equivalent to showing that [A(U ) − A(V )]U˜ X ≤ CR U − V Y U˜ H2 ,

U˜ ∈ H2N (Ω),

with a suitable space Y . We here notice that ˜ ∇ · [χ(Re u)∇ u] ˜ = χ(Re u)Δu˜ + χ (Re u)∇(Re u) · ∇ u. From this we observe that ˜ H2 ∇ · [χ(Re u)∇ u] ˜ L2 ≤ (χ(Re u)L∞ + χ (Re u)L∞ uH θ+1 )u ≤ (χ(Re u)H θ+1 + χ (Re u)H θ+1 uH θ+1 )u ˜ H2, (5.96)

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5 Quasilinear Evolution Equations

whatever 0 < θ < 1 is. In view of this fact, we set Y = H1+ε0 (Ω), where ε0 is a third exponent such that 0 < ε0 < ε1 < ε. Then, (5.5) is verified immediately with the aid of (1.89) and (1.94). By virtue of Proposition 5.5, (5.6) is then fulfilled with 1+ε1 0 α = 1+ε 2 and β = 2 . Relation (5.7) is now obvious. The nonlinear operator F is defined on W = H1+ε2 (Ω), where ε < ε2 < 12 . It is trivial that (5.44) is fulfilled. Conditions (5.46) and (5.47) are also verified, since 2 Proposition 5.5 provides that D(A(U )η ) = W with η = 1+ε 2 for every U ∈ KR . The 1+ε initial value U0 satisfies (5.30) if γ is taken as γ = 2 . Theorem 5.6 then provides that, for any U0 ∈ KR satisfying (5.94), (5.95) possesses a unique local solution in the function space: U ∈ C (0, TU0 ]; H2N (Ω) ∩ C([0, TU0 ]; H1+ε (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), (5.97) where TU0 > 0 is determined by the norm U0 H1+ε . By (5.66), A(U )U belongs to dU , dt

A(U )U ∈ F

1+ε 2 ,σ

((0, TU0 ]; L2 (Ω))

(5.98)

with any 0 < σ < min{ε1 − ε0 , 1−ε 2 }.

8.4 Nonnegativity of Local Solutions For U0 satisfying (5.94), let U (t) be the local solution of (5.95) constructed above in the function space (5.97) and (5.98). We want to prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . Let us notice first that U (t) is real valued. The complex conjugate U (t) of U (t) is also a local solution of (5.94) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. We then utilize the truncation method as in Sect. 7.3. Let H (u) be a C1,1 cutoff function given by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative

H (u)∇ · {[a + 2αχ(u)]∇u} dx − μ H (u)∇ · (u∇ρ) dx. ψ (t) = Ω

Ω

Using property (1.96), we have

H (u)∇ · {[a + 2αχ(u)]∇u} dx = − [a + 2αχ(u)]∇H (u) · ∇u dx Ω

=−

Ω

[a + 2αχ(u)]|∇H (u)|2 dx ≤ −a

Ω

By a similar reason,

|∇H (u)|2 dx. Ω

(5.99)

9 Nonautonomous Quasilinear Abstract Evolution Equations

−μ

H (u)∇ · (u∇ρ) dx = μ

Ω

247

u∇H (u) · ∇ρ dx

Ω

=μ

H (u)∇H (u) · ∇ρ dx

Ω

≤ μH (u)Lp ∇H (u)L2 ∇ρLq 2(1− p1 )

≤ CH (u)H 1 ≤

2

H (u)Lp 2 ∇ρLq

a p H (u)2H 1 + CH (u)2L2 ∇ρLq , 2

where 2 < p, q < ∞ are any numbers satisfying

1 p

+

1 q

= 12 . Therefore, it follows C

t

p

[∇ρ(s)

+1]ds

Lq that ψ (t) ≤ C[∇v(t)Lq +1]ψ(t). Consequently, ψ(t) ≤ ψ(0)e 0 for 0 < t < TU0 . p p From the regularity (5.98), we notice that ∇ρ(s)Lq ≤ Cq ρ(s)H 2 ≤ Cq CU0 ×

p

s p(ε−1)/2 . It is possible to choose p > 2 arbitrarily close to 2, which shows that p ∇ρ(·)Lq is integrable in (0, TU0 ). Then, ψ(0) = 0 implies ψ(t) ≡ 0. Thus, u(t) ≥ 0 for every 0 < t ≤ TU0 . It is similar (actually easier) for the proof of ρ(t) ≥ 0 for 0 < t ≤ TU0 . Hence, we have accomplished the proof of nonnegativity of U (t). Now, since u(t) ≥ 0, it holds that χ(Re u(t)) = u(t); this then means that the local solution of (5.95) is regarded as a local solution to the original problem (5.93).

8.5 Global Solutions We are able to build up a priori estimates for local solutions of (5.95) with initial values satisfying (5.94) and an additional condition U0 ∈ H2 (Ω). In fact, there exists a continuous increasing function p(·) such that, for any local solution of (5.95) in (5.97) (in which TU0 must be replaced by TU ) with initial value satisfying (5.94) and U0 ∈ H2 (Ω), it holds that U (t))H2 ≤ p(U0 H2 ),

0 < t ≤ TU .

(5.100)

For showing these estimates step-by-step, we need more complicated calculations for the norms of U (t) in each step. We will not describe the proof here, cf. Chap. 12, Sect. 3.2, Chap. 15, Sects. 3.1 and 3.2. It is of course immediate to conclude global existence of solutions for (5.95) utilizing (5.100) (cf. Corollary 5.2). Note also that, for any initial value satisfying (5.94), its local solution always satisfies U (t) ∈ H2 (Ω) for any t > 0.

9 Nonautonomous Quasilinear Abstract Evolution Equations In this section, we will show how to reduce nonautonomous problems to autonomous ones as in Chap. 4, Sect. 5. Consider the Cauchy problem for a quasilinear

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5 Quasilinear Evolution Equations

abstract evolution equation dU dt + A(t, U )U = F (t, U ) + G(t),

0 < t ≤ T,

U (0) = U0 ,

(5.101)

in a Banach X. Let Z be a second Banach space continuously embedded in X, and let K = {U ∈ Z; U Z < R}, 0 < R < ∞, be an open ball of Z. For (t, U ) ∈ [0, T ] × K, A(t, U ) is a sectorial operator of X with angle ωA(t,U ) < π2 . For each 0 ≤ t ≤ T , F (t, ·) is a nonlinear operator from W into X, where W is another Banach space continuously embedded in X. The function G is a given X-valued function. The initial value U0 is taken in K. For the sectorial operators A(t, U ), we make the same structural assumptions as in Sect. 1.1. We use three Banach spaces Z ⊂ Y ⊂ X. The operator A(t, U ) is a sectorial operator of X satisfying (5.2) and (5.3). The domain D(A(t, U )) is allowed to vary with (t, U ) ∈ [0, T ] × K, but it is assumed that there exists a fixed exponent 0 < ν ≤ 1 such that D(A(s, V )) ⊂ D A(t, U )ν for any pair (t, U ), (s, V ) ∈ [0, T ] × K (5.102) (analogously to (5.4)). The inverse A(t, U )−1 satisfies the Lipschitz condition A(t, U )ν A(t, U )−1 − A(s, V )−1 L(X) ≤ N (|t − s| + U − V Y ),

(t, U ), (s, V ) ∈ [0, T ] × K,

(5.103)

with a constant N > 0 (instead of (5.5)). Furthermore, the domains of fractional powers D(A(t, U )α ) and D(A(t, U )β ) are contained in Y and Z with estimate (5.6), respectively, and with exponents 0 ≤ α < β < 1 satisfying relations (5.7). Similarly, D(A(t, U )η ) is contained in W with estimate (5.46). When (5.47) is the case, i.e., β < ν, we assume the Lipschitz condition F (t, U ) − F (s, V )X ≤ ϕ(U Z + V Z )[U − V W + (U W + V W + 1)(|t − s| + U − V Z )], (t, U ), (s, V ) ∈ [0, T ] × W,

(5.104)

where ϕ(·) is some continuous increasing function. On the other hand, when (5.69) is the case, i.e., ν ≤ β, we have to assume not only the Lipschitz condition F (t, U ) − F (s, V )X ≤ ϕ(U Z + V Z )(|t − s| + U − V W ), (t, U ), (s, V ) ∈ [0, T ] × Z, but also the reflexivity of Z (see (5.36)). We now rewrite problem (5.101) as the following one: d τ 1 τ 0 0 τ +1 dt U + 0 A(χ(Re τ ),U ) U = F (χ(Re τ ),U ) + G(t) , 0 τ U (0) = U0 ,

(5.105)

(5.106)

Notes and Further Researches

249

in the product space X = C × X for the unknown function t (τ, U ), where χ(t) is a C0,1 cutoff function for −∞ < t < ∞ such that χ(t) ≡ 0 for −∞ < t < 0, χ(t) = t for 0 ≤ t ≤ T , and χ(t) ≡ T for T < t < ∞. Since τ (t) = t , (5.106) is certainly equivalent to (5.101). It is not difficult to verify that the conditions assumed above, including (5.102), (5.103), (5.104), and (5.105), imply all the structural conditions for the linear operators A(τ, U ) = diag{1, A(χ(Re τ ), U )},

(τ, U ) ∈ C × K,

of X and for the nonlinear operator F(τ, U ) = t (τ + 1, F (χ(Re τ ), U )) from C × W into X. Hence, Theorems 5.5, 5.6, 5.7, and others are available to (5.106).

Notes and Further Researches The abstract problems of the form (5.1) or (5.43) were first studied by Sobolevskii [Sob61b], where the independence of the domains D(A(t, U )ν ) is assumed for some 0 < ν ≤ 1. The analyticity of solutions was studied by Furuya [Fur81, Fur83], and the global existence of solutions was studied by Lunardi [Lun84, Lun85]. Under a similar assumption as (5.5), especially in the case where β ≤ ν (together with (5.36)), the problems were handled by Yagi [Yag91, Yag02]. For the nonautonomous problem (5.101), see [Yag04]. In Favini–Yagi [FY04], the Cauchy problem for the multivalued quasilinear equation dU + A(U )U F (U ), dt

0 < t ≤ T,

was handled. For the study of (5.1) or (5.43) by the variational methods, we quote Lions [Lio69, DL88]. Theorem 5.10 can be proved by similar arguments as in Ryu–Yagi [RY01]. By the methods of operational equations (see [DaPG75]), the abstract problems can also be studied; we quote Prüss [Pru02]. There is of course a great deal of important literature which deals with the quasilinear parabolic differential equations directly by the functional analytic methods. But we will here quote only Ladyzhenskaya–Solonnikov–Uraltseva [LSU68], Amann [Ama86, Ama93, Ama95], Quittner–Souplet [QS07], and so forth. The abstract results obtained in this chapter are available for various model equations presented from the real world. In Sects. 7 and 8, some of them were represented. The honeybee colony model (5.80) was introduced in Camazine–Sneyd– Jenkins–Murray [CSJM90] for describing pattern formation created spontaneously by the egg-laying queen. For more detailed observations and arguments for modeling, see Camazine et al. [CDFSTB, Chap. 16]. The chemotaxis system with selfdiffusion (5.93) is one of specific forms of the chemotaxis model presented by Keller–Segel [KS70]. Moreover, these abstract results will be utilized in Chaps. 14 and 15.

Chapter 6

Dynamical Systems

In this chapter, we study dynamical systems determined from the Cauchy problems for abstract semilinear and quasilinear evolution equations in Banach spaces. The theory of infinite-dimensional dynamical systems was developed by Ladyzhenskaya, Vishik, Temam, and many other mathematicians. The first half of the chapter is mainly devoted to studying two basic matters of the dynamical systems, namely, stable and unstable manifolds of the equilibria and attractors. We shall use the notation (S(t), X, X) to denote a dynamical system. Here, X is a Banach space and denotes the universal space in which a dynamical system is considered. The notation X denotes a subset of X and is a metric space. This is the space consisting of all possible states the system can take and is called the phase space. Meanwhile, S(t) (0 ≤ t < ∞) is a family of nonlinear operators acting on X with the semigroup property S(t)S(s) = S(t + s) and with S(0) = 1 (the identity operator on X). It is assumed that S(t) is continuous in both space and time variables in the sense that the correspondence (t, U0 ) → S(t)U0 is continuous from [0, ∞) × X into X. The operators S(t) give a propagation law of states of the system. For U0 ∈ X, the continuous function S(·)U0 for 0 ≤ t < ∞ with values in X denotes a transition of states starting from U0 at t = 0. The trace of points S(·)U0 which vary in the phase space X as time t goes on from 0 to ∞ is called the trajectory starting from U0 . The semigroup property implies S(t + s)U0 = S(t)S(s)U0 for any 0 ≤ s, t < ∞; this means that, s being fixed, a trajectory passing by S(s)U0 at time s coincides with that starting from S(s)U0 at time 0. The totality of such trajectories is called a dynamical system and is denoted by the triplet (S(t), X, X). A point U ∈ X which satisfies S(t)U = U is called a fixed point of S(t). A point U which is a fixed point for every S(t), that is, S(t)U = U for every 0 ≤ t < ∞, is called an equilibrium of dynamical system. For each equilibrium U , one can define a stable manifold M− (U ) and an unstable manifold M+ (U ) by (6.11) and (6.12), respectively. We then want to construct stable and unstable manifolds which are smooth in a neighborhood of U . If M− (U ) is a neighborhood of U , then U is stable. In the meantime, if M+ (U ) = {U }, then U is unstable. We shall indeed show that, if S(t) is Fréchet differentiable in a neighborhood of U with Hölder continuous derivative (6.19) and if the differential S(t) U satisfies a spectrum separation condition (6.20), then one can construct the desired smooth stable and unstable manifolds in a neighborhood of U . A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_6, © Springer-Verlag Berlin Heidelberg 2010

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For (S(t), X, X), a subset A of X is called an attractor if A is invariant (see (6.4)) and if A attracts, as t → ∞, all the trajectories starting in a neighborhood of A. Among others, an attractor A is called a global attractor of dynamical system if A is a compact set of X, if A is strictly invariant (see (6.5)), and if A attracts all trajectories in X. We know however that the global attractor is not necessarily robusts. Consider a dynamical system (Sξ (t), X, X) containing a parameter ξ and assume that (Sξ (t), X, X) possesses a global attractor Aξ , even if the semigroup Sξ (t) depends on ξ smoothly, Aξ may be not continuous in ξ . It is only upper semicontinuous in general. In addition, the global attractor may not be rich enough to describe the nature of the system, for, since A is strictly invariant, A can easily miss some important trajectories which correspond to a specific transition of states like pattern formations. As a more robust attractor, Foias–Sell–Temam introduced the notion of inertial manifolds which are hyperbolic (and thus robust), compact, invariant, finite-dimensional C1 manifolds that attract all the trajectories at exponential rates. But one can ensure the existence of inertial manifolds only under a very restrictive assumption called the spectral gap condition. Consequently, we do not know whether the dynamical systems constructed from many important models have inertial manifolds or not. In 1994, Eden–Foias–Nicolaenko–Temam have founded the new concept of exponential attractors (sometimes called inertial sets) to overcome this difficulty. An attractor M containing the global attractor is called an exponential attractor if M is a compact set of X with finite fractal dimension and if M attracts all the trajectories at exponential rates. (So, M need not to be a smooth manifold.) We notice that the exponential attractor is not uniquely determined and usually exists as a family. But the exponential attractor Mξ can depend continuously on a parameter ξ in a certain specific sense if the semigroup Sξ (t) depends on ξ smoothly, see Theorem 6.17. Two kinds of a sufficient condition on S(t) are known that ensure the existence of exponential attractors for (S(t), X, X). One is a squeezing property presented by the founders of the notion themselves. The squeezing property is meaningful only when X is a Hilbert space. When X is a Hilbert space, the operator S(t) is said to enjoy a squeezing property if, for any pair (U, V ) of points U, V ∈ X, it holds that either S(t)U − S(t)V ≤ δ U − V with some fixed exponent 0 ≤ δ < 14 or (1 − P )[S(t)U − S(t)V ] ≤ P [S(t)U − S(t)V ] with some fixed finite-rank orthogonal projection P (see (6.45)–(6.46)). The other is a compact perturbation of a contraction operator presented first by Efendiev–Miranville–Zelik. This property is meaningful in general Banach spaces, and there are two types. The operator S(t) is said to be a compact perturbation of a contraction operator of type I if the difference S(t)U − S(t)V is divided into two parts S0 (U, V ) and K(U, V ) such that S0 (U, V ) X ≤ δ U − V X with some fixed exponent 0 ≤ δ < 12 and that K(U, V ) Z ≤ L U − V X with some constant L > 0 and some fixed subspace Z which is compactly embedded in X (see (6.28)–(6.29)). Similarly, S(t) is said to be a compact perturbation of a contraction operator of type II if the norm S(t)U − S(t)V X is estimated by the sum of norms δ U − V X and KU −KV X with some fixed exponent 0 ≤ δ < 12 and with some Lipschitz continuous nonlinear operator K from X into a subspace Z which is compactly embedded in X (see (6.38)–(6.39)). As will be seen, the squeezing property provides sharper

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dimension estimates for exponential attractors than those of the compact perturbation condition. On the other hand, we have to notice that the property can easily be verified only when (S(t), X, X) are constructed from semilinear problems in Hilbert spaces of the form (6.53). Unfortunately, no convenient method is known so far for dynamical systems from quasilinear problems. In contrast, the method of using the compact perturbation condition on S(t) can easily be available to many concrete models. In the second half of the chapter, we shall consider dynamical systems which are determined by the Cauchy problems of semilinear and quasilinear abstract evolution equations. We are concerned with describing general methods for constructing stable and unstable manifolds for equilibria, i.e., stationary solutions. Similarly, we are concerned with general methods for constructing exponential attractors. For semilinear problems, stable–unstable manifolds are constructed under conditions (6.68), (6.69), and (6.76) and exponential attractors under (6.59) and (6.60), respectively. For quasilinear problems, stable–unstable manifolds are constructed under (6.115), (6.116) and (6.127) and exponential attractors under (6.103) and (6.104), respectively. We remark also that all these conditions are described mainly in terms of linear and nonlinear operators in the equations.

1 Basics of Dynamical Systems In this section, we shall prepare the basic matters of dynamical systems in Banach spaces. For details, see [Lad91, BV92, Tem97, Rob01] and others.

1.1 Equilibria and Attractors Let X be a Banach space with norm · . Let X be a subset of X, X being a metric space with the distance d(U, V ) = U − V induced by · . A family of nonlinear operators S(t), 0 ≤ t < ∞, from X into itself is called a (nonlinear) semigroup on X if S(t) satisfies (1) S(0) = 1 (identity mapping on X). (2) S(t)S(s) = S(t + s), 0 ≤ t, s < ∞ (semigroup property). When a semigroup S(t) on X has the property that the mapping G(t, U ) = S(t)U is continuous from [0, ∞) × X to X,

(6.1)

S(t) is called a continuous (nonlinear) semigroup on X. Let {Un }n=0,1,2,... be any sequence of X which converges to U in X. Then, for any fixed 0 < T < ∞, the set [0, T ] × {U0 , U1 , U2 , . . . , U } is a compact set of [0, ∞) × X. Therefore, if S(t) is a continuous semigroup, G is uniformly continuous on [0, T ] × {U0 , U1 , U2 , . . . , U }.

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6 Dynamical Systems

Thus, for any fixed 0 < T < ∞, Un → U

sup S(t)Un − S(t)U → 0.

implies

(6.2)

0≤t≤T

Let S(t) be a continuous semigroup on X. For each U0 ∈ X, S(·)U0 defines an X-valued continuous function for 0 ≤ t < ∞. The trace of the function S(·)U0 in X is called a trajectory starting from U0 . The totality of trajectories starting from the points in X are denoted by the triplet (S(t), X, X) and is called a dynamical system. The set X is called a phase space, and X a universal space. Let (S(t), X, X) be a dynamical system. A point U ∈ X is called a fixed point of S(t) if S(t)U = U . If U is a fixed point for all operators S(t), i.e., S(t)U = U

for every t ≥ 0,

(6.3)

then U is called an equilibrium of (S(t), X, X). More generally, a set A ⊂ X is said to be invariant under S(t) if S(t)A ⊂ A. If A is invariant under all operators S(t), i.e., S(t)A ⊂ A

for every t ≥ 0,

(6.4)

then A is called an invariant set of (S(t), X, X). Similarly, a set A ⊂ X is said to be strictly invariant under S(t) if S(t)A = A. If S(t)A = A

for every t ≥ 0,

(6.5)

then A is called a strictly invariant set of (S(t), X, X). Two important notions of absorbing and attraction are defined as follows. A set A is said to absorb a set B if there is a certain time t0 ≥ 0 such that S(t)B is contained in A for every t ≥ t0 , i.e., S(t)B ⊂ A

for every t ≥ t0 .

We call a subset A ⊂ X an absorbing set of (S(t), X, X) if A ⊂ X absorbs every bounded set of X, namely, if, for any bounded set B ⊂ X, there is a time tB ≥ 0 (depending on B) such that S(t)B ⊂ A

for every t ≥ tB .

In the meantime, a set A is said to attract a set B if for any ε > 0, the εneighborhood Wε (A) of A absorbs B. Using the Hausdorff pseudo-distance h(·, ·) (see (1.1)), this definition can be described as lim h(S(t)B, A) = 0.

t→∞

(6.6)

Indeed, if (6.6) holds true, then for any ε > 0, there exists some tε > 0 such that h(S(t)B, A) ≤ 2ε for t ≥ tε ; therefore, for any U ∈ S(t)B, d(U, A) ≤ 2ε ; hence, U ∈ Wε (A), i.e., S(t)B ⊂ Wε (A)

for every t ≥ tε .

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255

Conversely, this condition implies h(S(t)B, A) ≤ ε for every t ≥ tε . So, if A attracts B, then (6.6) is valid. An equilibrium U of (S(t), X, X) is said to be asymptotically stable if some neighborhood W of U is attracted by U , i.e., lim h(S(t)W, U ) = 0.

t→∞

(6.7)

We simply say that U is stable if for any positive number ε > 0, there exists a neighborhood W of U which is attracted by an ε-neighborhood of U in such a way that lim sup h(S(t)W, U ) ≤ ε. t→∞

Clearly, if U is asymptotically stable, it is stable. Asymptotic stability is therefore stronger than mere stability. When U is not stable, U is said to be unstable. In other words, U is unstable if and only if there exists a positive number ε0 > 0 such that, for any neighborhood W of U , it holds that lim sup h(S(t)W, U ) > ε0 .

(6.8)

t→∞

That is, for any neighborhood W , there are a time sequence tn → ∞ and a sequence of points Un ∈ W such that d(S(tn )Un , U ) ≥ ε0 for all n. An invariant set A of (S(t), X, X) is called an attractor if some open neighborhood W of A is attracted by A, namely, A is an attractor if S(t)A ⊂ A for every t and lim h(S(t)W, A) = 0

t→∞

for some neighborhood W of A. An attractor A is called a global attractor of (S(t), X, X) if (1) A is a compact subset of X. (2) S(t)A = A for every t > 0. (3) A attracts every bounded subset of X, i.e., for every bounded B ⊂ X, lim h(S(t)B, A) = 0.

t→∞

(6.9)

Let B be a bounded subset of X which contains the global attractor A. Then, A = S(t)A ⊂ S(t)B, i.e., h(A, S(t)B) = 0 for any t > 0; hence, limt→∞ d(S(t)B, A) = 0, where d(·, ·) is the distance of two sets. By definition, if it exists, the global attractor A is always unique.

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1.2 ω-limit Sets and Global Attractor Consider a dynamical system (S(t), X, X). For each U0 ∈ X, we put {S(τ )U0 ; t ≤ τ < ∞} (closure in X). ω(U0 ) = 0≤t 0. There exists a sequence tn → ∞ such that S(tn )U0 → U ; for tn ≥ t, S(tn )U0 = S(t)S(tn − t)U0 , and S(tn − t)U0 contains a convergent subsequence with a limit U ∈ ω(U0 ); hence, U = S(t)U . In this way, S(t)ω(U0 ) = ω(U0 ) for every t ≥ 0. Consider two open sets Oi (i = 1, 2) of X such that ω(U0 ) ⊂ O1 ∪ O2 and O1 ∩ O2 = ∅. Suppose that there exist points Ui ∈ ω(U0 ) ∩ Oi (i = 1, 2); then, there exist sequences sm → ∞ and tn → ∞ such that S(sm )U0 → U1 and S(tn )U0 → U2 , respectively; here, it is possible to assume that S(sm )U0 ∈ O1 and S(tn )U0 ∈ O2 . If sm < tn , then there exists a time τ ∈ (sm , tn ) such that S(τ )U0 ∈ / O1 ∪ O2 because of the connectedness of the arc {S(t)U0 ; sm ≤ t ≤ tn } of the trajectory S(t)U0 ; similarly, if tn < sm , then there is a time τ ∈ (tn , sm ) such that S(τ )U0 ∈ / O1 ∪ O2 ; therefore, we can take a sequence of times τk → ∞ for which S(τk )U0 ∈ / O1 ∪ O2 holds. Furthermore, this implies that there exists a point U ∈ ω(U0 ) such that U∈ / O1 ∪ O2 , which is a contradiction. Assume that (S(t), X, X) possesses an global attractor A. Then, for any U0 ∈ X, ω(U0 ) = ∅ and ω(U0 ) ⊂ A. In fact, since {U0 } is a bounded set, the trajectory S(t)U0 is attracted by A; therefore, for each integer n ≥ 1, there exists a time tn ≥ n such that d(S(tn )U0 , A) ≤ n1 and that d(S(tn )U0 , Vn ) ≤ n2 with some Vn ∈ A. Since the sequence Vn contains a subsequence which converges to a limit V ∈ A, it follows that S(tn )U0 → V and hence V ∈ ω(U0 ). Moreover, it is easy to see that ω(U0 ) ⊂ A. The following theorem is very convenient for constructing global attractors. Theorem 6.2 Let (S(t), X, X) be a dynamical system. Assume that there exists a compact absorbing set B.

(6.10)

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257

Then, (S(t), X, X) has a global attractor A. In addition, if B is a connected set, then it is the same for A. Proof Since B itself is absorbed by B, there exists a time tB ≥ 0 such that S(t)B ⊂ B for every t ≥ tB . We then consider the set ˜ = X

S(t)B ⊂ B (closure in X).

tB ≤t 0. There exist ˜ such that S(tn )Un → U . a time sequence tn → ∞ and a sequence Un of points of X We write S(tn )Un = S(t)S(tn − t)Un for tn ≥ t; then, there exists a subsequence ˜ i.e., U = S(t)V ; by the of S(tn − t)Un which is convergent to some point V ∈ X, verified fact, V must be in A; therefore, U ∈ S(t)A and hence A ⊂ S(t)A. Thus, we verify that S(t)A = A for every t > 0.

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˜ A) = 0. Indeed, suppose the contrary; Let us verify now that limt→∞ h(S(t)X, then, there would exist some α > 0 such that, for any T > 0, we can find some t ≥ T ˜ A) > α. This then means that there exist a time sequence tn → ∞ such that h(S(t)X, ˜ such that d(S(tn )Un , A) ≥ α. By the compactness of X, ˜ we and a sequence Un ∈ X ˜ can assume that S(tn )Un is convergent to some point V ∈ X. Clearly, d(V , A) ≥ α. Meanwhile, as noticed above, V must be a point of A; hence, d(V , A) = 0, a contradiction. Let us finally show that A is connected under the condition that B is. Suppose the contrary. Then, there would exist two open sets Oi (i = 1, 2) of X such that A ⊂ O1 ∪ O2 , O1 ∩ O2 = ∅, and A ∩ Oi = ∅ for i = 1, 2. Since A is compact, there exists an ε-neighborhood of A such that Wε (A) ⊂ O1 ∪ O2 . Applying (6.9) to B, we have that S(t)B ⊂ Wε (A) ⊂ O1 ∪ O2 for sufficiently large t. Since A ⊂ S(t)B, S(t)B ∩ Oi = ∅ for i = 1, 2. Therefore, for such t, S(t)B is neither connected. But this contradicts the fact that any image of a connected set by a continuous mapping is connected. In the proof of this theorem, we have already obtained the following result. Corollary 6.1 Let (S(t), X, X) possess a compact absorbing set B. Then, there ˜ ⊂ B. The asymptotic behavior of exists a compact, absorbing, and invariant set X the original dynamical system (S(t), X, X) is therefore reduced to that of a smaller ˜ X) having a compact phase space. one (S(t), X, Remark 6.1 In the proof of Theorem 6.2, it is possible to construct the global at˜ In fact, A can be set directly by tractor A directly without using the set X. A = ω(B) =

S(s)B

(closure in X).

0≤t 0, let t = nt ∗ + r with

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integer n ≥ 0 and 0 ≤ r < t ∗ . Then, S(t)U0 − U = S(r)S n U0 − U ≤ sup0≤r≤t ∗ S(r)S n U0 − S(r)U . As t → ∞, n also tends to ∞; therefore, (6.4) yields limt→∞ S(t)U0 − U = 0, which means that U0 ∈ M− (U ). Hence, W− (U ) ⊂ M− (U ). The coincidence of unstable manifolds is also verified in a similar way. By definition, it is obvious that M+ (U ) ⊂ W+ (U ). Let next U0 ∈ W+ (U ). From a sequence {U−n }n=1,2,3,... such that SU−n = U−n+1 , we define the function U : (−∞, 0] → X as U (−t) = S((n + 1)t ∗ − t)U−(n+1) for t ∈ (nt ∗ , (n + 1)t ∗ ], where n = 0, 1, 2, . . . , and as U (0) = U0 . Then, it is seen that the relation S(t)U (−τ ) = U (t − τ ) holds for any 0 ≤ t ≤ τ . Indeed, if τ = 0, then t = 0; so, the relation is clear. Let nt ∗ < τ ≤ (n + 1)t ∗ with some integer n ≥ 0. If τ − t = mt ∗ with some integer m ≥ 0, then S(t)U (−τ ) = S(t)S((n + 1)t ∗ − τ )U−(n+1) = S((n + 1 − m)t ∗ )U−(n+1) = U−m = U (t − τ ). Otherwise, mt ∗ < τ − t < (m + 1)t ∗ with some integer m ≥ 0 (it is clear that n ≥ m); then, S(t)U (−τ ) = S(t + (n + 1)t ∗ − τ )U−(n+1) = S((m + 1)t ∗ + t − τ )S n−m U−(n+1) = S((m + 1)t ∗ + t − τ )U−(m+1) = U (t − τ ). Therefore, in both cases, the function U (−t) satisfies the relation S(t)U (−τ ) = U (t − τ ). We now verify the convergence limt→∞ U (−t) = U . Let nt ∗ < t ≤ (n + 1)t ∗ . Then, U (−t) − U = S((n + 1)t ∗ − t)U−(n+1) − S((n + 1)t ∗ − t)U ≤ sup0≤r≤t ∗ S(r)U−(n+1) − S(r)U . Since n → ∞ as t → ∞, it is observed by (6.4) that limt→∞ U (−t) − U = 0. Therefore, U0 ∈ M+ (U ). Hence, W+ (U ) ⊂ M+ (U ). Let O be any neighborhood of U in X. Localizing the stable and unstable manifolds W∓ (U ) in this neighborhood, we will consider the sets W− (U ; O) = U0 ∈ O; S n U0 ∈ O for n ≥ 1 and lim S n U0 = U (6.13) n→∞

and

W+ (U ; O) = U0 ∈ O; ∃{U−n }n=1,2,3,... ⊂ O, SU−n = U−n+1 for n ≥ 1 and lim U−n = U , n→∞ (6.14) respectively. The set W− (U ; O) (resp. W+ (U ; O)) is called a local stable manifold (resp. a local unstable manifold) of U in a neighborhood O. By definition, it is clear that S(W− (U , O)) ⊂ W− (U , O))

and S(W+ (U ; O)) ⊃ W+ (U , O))

and that W− (U ; O) ⊂ W− (U )

and W+ (U ; O) ⊂ W+ (U ).

Moreover, it is verified without difficulty that W− (U ) =

∞ n=0

S −n (W− (U ; O)) and W+ (U ) =

∞ n=0

S n (W+ (U ; O)). (6.15)

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These facts, together with Theorem 6.3, then yield the following results. Theorem 6.4 We have M− (U ) =

∞

S −n (W− (U ; O)) and M+ (U ) =

n=0

∞

S n (W+ (U ; O)).

n=0

The global attractor, if it exists, always contains the unstable manifold. Theorem 6.5 Assume that (S(t), X, X) has a global attractor A. Then, M+ (U ) ⊂ A. Proof Let U0 ∈ M+ (U ). Then, there exists a negative trajectory U (−t) in X ending at U0 . The set B = {U (−t); 0 ≤ t < ∞} is a bounded set. Since A attracts B, we have limt→∞ h(S(t)B, A) = 0. In view of S(t)U (−t) = U (0) = U0 , we have d(U0 , A) = d(S(t)U (−t), A) ≤ h(S(t)B, A) → 0 as t → ∞. Hence, d(U0 , A) = 0, i.e., U0 ∈ A.

1.4 Exponential Attraction An equilibrium U of (S(t), X, X) is said to be exponentially stable if there exists some neighborhood W of U which U attracts exponentially in the sense that h(S(t)W, U ) ≤ C0 e−kt ,

0 ≤ t < ∞,

(6.16)

with some constant C0 > 0 and some exponent k > 0. In a neighborhood of an exponentially stable equilibrium, one can find an invariant set to define a small dynamical system. Theorem 6.6 Let U be an exponentially stable equilibrium of (S(t), X, X). Then, there exists some neighborhood W of U such that W is an invariant set (therefore, (S(t), W, X) defines a new dynamical system) and W is attracted exponentially by U in the sense (6.16). Proof There is some open ball W = B(U ; r), r > 0, for which (6.16) holds. Then r −kT take a time T > 0 such that C0 e = 2 . Then, for every t ≥ T , S(t)W ⊂ W . So, if we set W = 0≤t 0. (3) There exists some exponent k > 0 such that, for any bounded set B ⊂ X, it holds that h(S(t)B, M) ≤ CB e−kt

for 0 ≤ t < ∞

(6.17)

with a suitable constant CB > 0 depending on B. We must notice that, even if it exists, the exponential attractor is not determined uniquely in general. Usually, they exist as a family. To see this, assume that a dynamical system (S(t), X, X) possesses an exponential attractor M and that, for each t > 0, S(t) satisfies the Lipschitz condition S(t)U − S(t)V ≤ Lt U − V ,

U, V ∈ X.

(6.18)

For 0 < T < ∞, set MT = S(T )M. Then, MT is seen to be an exponential attractor of (S(t), X, X). In fact, since M is a compact set, its image MT by the continuous operator S(T ) is also a compact set of X. Since MT ⊂ M, MT has also a finite fractal dimension with dF (MT ) ≤ dF (M). By the semigroup property, MT is also an invariant set. Let us finally verify (6.17) for MT . For t ≥ T , it follows from (6.18) that d(S(t)U, S(T )V ) = d(S(T )S(t − T )U, S(T )V ) ≤ LT d(S(t − T )U, V ),

U ∈ X, V ∈ M.

Taking the infimum in the right-hand side with respect to V ∈ M, we have d(S(t)U, MT ) = d(S(t)U, S(T )M) ≤ LT d(S(t − T )U, M),

U ∈ X.

Let B be any bounded subset of X. Taking now the supremum in the left-hand side with respect to U ∈ B, we have h(S(t)B, MT ) ≤ LT h(S(t − T )B, M). Therefore, (6.17) implies that h(S(t)B, MT ) ≤ LT CB e−k(t−T ) = CB LT ekT e−kt ,

T ≤ t < ∞.

Meanwhile, since MT ⊂ M, h(S(t)B, MT ) ≤ h(S(t)B, M) + δ(M) ≤ CB + δ(M),

0 ≤ t ≤ T,

where δ(M) is the diameter of M. Therefore, MT also attracts B at an exponential rate. Hence, MT becomes a smaller exponential attractor than M.

2 Representation Theorem of Stable and Unstable Manifolds

263

Theorem 6.7 If (S(t), X, X) possesses an exponential attractor M and if S(t) satisfies (6.18), then in any ε neighborhood Wε (A) of the global attractor A, there exists another exponential attractor Mε , i.e., A ⊂ Mε ⊂ Wε (A). Proof Since M is a bounded set, limT →∞ h(S(T )M, A) = 0. So, for any ε > 0, there is a time T > 0 such that h(S(T )M, A) < ε, namely, MT = S(T )M ⊂ Wε (A). In addition, A = S(T )A ⊂ S(T )M = MT .

2 Representation Theorem of Stable and Unstable Manifolds Throughout this section, we handle a dynamical system whose phase space is a whole space X, namely, a dynamical system of the form (S(t), X, X).

2.1 Discrete Case Let (S n , X, X) be a discrete dynamical system in a Banach space X, where S is a continuous nonlinear operator acting on X. We assume that S has a fixed point U ∈ X, so U is an equilibrium of (S n , X, X). We assume also that S is Fréchet ˜ of U and the derivative satisfies the Hölder differentiable in an open neighborhood O condition S U − S V ≤ D U − V α ,

˜ U, V ∈ O,

(6.19)

with some exponent 0 < α ≤ 1 and constant D > 0. When the spectrum σ (S U ) of S U does not intersect with the unit circle, i.e., σ (S U ) ∩ {λ ∈ C; |λ| = 1} = ∅,

(6.20)

U is called a hyperbolic equilibrium. Let U be hyperbolic. According to Theorem 1.10, the space X and the operator S U are decomposed in a unique way into a direct sum X = Xi ⊕ Xe of two closed subspaces and into a direct sum S U = Si ⊕ Se in such a way that (1) Si and Se are bounded operators of Xi and Xe , respectively. (2) The spectra of Si and Se are given by σ (Si ) = σ (S U ) ∩ {λ ∈ C; |λ| < 1} and σ (Se ) = σ (S U ) ∩ {λ ∈ C; |λ| > 1}, respectively. The spectral radius of Si then satisfies r(Si ) = 1 − 2δi with some small δi > 0, and Se is invertible on Xe with r(Se −1 ) = 1 − 2δe with some small δe > 0. Using these numbers, we introduce the new norm on X by n Ui 0 = sup Si Ui /(1 − δi )n , Ui ∈ Xi , (6.21) n≥0

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6 Dynamical Systems

−n Ue 0 = sup Se Ue /(1 − δe )n ,

Ue ∈ X e ,

(6.22)

n≥0

Ui + Ue 0 = max{ Ui 0 , Ue 0 },

U = Ui + Ue ∈ X.

(6.23)

It is immediate to verify that · 0 defines a norm on X which is equivalent to the original one · . Furthermore, as seen in (1.14), it is true that S Ui ≤ (1 − δi ) Ui 0 , Ui ∈ Xi , i 0 −1 S Ue ≤ (1 − δe ) Ue 0 , Ue ∈ Xe . e 0 Under such a situation, we consider the local stable and unstable manifolds ˜ W∓ (U , O) defined by (6.13) and (6.14), respectively, in a neighborhood of O ⊂ O. When O is sufficiently small, we can establish the following representation theorem. Theorem 6.8 Let U be an equilibrium of (S n , X, X). Let S be Fréchet differentiable ˜ of U with derivative satisfying (6.19). Assume that U is in an open neighborhood O hyperbolic (see (6.20)) and let X be given by the continuous direct sum of two closed subspaces Xi and Xe with the properties described above. Then, for a sufficiently ˜ of U , there exists a one-to-one correspondence small open neighborhood O ⊂ O between the points of W− (U ; O) and the points of Oi ⊂ Xi such that W− (U ; O) U0 = U + Ui + Ge (Ui ), Xi ⊃ Oi Ui → Ge (Ui ) ∈ Xe , Ge ∈ C1+α

Ge (0) = 0, Ge (0) = 0,

when 0 < α < 1 and

Ge ∈ C1,1

when α = 1.

Similarly, there exists a one-to-one correspondence between the points of W+ (U ; O) and the points of Oe ⊂ Xe such that W+ (U ; O) U0 = U + Gi (Ue ) + Ue , Xe ⊃ Oe Ue → Gi (Ue ) ∈ Xi , Gi ∈ C1+α

Gi (0) = 0, Gi (0) = 0,

when 0 < α < 1 and

Gi ∈ C1,1

when α = 1.

Here, Oi and Oe are some suitable open neighborhoods of the origin of Xi and Xe , respectively. Proof Let us first prove the representation result for W+ (U ; O). For simplicity, we assume that U = 0, so that S0 = 0. Pi and Pe denote the projections from X onto Xi and Xe , respectively. By (6.23), we have Pi 0 ≤ 1 and Pe 0 ≤ 1. Let us consider the Banach space X = U = {Un }n=0,1,2,... ⊂ X; lim Un = 0 n→∞

2 Representation Theorem of Stable and Unstable Manifolds

265

with norm U X = sup Un 0 . n

The zero vector of X is denoted by 0. We define the linear operator L : X → X by (LU )0 = Si Pi U1 + 0e , (LU )n = Si Pi Un+1 + Se

−1

Pe Un−1 ,

n ≥ 1.

In addition, the nonlinear operator R : X → X is defined by (RU )0 = Pi (S − S 0)U1 + 0e , (RU )n = Pi (S − S 0)Un+1 − Se

−1

Pe (S − S 0)Un ,

n ≥ 1.

Obviously, R is an operator of class C1+α (resp. C1,1 ) when 0 < α < 1 (resp. α = 1) with R0 = 0. We finally put Φ = L + R. Contractibility of Φ. Let O denote any neighborhood of 0 in X. From O we can always define a neighborhood O of 0 = (0, 0, 0, . . .) in X in such a way that O = {U = {Un }n ∈ X; Un ∈ O}. Then, O is said to be a corresponding neighborhood of O. Let us show that, if O is sufficiently small, then Φ is a contraction mapping from O into X. By (6.21) and (6.22), L satisfies LU X ≤ max{1 − δi , 1 − δe } U X ,

U ∈ X.

In the meantime, R can satisfy RU − RV X ≤ C( U X + V X )α U − V X ,

U , V ∈ O ,

in a suitable neighborhood O , C > 0 being some constant. Indeed, this is readily verified by the following estimate: 1 (S − S 0)U − (S − S 0)V = {S (θ U + (1 − θ )V ) − S 0}(U − V ) dθ 0

≤ C sup θ U + (1 − θ )V α U − V ,

U, V ∈ O ;

0≤θ≤1

here O is a sufficiently small neighborhood of 0 in which any segment connecting ˜ appearing in (6.19). Theretwo points of O is contained in the neighborhood of O fore, we conclude that, if O is sufficiently small, then R satisfies RU − RV X ≤

1 min{δi , δe } U − V X , 2

in the corresponding neighborhood O.

U , V ∈ O,

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6 Dynamical Systems

In this way, Φ is shown to be a contraction on such a neighborhood. One-to-one correspondence. Let O be a sufficiently small open neighborhood of 0, and let O be the corresponding neighborhood on which Φ is a contraction. We can then apply the inverse mapping theorem, Theorem 1.14, to I − Φ, where I is the identity operator of X, to conclude that I − Φ is a homeomorphism from O onto some open neighborhood O of 0, both I − Φ and (I − Φ)−1 being of class C1+α (resp. C1,1 ) when 0 < α < 1 (resp. α = 1). Let U0 ∈ W+ (0; O), and let {Un }n=0,1,2,... be a sequence in O such that SUn = Un−1 for n ≥ 1 and limn→∞ Un = 0. Then, U = {Un }n is an element of O. So, let (I − Φ)U = U = {Un }n . Then, by definition, Pi (U0 − SU1 ) + Pe U0 = U0 Pi (Un − SUn+1 ) − Se

−1

for n = 0,

(6.24)

Pe (Un−1 − SUn ) = Un

for n ≥ 1.

(6.25)

Since SUn = Un−1 , it follows that U0 = Pe U0 ∈ Xe

and Un = 0 for n ≥ 1.

Conversely, let Ue ∈ Xe and U = (Ue , 0, 0, . . .) ∈ O . Then, there exists an element U = {Un }n ∈ O such that (I − Φ)U = U . As relations (6.24) (U0 = Ue ) and (6.25) (Un = 0) are still valid, it follows that Pi (U0 − SU1 ) = 0 and Pe U0 = Ue , Pi (Un − SUn+1 ) = 0

and Pe (Un−1 − SUn ) = 0 for n ≥ 1.

This indeed shows that U0 ∈ W+ (0; O) and Pe U0 = Ue . Set Oe = {Ue ∈ Xe ; (Ue , 0, 0, . . .) ∈ O }. Obviously, Oe is an open neighborhood of the origin 0e of Xe . As shown above, there is one-to-one correspondence between the points in W+ (0; O) and the points in Oe . Mapping Gi . We observe more precisely that the point Ue ∈ Oe corresponds to the point U0 = Ue + Gi (Ue ) ∈ W+ (0; O) in such a way that Gi (Ue ) = Pi Π0 (I − Φ)−1 (Ue , 0, 0, . . .),

Ue ∈ O e ,

where Π0 is the projection from X to X such that Π0 ({Un }n ) = U0 . Clearly, Gi is a C1+α (resp. C1,1 ) mapping when 0 < α < 1 (resp. α = 1) from Oe into Xi with Gi (0) = 0. Since (I − Φ) 0 = I − L, we have {(I − Φ)−1 } 0 = (I − L)−1 . Therefore, (Gi 0)Ue = Pi Π0 (I − L)−1 (Ue , 0, 0, . . .),

Ue ∈ X e .

2 Representation Theorem of Stable and Unstable Manifolds

267

We easily verify that (I − L)−1 (Ue , 0, 0, . . .) = {Un }n implies Pi Un = Si Pi Un+1 for n ≥ 0, and hence Pi Un = 0 for every n. This then shows that Gi 0 = 0. The representation result for W− (0; O) is also deduced in a quite analogous way. We shall use a contraction mapping Ψ instead of Φ which is given by Ψ = M + Q. Here, M : X → X is the linear operator given by (MU )0 = 0i + Se

−1

Pe U1 ,

(MU )n = Si Pi Un−1 + Se

−1

Pe Un+1 ,

n ≥ 1,

and Q : X → X is the nonlinear operator given by (QU )0 = 0i − Se

−1

Pe (S − S 0)U0 ,

(QU )n = Pi (S − S 0)Un−1 − Se

−1

Pe (S − S 0)Un ,

n ≥ 1.

The equivalent condition to (I − Ψ )U = U is written as Pi U0 + Se

−1

Pe (SU0 − U1 ) = U0

−Pi (SUn−1 − Un ) + Se

−1

The rest of proof is left to the reader.

for n = 0,

Pe (SUn − Un+1 ) = Un

for n ≥ 1.

Remark 6.2 Since X is the continuous direct sum of Xi and Xe , we can identify X as the product space Xi × Xe by the natural isomorphism U = Ui + Ue → (Ui , Ue ). If we use this identification, then the points Ui + Ge (Ui ) correspond to (Ui , Ge (Ui )). The theorem obtained above then shows that the local stable manifold W− (U ; O) can be represented as the sum of U and the graph of the mapping Ge : Oi → Xe . Therefore, W− (U ; O) is tangent to U + Xi × {0} at the equilibrium U . It is the same for the local unstable manifold W+ (U ; O). That is, W+ (U ; O) ⊂ Xi × Xe can be represented as the sum of U and the graph of the mapping Gi : Oe → Xi . Therefore, W+ (U ; O) is tangent to U + {0} × Xe at the U . ˜ This theorem shows that, for sufficiently small open neighborhoods O ⊂ O, W− (U ; O) and W+ (U ; O) are C1+α (resp. C1,1 ) manifolds, when 0 < α < 1 (resp. α = 1), with dimensions dim Xi and dim Xe , respectively. We also remark that the whole stable and unstable manifolds W− (U ) and W+ (U ) are given by the images of these localized manifolds by the operators S −n and S n , respectively, due to (6.15).

2.2 Continuous Case We shall now consider a continuous dynamical system (S(t), X, X), where S(t) is a continuous semigroup acting on the whole space X. We assume that, for some

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6 Dynamical Systems

fixed 0 < t ∗ < ∞, S = S(t ∗ ) is Fréchet differentiable in an open neighborhood ˜ of U and its derivative S U satisfies the same condition as in the discrete case O (6.19). In addition, we assume that U is a hyperbolic equilibrium of the discrete dynamical system (S n , X, X) (S = S(t ∗ )) in the sense described above. Let Xi = Xi (U ) and Xe = Xe (U ) be invariant subspaces of S U such that X = Xi + Xe and the parts (S U )|Xi and (S U )|Xe of S U have their spectra in {λ ∈ C; |λ| < 1} and in {λ ∈ C; |λ| > 1}, respectively. The following result is then a direct consequence of Theorem 6.8. Theorem 6.9 Let U be an equilibrium of (S(t), X, X). Let, for some fixed ˜ of U with 0 < t ∗ < ∞, S(t ∗ ) be Fréchet differentiable in an open neighborhood O derivative satisfying (6.19), and let U be hyperbolic with invariants subspaces Xi and Xe of the derivative [S(t ∗ )] U . Then, for sufficiently small open neighborhoods ˜ the local stable and unstable manifolds in O for (S(t ∗ )n , X, X) can be O ⊂ O, represented in such a way that W− (U ; O) U0 = U + Ui + Ge (Ui ), Xi ⊃ Oi Ui → Ge (Ui ) ∈ Xe , Ge ∈ C1+α

Ge (0) = 0, Ge (0) = 0,

when 0 < α < 1 and Ge ∈ C1,1

when α = 1

and W+ (U ; O) U0 = U + Gi (Ue ) + Ue , Xe ⊃ Oe Ue → Gi (Ue ) ∈ Xi , Gi ∈ C1+α

Gi (0) = 0, Gi (0) = 0,

when 0 < α < 1 and Gi ∈ C1,1

when α = 1,

respectively. Here, Oi and Oe are some suitable open neighborhoods of the origin of Xi and Xe , respectively. ˜ Theorem 6.9 shows that, for sufficiently small open neighborhoods O ⊂ O, 1+α 1,1 (resp. C ) manifolds when 0 < α < 1 (resp. W− (U ; O) and W+ (U ; O) are C α = 1), with dimensions dim Xi and dim Xe , respectively. In addition, as shown by Theorem 6.4, the whole stable and unstable manifolds M− (U ) and M+ (U ) are given by the images of these localized manifolds by the operators S(t ∗ )−n and S(t ∗ )n .

3 Exponentially Stable Equilibria Consider a dynamical system (S(t), X, X) in a Banach space X. Let U ∈ X be an equilibrium. We assume that S(t) is a contraction at the equilibrium in the following sense. There exist a time t ∗ > 0 and a constant 0 < δ < 1 such that S(t ∗ )U − U = S(t ∗ )U − S(t ∗ )U ≤ δ U − U , for some R > 0.

U ∈ B(U ; R) ∩ X, (6.26)

4 Exponential Attractors

269

Theorem 6.10 For an equilibrium U , let (6.26) be satisfied with some t ∗ > 0 and 0 < δ < 1. Then, there exists a radius r > 0 such that 1

S(t)U − U ≤ Rδ −1 e−( t ∗ log δ

−1 )t

,

U ∈ B(U ; r) ∩ X.

Proof We first notice that there exists an r > 0 such that {S(τ )U ; 0 ≤ τ ≤ t ∗ , U − U ≤ r} ⊂ B(U ; R). Indeed, suppose the contrary. Then, for any integer n ≥ 1, there exist some τn , 0 ≤ τn ≤ t ∗ , and some Un ∈ X such that Un − U ≤ n1 and S(τn )Un − U > R. We can then assume that τn → τ and Un → U as n → ∞. From (6.1) it follows that S(τ )U − U ≥ R, which clearly contradicts S(τ )U = U . Let next U ∈ B(U ; r) ∩ X. For any t > 0, there exists an integer n ≥ 0 such that t = nt ∗ + τ with some 0 ≤ τ < t ∗ . Then, S(τ )U − U ≤ R. Furthermore, using (6.26) repeatedly, we verify that S(t ∗ )n S(τ )U − U ≤ δ n R. Since S(t)U = S(t ∗ )n S(τ )U , we conclude that 1

S(t)U − U ≤ Rδ − t ∗ δ t ∗ ≤ Rδ −1 e−( t ∗ log δ τ

t

−1 )t

.

4 Exponential Attractors 4.1 Contraction Semigroups Consider a dynamical system (S(t), X, X) in a Banach space X, where the phase X is a closed bounded subset of X. We assume for some t ∗ > 0 that S(t ∗ ) is a contraction operator from X into itself, i.e., S(t ∗ )U − S(t ∗ )V ≤ δ U − V ,

U, V ∈ X,

(6.27)

with some exponent 0 < δ < 1. Let us consider under (6.27) a discrete dynamical system (S n , X, X), where S = S(t ∗ ). By virtue of the fixed point theorem for contraction mappings, Theorem 1.11, there exists a unique equilibrium U of (S n , X, X). Furthermore, for any integer n ≥ 0, S n U − U = S n U − S n U ≤ δ n U − U ,

U ∈ X.

We then verify that U is an equilibrium of (S(t), X, X), too. Indeed, for any t > 0, SS(t)U = S(t ∗ + t)U = S(t)SU = S(t)U ; that is, S(t)U is also a fixed point of S; by the uniqueness of fixed point, S(t)U = U . Furthermore, let t = nt ∗ + τ with integer n ≥ 0 and 0 ≤ τ < t ∗ . Then, S(t)U − U = S n S(τ )U − S n U ≤ Rδ

t−τ t∗

1

= Rδ −1 e−( t ∗ log δ

where R = δ(X) = supU,V ∈X U − V is the diameter of X. We have thus obtained the following result.

−1 )t

,

U ∈ X,

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6 Dynamical Systems

Theorem 6.11 Let X be a bounded closed subset of X, and let (6.27) be satisfied. Then, (S(t), X, X) has a unique equilibrium U which attracts every trajectory at the 1 −1 exponential rate d(S(t)X, U ) ≤ Rδ −1 e−( t ∗ log δ )t , where R is the diameter of X.

4.2 Compact Perturbation of Contraction Operator, I Let (S(t), X, X) be a dynamical system in a Banach space X, where X is a closed bounded subset of X. In the subsequent two subsections, we shall consider compact perturbations of condition (6.27) by two ways. In the first way, the semigroup S(t ∗ ) is divided into two parts in which the first part is a contraction like (6.27), and the second one is a Lipschitz continuous compact operator (see (6.29)). In the second way, we divide the norm S(t ∗ )U − S(t ∗ )V X into two parts in which the first part is again a contraction and the second one is estimated by the norm of a Lipschitz continuous compact operator (see (6.38)). In this subsection, we assume for some fixed t ∗ > 0 that S(t ∗ ) is divided into two parts as follows. There are two operators S0 : X × X → X and K : X × X → Z, where Z is a second Banach space compactly embedded in X and such that for each pair (U, V ) ∈ X2 , it holds that S(t ∗ )U − S(t ∗ )V = S0 (U, V ) + K(U, V ). In addition, S0 and K satisfy S0 (U, V ) X ≤ δ U − V X ,

U, V ∈ X,

(6.28)

K(U, V ) Z ≤ L U − V X ,

U, V ∈ X,

(6.29)

with some constants 0 ≤ δ < 12 and L > 0, respectively. When conditions (6.28)– (6.29) are satisfied, S(t ∗ ) is said to be a compact perturbation of contraction operator of type I. Putting S = S(t ∗ ), let us consider the discrete dynamical system (S n , X, X). We can prove the following theorem. Theorem 6.12 Let (6.28) and (6.29) be satisfied with 0 ≤ δ < 12 . Let θ be any ∗ exponent such that 0 < θ < 1−2δ 2L . Then, there exists an exponential attractor Mθ n for (S , X, X) with the following properties: (1) M∗θ contains a global attractor A∗ of (S n , X, X). (2) M∗θ is a compact set of X with finite fractal dimension dF (M∗θ ) ≤ log Nθ / log

1 . aθ

(6.30)

(3) M∗θ is invariant under S, i.e., SM∗θ ⊂ M∗θ . (4) h(S n X, M∗θ ) ≤ Raθn for all integers n ≥ 0. Here, R = δ(X) is the diameter of X, 0 < aθ < 1 is the exponent given by aθ = 2(δ + θ L), and Nθ is the minimal number of balls of X with radius θ which cover Z the closed unit ball B (0; 1) of Z.

4 Exponential Attractors

271

Proof The proof is divided into three steps. Step 1. For n = 0, 1, 2, . . . , let us construct inductively a finite covering of S n X such that n

SnX ⊂

Nθ

B(Wn,i ; Ra n ),

where a = aθ ,

(6.31)

i=1

with centers Wn,i ∈ S n X, 1 ≤ i ≤ Nθn . For n = 0, it is clear that S 0 X = X ⊂ B(W0 ; R) with an arbitrarily fixed W0 ∈ X. So, (6.31) is trivial. Assume that the covering (6.31) is defined for n − 1. Then, we have Nθn−1

S X = S(S n

n−1

X) ⊂

S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X).

i=1

So, it suffices to cover each set S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) by Nθ closed balls with centers in S n X and radius Ra n . We consider the image K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ). From (6.29) we see that Z

K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ) ⊂ B (0; LRa n−1 ). Then, by the compactness of closed bounded balls of Z in X, the last ball can be covered by Nθ closed balls B(V˜n−1,j ; θ LRa n−1 ), 1 ≤ j ≤ Nθ , of X with centers V˜n−1,j ∈ X and radius θ LRa n−1 . Therefore, we obtain that K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ) ⊂

Nθ

B(V˜n−1,j ; θ LRa n−1 ).

j =1

Furthermore, K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ) + SWn−1,i ⊂

Nθ

B(V˜n−1,j + SWn−1,i ; θ LRa n−1 ).

(6.32)

j =1

Let U ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, and let SU − SWn−1,i = S0 (U, Wn−1,i ) + K(U, Wn−1,i ). By (6.32), there is some j such that K(U, Wn−1,i ) + SWn−1,i ∈ B(V˜n−1,j + SWn−1,i ; θ LRa n−1 ).

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6 Dynamical Systems

Since SU = S0 (U, Wn−1,i )+K(U, Wn−1,i )+SWn−1,i and since S0 (U, Wn−1,i ) X ≤ δ U − Wn−1,i X ≤ δRa n−1 due to (6.28), it follows that SU ∈ B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ). Hence, S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂

Nθ

B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ).

j =1

We are here allowed to assume that S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ∩ B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ) = ∅ for every j , since, if not for some j , we can exclude such balls from the covering. So, we can choose for each j , a point Vn−1,i,j such that Vn−1,i,j ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, SVn−1,i,j ∈ B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ).

(6.33)

Substituting for each j , the central point SVn−1,i for V˜n−1,j +SWn−1,i and doubling the radius, we obtain the covering S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂

Nθ

B(SVn−1,i,j ; aθ Ra n−1 ).

(6.34)

j =1

From (6.33) we notice that SVn−1,i,j ∈ S n X. It is of course possible to construct the same covering as (6.34) for all other balls B(Wn−1,i ; Ra n−1 ), 1 ≤ i ≤ Nθn−1 . The desired covering (6.31) for n is thus obtained just locating central points as Wn,i ; 1 ≤ i ≤ Nθn = SVn−1,i,j ; 1 ≤ i ≤ Nθn−1 , 1 ≤ j ≤ Nθ ⊂ S n X.

Step 2. Let

P = Wn,i ; 0 ≤ n < ∞, 1 ≤ i ≤ Nθn

be a collection of all central points of the covering (6.31) and set M∗θ in such a way that M∗θ =

∞

S n P.

n=0

Then, as will be shown, M∗θ is an exponential attractor for (S n , X, X). Let us verify in this step that Mθ has a finite fractal dimension.

4 Exponential Attractors

273

We will first cover M∗θ by closed balls with radius Ra n . We know by (6.31) that is covered by Nθn balls with radius Ra n . We then decompose M∗θ as

n−1 ∗ p m S Wm,i ; 0 ≤ m ≤ n − 1, 1 ≤ i ≤ Nθ Mθ =

SnX

p=0

n−1

Sp

Wm,i ; n ≤ m < ∞, 1 ≤ i

≤ Nθm

∞

SpP

.

p=n

p=0

Then,

n−1

∞ S p Wm,i ; n ≤ m < ∞, 1 ≤ i ≤ Nθm S p P ⊂ S n X. p=n

p=0

In the meantime, the number of points of Nθm } is calculated by

n−1

p=0 S

p {W

m,i ;

0 ≤ m ≤ n − 1, 1 ≤ i ≤

n(Nθn − 1) nNθn < . n 1 + Nθ + Nθ2 + · · · + Nθn−1 = Nθ − 1 Nθ − 1 N n+1 +(n−1)N n

Therefore, M∗θ is covered at least by θ Nθ −1 θ balls of radius Ra n . Let next 0 < ε < R be any number. By N (ε) we denote the minimal number of balls with radius ε which cover M∗θ . Let Ra n+1 ≤ ε ≤ Ra n . Then, we have N (ε) ≤ N (Ra n+1 ) ≤

Nθn+2 + nNθn+1 . Nθ − 1

(6.35)

In particular, this means that M∗θ is a precompact set of X. As M∗θ is closed, M∗θ is a compact set of X. Furthermore, it follows from (6.35) and the estimate log Ra1 n ≤ log 1ε that log N (ε) log 1ε

≤

log(Nθn+2 + nNθn ) − log(Nθ − 1) n log a1 − log R1

.

Consequently, we obtain that dF (M∗θ ) = lim sup

log N (ε)

ε→0

log 1ε

≤

log Nθ log a1

.

Step 3. Let us verify that M∗θ enjoys the other properties as an exponential attractor. Since S is a continuous mapping, we have S

∞ n=0

SnP ⊂ S

∞ n=0

SnP ⊂

∞ n=0

S n P.

274

6 Dynamical Systems

Hence, M∗θ is an invariant set of S. If U ∈ S n X, we see from (6.31) that d U, M∗θ ≤ d(U, P) ≤ Ra n . Consequently,

h S n X, M∗θ ≤ Ra n .

(6.36)

∗ attracts the phase space X at an exponential rate. This shows that M θ ∗ n X. Since X ⊃ M∗ and since M∗ is an invariant comWe set A = ∞ n=0 S θ ∞θ n ∗ n pact set, it follows that ∞ n=0 S X ⊃ n=0 S Mθ = ∅. On the other hand, let n n+m U U∈ ∞ n+m = n=0 S X. Then, for any integers n, m ≥ 1, we can write U = S ∗ n m m S (S Un+m ) with Un+m ∈ X; since d(S Un+m , Mθ ) → 0 as m → ∞, we can assume that S m Un+m → V ∈ M∗θ as m → used the compactness of M∗θ ); ∞∞ (we ∞ ∗ n n n therefore, U =S V ∈ S Mθ . Hence, n=0 S X ⊂ n=0 S n M∗θ . So that A∗ = ∞ ∞ n n ∗ ∗ n=0 S X = n=0 S Mθ . In particular, by Theorem 6.2, A is a strictly invari-

ant compact set. Let us finally verify that A∗ attracts the phase space X. Let · X ≤ c · Z . Then, (6.28) and (6.29) give SU − SV X ≤ (δ + cL) U − V X , By (1.2),

U, V ∈ X.

(6.37)

h(S m+n X, A∗ ) ≤ h S m+n X, S m M∗θ + h S m M∗θ , A∗ .

Here, (6.36) and (6.37) yield that h S m+n X, S m M∗θ ≤ (δ + cL)m h S n X, M∗θ ≤ (δ + cL)m Ra n . Therefore,

h(S m+n X, A∗ ) ≤ R(δ + cL)m a n + h S m M∗θ , A∗ .

Letting n → ∞, we have lim supn→∞ h(S n X, A∗ ) ≤ h(S m M∗θ , A∗ ). By Theorem 6.2, we already know that limm→∞ h(S m M∗θ , A∗ ) = 0. Hence, it follows that limn→∞ h(S n X, A∗ ) = 0. Thus, we have shown that A∗ is a global attractor of (S n , X, X).

4.3 Compact Perturbation of Contraction Operator, II Let (S(t), X, X) be a dynamical system in a Banach space X, where X is a closed bounded subset of X. In this subsection, we assume for some fixed t ∗ > 0 that S = S(t ∗ ) is a compact perturbation of a contraction operator in the sense that SU − SV X ≤ δ U − V X + KU − KV X ,

U, V ∈ X.

(6.38)

4 Exponential Attractors

275

Here, δ is a constant such that 0 ≤ δ < 12 , and K is an operator from X into a second Banach Z which is compactly embedded in X and satisfies the Lipschitz condition KU − KV Z ≤ L U − V X ,

U, V ∈ X,

(6.39)

with some L > 0. When conditions (6.38)–(6.39) are satisfied, S(t ∗ ) is said to be a compact perturbation of contraction operator of type II. We are then able to prove a similar result as Theorem 6.12 for the discrete dynamical system (S n , X, X). Theorem 6.13 Let (6.38) and (6.39) be satisfied with 0 ≤ δ < 12 . Let θ be any ∗ exponent such that 0 < θ < 1−2δ 2L . Then, there exists an exponential attractor Mθ n for (S , X, X) having the same properties as those announced in Theorem 6.12. Proof We can follow the similar procedure as in the proof of Theorem 6.12. Let us construct under (6.38)–(6.39) the same covering as before. For n = 0, 1, 2, . . . , there is a covering of S n X such that n

Nθ

S X⊂ n

B(Wn,i ; Ra n ),

where a = aθ ,

(6.40)

i=1

with centers Wn,i ∈ S n X, 1 ≤ i ≤ Nθn . Here, as before, R = δ(X) is the diameter of X, 0 < aθ < 1 is the exponent given by aθ = 2(δ + θ L), and Nθ is the minimal Z number of balls of X with radius θ which cover the closed unit ball B (0; 1) of Z. For n = 0, it is clear that S 0 X = X ⊂ B(W0 ; R) with arbitrarily fixed W0 ∈ X. Assume that the covering (6.40) is defined for n − 1. Then, we have Nθn−1

S X = S(S n

n−1

X) ⊂

S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X).

i=1

So, it suffices to cover each set S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) by Nθ closed balls with centers in S n X and the radius Ra n . From (6.39) we see that Z

K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂ B (KWn−1,i ; LRa n−1 ). Then, by the compactness of closed bounded balls of Z in X, the last ball can be covered by Nθ closed balls of X in such a way that Z

B (KWn−1,i ; LRa n−1 ) ⊂

Nθ j =1

B(V˜n−1,i,j ; θ LRa n−1 )

276

6 Dynamical Systems

with centers V˜n−1,i,j ∈ X, 1 ≤ j ≤ Nθ , and radius θ LRa n−1 . Therefore, K(B(Wn−1,i ; Ra

n−1

)∩S

n−1

X) ⊂

Nθ

B(V˜n−1,i,j ; θ LRa n−1 ).

(6.41)

j =1

We are here allowed to assume that K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ∩ B(V˜n−1,i,j ; θ LRa n−1 ) = ∅ for every j , since, if not for some j , we can exclude these balls from the covering. So, we can choose for each j , a point Vn−1,i,j such that Vn−1,i,j ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, KVn−1,i,j ∈ B(V˜n−1,i,j ; θ LRa n−1 ).

(6.42)

Therefore, from (6.41) it is deduced that K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂

Nθ

B(KVn−1,i,j ; 2θ LRa n−1 ).

j =1

Let now U ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X. Then, there is some j such that KU ∈ B(KVn−1,i,j ; 2θ LRa n−1 ). As a consequence, it follows from (6.38) that SU − SVn−1,i,j X ≤ δ U − Vn−1,i,j X + KU − KVn−1,i,j X ≤ δ U − Vn−1,i,j X + 2θ LRa n−1 . In addition, by (6.42), U − Vn−1,i,j X ≤ U − Wn−1,i X + Wn−1,i,j − Vn−1,i,j X ≤ 2Ra n−1 , so that SU − SVn−1,i,j X ≤ 2δRa n−1 + 2Lθ Ra n−1 = Ra n . Hence, it holds that S(B(Wn−1,i ; Ra

n−1

)∩S

n−1

X) ⊂

Nθ

B(SVn−1,i,j ; Ra n ).

(6.43)

j =1

We observe that SVn−1,i,j ∈ S n X due to (6.42). Covering of the form (6.43) can of course be constructed for all other balls B(Wn−1,i ; Ra n−1 ), 1 ≤ i ≤ Nθn−1 . Therefore, the desired covering (6.40) for n is obtained by locating central points as Wn,i ; 1 ≤ i ≤ Nθn = SVn−1,i,j ; 1 ≤ i ≤ Nθn−1 , 1 ≤ j ≤ Nθ ⊂ S n X. We here recall that in Steps 2 and 3, the proof of Theorem 6.12 was carried out only on the basis of the covering (6.31) and conditions (6.28)–(6.29) were never used. As (6.40) has been established, we shall repeat the same arguments as in Steps 2 and 3 of the proof of the previous theorem to complete our proof.

4 Exponential Attractors

277

4.4 Squeezing Property In this subsection, let X be a Hilbert space. We consider a dynamical system (S(t), X, X) in X, where X is a compact subset of X. For some fixed 0 < t ∗ < ∞, assume that S = S(t ∗ ) satisfies the Lipschitz condition SU − SV ≤ L U − V ,

U, V ∈ X,

(6.44)

and the following condition: either SU − SV ≤ δ U − V

(6.45)

or (1 − P )(SU − SV ) ≤ P (SU − SV )

(6.46)

holds for any pair (U, V ) in X × X with some fixed constant 0 ≤ δ < and some fixed orthogonal projection P from X onto a finite-dimensional subspace Z in X, say dim Z = N < ∞. Especially, the second condition that either (6.45) or (6.46) is valid is called a squeezing property for S. Since 1 4

SU − SV 2 = P (SU − SV ) 2 + (1 − P )(SU − SV ) 2 , it is easy to observe from the squeezing property that √ SU − SV ≤ δ U − V + 2 P (SU − SV ) for any pair (U, V ) ∈ X × X. In addition, (6.44) implies that P (SU − SV ) ≤ L U − V ,

U, V ∈ X.

As the finite-dimensional space Z is compactly embedded in X, we notice that the squeezing property (6.45)–(6.46), together with (6.44), √ implies that S fulfills the compact perturbation condition (6.38)–(6.39) with K = 2P S. As a consequence √ , there exists an exponential attractor M∗ for of Theorem 6.13, for 0 < θ < 1−2δ θ 2 2L the discrete dynamical system (S n , X, X). It is, however, known that we can acquire a more precise dimension estimate for M∗θ than (6.30) without using the covering number Nθ . Theorem 6.14 Let X be a Hilbert space, and let X be a compact phase space. Let (6.44) and (6.45)–(6.46) be satisfied with 0 ≤ δ < 14 . Then, for any 0 < θ < 1 − 2δ, there exists an exponential attractor M∗θ for (S n , X, X) with fractal dimension 3L 1 ∗ +1 log , 1 dF (Mθ ) ≤ N max log θ aθ and the same properties as (1), (3), and (4) announced in Theorem 6.12, where aθ = 2δ + θ .

278

6 Dynamical Systems

The proof of this theorem requires us more refined techniques than those for proving Theorem 6.13. We refer the reader to Eden, Foias, Nicolaenko, and Temam [EFNT94, Theorem 2.1]. We here remark only the following fact. Remark 6.3 The squeezing property implies a sharper covering than (6.31) or (6.40). Let W ∈ X and r > 0. We are concerned with the image S(B(W ; r) ∩ X) of a closed ball B(W ; r) by S. So, let U ∈ B(W ; r) ∩ X. If (6.45) takes place for the pair (U, W ), then SU ∈ B(SW ; δr). On the other hand, if (6.46) takes place, then it follows that SU − SW 2 ≤ 2 Re(SU − SW, P (SU − SW )). Therefore, SU − SW P (SU − SW ) ≤ 2 Re(SU − SW, P (SU − SW )).

(6.47)

In view of this, we consider the cone of X defined by C = U˜ ∈ X − {0};

max

V ∈Z, V Z =1

Re

1 (U˜ , V ) ≥ . U˜ V 2

Then, (6.47) shows that SU − SW ∈ C. Hence, (6.46), together with (6.44), implies that SU ∈ (SW + C) ∩ B(SW ; Lr). We thus observed that (6.44) and (6.45)–(6.46) imply that S(B(W ; r) ∩ X) ⊂ B(SW ; δr) ∪ [(SW + C) ∩ B(SW ; Lr)].

4.5 Continuous Dynamical System Let us return to the continuous dynamical system (S(t), X, X) in a Banach space X, where X is a closed bounded subset of X. We assume for some fixed t ∗ > 0 that S(t ∗ ) satisfies the compact perturbation condition either (6.28)–(6.29) or (6.38)– (6.39). In addition, we assume that G(t, U0 ) = S(t)U0 satisfies a Lipschitz condition on [0, t ∗ ] × X, i.e., G(t, U ) − G(s, V ) ≤ L1 (|t − s| + U − V ),

t, s ∈ [0, t ∗ ], U, V ∈ X. (6.48)

Then, on the basis of the exponential attractors M∗θ for (S(t ∗ )n , X, X), we can readily construct exponential attractors for (S(t), X, X). Theorem 6.15 Let (S(t), X, X) be a dynamical system in a Banach space X with closed bounded phase space X. For some fixed t ∗ > 0, let S(t ∗ ) satisfy either (6.28)– (6.29) or (6.38)–(6.39), and let (6.48) be satisfied. Then, for any 0 < θ < 1−2δ 2L , there exists an exponential attractor Mθ for (S(t), X, X) with the following properties: (1) Mθ contains a global attractor A of (S(t), X, X).

4 Exponential Attractors

279

(2) Mθ is a compact set of X with finite fractal dimension dF (Mθ ) ≤ dF M∗θ + 1. (3) Mθ is an invariant set, i.e., S(t)Mθ ⊂ Mθ for every 0 < t < ∞. −1 1 (4) h(S(t)X, Mθ ) ≤ Raθ−1 e−( t ∗ log aθ )t for every 0 < t < ∞. Here, M∗θ is the exponential attractor of (S(t ∗ )n , X, X) constructed in Theorem 6.12 or Theorem 6.13, R = δ(X) is the diameter of X, and aθ = 2(δ + θ L). Proof Let M∗θ be the exponential attractor of (S(t ∗ )n , X, X) constructed by Theorem 6.12 or 6.13. We consider the image G([0, t ∗ ] × M∗θ ) of a set [0, t ∗ ] × M∗θ by G and denote it by Mθ , i.e., S(τ )M∗θ . Mθ = G [0, t ∗ ] × M∗θ = 0≤τ ≤t ∗

Our goal is then to show that Mθ is the desired exponential attractor. Since [0, t ∗ ]×M∗θ is a compact set of C×X and since G is a continuous mapping on [0, t ∗ ] × M∗θ , its image Mθ is also a compact set of X. In addition, since G is Lipschitz continuous, we obtain by Theorem 1.1 that dF (Mθ ) ≤ dF [0, t ∗ ] × M∗θ = dF M∗θ + 1. Let 0 < t < ∞ and write t = nt ∗ + τ with integer n ≥ 0 and 0 ≤ τ < t ∗ . Then, S(t)M∗θ = S(τ )S(nt ∗ )M∗θ = S(τ )S(t ∗ )n M∗θ ⊂ S(τ )M∗θ . ∗ S(τ )M∗θ = Mθ . Conversely, it is clear that Therefore, 0≤t 0 (depending only on U (s; U0 ) Dβ and hence only on p( U0 Dβ )) and a continuous increasing function p1 (·) such that AU (t) X ≤ (t − s)β−1 p1 ( U (s; U0 ) Dβ ),

s < t ≤ s + τ.

First, applying this with s = 0, we have AU (t) X ≤ t β−1 p1 ( U0 Dβ ),

0 < t ≤ τ.

Secondly, taking s = t − τ , we have AU (t) X ≤ τ β−1 p1 ( U (t − τ ) Dβ ) ≤ τ β−1 p1 (p0 ( U0 Dβ )),

τ < t < ∞.

Combining these estimates, we obtain the desired one.

Under (6.55), (6.53) possesses a unique global solution U (t; U0 ) in the function space: U (·; U0 ) ∈ C((0, ∞); D(A)) ∩ C([0, ∞); Dβ ) ∩ C1 ((0, ∞); X),

U0 ∈ D β ,

and (6.56) and (6.57) hold true. We next set S(t)U0 = U (t; U0 ), 0 ≤ t < ∞. Then, the uniqueness of solutions readily implies that S(t) is a nonlinear semigroup acting on Dβ . Let us establish

5 Dynamical Systems for Semilinear Evolution Equations

283

its continuity on Dβ . For any 0 < R < ∞, let KR = {U ∈ Dβ ; U Dβ < R} be an open ball. By Corollary 4.2 (when β > 0) and Theorem 4.5 (when β = 0), we know that there exists tR > 0 such that S(t)U0 − S(t)V0 Dβ ≤ LR U0 − V0 Dβ , 0 ≤ t ≤ tR ; U0 , V0 ∈ KR . In view of 0≤t 0, it is often very important to construct a dynamical system in the universal space X (instead of Dβ ). To this end, we introduce for any 0 < R < ∞, the phase space {S(t)U0 ; 0 ≤ t < ∞} ⊃ KR . XR = U0 ∈KR

Clearly, S(t) maps XR into itself, namely, XR is invariant under S(t). In addition, since XR ⊂ Kp(R) due to (6.56), XR is a bounded set of Dβ . With the aid of Theorem 4.3, we can argue in a similar way as for the proof of (6.58) to deduce the following proposition. Proposition 6.3 For any 0 < R < ∞, it holds that S(t)U0 − S(t)V0 X ≤ Ln+1 p(R) U0 − V0 X , t ∈ [ntp(R) , (n + 1)tp(R) ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ XR ; with some constant Lp(R) > 0. This ultimately implies that G : [0, ∞) × XR → X is continuous, where XR is equipped with the induced distance from X. Thus, problem (6.53) determines a dynamical system (S(t), XR , X) even in the universal space X, too.

284

6 Dynamical Systems

5.2 Absorbing and Invariant Compact Set We shall now make (in addition to the above) the assumptions that X is a reflexive Banach space; D(A) is compactly embedded in X.

(6.59)

As a consequence, we observe the following fact. Proposition 6.4 Let D = D(A) be a Banach space with the graph norm · D = D A · . Then, any bounded closed ball B (0; R), 0 < R < ∞, of D is a compact subset of X. D

Proof Indeed, it suffices to prove that the ball is closed in X. Let Un ∈ B (0; R) and Un → U in X. Since D is reflexive (note that D and X are isomorphic), D B (0; R) is weakly compact; therefore, we can assume that Un → V weakly in D with V D ≤ R. Since D ⊂ X densely and continuously, we have X ⊂ D ; D hence, also Un → V weakly in X. Consequently, U = V ∈ B (0; R). We assume also a dissipative condition, that is, there exists a number C˜ > 0 for which the following assertion is valid. For any bounded subset B of Dβ , there is a time tB such that ˜ sup sup S(t)U0 Dβ ≤ C.

U0 ∈B t≥tB

(6.60)

A stronger dissipative condition is then deduced by the same argument as for (6.57). In fact, there exists a number C˜ 1 > 0 such that, for any bounded subset B of Dβ , it holds that sup sup S(t)U0 D ≤ C˜ 1

U0 ∈B t≥tB

(6.61)

with a suitable time tB depending on B. Using this constant C˜ 1 , we introduce the closed ball D B = B (0; C˜ 1 ).

Estimate (6.61) obviously means that, for any bounded subset B of Dβ , there exists a time tB such that S(t)B ⊂ B for all t ≥ tB , that is, B is an absorbing set. As noticed above, B is a compact set of X. We furthermore consider the set X=

S(t)B ⊂ B

(closure in X).

tB ≤t 0 such that the images S(t)B are also absorbed by itself for all t ≥ tB . As B is an absorbing set, it is the

5 Dynamical Systems for Semilinear Evolution Equations

285

same for X. In addition, since S(t) is continuous with respect to the X norm (recall Proposition 6.3), it holds that S(t) S(τ )B ⊂ S(t)S(τ )B ⊂ S(τ )B. tB ≤τ 0. As X is a bounded set of D, t S(t)V0 − S(s)V0 = [F (S(τ )V0 ) − AS(τ )V0 ] dτ s

≤ CX (t − s),

0 ≤ s < t ≤ t ∗ , V0 ∈ X.

These two estimates obviously yield (6.48). We have thus deduced, under (2.92), (2.93), either (4.2) or (4.21), (6.55), (6.59), and (6.60), that the dynamical system (S(t), X, X) possesses a family of exponential attractors M. Let us finally verify that each M is an exponential attractor of the dynamical system (S(t), X, Dβ ), too, where X is equipped with the induced distance from Dβ . By the moment inequality (2.117), it is readily seen that hDβ (S(t)X, M) ≤ CX h(S(t)X, M)1−β ≤ CX e−(1−β)kt ,

0 ≤ t < ∞,

where hDβ (·, ·) denotes the Hausdorff pseudo-distance in Dβ . This shows that M attracts S(t)X at an exponential rate even in Dβ . Meanwhile, by Theorem 1.3, the fractal dimension dDβ (M) of M in Dβ is estimated by (1 − β)−1 dX (M). Hence, M keeps its properties as an exponential attractor in (S(t), X, Dβ ).

286

6 Dynamical Systems

5.4 Squeezing Property of S(t) Let X be a Hilbert space. For the operator A in (6.53), we now assume that A is a positive definite self-adjoint operator of X; (6.62) D(A) is compactly embedded in X. For the nonlinear operator F , we assume (4.2) (when β > 0) or (4.21) (when β = 0). We assume also that problem (6.53) determines a dynamical system (S(t), X, X) in which X is a compact set of X and is a bounded set of D(A). Consequently, F satisfies the Lipschitz condition F (U ) − F (V ) ≤ D Aη (U − V ) ,

U, V ∈ X, 0 ≤ η < 1.

(6.63)

On account of (6.62), there exist positive eigenvalues 0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · → ∞ of A, and the corresponding eigenvectors U1 , U2 , U3 , . . . compose an orthonormal basis of X. We consider a finite-dimensional subspace Z = Span{U1 , U2 , U3 , . . . , UN }. Let P : X → Z be the orthogonal projection. We can show that the squeezing property (6.45)–(6.46) is fulfilled on X if N is sufficiently large. Let t ∗ > 0 be a time such that the estimate t η Aη [S(t)U − S(t)V ] + S(t)U − S(t)V ≤ C U − V ,

U, V ∈ X,

holds for any 0 < t ≤ t ∗ (see Theorem 4.3 or 4.5) and set S = S(t ∗ ). In view of (6.45) and (6.46), it suffices to prove that, if (1 − P )[SU − SV ] > P [SU − SV ] for U, V ∈ X, then (6.45) is valid with a suitable constant 0 ≤ δ < 14 . So, let (1 − P )[SU − SV ] > SU − SV be satisfied. Then, SU − SV 2 = P [SU − SV ] 2 + (1 − P )[SU − SV ] 2 < 2 (1 − P )[SU − SV ] 2 . In the meantime, we have (1 − P )SU = e

−t ∗ A

(1 − P )U +

t∗

e−(t

∗ −s)A

(1 − P )F (S(s)U ) ds.

0

It is the same for (1 − P )SV . We then observe that e−t

∗A

(1 − P )(U − V ) ≤ e−t

∗λ N+1

1 U − V ≤ √ U − V 2 34

if N is sufficiently large. Similarly, due to (6.63),

6 Stationary Solutions to Semilinear Equations

t∗

e−(t

0

∗ −s)A

≤D

t∗

287

(1 − P )[F (S(s)U ) − F (S(s)V )] ds

e−(t

∗ −s)λ N+1

F (S(s)U ) − F (S(s)V ) ds

0

t∗

1 s −η ds U − V ≤ √ U − V 2 34 0 t∗ ∗ if N is sufficiently large. We here used the fact that limλ→∞ 0 e−(t −s)λ s −η ds = 0. 1 Hence, it follows that SU − SV 2 ≤ 17 U − V 2 , namely, (6.45) is fulfilled. We have thus shown under (6.62) that S(t) enjoys the squeezing property. ≤D

e−(t

∗ −s)λ N+1

6 Stationary Solutions to Semilinear Equations 6.1 Equilibria of Dynamical System Consider problem (6.53) in a Banach space X. We assume that the linear operator A and the nonlinear operators F satisfy (2.92), (2.93), and either (4.2) (when β > 0) or (4.21) (when β = 0). In addition, we assume that the a priori estimate (6.55) holds for any U0 ∈ Dβ . As shown, a dynamical system (S(t), Dβ , Dβ ) is constructed, where Dβ = D(Aβ ) (and D0 = X when β = 0). Let (6.53) have a stationary solution U (t) ≡ U , i.e., AU = F (U ). Clearly, U is an equilibrium of (S(t), Dβ , Dβ ). Let R > 0 be such that U Dβ < R. In this section, we shall investigate the stability and instability of U by applying Theorems 6.9 and 6.10, respectively, in which the Fréchet differential of S(t) at U played an essential role.

6.2 Exponential Stability of U Let us assume that F : D(Aη ) → X is Fréchet differentiable at U , i.e., ΔF (h) = F (U + h) − F (U ) − F (U )h, F (U ) ∈ L(D(Aη ), X), limh→0 oF (h)/ Aη h = 0. oF (h) = ΔF (h) ,

(6.64)

Here, oF (h) is defined for all h ∈ D(Aη ). In view of (4.2) or (4.21), we observe that oF (h) ≤ C Aη h for all h such that Aη h ≤ 1. Let now h ∈ Dβ denote a small variable satisfying h Dβ < R − U Dβ . For such an h, let Uh (t) = S(t)(U + h). By Theorem 4.3 or 4.5, there exists T > 0 such that t η Aη [Uh (t) − U ] + t β Aβ [Uh (t) − U ] ≤ C h ,

0 < t ≤ T.

(6.65)

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6 Dynamical Systems

We here introduce the linear problem dV h dt + AVh = F (U )Vh ,

0 < t ≤ T,

Vh (0) = h, in X. Since F (U ) ∈ L(D(Aη ), X), we know by Theorem 3.3 that A − F (U ) generates an analytic semigroup on X (cf. Remark 3.7). By Theorem 3.14 the solution of the problem is given by Vh (t) = e−tA h, where A = A − F (U ). Put ΔU (t) = Uh (t) − U − Vh (t). Then, since F (Uh (t)) − F (U ) − F (U )Vh (t) = F (U )ΔU (t) + ΔF (Uh (t) − U ), it is observed that dΔ U dt + AΔU = F (U )ΔU + ΔF (Uh (t) − U ),

0 < t ≤ T,

ΔU (0) = 0. Using the semigroup e−tA , ΔU (t) is written by

t e−(t−s)A ΔF (Uh (s) − U ) ds, ΔU (t) =

0 < t ≤ T.

0

Note that ΔF (Uh − U ) ∈ Fβ,σ ((0, T ]; X) due to (4.15) (when β > 0) and ΔF (Uh − U ) ∈ F1−η,σ ((0, T ]; X) due to (4.27) (when β = 0). Therefore, for any 0 ≤ θ < 1,

t (t − s)−θ oF (Uh (s) − U ) ds. Aθ ΔU (t) ≤ C 0

Furthermore, by (6.65),

t θ A ΔU (t) ≤ C (t − s)−θ s −η oF (Uh (s) − U )/ Aη (Uh (s) − U ) ds h . 0

Since Aη (Uh (t) − U ) → 0 as h → 0, we conclude by the dominate convergence theorem that, as h → 0 (a fortiori as h Dβ → 0), 0 ≤ Aθ ΔU (t) / h Dβ ≤ Aθ ΔU (t) / h → 0.

(6.66)

Thus, under (6.64), the operator S(t) : Dβ → D(Aθ ), where 0 < t ≤ T and 0 ≤ θ < 1, is Fréchet differentiable at U with differential S(t) U = e−tA ,

where A = A − F (U ).

Moreover, since S(2t)U = S(t)S(t)U and S(t)U = U , we verify that S(2t) is also Fréchet differentiable at U . Repeating this argument, S(t) : Dβ → D(Aθ ) is Fréchet differentiable at U for every time 0 < t < ∞.

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289

We can now give a sufficient condition for exponential stability of U . Let us assume that the spectrum of A is contained in the right half-plane, i.e., σ (A) = σ (A − F (U )) ⊂ {λ ∈ C; Re λ > 0}.

(6.67)

This condition, together with Theorem 3.3, yields that A is a sectorial operator, namely, A satisfies (2.92) and (2.93). Proposition 2.5 then provides that e−tA ≤ Ce−dt , 0 ≤ t < ∞, with some positive exponent d > 0. As a consequence, e−tA U Dβ ≤ Aβ e−A e−(t−1)A U ≤ Ce−d(t−1) U Dβ . ∗

This implies that e−t A L(Dβ ) ≤ δ < 1 for any sufficiently large t ∗ > 0. Let K˜ r = B Dβ (0; r) be an open ball of Dβ . Then, (6.66) (θ = β) implies that S(t ∗ )(U + h) − S(t ∗ )U − S(t ∗ ) h Dβ = ΔU (t ∗ ) Dβ ≤

1−δ h Dβ , 2

if r > 0 is sufficiently small. Hence, 1−δ δ+1 S(t ∗ )(U + h) − U Dβ ≤ δ + h Dβ = h Dβ , 2 2

h ∈ K˜ r ,

h ∈ K˜ r .

This shows that (6.26) holds true, that is, S(t ∗ ) is a contraction at U . We have thus proved by Theorem 6.10 that, under (6.64) and (6.67), U is an exponentially stable equilibrium of (S(t), Dβ , Dβ ).

6.3 Unstable Manifold of U Let U ∈ D be again a stationary solution of (6.53) and be an equilibrium of (S(t), Dβ , Dβ ). In this subsection, we assume that F : D(Aη ) → X is of class C1,1 in a neighborhood of U . More precisely, there is a number r > 0 such that, for any U ∈ D(Aη ) ∩ B Dβ (U ; r), F is Fréchet differentiable and the derivative satisfies F (U )h ≤ L1 Aη h ,

U ∈ D(Aη ) ∩ B Dβ (U ; r), h ∈ D(Aη ),

(6.68)

with some constant L1 > 0. In addition, [F (U1 ) − F (U2 )]h ≤ L2 Aβ (U1 − U2 ) Aη h , U1 , U2 ∈ D(Aη ) ∩ B Dβ (U ; r), h ∈ D(Aη ),

(6.69)

with some constant L2 > 0. Put ΔF (U ; h) = F (U + h) − F (U ) − F (U )h. Then, (6.69) implies that ΔF (U ; h) ≤ L2 Aβ h Aη h ,

U + h, U ∈ D(Aη ) ∩ B Dβ (U ; r).

(6.70)

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Consider an open neighborhood B Dβ (U ; r1 ) of U . Let U0 ∈ B Dβ (U ; r1 ) be an initial value, and let U (t) = S(t)U0 . By Corollary 4.2 (when β > 0) or Theorem 4.5 (when β = 0), we have t η−β Aη (U (t) − U ) + Aβ (U (t) − U ) ≤ C Aβ (U0 − U ) , 0 ≤ t ≤ T1 , U0 ∈ B Dβ (U ; r1 ), for sufficiently small time T1 > 0. So, if r1 > 0 and T1 > 0 are sufficiently small, then U (t) ∈ B Dβ (U ; r) for 0 ≤ t ≤ T1 . Consequently, F (U (t)) is defined. Consider also a small vector variable h ∈ B Dβ (0; r2 ) and let Uh (t) = S(t)(U0 + h). By Theorem 4.3 or 4.5, we have t η Aη [Uh (t) − U (t)] + t β Aβ [Uh (t) − U (t)] ≤ C h , 0 < t ≤ T2 , U0 ∈ B Dβ (U ; r1 ), h ∈ B Dβ (0; r2 ),

(6.71)

for sufficiently small time T2 > 0. We here introduce the linear problem dV

h

dt

+ AVh = F (U (t))Vh ,

0 < t ≤ T,

Vh (0) = h,

(6.72)

in X, where T = min{T1 , T2 }. To solve this problem, we will employ Theorem 3.14. Let B(t) = −F (U (t)), 0 < t ≤ T . Then, it is obvious from (6.68) that B(t)A−η ≤ L1 ,

0 < t ≤ T.

(6.73)

Furthermore, noticing (4.13), we have [B(t) − B(s)]A−η ≤ L2 Aβ [U (t) − U (s)] ≤ Cs −σ˜ (t − s)σ˜ ,

0 < s ≤ t ≤ T,

(6.74) (6.75)

with any exponent σ˜ such that 0 < σ˜ < 1 − η. These estimates (6.73) and (6.74) show that (3.117), (3.118), and (3.119) are satisfied with μ˜ = 1 and ν˜ = 1 − η. Therefore, Theorems 3.13 and 3.14 provide the existence of the evolution operator U˜ (t, s) for the family of operators A − F (U (t)) which describes the solution of (6.72) in the form Vh (t) = U˜ (t, 0)h. Let ΔU (t) = Uh (t) − U (t) − Vh (t). Since F (Uh (t))−F (U (t))−F (U (t))Vh (t) = F (U (t))ΔU (t)+ΔF (U (t); Uh (t)−U (t)), we have dΔ

U

dt

+ AΔU = F (U (t))ΔU + ΔF (U (t); Uh (t) − U (t)),

ΔU (0) = 0.

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291

Then, ΔU is also described by U˜ (t, s) in the form

t ΔU (t) = U˜ (t, s)ΔF (U (s); Uh (s) − U (s)) ds. 0

Note that ΔF (U ; Uh − U ) ∈ Fβ,σ ((0, T ]; X) (when β > 0) due to (4.15) and ΔF (U ; Uh − U ) ∈ F1−η,σ ((0, T ]; X) (when β = 0) due to (4.27). Therefore, on account of (3.124) and (6.70), for any 0 ≤ θ < 1,

t θ A ΔU (t) ≤ C (t − s)−θ Aβ [Uh (s) − U (s)] Aη [Uh (s) − U (s)] ds. 0

Hence, by (6.71),

t

Aθ ΔU (t) ≤ C

(t − s)−θ s −η Aβ [Uh (s) − U (s)] ds h .

0

Aβ [U

Since h (s) − U (s)] → 0 as h → 0, it follows that, as h → 0 (a fortiori as h Dβ → 0), 0 ≤ Aθ ΔU (t) / h Dβ ≤ Aθ ΔU (t) / h → 0. Thus, under (6.68) and (6.69), S(t) : Dβ → D(Aθ ) is Fréchet differentiable in the neighborhood B Dβ (U ; r1 ) for 0 ≤ t ≤ T , where r1 > 0 and T > 0 are sufficiently small. Furthermore, its derivative is given by S(t) U0 = U˜ (t, 0),

0 < t ≤ T , U0 ∈ B Dβ (U ; r1 ),

where U˜ (t, s) is the evolution operator for the family of operators A − F (S(t)U0 ). Let U1 (t, 0) = S(t) U1 and U2 (t, 0) = S(t) U2 . Then,

t Aθ U2 (t, s)[F (S(s)U1 ) − F (S(s)U2 )]A−η Aθ [U1 (t, 0) − U2 (t, 0)]h = 0

× Aη U1 (s, 0)A−β Aβ h ds. By (3.124), (3.125), and (6.69), it follows that

t θ (t − s)−θ s β−η ds h Dβ A [U1 (t, 0) − U2 (t, 0)]h ≤ C 0

× sup Aβ [S(s)U1 − S(s)U2 ] . 0≤s≤T

Therefore, on account of (6.58), S(t) U1 − S(t) U2 L(Dβ ,D(Aθ )) = U1 (t, 0) − U2 (t, 0) L(Dβ ,D(Aθ )) ≤ Ct 1+β−θ−η U1 − U2 Dβ ,

0 ≤ t ≤ T ; U1 , U2 ∈ B Dβ (U ; r1 ).

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6 Dynamical Systems

Let us assume now that the spectrum of A = A − F (U ) is separated by the imaginary axis, i.e., σ (A) ∩ {λ ∈ C; Re λ = 0} = ∅.

(6.76)

Denote σ− (A) = σ (A) ∩ {λ ∈ C; Re λ < 0} and σ+ (A) = σ (A) ∩ {λ ∈ C; Re λ > 0}. Since F (U ) satisfies (6.68), A satisfies (3.2). This necessarily implies that σ− (A) is a bounded closed set. We can then decompose the space X into X = X− ⊕ X+ and the operator A into A = A− ⊕ A+ by applying Theorem 1.10. Here, X∓ are closed subspaces of X with continuous projection, and A∓ are the parts of A in X∓ , respectively. The operator A− is a bounded operator of X− with σ (A− ) = σ− (A); in particular, X− ⊂ D(A). In the meantime, A+ is a densely defined, closed linear operator of X+ with D(A+ ) = D(A) ∩ X+ and σ (A+ ) = σ+ (A). For λ ∈ ρ(A), the resolvent (λ − A)−1 is also decomposed into (λ − A)−1 = (λ − A− )−1 ⊕ (λ − A+ )−1 ; in particular, we verify that A+ is a sectorial operator of X+ . As a consequence, we obtain a decomposition of semigroup e−tA of the form e−tA = e−tA− ⊕ e−tA+ . Since A− is a bounded operator, we can use the spectral mapping theorem, Theorem 1.9, to conclude that σ (e−tA− ) = e−tσ− (A) ⊂ {λ ∈ C; |λ| > 1} for any 0 < t < ∞. In the meantime, on account of Proposition 2.5, we have e−tA+ ≤ Ce−δt with some δ > 0. Therefore, 1 the spectral radius of e−tA+ is estimated by limn→∞ e−ntA+ n ≤ e−δt ; hence, σ (e−tA+ ) ⊂ {λ ∈ C; |λ| < 1}. In this way, we have verified from (6.76) that σ (e−tA ) ∩ {λ ∈ C; |λ| = 1} = ∅,

0 < t < ∞,

and σ (e−tA ) ∩ {λ ∈ C; |λ| > 1} = e−tσ− (A) ,

0 < t < ∞.

Moreover, the subspace Xe of X corresponding to the exterior spectrum σ (e−tA ) ∩ {λ ∈ C; |λ| > 1} is the space X− . When β > 0, let us consider e−tA as a bounded operator from Dβ into itself. We observe the following fact. Lemma 6.1 Let t > 0. For λ = 0, λ ∈ σ (e−tA ) if and only if λ ∈ σ Dβ (e−tA ), where σ Dβ (·) denotes the spectrum of an operator in the space Dβ . Proof Let λ ∈ ρ(e−tA ). Then, (λ − e−tA )−1 = λ−1 e−tA (λ − e−tA )−1 + 1 . From this it is deduced that (λ − e−tA )−1 is a bounded operator from Dβ into itself. Therefore, (λ − e−tA )−1 is a bounded inverse of λ − e−tA in the space Dβ , too, i.e., λ ∈ ρ Dβ (e−tA ). Conversely, let λ ∈ ρ Dβ (e−tA ). Taking a λ0 ∈ ρ(A), we have (λ − e−tA )−1 = λ−1 A−β Aβ (λ − e−tA )−1 A−β Aβ (λ0 − A)−1 (λ0 − A)e−tA + 1 .

6 Stationary Solutions to Semilinear Equations

293

Since Aβ (λ − e−tA )−1 A−β ∈ L(X), this shows that (λ − e−tA )−1 is continuous in the norm of X, too. Since Dβ is dense in X, it follows that (λ − e−tA )−1 is extended over X as a bounded operator of X, and hence λ ∈ ρ(e−tA ). Thus, the assertion of lemma is verified. In view of this lemma, we conclude that (6.76) equally implies σ Dβ (e−tA ) ∩ {λ ∈ C; |λ| = 1} = ∅,

0 < t < ∞,

and σ Dβ (e−tA ) ∩ {λ ∈ C; |λ| > 1} = e−tσ− (A) ,

0 < t < ∞.

Moreover, the subspace Dβ,e of Dβ which corresponds to the exterior spectrum σ Dβ (e−tA ) ∩ {λ ∈ C; |λ| > 1} is nothing more than X− . Indeed, the last assertion is verified as follows. Let X = Xe ⊕ Xi and Dβ = Dβ,e ⊕ Dβ,i be the decompositions of X and Dβ , respectively, corresponding to the spectrum separation of the semigroup e−tA by the unit circle. Let P and Pβ be the projections from X to Xi and from Dβ to Dβ,i . By (1.15), if U ∈ Dβ , then obviously Pβ U = P U ; consequently, (1 − Pβ )U = (1 − P )U for U ∈ Dβ ; in particular, Dβ,e ⊂ Xe . While, as shown above, Xe = X− ⊂ D(A) ⊂ Dβ ; hence, Xe ⊂ Dβ,e , i.e., Xe = Dβ,e . In this way, under (6.68), (6.69), and (6.76), we have shown that S(t) U satisfies (6.20), that is, U is a hyperbolic equilibrium of (S(t), Dβ , Dβ ). The exterior spec trum of S(t) U is given by e−tσ− (A−F (U )) . By virtue of Theorem 6.9, there exists a smooth local unstable manifold W+ (U ; O) of U which has a dimension dim X− and is tangent to the subspace U + X− of Dβ at U .

6.4 Reaction–Diffusion Systems Let us apply our general results obtained above to the initial-boundary-value problem ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu + f (u, v) ⎪ ⎪ ⎪ ⎨ ∂v = bΔv + g(u, v) in Ω × (0, ∞), ∂t (6.77) ∂u ∂v ⎪ = ∂n = 0 on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x) in Ω, 0

0

in a C2 or convex, bounded domain Ω ⊂ Rd (d = 1, 2, 3). Here, a > 0 and b > 0 are diffusion constants. The kinetic functions f (u, v) and g(u, v) are real smooth functions defined for (u, v) ∈ R2 . Let (u, v) ≡ (u, v) be a stationary solution to (6.77), that is, (u, v) ∈ R2 satisfies f (u, v) = g(u, v) = 0.

(6.78)

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6 Dynamical Systems

We assume that f (u, v) and g(u, v) can be extended over a complex neighborhood D ⊂ C2 of (u, v) and the extended functions, still denoted by f (u, v) and g(u, v), are both differentiable as complex variable functions from D into C: f (u, v) and g(u, v) are complex differentiable functions in D.

(6.79)

We set the underlying space by f X= ; f ∈ L2 (Ω) and g ∈ L2 (Ω) , g and formulate problem (6.77) as the Cauchy problem for an abstract equation of the form (6.53). The linear operator A of X is defined by u A1 0 2 2 , D(A) = ; u ∈ HN (Ω) and v ∈ HN (Ω) , A= v 0 A2 where A1 and A2 are realizations of the differential operators −aΔ+1 and −bΔ+ 1 in L2 (Ω), respectively, under the homogeneous Neumann boundary conditions on ∂Ω. Then, A is a positive definite self-adjoint operator of X. By virtue of Theorems 16.7 and 16.9, the domains of its fractional powers are given by 3 u θ 2θ 2θ ; u ∈ H (Ω) and v ∈ H (Ω) for 0 ≤ θ < D(A ) = v 4 and D(Aθ ) =

u ; u ∈ HN2θ (Ω) and v ∈ HN2θ (Ω) v

for

3 < θ ≤ 1. 4

The nonlinear operator F from D(Aη ) into X is defined by u + f (u, v) u F (U ) = , U= ∈ D(Aη ), v + g(u, v) v where the exponent η is taken in such a way that d4 < η < 1. Since H s (Ω) ⊂ C(Ω) for s > d2 (due to (1.76)), we observe that D(Aη ) ⊂ C(Ω). We take another exponent β so that d4 < β < η < 1. Then, D(Aη ) ⊂ D(Aβ ) ⊂ C(Ω). By direct calculations, it is easily verified from (6.79) that F is Fréchet differentiable in a neighborhood of U = t (u, v) with derivative 1 + fu (u, v) fv (u, v) h, F (U )h = 1 + gv (u, v) gu (u, v) u U= ∈ B Dβ (U ; r), h ∈ D(Aη ), v where Dβ = D(Aβ ) and r > 0. Furthermore, the derivative fulfills conditions (6.68) and (6.69). Let us then investigate how the spectral condition (6.67) or the separation condition (6.76) is fulfilled by A = A − F (U ).

6 Stationary Solutions to Semilinear Equations

295

We here consider the self-adjoint operator Λ of L2 (Ω) which is a realization of −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. As is well known, Λ possesses denumerable nonnegative eigenvalues, and the corresponding real eigenfunctions can constitute an orthonormal basis of L2 (Ω), see Corollary 2.1. So, let 0 = μ 0 < μ1 ≤ μ 2 ≤ · · · → ∞ 1

be eigenvalues of Λ, and let φ0 = |Ω|− 2 , φ1 , φ2 , . . . , be the corresponding real eigenfunctions which constitute an orthonormal basis. We notice that X is decomposed into an infinite orthogonal direct sum of subspaces Xk , where each Xk is a two-dimensional subspace given by φk 0 + ηk ; ξk , ηk ∈ C , k = 0, 1, 2, . . . , Xk = U = ξ k 0 φk 0 φk X= ∞ k=0 Xk . Obviously, Φk = 0 and Ψk = φk are orthonormal bases of Xk with components of real functions. Since aΛ + 1 0 1+fu fv A= and F (U ) = , 0 bΛ + 1 gu 1 + gv where f u = fu (u, v), f v = fv (u, v), g u = gu (u, v), and g v = gv (u, v), we notice each Xk is invariant also that A maps the subspace Xk into itself. In other words, under A. In this way, A is also decomposed into the sum A = ∞ k=0 Ak , where Ak is the part of A in Xk . Let us consider the part Ak in Xk . Obviously, Ak Φk = (aμk − f u )Φk − g u Ψk ,

Ak Ψk = −f v Φk + (bμk − g v )Ψk .

The transformation matrix M k of Ak with respect to the bases Φk , Ψk is given by aμk − f u −g u Mk = . −f v bμk − g v So, the characteristic equation of M k is given by |λ − M k | ≡ λ2 − [(a + b)μk − (f u + g v )]λ + abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0. If the discriminant Δk is nonnegative, i.e., Δk = (a − b)2 μ2k + 2(a − b)(g v − f u )μk + (f u − g v )2 + 4f v g u ≥ 0, then the characteristic equation has two real solutions λk ≤ λk satisfying 2[abμ2k − (ag v + bf u )μk + (f u g v − f v g u )] ≤ λk ≤ λk . √ (a + b)μk − (f u + g v ) + Δk

(6.80)

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6 Dynamical Systems

On the other hand, if Δk < 0, then the solutions λk and λk are complex and satisfy Re λk = Re λk = [(a + b)μk − (f u + g v )]/2, √ |Im λk | = |Im λk | = |a − b|μk + O( μk ). Both cases, as k → ∞, λk and λk lie in Σω + 1 with some fixed angle ω such π that arctan |a−b| a+b < ω < 2 ; in addition, |λk | = O(μk ) and |λk | = O(μk ). Let ψk : Xk → C2 be a natural isomorphism such that ψk (ξk Φk + ηk Ψk ) = t (ξk , ηk ) ( ψk = ψk−1 = 1); then, Ak and its transformation matrix M k satisfy the relation Ak = ψk−1 M k ψk . Therefore, (λ − Ak )−1 = ψk−1 (λ − M k )−1 ψk

for λ = λk , λk .

(6.81)

Hence, σ (Ak ) = {λk , λk } for each k = 0, 1, 2, . . . . Let us now consider (λ − A)−1 . Take an angle ω such that ω < ω < π2 , and let λ∈ / Σω + 1. For all sufficiently large k such that λk , λk ∈ Σω + 1, we have 1 λ − bμk + g v −g u (λ − M k )−1 = . −f v λ − aμk + f u (λ − λk )(λ − λk ) In addition, |λ − λk |2 ≥ |λ|2 + |λk |2 − 2|λ||λk | cos(ω − ω), |λ − λk |2 ≥ |λ|2 + |λk |2 − 2|λ||λk | cos(ω − ω). Therefore, in view of (6.81), we obtain that (λ − Ak )−1 L(Xk ) ≤ Cλ for all such k, where Cλ > 0 are independent of k. This then means that λ ∈ ρ(A) if and only if λ ∈ ρ(Ak ) for each k = 0, 1, 2, . . . . By these considerations, we can now state the following. The operator A fulfills (6.67) or (6.76) if and only if the part Ak fulfills the same for each k = 0, 1, 2, . . . . Condition (6.67). A necessary and sufficient condition that A0 (μ0 = 0) satisfies (6.67) is given by f u + g v < 0 and f u g v > f v g u . Therefore, a necessary and sufficient condition that every Ak satisfies (6.67) is that f u + g v < 0, f u g v > f v g u , and that (ag v + bf u )μk < abμ2k + (f u g v − f v g u ) for every μk . As (ag v + bf u )μ < abμ2 + (f u g v − f v g u ) for all μ ≥ 0, provided that ag v + bf u < 2 ab(f u g v − f v g u ), we find that the triplet of inequalities f u + g v < 0, f u g v > f v g u , and ag v + bf u < 2 ab(f u g v − f v g u ) (6.82) is a sufficient condition for (6.67).

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297

Condition (6.76). A necessary and sufficient condition that equation (6.80) has a purely imaginary solution is given by abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0, or by (a + b)μk = f u + g v and abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) > 0. So, it is equivalent to the condition that abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0, or μk = (f u + g v )/(a + b) and −(a + b)2 f u g v > (ag v − bf u )2 . Therefore, a necessary and sufficient condition in order that (6.76) is fulfilled (in other words, (6.80) has no purely imaginary solution) is that abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0 for all μk , and μk = (f u + g v )/(a + b) for all μk ’s or − (a + b)2 f v g u ≤ (ag v − bf u )2 . (6.83)

Condition σ (A) ∩ {λ ∈ C; Re λ < 0} = ∅. Assume that, for some μk , abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) < 0.

(6.84)

For such a μk , the characteristic equation (6.80) has solutions λk < 0 < λk . There ξ fore, λk ∈ σ (A) ∩ {λ ∈ C; Re λ < 0}. Let ηk be an eigenvector of M k correspondk

ing to the negative eigenvalue λk . Since M k is a real matrix, ξk and ηk are real num ξ bers. It then follows that ψk−1 ηk ∈ Xk is an eigenfunction of Ak corresponding to k ξ ξ φ the eigenvalue λk . Since the eigenfunction is given by ψk−1 ηk = ηk φk , we observe k

k k

that the eigenfunction of Ak corresponding to the negative eigenvalue λk is a realvalued function. Thus, under (6.84), we deduce that A has a negative eigenvalue and the corresponding eigenfunction is a real-valued function componentwise.

7 Dynamical Systems for Quasilinear Evolution Equations 7.1 Structural Assumptions We consider the Cauchy problem for a quasilinear abstract parabolic evolution equation dU dt + A(U )U = F (U ), 0 < t < ∞, (6.85) U (0) = U0 , in a Banach space X. Let Z be a second Banach space continuously embedded in X. For U ∈ Z, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 . The domain D(A(U ) is uniform, i.e., D(A(U )) ≡ D,

U ∈ Z.

(6.86)

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6 Dynamical Systems

The domain D is a Banach space equipped with the graph norm A(0) · X of the operator A(0). The operator F is a nonlinear operator from a third Banach space W into X, where W ⊂ Z ⊂ X with continuous embeddings. We make the following structural assumptions. For each 0 < R < ∞, let KR = {U ∈ Z; U Z < R} denote an open ball of Z. For U ∈ KR , A(U ) satisfies σ (A(U )) ⊂ ΣR = {λ ∈ C; | arg λ| < ωR },

U ∈ KR ,

(6.87)

with some uniform angle 0 < ωR < π2 , and the resolvent satisfies (λ − A(U ))−1 L(X) ≤ MR |λ|,

λ∈ / ΣR , U ∈ KR ,

(6.88)

with some uniform constant MR ≥ 1. In addition, A(U ) satisfies the Lipschitz condition A(U ) A(U )−1 − A(V )−1 ≤ NR U − V Y , U, V ∈ KR , (6.89) L(X) with some constant NR > 0, where Y is another Banach space such that Z ⊂ Y ⊂ X with continuous embeddings. It is verified from (6.89) that there is a constant CR > 0 such that CR−1 A(0)U˜ X ≤ A(U )U˜ X ≤ CR A(0)U˜ X ,

U ∈ KR , U˜ ∈ D.

(6.90)

For the nonlinear operator F : W → X, we assume the Lipschitz condition F (U ) − F (V ) X ≤ ϕ( U Z + V Z )[ U − V W + ( U W + V W ) × U − V Z ],

U, V ∈ W,

(6.91)

with some continuous increasing function ϕ(·). We may recall that (6.91) implies (5.45). For the spaces W ⊂ Z ⊂ Y ⊂ X, we assume that there exist exponents 0 ≤ α < β < η < 1 such that, for every U ∈ Z, D(A(U )α ) ⊂ Y, D(A(U )β ) ⊂ Z, D(A(U )η ) ⊂ W with continuous embeddings such that ⎧ α α ⎪ ⎨ U˜ Y ≤ D1,R A(U ) U˜ X , U˜ ∈ D(A(U ) ), U ∈ KR , (6.92) U˜ Z ≤ D2,R A(U )β U˜ X , U˜ ∈ D(A(U )β ), U ∈ KR , ⎪ ⎩ ˜ U W ≤ D3,R A(U )η U˜ X , U˜ ∈ D(A(U )η ), U ∈ KR , with some constants Di,R > 0 (i = 1, 2, 3). We furthermore introduce another exponent γ such that β < γ < η and assume that the domain D(A(U )γ ) of the fractional power is independent of U ∈ Z with the equivalence estimate CR−1 A(0)γ U˜ X ≤ A(U )γ U˜ X ≤ CR A(0)γ U˜ X , U˜ ∈ D(A(0)γ ), U ∈ KR , (6.93)

7 Dynamical Systems for Quasilinear Evolution Equations

299

with some constant CR > 0. The domain D(A(U )γ ) is denoted by Dγ , which is a Banach space with the graph norm · Dγ = A(0)γ · X . This space is then used as a space of initial values for problem (6.85). Since D(A(0)γ ) ⊂ D(A(0)β ) ⊂ Z due to (6.92), Dγ is continuously embedded in Z; so, let · Z ≤ c · Dγ . For ˜ be an open ball of Dγ . Then, 0 < R˜ < ∞, let K˜ R˜ = {U ∈ Dγ ; U Dγ < R} K˜ R˜ ⊂ KcR˜ ,

0 < R˜ < ∞.

(6.94)

7.2 Dynamical System Applying the results obtained in Chap. 5, we can see the local existence of solution ˜ to (6.85) for each initial value U0 ∈ Dγ . In fact, let U0 ∈ Dγ with U0 Dγ < R; ˜ then, U0 ∈ Z with U0 Z < cR = R due to (6.94). The family of operators A(U ), U ∈ KR , and the nonlinear operator F satisfy all the structural assumptions (5.2), (5.3), (5.4), (5.5), (5.6), (5.7), (5.46), and (5.47) with ν = 1. In view of U0 ∈ D(A(0)γ ) = D(A(U0 )γ ), we get by Theorem 5.6 the existence of a unique local solution U (t) on some interval [0, TU0 ] in the function space: U ∈ C((0, TU0 ]; D) ∩ C([0, TU0 ]; Dγ ) ∩ C1 ((0, TU0 ]; X),

(6.95)

where TU0 > 0 is determined by the norm A(0)γ U0 X . In order to obtain the global existence, we have to establish a priori estimates for any local solution with the initial value U0 . We here assume that there exists some fixed continuous increasing function p(·) > 0 such that the estimate U (t) Dγ ≤ p( U0 Dγ ),

0 ≤ t ≤ T U , U0 ∈ D γ ,

(6.96)

holds for any local solution U on [0, TU ] to (6.85). This means that any local solution lies in a fixed open ball Kcp( U0 Dγ ) of Z. Since the family of operators A(U ), U ∈ Kcp( U0 Dγ ) , also satisfies all the structural assumptions and since A(U (t))γ U (t) ≤ Ccp( U0 Dγ ) p( U0 Dγ ) due to (6.93), we can use Corollary 5.2 to conclude the existence of a global solution U (t; U0 ) in the function space (6.95) in which TU0 is an arbitrary finite time. Clearly, (6.96) implies that U (t; U0 ) Dγ ≤ p( U0 Dγ ),

0 ≤ t < ∞, U0 ∈ Dγ .

(6.97)

Furthermore, by the same arguments as in Proposition 6.1, the following stronger estimate is verified. Proposition 6.5 For U (t; U0 ), it holds that U (t; U0 ) D ≤ (t γ −1 + 1)p1 ( U0 Dγ ), with some continuous increasing function p1 (·).

0 < t < ∞, U0 ∈ Dγ ,

(6.98)

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6 Dynamical Systems

In view of γ < η < 1, we can use the moment inequality (2.119). Then, (6.97) and (6.98) yield that U (t; U0 ) W ≤ (t γ −η + 1)pη ( U0 Dγ ),

0 ≤ t < ∞, U0 ∈ Dγ ,

(6.99)

with some continuous increasing function pη (·). For U0 ∈ Dγ , we set S(t)U0 = U (t; U0 ). The uniqueness of solutions implies that S(t) is a nonlinear semigroup acting on Dγ . Let us show that S(t) is continuous on Dγ . Let 0 < R˜ < ∞. For U0 ∈ K˜ R˜ , it holds that U0 Z < cR˜ and A(U0 )γ U0 X ≤ CcR˜ . So, Corollary 5.4 is applicable for the solutions S(t)U0 , U0 ∈ K˜ R˜ , to see that t η−α S(t)U0 − S(t)V0 W + t β−α S(t)U0 − S(t)V0 Z + S(t)U0 − S(t)V0 Y 0 < t ≤ tR˜ , U0 , V0 ∈ K˜ R˜ .

≤ CR˜ U0 − V0 Dγ ,

(6.100)

On the basis of this estimate, we can deduce a Hölder condition of S(t). Proposition 6.6 With the exponent δ = min{ γ1−η −α , 1}, S(t) satisfies S(t)U0 − S(t)V0 Dγ ≤ LR˜ U0 − V0 δDγ ,

0 ≤ t ≤ tR˜ , U0 , V0 ∈ K˜ R˜ .

Proof For simplicity, let us use the notation U (t) = S(t)U0 and V (t) = S(t)V0 , and let UU (t, s) and UV (t, s) denote the evolution operators for the families of operators AU (t) = A(U (t)) and AV (t) = A(V (t)), respectively. We repeat the similar argument as in the proof of Theorem 5.9. Indeed, as shown in (5.71), U (t) − V (t) = UU (t, 0)(U0 − V0 ) + [UU (t, 0) − UV (t, 0)]V0

t + [UU (t, s) − UV (t, s)]F (V (s)) ds 0

+

t

UU (t, s)[F (U (s)) − F (V (s))] ds.

0

Here, by (6.93), AU (t)γ UU (t, 0)(U0 − V0 ) ≤ C ˜ A(U0 )γ (U0 − V0 ) ≤ C ˜ U0 − V0 D . γ R R X X By the same arguments as for (5.39) but substituting γ for β, we verify that AU (t)γ [UU (t, 0) − UV (t, 0)]V0 X

t ≤ CR˜ V0 Dγ (t − s)−γ s γ −1 U (s) − V (s) Y ds. 0

Hence, by (3.83), AU (t)γ [UU (t, 0) − UV (t, 0)]V0 ≤ C ˜ U0 − V0 D . γ R X

7 Dynamical Systems for Quasilinear Evolution Equations

301

Similarly, by the same arguments as for (5.40), we verify that

t AU (t)γ [UU (t, s) − UV (t, s)]F (V (s)) ds 0

≤ CR˜ F (V (·)) Fγ ,σ

X

t

(t − s)−γ s γ −1 U (s) − V (s) Y ds.

0

Hence, by (6.100),

t AU (t)γ [UU (t, s) − UV (t, s)]F (V (s)) ds ≤ CR˜ U0 − V0 Dγ . 0

X

In the meantime, by (6.91),

t AU (t)γ U (t, s)[F (U (s)) − F (V (s))] ds U 0

≤ CR˜

t

X

(t − s)−γ [ U (s) − V (s) W

0

+ ( U (s) W + V (s) W ) U (s) − V (s) Z ] ds. We notice from (6.99) and (6.100) that U (s) − V (s) W ≤ ( U (s) W + V (s) W )1−δ U (s) − V (s) δW ≤ CR˜ s γ (1−δ)+αδ−η U0 − V0 δDγ with any 0 < δ ≤ 1. So, taking δ = min{ γ1−η −α , 1}, we have

t

(t − s)−γ s γ (1−δ)+αδ−η ds ≤ C.

0

Similarly, ( U (s) W + V (s) W ) U (s) − V (s) Z

≤ CR˜ ( U (s) W + V (s) W )( U (s) Z + V (s) Z )1−δ U (s) − V (s) δZ

≤ CR˜ s γ −η+(α−β)δ U0 − V0 δDγ 1−η with any 0 < δ ≤ 1. So, taking δ = min{ β−α , 1}, we have

t

(t − s)−γ s γ −η+(α−β)δ ds ≤ C.

0

Hence, we conclude that

t AU (t)γ ≤ C ˜ U0 − V0 δ U (t, s)[F (U (s)) − F (V (s))] ds U Dγ R 0

X

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6 Dynamical Systems

with the exponent δ = min{δ, δ }. Summing up these estimates, we obtain the desired one.

The a priori estimate (6.96) implies that

{S(t)U0 ; 0 ≤ t < ∞} ⊂ K˜ p(R) ˜ .

U0 ∈K˜ R˜

We now apply Proposition 6.6 on the ball K˜ p(R) ˜ . Proposition 6.7 For any 0 < R˜ < ∞, it holds that n+1

S(t)U0 − S(t)V0 Dγ ≤ Ln+1˜ U0 − V0 δDγ , p(R)

˜ t ∈ [ntp(R) ˜ , (n + 1)tp(R) ˜ ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ KR˜ .

(6.101)

Proof In fact, when n = 0, this is already verified by Proposition 6.6 because of ˜ Assume that this is true for n − 1. For nt ˜ < t ≤ (n + 1)t ˜ , we R˜ ≤ p(R). p(R) p(R) have S(t) = S(t − ntp(R) ˜ )S(ntR˜ ); then (6.101) is easily obtained since S(ntp(R) ˜ )U0 and S(nt ˜ )V0 are in K˜ ˜ . p(R)

p(R)

Estimate (6.101) means that S(t) is Hölder continuous on the ball K˜ R˜ and the Hölder exponent is uniform in t on any finite time interval. Meanwhile, for each U0 ∈ K˜ R˜ , S(·)U0 is continuous from [0, ∞) to Dγ . Then, the mapping G : [0, ∞) × K˜ R˜ → Dγ , where G(t, U0 ) = S(t)U0 , is continuous. As R˜ can be taken arbitrarily, we conclude that S(t) is a continuous semigroup on Dγ . Thus, under (6.87), (6.88), (6.89), (6.91), (6.92), (6.93), and (6.96), problem (6.85) determines a dynamical system (S(t), Dγ , Dγ ), where Dγ = D(A(0)γ ). It is often important to construct a dynamical system in the universal space X instead of Dγ . To define such a dynamical system, we introduce the phase space XR˜ =

{S(t)U0 ; 0 ≤ t < ∞},

0 < R˜ < ∞.

(6.102)

U0 ∈K˜ R˜

Clearly, XR˜ is invariant under S(t) for every 0 < t < ∞, and XR˜ ⊂ K˜ p(R) ˜ due to (6.96). With the aid of Theorem 5.9, we can then repeat the same arguments as in Proposition 6.3 to deduce the Lipschitz continuity of S(t) in the X norm. Proposition 6.8 For any 0 < R˜ < ∞, it holds that S(t)U0 − S(t)V0 X ≤ Ln+1˜ U0 − V0 X , p(R)

t ∈ [ntp(R) ˜ , (n + 1)tp(R) ˜ ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ XR˜ .

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303

Consequently, G : [0, ∞) × XR˜ → X is continuous, where XR˜ is equipped with the induced distance from X. Thus, problem (6.85) defines a dynamical system (S(t), XR˜ , X) in the space X, too. Clearly, K˜ R˜ ⊂ XR˜ for 0 < R˜ < ∞.

7.3 Absorbing and Invariant Compact Set We now add two assumptions that X is a reflexive Banach space, D is compactly embedded in X.

(6.103)

As for Proposition 6.4, these assumptions immediately imply the following result. D

Proposition 6.9 Any closed bounded ball B (0; R), 0 < R < ∞, of D is a compact set of X. We assume also the dissipative condition that there exists a number C˜ > 0 for which the following estimate is valid. For any bounded subset B of Dγ , there is a time tB such that ˜ sup sup S(t)U0 Dγ ≤ C.

U0 ∈B t≥tB

(6.104)

As for Proposition 6.5 (actually Proposition 6.1), this condition implies a stronger one. There exists a number C˜ 1 > 0 such that, for any bounded subset B of Dγ , it holds that sup sup S(t)U0 D ≤ C˜ 1

U0 ∈B t≥tB

(6.105)

with a suitable time tB depending on B. Using this constant C˜ 1 , we introduce the closed ball D B = B (0; C˜ 1 ).

Then (6.105) obviously means that B is an absorbing set. As noticed above, B is a compact set of X. We next introduce the set X=

S(t)B ⊂ B

(closure in X).

(6.106)

tB ≤t 0 that the operator S(t ∗ ) from X into Dγ satisfies the compact perturbation condition (6.38)–(6.39) (with δ = 0 and Z = Dγ ). Similarly, by Proposition 6.8, S(t)U0 − S(t)V0 X ≤ CX U0 − V0 X ,

0 ≤ t ≤ t ∗ , U0 , V0 ∈ X,

with some constant CX > 0. Meanwhile, by (5.45) and (6.90), t [F (S(τ )V ) − A(S(τ )V )S(τ )V ] dτ S(t)V0 − S(s)V0 X = 0 0 0 s

≤ CX (t − s),

X

∗

0 ≤ s < t ≤ t , V0 ∈ X.

Hence, condition (6.48) is fulfilled. Under (6.87), (6.88), (6.89), (6.91), (6.92), (6.93), (6.96), (6.103), and (6.104), we have proved that (S(t), X, X) possesses a family of exponential attractors M. Let us equally verify that M are exponential attractors of the dynamical system (S(t), X, Dγ ), too, where X is equipped with the induced distance from Dγ . Let M be an exponential attractor of (S(t), X, X) satisfying (6.17). By the moment inequality, we have hDγ (S(t)X, M) ≤ CX h(S(t)X, M)1−γ ≤ CX e−(1−γ )kt ,

0 ≤ t < ∞,

where hDγ (·, ·) denotes the Hausdorff pseudo-distance in Dγ . In addition, by Theorem 1.3, the fractal dimension dDγ (M) of M in Dγ is estimated by (1 − γ )−1 dX (M), where dX (M) denotes that of M in X. These facts show that M is an exponential attractor of (S(t), X, Dγ ), too.

8 Stationary Solutions to Quasilinear Equations 8.1 Equilibria of Dynamical System Let U be a stationary solution of (6.85), i.e., U ∈ D and A(U )U = F (U ). We assume that (6.85) determines a dynamical system (S(t), Dγ , Dγ ). More precisely, all the structural conditions (6.87), (6.88), (6.89), (6.91), (6.92), and (6.93) are satisfied and the a priori estimates (6.96) hold for all initial values in Dγ . So, U is an equi˜ Then, U Z < R, librium of (S(t), Dγ , Dγ ). Let R˜ > 0 be such that U Dγ < R. Z where R = cR˜ (see (6.94)), and let KR be the open ball B (0; R) of Z. In this section, we shall investigate the stability and instability of U by applying Theorems 6.9 and 6.10, respectively.

8 Stationary Solutions to Quasilinear Equations

305

8.2 Exponential Stability of U Let us assume the following conditions. The operator A : Z → L(D, X) is Fréchet differentiable at U with respect to the Y norm. Its differential A (U ) is a bounded operator from Y to L(D, X) and, equivalently, is a bilinear operator from Y × D into X. So, the derivative is written by A (U )[U, U˜ ] ∈ X for [U, U˜ ] ∈ Y × D. Therefore, ΔA (h) = A(U + h) − A(U ) − A (U )[h, ·], A (U ) ∈ L(Y, L(D, X)), (6.107) limh→0 oA (h)/ h Y = 0. oA (h) = ΔA (h) L(D,X) , Here, oA (h) is defined for h ∈ Z such that U + h ∈ KR , and oA (h) is uniformly bounded for h such that h ∈ KR − U . The nonlinear operator F : W → X is also Fréchet differentiable at U , i.e., ΔF (h) = F (U + h) − F (U ) − F (U )h, F (U ) ∈ L(W, X), (6.108) limh→0 oF (h)/ h W = 0. oF (h) = ΔF (h) X , Here, oF (h) is defined for all h ∈ W . In view of (6.91), oF (h) ≤ C h W for all h ∈ W such that h W < 1. From now on, let h denote a small variable in Dγ such that h Dγ < R˜ − U Dγ , and let Uh (t) = S(t)(U + h) be the solution to (6.85) with initial value U + h. According to Theorem 5.9, if T > 0 is sufficiently small, then t η Uh (t) − U W + t β Uh (t) − U Z + t α Uh (t) − U Y ≤ C h X ,

0 ≤ t ≤ T. (6.109)

We consider the linear problem dV

h

dt

+ A(U )Vh + A (U )[Vh , U ] = F (U )Vh ,

0 < t ≤ T,

Vh (0) = h,

(6.110)

in X with the initial value h. The linear operator of this problem can be regarded as a perturbation of A(U ) in a sense that condition (3.8) is fulfilled with ν˜ = 1 − η. Therefore, by Theorem 3.3, A(U ) + A (U )[·, U ] − F (U ) generates an analytic semigroup e−tA on X, where A = A(U ) + A (U )[·, U ] − F (U ). In addition, by Theorem 3.14 (cf. Remark 3.7), the solution of the problem is given by Vh (t) = e−tA h. In particular, we notice by maximal regularity (3.130) that ˜

A(U )Vh ∈ Fβ,σ˜ ((0, T ]; X) ˜ σ˜ such that 0 < σ˜ < β˜ < min{1 − η, γ } and that with some exponents β, Vh (t) D ≤ Ct β−1 A(U )β h , 0 < t ≤ T .

(6.111)

(6.112)

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6 Dynamical Systems

We set ΔU (t) = Uh (t) − U − Vh (t), 0 < t ≤ T . It is observed that A(Uh )Uh − A(U )U − A(U )Vh − A (U )[Vh , U ] = A(Uh )ΔU + A (U )[ΔU , U ] + A (U )[Uh − U , Vh ] + ΔA (Uh − U )(U + Vh ) and F (Uh ) − F (U ) − F (U )Vh = F (U )ΔU + ΔF (Uh − U ). This then means that ΔU (t) is regarded as a solution to the linear problem dΔ U dt + A(Uh (t))ΔU + A (U )[ΔU , U ] − F (U )ΔU = G(t), ΔU (0) = 0, where the external force function is given by G(t) = −A (U )[Uh (t) − U , Vh (t)] − ΔA (Uh (t) − U )(U + Vh (t)) + ΔF (Uh (t) − U ). The linear operators in the equation are regarded as perturbations of the operators A(t) = A(Uh (t)), 0 ≤ t ≤ T . More precisely, we can show that A(t) = A(Uh (t)) and B(t) ≡ B = A (U )[·, U ] − F (U ) satisfy the assumptions in Theorems 3.13 and 3.14. In fact, it is already known that A(t) satisfy assumptions (3.27), (3.28), (3.29), (3.30), and (3.31) with ν = 1 and with some exponent μ = β − α > 0 (due to (6.89) and (5.48)). In the meantime, B is a bounded operator from W into X. Therefore, it satisfies (3.117) and (3.118) with μ˜ = 1 and with some exponent 0 < ν˜ < 1 − η. Moreover, by using (3.92) with θ = η and ϕ = 1 − ν˜ , we observe that B A(t)ν˜ −1 − A(s)ν˜ −1 ≤ B L(W,X) A(t)η A(t)ν˜ −1 − A(s)ν˜ −1 ≤ C|t − s|η . Hence, (3.119) and (3.120) are satisfied. Consequently, there exists an evolution operator U˜ (t, s) for the operators A(t) + B. In addition, ΔU is written by

t ΔU (t) = U˜ (t, s)G(s) ds, 0 < t ≤ T . 0 ˜

Here we used a fact that G ∈ Fβ,σ˜ ((0, T ]; X) due to (6.111). Utilizing this expression, we want to estimate the norm ΔU (t) Dγ . By (3.124) and (6.93),

t ΔU (t) Dγ ≤ C A(t)γ ΔU (t) X ≤ C (t − s)−γ G(s) X ds. 0

Here, it follows from (6.109) and (6.112) that A (U )[Uh (s) − U , Vh (s)] X ≤ C Uh (s) − U Y Vh (s) D ≤ Cs −α h X s β−1 A(U )β h X ,

8 Stationary Solutions to Quasilinear Equations

307

ΔA (Uh (s) − U )(U + Vh (s)) X ≤ oA (Uh (s) − U ) U + Vh (s) D −α ≤ oA (Uh (s) − U ) Uh (s) − U −1 h X Y s β−1 A(U )β h + 1 , × s X

and −η ΔF (Uh (s) − U ) X ≤ oF (Uh (s) − U ) Uh (s) − U −1 W s h X .

Hence, G(s) X ≤ C(s β−α−1 + s −η ) A(U )β h X + oA (Uh (s) − U ) Uh (s) − U −1 Y

+ oF (Uh (s) − U ) Uh (s) − U −1 W h X . As a consequence,

ΔU (t) Dγ / h X ≤ C

0

t

(t − s)−γ (s β−α−1 + s −η ) A(U )β h X

+ oA (Uh (s) − U ) Uh (s) − U −1 Y

+ oF (Uh (s) − U ) Uh (s) − U −1 W ds. β/γ

1−β/γ

, h X → 0 implies Since A(U )β h X ≤ C A(U )γ h X h X A(U )β h X → 0. Similarly, h X → 0 implies Uh (s) − U W → 0 and Uh (s) − U Y → 0 due to (6.109). Therefore, by virtue of the dominate convergence theorem, we conclude that, as h X → 0 (a fortiori as h Dγ → 0), 0 ≤ ΔU (t) Dγ / h Dγ ≤ ΔU (t) Dγ / h X → 0.

(6.113)

Thus, under (6.107) and (6.108), the operator S(t) : Dγ → Dγ is Fréchet differentiable at U for any time t such that 0 < t ≤ T . Its differential is given by S(t) U = e−tA

where A = A(U ) + A (U )[·, U ] − F (U ).

As S(t)U = U , we verify that S(2t) = S(t)S(t) is also Fréchet differentiable at U . Repeating this argument, we conclude that S(t) is Fréchet differentiable at U for every time 0 < t < ∞. Let us now assume that U satisfies the spectral condition σ (A) = σ (A(U ) + A (U )[·, U ] − F (U )) ⊂ {λ ∈ C; Re λ > 0}.

(6.114)

This condition, together with Theorem 3.3, implies that A is a sectorial operator, i.e., A satisfies (2.92) and (2.93). Proposition 2.5 then provides the exponentially decaying estimate e−tA L(X) ≤ Ce−dt , 0 ≤ t < ∞. Then, we can repeat the same

308

6 Dynamical Systems

arguments as in Sect. 6.2 for verifying that S(t ∗ ) is a contraction at U . Hence, by Theorem 6.10, we conclude that U is exponentially stable. We have thus proved under (6.107), (6.108), and (6.114) that U is an exponentially stable equilibrium of (S(t), Dγ , Dγ ).

8.3 Unstable Manifold of U For simplicity, we will consider only the case where the semilinear term is given by F (U ) = kU with some complex number k. (For the general case, see Remark 6.4 below.) So, U is a stationary solution such that A(U )U = kU . We assume the following conditions. The operator A : Z → L(D, X) is Fréchet differentiable in a neighborhood B Z (U ; r) of U with respect to the Y norm, and the derivative satisfies the following estimates: A (U )[h, U˜ ] X ≤ N1 h Y U˜ D ,

U ∈ B Z (U ; r), h ∈ Z, U˜ ∈ D, (6.115)

{A (U1 ) − A (U2 )}[h, U˜ ] X ≤ N2 U1 − U2 Z h Y U˜ D ,

U1 , U2 ∈ B Z (U ; r), h ∈ Z, U˜ ∈ D,

(6.116)

with some constants Ni > 0 (i = 1, 2). Put ΔA (U ; h) = A(U + h) − A(U ) − A (U )[h, ·]. Then, (6.116) implies that ΔA (U ; h) L(D,X) ≤ C h Z h Y ,

U + h, U ∈ B Z (U ; r).

(6.117)

Consider an open neighborhood B Dγ (U ; r1 ) of U . Let U0 ∈ B Dγ (U ; r1 ) be an initial value, and let U (t) = S(t)U0 . By Proposition 6.6 and (6.92), we have U (t) − U Z ≤ C U0 − U δDγ ,

0 ≤ t ≤ T1 , U0 ∈ B Dγ (U ; r1 ),

for sufficiently small time T1 > 0. So, if r1 > 0 and T1 > 0 are sufficiently small, then U (t) ∈ B Z (U ; r) for 0 ≤ t ≤ T1 . Consequently, A (U (t)) is defined. Consider also a small vector variable h ∈ B Dγ (0; r2 ) and let Uh (t) = S(t)(U0 + h). By Theorem 5.9, we have t α Uh (t) − U (t) Y ≤ C h X ,

0 < t ≤ T2 , U0 ∈ B Dγ (U ; r1 ), h ∈ B Dγ (0; r2 ), (6.118)

for sufficiently small T2 > 0. We introduce the linear problem dV h dt + A(U (t))Vh + A (U (t))[Vh , U (t)] = kVh , Vh (0) = h,

0 < t ≤ T,

(6.119)

8 Stationary Solutions to Quasilinear Equations

309

in X with T = min{T1 , T2 }. The linear operators in the equation can then be treated as perturbations of operators A(t) = A(U (t)). In fact, we verify the following lemma. Lemma 6.2 Set B(t) = A (U (t))[·, U (t)] − k. Then, B(t)A(t)−β ≤ Ct γ −1 , 0 < t ≤ T , L(X) B(t)A(t)−β − B(s)A(s)−β ≤ Cs γ −1−σ (t − s)σ , L(X)

(6.120) 0 < s < t ≤ T, (6.121)

with any σ such that 0 < σ < β − α. Proof From (5.66), (6.92), and (6.115) it follows that B(t)A(t)−β ≤ B(t)A(t)−α ≤ C U (t) D ≤ Ct γ −1 . Hence, (6.120) is verified. In the meantime, B(t)A(t)−β − B(s)A(s)−β ≤ [B(t) − B(s)]A(t)−β + B(s) A(t)−β − A(s)−β . By (6.116), [B(t) − B(s)]A(t)−β ≤ C[ U (t) − U (s) Z U (t) D + U (t) − U (s) D ]. Here, by (6.92), U (t) − U (s) Z ≤ C A(t)β [U (t) − U (s)] X β

1−β

≤ C U (t) − U (s) D U (t) − U (s) X . From (5.66), in particular from A(t)U (t) ∈ Fγ ,σ ((0, T ]; X), it follows that U (t) − U (s) D ≤ C A(t)U (t) − A(s)U (s) X + A(t) A(t)−1 − A(s)−1 A(s)U (s) X

≤ Cs

γ −1−σ

with any 0 < σ < β − α. In addition,

(t − s)

dU dt

σ

∈ Fγ ,σ ((0, T ]; X) implies that

U (t) − U (s) ≤ Cs γ −1 (t − s). Therefore,

[B(t) − B(s)]A(t)−β ≤ Cs γ −1−σ (t − s)σ .

(6.122)

310

6 Dynamical Systems

We obtain from Theorem 2.25, (5.66), and (6.115) that B(s) A(t)−β − A(s)−β ≤ C B(s)A(0)−α A(0)α A(t)−β − A(s)−β ≤ C U (s) D U (t) − U (s) Y ≤ Cs γ −1 (t − s)σ .

This, together with (6.122), then yields (6.121).

We have just verified that A(t) = A(U (t)) and B(t) = A (U (t))[·, U (t)] − k satisfy (3.117), (3.118), and (3.119) with μ˜ = γ and ν˜ = 1 − β. Consequently, there exists an evolution operator U˜ (t, s) for the family of operators A(t) + B(t) by which the solution of (6.119) is written as Vh (t) = U˜ (t, 0)h. By the maximal regularity (3.130), ˜

A(U )Vh ∈ Fβ,σ˜ ((0, T ]; X)

(6.123)

˜ σ˜ such that 0 < σ˜ < β˜ < γ − β, and with some exponents β, Vh (t) D ≤ Ct β−1 A(U0 )β h , 0 < t ≤ T .

(6.124)

We now put ΔU (t) = Uh (t) − U (t) − Vh (t),

0 < t ≤ T.

It is observed that A(Uh )Uh − A(U )U − A(U )Vh − A (U )[Vh , U ] = A(Uh )ΔU + A (U )[ΔU , U ] + A (U )[Uh − U, Vh ] + ΔA (U ; Uh − U )(U + Vh ). This then means that ΔU (t) is a solution to the linear problem dΔU dt + A(Uh (t))ΔU + A (U )[ΔU , U ] − kΔU = G(t), ΔU (0) = 0,

0 < t ≤ T,

where G(t) = −A (U (t))[Uh (t) − U (t), Vh (t)] − ΔA (U (t); Uh (t) − U (t))[U (t) + Vh (t)]. By the same arguments as before, the linear operators are seen to be perturbations of the operators A(Uh (t)) with μ˜ = γ and ν˜ = 1 − β. Therefore, by using an evolution operator U˜ h (t, s) for the linear operators A(Uh (t)) + A (U (t))[·, U (t)] − k, we obtain the expression

t

ΔU (t) = 0

U˜ h (t, s)G(s) ds,

0 < t ≤ T.

8 Stationary Solutions to Quasilinear Equations

311

Then, in view of (6.117), (6.118), (6.123), and (6.124), we have

t ΔU (t) Dγ / h X ≤ C (t − s)−γ s β−α−1 A(U0 )β h X 0

+ ΔA (U (s); Uh (s) − U (s)) L(D,X) Uh (t) − U (t) −1 Y ds. We can now repeat the same arguments as in the preceding subsection to verify that, as h X → 0 (a fortiori as h Dγ → 0), 0 ≤ ΔU (t) Dγ / h Dγ ≤ ΔU (t) Dγ / h X → 0. Thus, under (6.115) and (6.116), S(t) : Dγ → Dγ is Fréchet differentiable in a neighborhood B Dγ (U ; r1 ) of U for any time t such that 0 < t ≤ T , T > 0 being sufficiently small, with the derivative S(t) U0 = U˜ (t, 0), where U˜ (t, s) is an evolution operator for a family of the linear operators A(S(t)U0 ) + A (S(t)U0 )[·, S(t)U0 ] − k,

0 ≤ t ≤ T , U0 ∈ B Dγ (U ; r1 ).

In particular, S(t) U = e−tA , where A = A(U ) + A (U )[·, U ] − k. Let us investigate next the regularity of the derivative S(t) U0 with respect to U0 . For this purpose, let Ui ∈ B Dγ (U ; r1 ), Ui (t) = S(t)Ui , and Ai (t) ≡ A(Ui (t)) for i = 1, 2. We denote also by Ui (t, s) the evolution operator for the operators Ai (t). Similarly, let A˜ i (t) ≡ A(Ui (t)) + A (Ui (t))[·, Ui (t)] − k and denote by U˜ i (t, s) the evolution operator for the operators A˜ i (t). Then, it is derived that U˜ 2 (t, 0) − U˜ 1 (t, 0) L(Dγ ) ≤ C U2 − U1 δDγ ,

0 ≤ t ≤ T,

(6.125)

where δ = min{ γ1−η −α , 1} (cf. Proposition 6.6). That is, S(t) U2 − S(t) U1 L(Dγ ) ≤ C U2 − U1 δDγ ,

0 ≤ t ≤ T ; U1 , U2 ∈ B Dγ (U ; r1 ).

In order to prove (6.125), we write A1 (t)γ [U˜ 2 (t, 0) − U˜ 1 (t, 0)]h

t = A1 (t)γ U˜ 1 (t, τ )[A(U1 (τ )) − A(U2 (τ ))] 0

× A(U2 (τ ))−1 A(U2 (τ ))U˜ 2 (τ, s)h dτ

t + A1 (t)γ U˜ 1 (t, τ ){A (U1 (τ )) − A (U2 (τ ))}[U˜ 2 (τ, 0)h, U1 (τ )] dτ 0

+

t

A1 (t)γ U˜ 1 (t, τ )A (U2 (τ ))[U˜ 2 (τ, 0)h, U1 (τ ) − U2 (τ )] dτ.

0

Then, by (6.89), (6.116), and Proposition 6.6, we immediately verify that the first and second integrals in the right-hand side are estimated by C U2 − U1 δDγ h Dγ .

312

6 Dynamical Systems

Meanwhile, the estimate for the third one is more profound (it is necessary to extend the known estimate (5.39) or (5.23) to the critical case θ = ν = 1). We can in fact show the following proposition. The proof of this proposition will be described in the next subsection. Proposition 6.10 We have U2 (t) − U1 (t) D ≤ Ct γ −1 U2 − U1 δDγ ,

0 < t ≤ T.

(6.126)

Clearly, (6.126) implies that the third integral is also estimated by C U2 − U1 δDγ h Dγ ; hence, (6.125) has been proved. Let us now make the assumption that the operator A = A(U ) + A (U )[·, U ] − k as an operator of X satisfies the spectrum separation condition σ (A) ∩ {λ ∈ C; Re λ = 0} = ∅.

(6.127)

Let σ− (A) = σ (A) ∩ {λ ∈ C; Re λ < 0} and σ+ (A) = σ (A) ∩ {λ ∈ C; Re λ > 0}, and let X = X− ⊕ X+ be the decomposition of X corresponding to the spectrum separation. By the same arguments as in Sect. 6.3, we can verify that the spectrum σ Dγ (S(t) U ) is separated by the unit circle as (6.20) and the exterior spectrum is given by e−tσ− (A) . We have thus shown that, under (6.115) and (6.116), if U satisfies (6.127), then U is a hyperbolic equilibrium of (S(t), Dγ , Dγ ). The exterior spectrum σ Dγ (S(t) U ) ∩ {λ ∈ C; |λ| > 1} is given by e−tσ− (A) . Therefore, by virtue of Theorem 6.9, there exists a smooth local unstable manifold W+ (U ; O) of U with dimension dim X− which is tangent to the subspace U + X− of Dγ at U . Remark 6.4 It is possible to derive analogous results for the general case where the term F (U ) is not necessarily of the form F (U ) = kU under the conditions F (U )h X ≤ L1 h W ,

U ∈ W ∩ B Z (U ; r), h ∈ W ;

[F (U1 ) − F (U2 )]h X ≤ L2 U1 − U2 Z h W , U1 , U2 ∈ W ∩ B Z (U ; r), h ∈ W, where Li > 0 (i = 1, 2) are some constants. Instead of (6.119), however, we have to handle the more complicated problem dV h dt + A(U (t))Vh + A (U (t))[Vh , U (t)] = F (U (t))Vh , 0 < t ≤ T , Vh (0) = h. For constructing an evolution operator for this problem, we must go back to the study of perturbed problems in Chap. 3, Sect. 10 and in fact must generalize the main theorems for the operators B(t) ≡ B1 (t) + B2 (t), each family Bi (t) satisfying conditions (3.117), (3.118), and (3.119) with some exponents 0 < μ˜ i , ν˜ i ≤ 1 depending on i = 1, 2.

8 Stationary Solutions to Quasilinear Equations

313

8.4 Proof of Proposition 6.10 We easily observe that Ui (t) = ekt Ui (t, 0)Ui for i = 1, 2. So, A1 (t)[U2 (t) − U1 (t)] = ekt A1 (t){U2 (t, 0)[U2 − U1 ] + [U2 (t, 0) − U1 (t, 0)]U1 }. Clearly, the first term in the right-hand side is estimated by A1 (t)U2 (t, 0)A2 (0)−γ A2 (0)γ [U2 − U1 ] ≤ Ct γ −1 U2 − U1 D . γ X In the same way as in the proof of Proposition 5.1, the second term is written by

t A1 (t)[U2 (t, 0) − U1 (t, 0)]U1 = A1 (t)U1 (t, τ )[A1 (t) − A2 (t)]U2 (τ, 0)U1 dτ 0

+

t

A1 (t)U1 (t, τ )[A1 (τ ) − A2 (τ )

0

− A1 (t) + A2 (t)]U2 (τ, 0)U1 dτ = J1 (t) + J2 (t). Furthermore, J1 (t) is decomposed into the form

t J1 (t) = [A1 (t)U1 (t, τ ) − A1 (t)e−(t−τ )A1 (t) ][A2 (t) − A1 (t)]U2 (τ, 0)U1 dτ 0

t

+

A1 (t)e−(t−τ )A1 (t) [A2 (t) − A1 (t)][1 − e−(t−τ )A2 (τ ) ]U2 (τ, 0)U1 dτ

0 t

+

A1 (t)e−(t−τ )A1 (t) [A2 (t) − A1 (t)][e−(t−τ )A2 (τ ) − U2 (t, τ )]

0

× U2 (τ, 0)U1 dτ + [1 − e−tA1 (t) ][A2 (t) − A1 (t)]U2 (t, 0)U1 . t Here, we used the formula 0 A1 (t)e−(t−τ )A1 (t) dτ = 1 − e−tA1 (t) . Then, by applying the estimates for the evolution operator and (6.100), we deduce the estimate J1 (t) X ≤ Ct γ −1 U2 − U1 Dγ . For estimating J2 (t), we write [Ai (t) − Ai (τ )]U˜ =

1

A (θ Ui (t) + (1 − θ )Ui (τ ))[Ui (t) − Ui (τ ), U˜ ] dθ

0

for i = 1, 2. Then, {[A2 (t) − A2 (τ )] − [A1 (t) − A1 (τ )]}U˜

1 = A (θ U1 (t) + (1 − θ )U1 (τ ))[(U2 (t) − U2 (τ )) − (U1 (t) − U1 (τ )), U˜ ] dθ 0

314

6 Dynamical Systems

+

1

{A (θ U2 (t) + (1 − θ )U2 (τ ))

0

− A (θ U1 (t) + (1 − θ )U1 (τ ))}[U2 (t) − U2 (τ ), U˜ ] dθ. Then, by (6.115) and (6.116), there exists a positive exponent ε > 0 such that {[A2 (t) − A2 (τ )] − [A1 (t) − A1 (τ )]}U˜ X ≤ C(t − τ )ε U2 − U1 δDγ U˜ D . This estimate furthermore implies that J2 (t) X ≤ Ct γ −1 U2 − U1 δDγ . We have thus accomplished the proof of Proposition 6.10.

Notes and Further Researches We refer the reader for the general matters of dynamical systems in infinitedimensional spaces to the books by Ladyzhenskaya [Lad91], Babin–Vishik [BV92], Temam [Tem97], and Robinson [Rob01]. Especially, for the attractors, we cite the literature Cholewa–Doltko [CD00], Chepyzhov–Vishik [CV02], and Miranville– Zelik [MZ08]. For the inertial manifolds, see Foias–Sell–Temam [FST88] and [Tem97, Chap. 8]. In constructing smooth stable and unstable manifolds in Theorem 6.8, we followed the methods due to Wells [Wel76], cf. also [BV92, Chap. 5] and [Tem97, Chap. VII: Part B]. The notion of exponential attractors was founded by Eden–Foias–Nicolaenko– Temam [EFNT94]. The squeezing property (6.45)–(6.46) was also introduced by the founders. Efendiev–Miranville–Zelik [EMZ00] introduced the condition of compact perturbation of a contraction operator for constructing exponential attractors. (Actually in [EMZ00], δ was taken as δ = 0. For the general case, see Takei and Yagi [TY06].) The two types of conditions (6.28)–(6.29) and (6.38)–(6.39) may seem to be very close. We see, however, that the squeezing property (6.45)–(6.46) implies (6.28)–(6.29) but not (6.38)–(6.39). In the meantime, there exists a semigroup S(t) which fulfills (6.28)–(6.29) but does not (6.38)–(6.39) (see Chap. 13). It is in fact possible to verify the existence of exponential attractors in various mathematical models. After the notion had been introduced by Eden et al., almost hundred papers were already devoted to constructing exponential attractors for dissipative nonlinear systems. We here quote only some results concerning the reaction– diffusion equations [BN95, EMZ04, GP07], the damped wave equations [FGMZ04, MPZ07], Navier–Stokes equations [EFN93, EFNS93, KB06], Ginzburg–Landau equation [Gao95, LL08], Cahn–Hilliard equations [Gal08, MZ04], the phase field systems [BHC93, MZ02, GP05], and others [Du03, EM93, FaN96, Gal98]. In Chaps. 8, 9, 10, 12, 13, 14, and 15, we will also utilize the squeezing property and the two types of compact perturbation of a contraction operator in order to construct exponential attractors for the dynamical systems determined from the model equations under consideration.

Notes and Further Researches

315

The continuous dependence of exponential attractors on a parameter was also investigated by the founders Eden–Foias–Nicolaenko–Temam in [EFNT94, Chap. 4]. Theorem 6.17 then provides us a very general result on this parameter dependence. Indeed, the theorem indicates that, if the semigroup Sξ (t) satisfies the compact perturbation condition of type I uniformly for the parameter ξ and if Sξ (t) converges to S0 (t) only on a finite time interval [0, t ∗ ], then the corresponding exponential attractor Mξ converges to M0 . This result was obtained by Efendiev–Yagi [EY05]. A similar result can be proved under the compact perturbation condition of type II, too.

Chapter 7

Numerical Analysis

This chapter is devoted to numerical analysis for nonlinear abstract parabolic evolution equations. We consider approximate problems by discretizing the spatial variable in the Cauchy problems of the forms (4.1) and (5.43) in Banach spaces. We investigate the order of convergence for the approximation problems. In the first half, we show local estimates of convergence. We introduce the approximate equations ξ dU ξ ), ξ = Fξ (U + Aξ U dt

0 < t < ∞,

and ξ dU ξ )U ξ = Fξ (U ξ ), 0 < t < ∞, + Aξ (U dt in Banach spaces Xξ , where 0 ≤ ξ ≤ 1, for the semilinear and quasilinear equations in (4.1) and (5.43), respectively. Here, Xξ denote a family of approximate spaces such that X0 = X is an underlying Banach space in which (4.1) and (5.43) are considered, and for 0 < ξ ≤ 1, Xξ are finite-dimensional subspaces of X. Under suitable structural assumptions on Xξ , Aξ , Aξ (U ), and Fξ (U ), we derive order estimates of convergence of approximate solutions in a finite interval [0, T ]. In the second half, we show some global estimates. We assume that the Cauchy problems for the approximate equations mentioned above have global solutions and determine dynamical systems (Sξ (t), Xξ , Xξ ) for 0 ≤ ξ ≤ 1. Let us assume furthermore that Sξ (t) satisfy conditions (6.38)–(6.39) uniformly for 0 ≤ ξ ≤ 1. Consequently, each dynamical system (Sξ (t), Xξ , Xξ ) has exponential attractors. We then establish that the convergence of semigroups Sξ (t) → S0 (t) as ξ → 0, together with that of phase spaces Xξ → X0 , implies the convergence of exponential attractors with respect to the distance (1.3).

1 Semilinear Evolution Equations We consider the Cauchy problem for a semilinear equation A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_7, © Springer-Verlag Berlin Heidelberg 2010

317

318

7 Numerical Analysis

dU dt

+ AU = F (U ),

0 < t < ∞,

(7.1)

U (0) = U0 ,

in a Banach space X. Here, A is a sectorial operator of X with angle ωA < π2 , that is, satisfies (2.92) and (2.93). The operator F is a nonlinear operator from D(Aβ ) into X, where 0 < β < 1. We assume that F satisfies the Lipschitz condition F (U ) − F (V ) ≤ ϕ(Aβ U + Aβ V )Aβ (U − V ),

U, V ∈ D(Aβ ), (7.2)

where ϕ(·) is some continuous increasing function. This means that (4.2) is satisfied with β = η. The initial value U0 is taken in D(Aγ ), where γ is such that β ≤ γ ≤ 1. By Theorems 4.1 and 4.2, there exists a unique solution to (7.1) on an interval [0, T ] in the function space: U ∈ C((0, T ]; D(A)) ∩ C([0, T ]; D(Aγ )) ∩ C1 ((0, T ]; X), (7.3) AU ∈ Fγ ,σ ((0, T ]; X), with 0 < σ < 1 − β.

1.1 Approximate Problems Let {Xξ }0dist(x,∂σ )

dy π = dist(x, ∂σ )−2ε , 2+2ε ε |x − y|

,

3 Two-Dimensional Finite Element Methods

we see that σ

Ω\σ

dy |x − y|2+2ε

q

333

dx ≤ σ

≤ Cε

Cε dist(x, ∂σ )−2εq dx

rσ

−rσ

rσ

η−2εq dη dξ = Cε,q rσ2−2εq .

0

uξ − ∂j uL2p (σ ) , we notice by Lemma 7.2 below and (7.49) In order to estimate ∂j that 1 1 1− p1 p p −1 ∂j uξ − ∂j uL2p (σ ) ≤ Cp uH 2 (σ u − u + r u − u 1 ξ ξ σ H (σ ) ) H 1 (σ ) 1 1 1− p1 −1 p +rp (r u ) · r u ≤ Cp uH 2 (σ 2 2 σ σ σ H (σ ) H (σ ) ) 1

= Cp rσp uH 2 (σ ) . Hence, we have σ ×(Ω\σ )

σ ∈τξ

≤

|∂j uξ (x) − ∂j u(x)|2 dx dy |x − y|2+2ε

2

2

Cε rσp u2H 2 (σ ) · rσq

−2ε

≤ Cε ξ 2−2ε u2H 2 .

σ

In view of (1.70), we have thus proved the proposition. Lemma 7.2 For 1 ≤ p < ∞, wL2p (σ ) ≤ Cp

rσ ρσ

1− 1

p

1− 1

1

1

∇wL2 (σp ) wLp 2 (σ ) + ρσp

−1

wL2 (σ ) ,

w ∈ H 1 (σ ), σ ∈ τξ ,

(7.54)

Cp > 0 being a constant independent of σ and ξ . Proof Let σ0 = {(z1 , z2 ) ∈ R2 ; z1 > 0, z2 > 0, z1 + z2 < 1} be the reference triangle, and φσ : σ0 → σ an affine mapping. Then, we have 1

1

Cq−1 |σ | q w ◦ φσ Lq (σ0 ) ≤ wLq (σ ) ≤ Cq |σ | q w ◦ φσ Lq (σ0 ) and − q1

∇(w ◦ φσ )Lq (σ0 ) ≤ Cq rσ |σ |

∇wLq (σ )

for any 1 ≤ q ≤ ∞. Now, (7.54) is readily verified by applying Theorem 1.37 (with p = n = 2) to w ◦ φσ .

334

7 Numerical Analysis

3.3 Projection Operator Pξ Consider Xξ = (Cξ (Ω), · L2 ) as a closed subspace of L2 (Ω). Let Pξ : L2 (Ω) → Xξ be the orthogonal projection. By the definition, if u ∈ C(Ω), then (1 − Pξ )uL2 = min u − vL2 ≤ u − Πξ uL2 . v∈Xξ

Therefore, by (7.53) (s = 0), if u ∈ H 2 (Ω), then (1 − Pξ )uL2 ≤ Cξ 2 uH 2 . More generally, for 0 ≤ s < 32 , we obtain by (7.50) and (7.53) that (1 − Pξ )uH s ≤ (1 − Πξ )uH s + (Πξ − Pξ )uH s ≤ Cs [ξ 2−s uH 2 + ξ −s (Πξ − Pξ )uL2 ] ≤ Cs ξ 2−s uH 2 , u ∈ H 2 (Ω).

(7.55)

As is seen, we have (1 − Pξ )uL2 ≤ Cξ 2 AuL2 for u ∈ D(A). We then use the Heinz and Kato inequality, Theorem 2.31, with X = Y = L2 (Ω) and T = 1 − Pξ , B = 1. Then, 1 − Pξ is a bounded operator from D(Aθ ) into L2 (Ω) whose operator norm is estimated by θ θ (1 − Pξ )uL2 ≤ C1 − Pξ 1−θ L(L2 ,L2 ) 1 − Pξ L(D(A),L2 ) A uL2 ,

u ∈ D(Aθ ).

Therefore, for 0 ≤ θ ≤ 1, (1 − Pξ )uL2 ≤ Cξ 2θ Aθ uL2 ,

u ∈ D(Aθ ).

(7.56)

In view of (7.48), (7.55) shows that assumptions (7.4) and (7.23) are actually fulfilled for any 0 < β < 34 with Oξ = O(ξ 2 ).

3.4 Ritz Operator Rξ Let Zξ = (Cξ (Ω), · H 1 ). Since a(u, v) is symmetric, a(u, v) can be regarded as an inner product of Zξ . Hence, for each u ∈ H 1 (Ω), there exists a unique vector Rξ u ∈ Zξ such that a(Rξ u, vξ ) = a(u, vξ ) for all v ξ ∈ Zξ . Then, Rξ uH 1 ≤ (M/ δ )uH 1 , that is, Rξ is a bounded operator from H 1 (Ω) into Zξ . Operator Rξ is called the Ritz operator. If u ∈ D(A), then a(u, vξ ) = (Au, vξ ) = (Pξ Au, vξ ) = Aξ A−1 vξ = a A−1 vξ . ξ Pξ Au, ξ Pξ Au, Therefore, Rξ u = A−1 ξ Pξ Au.

3 Two-Dimensional Finite Element Methods

335

For u ∈ H 1 (Ω), since a((1 − Rξ )u, (Πξ − Rξ )u) = 0, we have a((1 − Rξ )u, (1 − Rξ )u) = a((1 − Rξ )u, (1 − Πξ )u). By (2.6) and (2.7), − Rξ )uH 1 (1 − Πξ )uH 1 . δ (1 − Rξ )u2H 1 ≤ a((1 − Rξ )u, (1 − Rξ )u) ≤ M(1 Therefore, by (7.53) (s = 1), (1 − Rξ )uH 1 ≤ C(1 − Πξ )uH 1 ≤ Cξ uH 2 ,

u ∈ H 2 (Ω).

Furthermore, it is seen (Aubin–Nitsche’s trick) that (1 − Rξ )u2L2 = a(A−1 (1 − Rξ )u, (1 − Rξ )u) = a((1 − Rξ )A−1 (1 − Rξ )u, (1 − Rξ )u) ≤ C1 − Rξ L(H 2 ,H 1 ) A−1 L(L2 ,H 2 ) × (1 − Rξ )uL2 (1 − Rξ )uH 1 . Hence, (1 − Rξ )uL2 ≤ Cξ (1 − Rξ )uH 1 ≤ Cξ 2 uH 2 ,

u ∈ H 2 (Ω).

More generally, for 0 ≤ s < 32 , (1 − Rξ )uH s ≤ (1 − Πξ )uH s + (Πξ − Rξ )uH s ≤ Cs [ξ 2−s uH 2 + ξ −s (Πξ − Rξ )uL2 ] ≤ Cs ξ 2−s uH 2 , In addition, for 0 ≤ θ

0 being a constant independent of ξ .

u ∈ D(Aθ ),

(7.58)

336

7 Numerical Analysis

We have thus verified that assumptions (7.6) and (7.29) are fulfilled with Oξ = O(ξ 2 ).

3.5 Domains of Fractional Powers of Aξ 1

1

Since a(u, v) is symmetric, Theorem 2.34 yields that D(Aξ2 ) = Zξ and Aξ2 uξ L2 ≤ C uξ H 1 . Therefore, in view of (7.50), we have 1 1 uξ L2 ≤ C Aξ2 uξ H 1 ≤ Cξ −1 Aξ2 uξ L ≤ Cξ −1 uξ H 1 ≤ Cξ −2 uξ L2 . Aξ 2

Therefore, the moment inequality (2.117) yields that, for 0 ≤ θ ≤ 1, θ −2θ A uξ L2 , uξ ∈ Xξ . ξ uξ L ≤ Cξ 2

(7.59)

For u ∈ D(A), we have Aξ Pξ u = Aξ (Pξ − Rξ )u + Aξ Rξ u = [Aξ (Pξ − Rξ )A−1 + Pξ ]Au. Hence, by (7.56), (7.58), and (7.59), we have Aξ Pξ uL2 ≤ CAuL2 , u ∈ D(A). The Heinz and Kato inequality then provides that, for 0 ≤ θ ≤ 1, θ A Pξ u ≤ CAθ uL , u ∈ D(Aθ ). (7.60) 2 ξ L 2

Conversely, for 0 ≤ θ < 34 , we observe that

θ−1 uξ , v = a uξ , Aθ−1 v = a uξ , Rξ Aθ−1 v = Aθξ uξ , A1−θ v Aθ ξ Rξ A θ−1 θ−1 = Aθξ uξ , A1−θ v + Aθξ uξ , A1−θ v . ξ (Rξ − Pξ )A ξ Pξ A

Therefore, by (7.56), (7.58), (7.59), and (7.60), θ A uξ , v ≤ C Aθξ uξ L ξ 2(θ−1) (Rξ − Pξ )Aθ−1 vL2 + A1−θ Pξ Aθ−1 v L ξ 2 2 θ ≤ Cθ Aξ uξ L vL2 . 2

uξ L2 ≤ Cθ Aθξ uξ L2 . Thus, for 0 ≤ θ < 34 , This implies that Aθ θ A uξ

L2

≤ Cθ Aθξ uξ L , 2

uξ ∈ Xξ .

(7.61)

Estimate (7.61) then shows that assumptions (7.5) and (7.28) are fulfilled for α < β < 34 . Similarly, (7.59) shows that assumptions (7.7) and (7.31) are fulfilled with Oξ−1 = O(ξ −2 ). Finally, (7.60) means that, if u0 ∈ D(Aβ ), then β

sup0 0 being some constant. Here, d(X1 , X2 ) is the distance defined by (1.3), and Oξ stands for the rate of approximations and tends to 0 as ξ → 0. For 0 ≤ ξ ≤ 1, the operator Sξ is a nonlinear operator from Xξ into itself which satisfies uniformly the compact perturbation conditions (6.38)–(6.39) of type II, i.e., Sξ U − Sξ V X ≤ δU − V X + Kξ U − Kξ V X , Kξ U − Kξ V Z ≤ LU − V X ,

U, V ∈ Xξ , (7.63)

U, V ∈ Xξ ,

(7.64)

with some fixed 0 ≤ δ < 12 and some uniform constant L > 0. In addition, Kξ is an operator from Xξ into a second Banach space Z with norm · Z which is compactly embedded in X. Clearly, (7.63) and (7.64) imply the uniform Lipschitz condition Sξ U − Sξ V X ≤ (δ + cL)U − V X ,

U, V ∈ Xξ ,

(7.65)

where c > 0 is an embedding constant such that · X ≤ c · Z . We assume also that, as ξ → 0, Sξ converges to S0 at a rate sup Sξ U − S0 U X ≤ D2 Oξ ,

U ∈Xξ

0 < ξ ≤ 1,

(7.66)

D2 > 0 being some constant. We already know by Theorem 6.13 that, under (7.63) and (7.64), each system (Sξn , Xξ , X) has exponential attractors. We will then show that the convergence (7.62) and (7.66) provide continuous dependence of the exponential attractors as ξ → 0. Theorem 7.3 Under (7.63) and (7.64), let (7.62) and (7.66) be satisfied. Then, there exist exponential attractors M∗ξ for (Sξn , Xξ , X), 0 ≤ ξ ≤ 1, respectively, for which the estimate (7.67) d M∗ξ , M∗0 ≤ C ∗ Oξκ , 0 < ξ ≤ 1, holds with some exponent 0 < κ < 1 and some constant C ∗ > 0.

338

7 Numerical Analysis

Proof We will remember the method employed for proving Theorem 6.13 in constructing exponential attractors of the dynamical system (S0n , X0 , X). Let 0 < θ < 1−2δ 2L , and let a = 2(δ +Lθ ) < 1. Let us introduce for n = 0, 1, 2, . . . , an Ra n -radius covering of S0n X0 by a finite number of closed balls of X with centers in S0n X0 , where R is the diameter of X0 . More precisely, let n

S0n X0

⊂

Nθ

X

B (Wn,i ; Ra n ),

Wn,i ∈ S0n X0 ,

(7.68)

i=1

where Nθ is the minimal number of balls of X with radii θ which can cover the unit Z closed ball B (0; 1) of Z centered at the origin. Then, the set defined by M∗0 =

∞

S0n P0 ,

where P0 = Wn,i ; 0 ≤ n < ∞, 1 ≤ i ≤ Nθn ,

n=0

gives an exponential attractor for (S0n , X0 , X). We are then going to construct an exponential attractor for (Sξn , Xξ , X), 0 < ξ ≤ 1, on the basis of the set P0 . Since Wn,i ∈ S0n X0 , there exists Vn,i ∈ X0 such that Wn,i = S0n Vn,i . Furthermore, by (7.62), there exists Vn,i,ξ ∈ Xξ such that Vn,i − Vn,i,ξ X ≤ D1 Oξ . Setting Wn,i,ξ = Sξn Vn,i,ξ , we use Lemma 7.3 be presented below. Then, in view of (7.66), Wn,i − Wn,i,ξ X ≤ L˜ n Oξ with some constant L˜ > 1. Therefore, we obtain from (7.68) the new covering n

S0n X0 ⊂

Nθ

X B (Wn,i,ξ ; Ra n + L˜ n Oξ ),

Wn,i,ξ ∈ Sξn Xξ .

(7.69)

i=1

In the meantime, Lemma 7.3, together with (7.62) and (7.66), yields also that Sξn Xξ ⊂ W(L˜ n Oξ ) S0n X0 , (7.70) where W(L˜ n Oξ ) (S0n X0 ) denotes the L˜ n Oξ -neighborhood of S0n X0 in X. Therefore, it follows that n

Sξn Xξ ⊂

Nθ

X B (Wn,i,ξ ; Ra n + 2L˜ n Oξ ),

Wn,i,ξ ∈ Sξn Xξ .

i=1

Lemma 7.3 Let L0 = δ + cL. Then, for every n = 0, 1, 2, . . . , n S Uξ − S n U0 ≤ Ln+1 − 1 /(L0 − 1) sup Sξ V − S0 V X ξ 0 0 X

+ Uξ − U0 X ,

V ∈Xξ

Uξ ∈ Xξ , U0 ∈ X0 .

(7.71)

4 Convergence of Exponential Attractors

339

Proof We have Sξn Uξ − S0n U0 = Sξn Uξ − S0n Uξ + S0n Uξ − S0n U0 =

n

S0n−i Sξi Uξ − S0n−i+1 Sξi−1 Uξ + S0n Uξ − S0n U0 .

i=1

By the Lipschitz condition (7.65) (ξ = 0), n n S U ξ − S n U 0 ≤ Sξ S i−1 Uξ − S0 S i−1 Uξ + Ln Uξ − U0 X Ln−i ξ 0 0 ξ ξ 0 X X i=1

≤

n

Ln−i 0

i=0

sup Sξ V − S0 V X + Uξ − U0 X .

V ∈Xξ

Hence, the desired estimate is obtained.

ξ be the largest Let us accomplish the proof of theorem. For each 0 < ξ ≤ 1, let N n n n n ˜ ˜ ξ and Ra n < integer satisfying 2L Oξ ≤ Ra , i.e., 2L Oξ ≤ Ra for all 0 ≤ n ≤ N n ˜ 2L Oξ for all Nξ < n < ∞. If we use (7.71) for 0 ≤ n ≤ Nξ , then n

Sξn Xξ

⊂

Nθ

X

B (Wn,i,ξ ; 2Ra n ),

Wn,i,ξ ∈ Sξn Xξ .

i=1

ξ with central points WNξ ,i,ξ , 1 ≤ i ≤ N Nξ , and following the Starting from n = N θ same procedure as in Step 1 of the proof of Theorem 6.12, we can construct inducξ < n < ∞, a 2Ra n -radius covering of S n Xξ such that tively for every N ξ n

Sξn Xξ

⊂

Nθ

X B (W˜ n,i,ξ ; 2Ra n ),

W˜ n,i,ξ ∈ Sξn Xξ .

i=1

Put ξ , 1 ≤ i ≤ Nθn ∪ W˜ n,i,ξ ; N ξ < n < ∞, 1 ≤ i ≤ Nθn . Pξ = Wn,i,ξ ; 0 ≤ n ≤ N Then, for all n = 0, 1, 2, . . . ,

h Sξn Xξ , Pξ ≤ 2Ra n .

(7.72)

n Moreover, by the same reason as for M∗0 , the set M∗ξ = ∞ n=0 Sξ Pξ gives an expon nential attractor for (Sξ , Xξ , X). It now remains to estimate the distance d(M∗ξ , M∗0 ). By definition, M∗ξ

⊂

Nξ n=0

Sξn

Nξ +1 k ξ − n, 1 ≤ i ≤ Nθ Sξ Wk,i,ξ ; 0 ≤ k ≤ N Xξ .

340

7 Numerical Analysis

ξ + 1), together with (7.68) (n = N ξ + 1), yields that The inclusion (7.70) (n = N N +1 h Sξ ξ Xξ , M∗0 ≤ h W ≤h W

Nξ +1 ∗ X 0 , M0 S0

Nξ +1 S0 X0 , P0 ≤ L˜ Nξ +1 Oξ + Ra Nξ +1

(L˜ Nξ +1 Oξ ) (L˜ Nξ +1 Oξ )

≤ 3L˜ Nξ +1 Oξ . Meanwhile, due to Lemma 7.3, n S Wk,i,ξ − S n Wk,i ≤ Ln+1 − 1 /(L0 − 1) ξ 0 0 X × D2 Oξ + L˜ k Oξ ≤ C L˜ Nξ +1 Oξ ; therefore, it follows that d Sξn Wk,i,ξ , M∗0 ≤ C L˜ Nξ +1 Oξ ,

(7.73)

ξ − n, 1 ≤ i ≤ Nθk . 0≤k≤N

Thus, we have proved that h(M∗ξ , M∗0 ) ≤ C L˜ Nξ +1 Oξ . ξ = log(R/2)−log Oξ , we easily observe that L˜ Nξ +1 Oξ ≤ CO κ with the As N ξ ˜ exponent κ =

log L−log a − log a . ˜ log L−log a

The converse estimate h(M∗0 , M∗ξ ) ≤ COξκ is also verified in a similar way. Indeed, by definition, M∗0

⊂

Nξ

S0n

Nξ +1 k ξ − n, 1 ≤ i ≤ Nθ S0 Wk,i ; 0 ≤ k ≤ N X0 .

n=0

ξ ) to obtain that We use (7.69) (n = N N +1 N N h S0 ξ X0 , M∗ξ ≤ h S0 ξ X0 , M∗ξ ≤ h S0 ξ X0 , Pξ

≤ L˜ Nξ Oξ + Ra Nξ ≤ (1 + 2a −1 )L˜ Nξ +1 Oξ . Meanwhile, from (7.73) we conclude that ξ − n, 1 ≤ i ≤ Nθk . d S0n Wk,i , M∗ξ ≤ C L˜ Nξ +1 Oξ , 0 ≤ k ≤ N Hence, h(M∗0 , M∗ξ ) ≤ COξκ with the same exponent κ as above.

4.2 Continuous Case Let (Sξ (t), Xξ , X) be a family of continuous dynamical systems parameterized by 0 ≤ ξ ≤ 1 in a Banach space X. The phase spaces Xξ are all closed subsets of X with

4 Convergence of Exponential Attractors

341

the relation Xξ ⊂ X0 for 0 < ξ ≤ 1, and Sξ (t) are continuous nonlinear semigroups acting on Xξ , respectively. We assume that there exists a compact set B of X for which the set Xξ ∩ B is an absorbing set of (Sξ (t), Xξ , X) for every 0 ≤ ξ ≤ 1. Furthermore, we assume that Xξ ∩ B absorbs itself uniformly for ξ , that is, there exists a uniform time t ∗ > 0 such that Sξ (t)(Xξ ∩ B) ⊂ Xξ ∩ B

for every t ∗ ≤ t < ∞ and 0 ≤ ξ ≤ 1.

(7.74)

We then set Bξ = Xξ ∩ B. Clearly, Bξ is a compact set of X with the relation Bξ ⊂ B0 . Furthermore, if Sξ is set by Sξ = Sξ (t ∗ ), then (Sξn , Bξ , X) defines a discrete dynamical system for each 0 ≤ ξ ≤ 1. Let us assume that conditions (7.62), (7.63), and (7.64) are satisfied for (Sξn , Bξ , X). In addition, let Sξ (t) satisfy the Lipschitz condition Sξ (t)U − Sξ (s)V X ≤ L1 (|t − s| + U − V X ),

0 ≤ s, t ≤ t ∗ , U, V ∈ Bξ , (7.75)

with some uniform constant L1 > 0. Finally, instead of (7.66), let the convergence condition sup

sup Sξ (τ )U − S0 (τ )U X ≤ D3 Oξ ,

0≤τ ≤t ∗ U ∈Bξ

0 < ξ ≤ 1,

(7.76)

be satisfied with some constant D3 > 0. We can then prove a convergence theorem of exponential attractors for the continuous case, namely, Theorem 7.4 below. In fact, by applying Theorem 7.3 to (Sξn , Bξ , X), we deduce that there exist exponential attractors M∗ξ , 0 ≤ ξ ≤ 1, enjoying property (7.67). For each 0 ≤ ξ ≤ 1, we then set Mξ =

Sξ (τ )M∗ξ .

(7.77)

0≤τ ≤t ∗

It is seen that this Mξ gives an exponential attractor for the continuous system (Sξ (t), Xξ , X). Indeed, by the same arguments as in the proof of Theorem 6.15, Mξ is a compact set of X with finite fractal dimension less than dF (M∗ξ ) + 1 and contains the global attractor Aξ of (Sξ (t), Xξ , X), i.e., Aξ ⊂ Mξ ⊂ Xξ . Similarly, Mξ is observed to be invariant under Sξ (t) for every 0 < t < ∞. Finally, the exponential attraction is verified by the following proposition. Proposition 7.3 There exist an exponent k > 0 such that, for any bounded set B of Xξ , h(Sξ (t)B, Mξ ) ≤ CB e−kt ,

tB ≤ t < ∞,

with suitable time tB ≥ 0 and a constant CB > 0 depending on B.

(7.78)

342

7 Numerical Analysis

Proof First, let us consider the case where B = Bξ . Let 0 < t < ∞, and let t = nt ∗ + τ , where n ≥ 0 is an integer and 0 ≤ τ < t ∗ . Since Mξ ⊃ Sξ (τ )M∗ξ , we have h(Sξ (t)Bξ , Mξ ) ≤ h Sξ (t)Bξ , S(τ )M∗ξ . Using (7.75) (t = s = τ ) and noting (7.72), we see that h(Sξ (t)Bξ , Mξ ) ≤ h Sξ (τ )Sξ (t ∗ )n Bξ , S(τ )M∗ξ ≤ L1 h Sξ (t ∗ )n Bξ , M∗ξ ≤ 2L1 Ra n ≤ 2L1 Ra −1 e−(log a

−1 /t ∗ )t

,

0 ≤ t < ∞.

This shows that (7.78) is valid for Bξ with k = log a −1 /t ∗ , tBξ = 0 and CBξ = 2L1 R/a. Secondly, let us consider the general case. Let B be any bounded set of Xξ . By definition, Sξ (t)B ⊂ Bξ for any t such that t ≥ tB , tB ≥ 0 being a suitable time. Then, h(Sξ (t)B, Mξ ) = h(Sξ (t − tB )Sξ (tB )B, Mξ ) ≤ h(Sξ (t − tB )Bξ , Mξ ) ≤ CBξ e−k(t−tB ) = CBξ ektB e−kt ,

tB ≤ t < ∞.

Hence, the assertion of proposition is verified.

In this way, the set Mξ given by (7.77) has been shown to be an exponential attractor of (Sξ (t), Xξ , X) for each ξ . We can then verify that Mξ also enjoys the same property as (7.67). Theorem 7.4 Under (7.63), (7.64), (7.74), and (7.75), assume (7.62) and (7.76) for the phase spaces Bξ and the semigroups Sξ (t), respectively. Then, there exists an exponential attractor Mξ for (Sξ (t), Xξ , X) for which the estimate d(Mξ , M0 ) ≤ COξκ ,

0 < ξ ≤ 1,

(7.79)

holds with some exponent 0 < κ < 1 and constant C > 0. Proof It now suffices to verify under (7.67) that the set Mξ given by (7.77) has property (7.79). Let Uξ ∈ Mξ be any element. Then, Uξ = Sξ (τ )Uξ∗ with some 0 ≤ τ ≤ t ∗ and some Uξ∗ ∈ M∗ξ . By (7.67), there exists an element U0∗ ∈ M∗0 such that d(Uξ∗ , U0∗ ) ≤ C ∗ Oξκ . Put U0 = S0 (τ )U0∗ ∈ M0 . Then, by (7.76) and (7.75), d(Uξ , M0 ) ≤ d(Uξ , U0 ) = d Sξ (τ )Uξ∗ , S0 (τ )U0∗ ≤ d Sξ (τ )Uξ∗ , S0 (τ )Uξ∗ + d S0 (τ )Uξ∗ , S0 (τ )U0∗

Notes and Further Researches

343

≤ D3 Oξ + L1 C ∗ Oξκ . Thus, we obtain that h(Mξ , M0 ) ≤ COξκ . By the same argument, we can observe also h(M0 , Mξ ) ≤ COξκ . Hence, (7.79) has been verified.

Notes and Further Researches The local convergence theorems, Theorems 7.1 and 7.2, for semilinear and quasilinear equations, respectively, have been obtained by the semigroup methods. The methods are based on the fact that the sectorial operators with angles < π2 acting in infinite-dimensional spaces can be approximated by those with the similar angles < π2 in finite-dimensional spaces. Those were founded by Fujita–Mizutani [FM76], Baker–Bramble–Thomée [BBT77], Brenner–Thomée [BT79], and Piskarev [Pis79], cf. also Ushijima [Ush75, Ush79], and were developed by Crouzeix–Larsson– Piskarev–Thomée [CLPT93], Palencia [Pal93, Pal95], and others. In the meanwhile, the methods were applied to nonautonomous linear parabolic equations by Suzuki [Suz78, Suz82] and Sammon [Sam82, Sam83] and to semilinear parabolic equations by Keeling [Kee90]. Theorems 7.1 and 7.2 were obtained by Nakaguchi–Yagi [NY99b, NY01, NY02]. The alternative way for constructing approximate solutions for parabolic equations is the variational methods, see the books by Dautray–Lions [DL91]. For the linear problems, we quote Helfrich [Hel74], Bramble–Schatz–Thomée–Wahlbin [BSTW77] and Savaré [Sav93], and for the nonlinear problems, Lubich–Ostermann [LO93, LO95] and Nakaguchi–Yagi [NY03]. In Sects. 1 and 2, we handled only semidiscretization on the spatial variables. However, it is known that the usual discretization schemes for ordinary differential equations in finite-dimensional spaces can be properly extended for ordinary evolution equations in infinite-dimensional spaces, too, cf. Le Roux [LeR79] and Ostermann–Roche [OR92]. In the papers Lubich–Ostermann [LO95] and Nakaguchi–Yagi [NY99a], the Runge–Kutta methods (cf. Butcher [But87] and Iserles [Ise96]) were applied to the Cauchy problem for quasilinear abstract evolution equations in Banach spaces, and some convergence theorems were obtained. These results show that the Runge–Kutta methods are available for the approximate problems (7.32) in order to build full-discretization schemes for the continuous problem (7.15) as done in [NY99b, NY01, NY02]. The structural assumptions made in Sects. 1 and 2 can actually be verified by concrete discretization schemes. Section 3 was devoted to verifying some of them for the finite element methods in two-dimensional domains. Propositions 7.1 and 7.2 were first verified in [NY02]. In higher-dimensional domains, however, the verification is left to be studied. We may have to use higher-order finite element methods for obtaining a similar result as Proposition 7.1. For the finite element methods, we quote the monographs Ciarlet [Cia78], Fujita–Suzuki [FS91], and Thomée [Tho97].

344

7 Numerical Analysis

Theorems 7.3 and 7.4 were proved by modifying the arguments in Efendiev–Yagi [EY05] (cf. also [AY04b]). As seen in the subsequent chapters, the positivity of solutions of the model equations is extremely important in constructing exponential attractors. Nevertheless, the approximate equations do not admit positivity in general (cf. Aida–Yagi [AY03]). Recently, Saito [Sai04, Sai07] and Efendiev–Nakaguchi– Wendland [ENW09] have presented discretization schemes which conserve positivity. Using them, we can show that the approximate problems generate dynamical systems in which exponential attractors equally exist. Theorems 7.3 and 7.4 are then applicable to the family of dynamical systems.

Chapter 8

Semiconductor Models

In 1950, Shockley presented the drift-diffusion model ⎧ ∂u ⎪ ⎨ ∂t = aΔu − ∇ · [μu∇χ] + f (1 − uv) + g(x) ∂v = bΔv + ∇ · [νv∇χ] + f (1 − uv) + g(x) ⎪ ⎩ ∂t 0 = cΔχ − u + v + h(x)

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞),

for describing the flows of electrons and holes in a semiconductor system. Here, the unknown functions u(x, t) and v(x, t) denote the densities of electrons and holes, respectively, in a semiconductor device Ω at time t. Electrons and holes diffuse with positive diffusion rates a > 0 and b > 0. The terms −∇ · [μu∇χ] and ∇ · [νv∇χ] denote the drift-diffusions of electrons and holes, where μ and ν denote their mobilities, respectively. The function χ(x, t) denotes electrostatic potential and is determined by the Poisson equation, where c > 0 is the dielectric constant. Meanwhile, the reaction term f (1 − uv) denotes the effects of generation and recombination of electrons and holes. The functions g(x) ≥ 0 and h(x) are given, and both denote external constraints to the system. In this chapter, we show the global existence of solutions and construct a dynamical system and exponential attractors.

1 Prototype Semiconductor Equations 1.1 Model Equations

We consider the initial-boundary-value problem for a semiconductor model A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_8, © Springer-Verlag Berlin Heidelberg 2010

345

346

8 Semiconductor Models

⎧ ∂u ⎪ ⎪ ∂t = aΔu − μ∇ · [u∇χ] + f (1 − uv) + g(x) ⎪ ⎪ ⎪ ∂v ⎪ ⎪ ⎪ ∂t = bΔv + ν∇ · [v∇χ] + f (1 − uv) + g(x) ⎪ ⎪ ⎪ ⎨0 = cΔχ − u + v + h(x) ⎪ u=v=χ =0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂u = ∂v = ∂χ = 0 ⎪ ∂n ∂n ∂n ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x)

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞), on ΓD × (0, ∞),

(8.1)

on ΓN × (0, ∞), in Ω,

in a two-dimensional bounded domain Ω with Lipschitz boundary ∂Ω. We make the following assumptions. The boundary ∂Ω of Ω is split into two parts ΓD and ΓN in such a way that (2.41) and (2.42) are satisfied. In particular, ΓD is a nonempty open set of ∂Ω. On ΓD , the Dirichlet boundary conditions are imposed on u, v, and χ ; on the other hand, on ΓN , the Neumann boundary conditions are imposed. At x ∈ ΓN , n(x) denotes the outer normal vector of the boundary. The mobilities μ and ν of drift-diffusions are both positive constants (> 0). Physically these mobilities are determined by the scattering process of electrons and holes by the crystal lattices, the impurities, and so on, and may depend on u, v and ∇u, ∇v. In this chapter, however, we are concerned with the simplest case that μ and ν are constants. From a physical point of view, this means that the densities of electrons and holes are not so high. The rates of generation and recombination of electrons and holes are nonnegative constants, and the two rates coincide. These rates are denoted by f ≥ 0. If the two rates are positive constants, then it is always possible to make them equal by suitable normalization. The function g(x) is a given nonnegative function such that 0 ≤ g ∈ L2 (Ω),

(8.2)

and h(x) is a given real function such that h ∈ L∞ (Ω; R).

(8.3)

1.2 Some Preliminary Results Let H˚ D1 (Ω) be the space given by (2.25). We consider the sesquilinear form ∇u · ∇v dx, u, v ∈ H˚ D1 (Ω). a(u, v) = Ω

It is easy to see that this form satisfies (2.6) and (2.7) on H˚ D1 (Ω). For (2.7), we may recall Remark 2.3. Let H˚ D1 (Ω) ⊂ L2 (Ω) ⊂ H˚ D1 (Ω)∗ = HD−1 (Ω) be a triplet of spaces, and let Λ be the associated sectorial operator of HD−1 (Ω) which is an isomorphism from H˚ D1 (Ω) onto HD−1 (Ω) and whose part in L2 (Ω) is a positive

1 Prototype Semiconductor Equations

347

definite self-adjoint operator. As noticed by Theorem 2.5, Λ is considered as a realization of the Laplace operator −Δ in HD−1 (Ω) under the Dirichlet boundary conditions on ΓD and the Neumann boundary conditions on ΓN . 1 By Theorem 2.35, the domain of the square root Λ 2 coincides with L2 (Ω), i.e., 1 D Λ 2 = L2 (Ω). Moreover, for

1 2

(8.4)

≤ θ ≤ 1, it holds that

D(Λθ ) = L2 (Ω), H˚ D1 (Ω) 2θ −1 ⊂ [L2 (Ω), H 1 (Ω)]2θ−1 = H 2θ−1 (Ω).

(8.5)

The part Λ|L2 may not have the optimal shift property that Λu ∈ L2 (Ω) always implies u ∈ H 2 (Ω). According to Theorem 2.10, however, it holds that D(Λ|L2 ) ⊂ Wp10 (Ω) with the estimate u Wp1 ≤ C Λu L2 , 0

u ∈ D(Λ|L2 ),

(8.6)

where p0 is a certain fixed number such that p0 > 2.

2

We finally extend the definition of the drift term ∇ · [u∇χ] for all u ∈ H p (Ω) and all χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω), provided that 2 < p < ∞. Proposition 8.1 Let 2 < p < ∞ be arbitrarily fixed. The correspondence (u, χ) → ∇ · [u∇χ] is continuous from H 2 (Ω) × [D(Λ|L2 ) ∩ Wp1 (Ω)] into L2 (Ω). It is possi2

ble to extend this correspondence continuously from H p (Ω) × [D(Λ|L2 ) ∩ Wp1 (Ω)] into HD−1 (Ω). Proof Let u ∈ H 2 (Ω) and χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω). Then, we observe directly that ∇ · [u∇χ] = ∇u · ∇χ − uΛχ ∈ L2 (Ω) in the sense of distribution. In addition, by the definition of Λ, ∇χ · ∇(uv) dx = ∇χ · (v∇u + u∇v) dx, (Λχ, uv)L2 = Ω

Ω

v ∈ H˚ D1 (Ω),

since u ∈ H 2 (Ω) and v ∈ H˚ D1 (Ω) imply that uv ∈ H˚ D1 (Ω). Noting (1.26), we obtain that u∇χ · ∇v dx, v ∈ H˚ D1 (Ω).

∇ · [u∇χ], vH −1 ×H˚ 1 = (∇ · [u∇χ], v)L2 = − D

D

Ω

2

For u ∈ H p (Ω) and χ ∈ D(ΛL2 ) ∩ Wp1 (Ω), it is observed (due to (1.86)) that |u∇χ∇v| dx ≤ u L Ω

2p p−2

∇χ Lp ∇v L2 ≤ Cp u

2

Hp

χ Wp1 v H˚ 1 . D

348

8 Semiconductor Models 2

Since H 2 (Ω) is a dense subspace of H p (Ω), we are led to extend ∇ · [u∇χ] ∈ 2

HD−1 (Ω) continuously for all u ∈ H p (Ω) and χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω) by the formula

∇ · [u∇χ], vH −1 ×H˚ 1 = − u∇χ · ∇v dx, v ∈ H˚ D1 (Ω). (8.7) D

D

Ω

We also have the same estimate ∇ · [u∇χ] H −1 ≤ C u D

2

Hp

χ Wp1 ,

2

u ∈ H p (Ω), χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω).

(8.8)

Of course, this proposition is available with p = p0 . However, in this case it holds that D(Λ|L2 ) ∩ Wp10 (Ω) = D(Λ|L2 ) due to (8.6). We choose as an underlying space for dealing with (8.1) the product space of HD−1 (Ω), i.e., ϕ X= ; ϕ ∈ HD−1 (Ω) and ψ ∈ HD−1 (Ω) . (8.9) ψ For electrons and holes, the initial densities are taken in the space u0 ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ v0 ∈ L2 (Ω) . K= v0

(8.10)

2 Local Solutions 2.1 Construction of Local Solutions We formulate problem (8.1) as the Cauchy problem for an abstract equation dU 0 < t < ∞, dt + AU = F (U ), (8.11) U (0) = U0 , in the space X. Here, A is a diagonal matrix operator A = diag{aΛ, bΛ} in X with domain u 1 1 ˚ ˚ D(A) = ; u ∈ HD (Ω) and v ∈ HD (Ω) , v where Λ is the operator introduced above. From (8.4) and (8.5) we have 1 D(A 2 ) = L2 (Ω), D(Aθ ) ⊂ H2θ−1 (Ω), 12 < θ ≤ 1

(8.12)

2 Local Solutions

349

(see (1.84) and (1.85)). The nonlinear operator F is given by F (U ) =

−μ∇ · [u∇(cΛ)−1 (−u + v + h(x))] + f (1 − uv) + g(x) ν∇ · [v∇(cΛ)−1 (−u + v + h(x))] + f (1 − uv) + g(x)

with domain D(F ) =

2 2 u ; u ∈ H p0 (Ω) and v ∈ H p0 (Ω) , v

where p0 > 2 is the number for which (8.6) holds. In view of (8.7), F (U ) is well defined. Let η = 12 + p10 < 1. Then, by (8.12) we have D(Aη ) ⊂ D(F ). The initial value U0 is taken in K. We are going to apply Theorem 4.1 with β = 12 and η = 12 + p10 . By virtue of (8.6) and (8.8),

∇ · [u∇Λ−1 (−u + v + h(x))] H −1 ≤ C u H 2/p0 − u + v + h L2 , D

∇ · [v∇Λ−1 (−u + v + h(x))] H −1 ≤ C v H 2/p0 − u + v + h L2 . D

Similarly, uv H −1 = D

sup uvw dx ≤ C

w H˚ 1 ≤1

Ω

D

sup

w H˚ 1 ≤1 D

u L

2p0 p0 −2

v L2 w Lp0

≤ C u H 2/p0 v L2 . Therefore, we verify by (8.12) that 1 1 F (U ) − F (V ) X ≤C A 2 U X + A 2 V X + 1 Aη (U − V ) X 1

+ ( Aη U X + Aη V X + 1)A 2 (U − V )X , U, V ∈ D(Aη ). This shows that (4.2) is fulfilled with β = 12 . Consequently, Theorem 4.1 is available to (8.11). We thus conclude that, for any U0 ∈ K, there exists a unique local solution U to (8.11) in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; X), where TU0 > 0 is determined by the norm U0 L2 only. Moreover, √ t AU (t) X + U (t) L2 ≤ CU0 ,

0 < t ≤ TU 0 ,

where CU0 > 0 is also determined by the norm U0 L2 only.

(8.13)

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8 Semiconductor Models

2.2 Nonnegativity of Solutions For U0 ∈ K, let U = t (u, v) be the local solution obtained above in the function space (8.13). We will prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . Let us first verify that U (t) is real valued. Indeed, it is clear that the complex conjugate U (t) of U (t) is also a local solution of (8.11) with the same initial value; so, by the uniqueness of solution, U (t) = U (t); hence, U (t) is real valued. Let us next introduce auxiliary linear problems. It is not difficult to construct sequences {uk }, {vk } of functions such that uk , vk ∈ C0,1 ([0, TU0 ]; L∞ (Ω))

(8.14)

and that, as k → ∞, uk → u and vk → v in C([0, TU0 ]; L2 (Ω)). (For example, we cut off the values of |u(t)| ≥ k and |v(t)| ≥ k and use the mollifier concerning the variable t.) Using uk and vk , we formulate the linear problem ⎧ ⎨ d U˜ k + AU˜ = B (t)U˜ + F, 0 < t ≤ T , k k k U0 dt (8.15) ⎩U˜ k (0) = U0 , in X. Here, Bk (t) is the family of closed linear operators given by

−μ∇ · [u˜ k ∇χk (t)] − u˜ k vk (t) u˜ Bk (t)U˜ k = , U˜ k = k ∈ D(Bk (t)), v˜k ν∇ · [v˜k ∇χk (t)] − uk (t)v˜k where χk (t) = (cΛ)−1 [−uk (t) + vk (t) + h], with domains D(Bk (t)) ≡ D(Aη ), and F is the vector of X given by F = [f + g(x)]e, where e = t (1, 1). The operators Bk (t) are weaker linear operators than A in the sense that D(A) ⊂ D(Aη ) ≡ D(Bk (t)) for all 0 < t ≤ TU0 . In addition, they are seen from (8.7) to have regularity [Bk (t) − Bk (s)]A−η L(X) ≤ Ck |t − s|,

0 ≤ s, t ≤ TU0 ,

due to (8.14). Then, we can apply Theorem 3.14 with μ˜ = 1 − η and ν˜ = 1 to this problem. It is then deduced that problem (8.15) possesses a unique solution U˜ k in the function space: U˜ k ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; X). Furthermore, by direct calculations (cf. Theorem 4.3), it is possible to verify that t η Aη [U˜ k (t) − U (t)] X ≤ C

1 sup A 2 [Uk (t) − U (t)]X ,

0≤t≤TU0

that is, as k → ∞, U˜ k (t) converges to U (t) for every t ∈ (0, TU0 ] in D(Aη ). Therefore, if we show that u˜ k (t) ≥ 0 and v˜k (t) ≥ 0, then it follows that u(t) ≥ 0 and v(t) ≥ 0.

2 Local Solutions

351

Our goal is therefore to prove that u˜ k (t) ≥ 0 and v˜k (t) ≥ 0 for every 0 < t ≤ TU0 . To this end, we use the truncation method. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. Since u˜ k ∈ C (0, TU0 ]; H˚ D1 (Ω) ∩ C1 (0, TU0 ]; HD−1 (Ω) , we get by (1.101) that the function ψk (t) = Ω H (u˜ k (x, t)) dx is continuously differentiable with the derivative ψk (t) = H (u˜ k ), −aΛu˜ k − μ∇ · [u˜ k ∇χk ]H˚ 1 ×H −1 D D dt + (H (u˜ k ), −u˜ k vk + f + g(x))L2 . By property (1.96), we have

H (u˜ k ), −aΛu˜ k H˚ 1 ×H −1 = −a D

D

∇H (u˜ k ) · ∇ u˜ k dx

Ω

|∇H (u˜ k )|2 dx ≤ 0.

= −a Ω

Similarly, by (8.7) and Lemma 8.1 presented below, u˜ k ∇H (u˜ k ) · ∇χk dx

H (u˜ k ), −μ∇ · [u˜ k ∇χk ]H˚ 1 ×H −1 = μ D

D

Ω

H (u˜ k )∇H (u˜ k ) · ∇χk dx

=μ Ω

=

μ 2c

H (u˜ k )2 (−uk + vk + h) dx.

Ω

Finally, due to (8.2) and f ≥ 0, H (u˜ k )[−u˜ k vk + f + g(x)] dx ≤ − vk H (u˜ k )2 dx. Ω

Ω

Thus, in view of (8.3) and (8.14), ψk (t) ≤ Ck ψk (t); consequently, ψk (t) ≤ ψk (0)eCk t . Therefore, ψk (0) = 0 implies ψk (t) ≡ 0, that is, u˜ k (t) ≥ 0 for every 0 < t ≤ TU0 . It is the same for v˜k (t) ≥ 0 for every 0 < t ≤ TU0 . Lemma 8.1 For u ∈ H˚ D1 (Ω) and χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω), where 2 < p < ∞, we have 1 u∇u · ∇χ dx = u2 Λχ dx. 2 Ω Ω Proof If u ∈ H˚ D1 (Ω) ∩ L∞ (Ω), then u2 ∈ H˚ D1 (Ω). Therefore, the result is verified directly noting that u∇u = 12 ∇u2 . For general u ∈ H˚ D1 (Ω), it is sufficient to approximate u by cutoff functions Ψk (u) ∈ H˚ D1 (Ω) ∩ L∞ (Ω), where Ψk (u) ≡ −k for

352

8 Semiconductor Models

u ≤ −k, Ψk (u) = u for |u| ≤ kd, and Ψk (u) ≡ k for u ≥ k. Owing to (1.96), we verify that Ψ (uk ) → u in H˚ D1 (Ω) as k → ∞. Meanwhile, note by (1.79) that it follows from u ∈ H˚ D1 (Ω) ⊂ H 1 (Ω) that u ∈ L 2p (Ω). p−2

3 Global Solutions 3.1 A Priori Estimates for Local Solutions For U0 ∈ K, let U = t (u, v) denote a local solution of (8.11) on [0, TU ] in the function space: 0 ≤ u ∈ C((0, TU ]; H˚ D1 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; HD−1 (Ω)), 0 ≤ v ∈ C((0, TU ]; H˚ D1 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; HD−1 (Ω)). (8.16) Proposition 8.2 There exists a constant C > 0 such that, for any U0 ∈ K and any local solution of (8.11) in the function space (8.16), it holds that U (t) L2 ≤ C( U0 L2 + 1),

0 ≤ t ≤ TU , U0 ∈ K.

(8.17)

Proof Consider the duality product of HD−1 (Ω) × H˚ D1 (Ω) between the first equation of (8.11) and u. Then, 1 d 2 2 u dx + a |∇u| dx − μ u∇u · ∇χ dx + f u2 v dx 2 dt Ω Ω Ω Ω = [f + g(x)]u dx, Ω

where χ = (cΛ)−1 [−u + v + h(x)]. Similarly, by the second equation of (8.11), 1 d 2 2 v dx + b |∇v| dx + ν v∇v · ∇χ dx + f uv 2 dx 2 dt Ω Ω Ω Ω = [f + g(x)]v dx. Ω

We sum up these two equalities after multiplying the first one by ν and the second one by μ, respectively. Then, since Lemma 8.1 yields that μν μν (−u∇u · ∇χ + v∇v · ∇χ) dx = (v 2 − u2 )Λχ dx 2 Ω Ω μν = (u − v)2 (u + v) dx 2c Ω μν + h(x)(v 2 − u2 ) dx, 2c Ω

3 Global Solutions

353

we have 1 d 2 dt

(νu + μv ) dx + 2

2

Ω

aν|∇u|2 + bμ|∇v|2 dx

Ω

μν + (u − v)2 (u + v) dx + f (νu + μv)uv dx 2c Ω Ω μν = [f + g(x)](νu + μv) dx + h(x)(u2 − v 2 ) dx. 2c Ω Ω

Furthermore, by (8.2), [f + g(x)](νu + μv) dx ≤ f + g L2 μu + νv L2 Ω

≤ f + g L2 ζ u 2L2 + v 2L2 + Cζ

with an arbitrarily small parameter ζ > 0. Similarly, by (8.3),

ζ (u − v)2 (u + v) + Cζ (u + v) dx h(x)(u2 − v 2 ) dx ≤ h L∞ Ω

Ω

≤ h L∞

ζ (u − v)2 (u + v) + ζ (u2 + v 2 ) + Cζ dx.

Ω

Meanwhile, by Poincaré’s inequality (see Remark 2.3), 2 2 aν|∇u| + bμ|∇v| dx ≥ α (u2 + v 2 ) dx Ω

Ω

with some positive number α > 0. Therefore, taking ζ > 0 sufficiently small, we obtain that α 1 d 2 2 (νu + μv ) dx + (u2 + v 2 ) dx ≤ C 2 dt Ω 2 Ω with some constant C > 0. Hence (cf. (1.58)),

u(t) 2L2 + v(t) 2L2 ≤ C e−δt u0 2L2 + v0 2L2 + 1 , with some positive exponent δ > 0 and constant C > 0.

0 ≤ t ≤ TU ,

(8.18)

3.2 Global Solutions As the a priori estimates (8.17) are verified for local solutions, we can apply Corollary 4.1 to conclude the global existence of solutions. Thus, for any U0 ∈ K, (8.11)

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8 Semiconductor Models

possesses a unique global solution in the function space: 0 ≤ u ∈ C((0, ∞); H˚ D1 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); HD−1 (Ω)), 0 ≤ v ∈ C((0, ∞); H˚ D1 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); HD−1 (Ω)). (8.19)

3.3 Global Norm Estimates For U0 ∈ K, let U (t; U0 ) be the global solution of (8.11) with the initial value U0 in the function space (8.19). By (8.18), there exist an exponent δ > 0 and a constant C˜ > 0 such that, for any U0 ∈ K, it holds that ˜ −δt U0 L2 + 1), U (t; U0 ) L2 ≤ C(e

0 ≤ t < ∞, U0 ∈ K.

(8.20)

4 Dynamical System 4.1 Construction of Dynamical System We are now in a position to construct a dynamical system determined from problem (8.11) in the universal space X by following the general methods described in Chap. 6, Sect. 5, where the exponent β will be taken as β = 12 . We already know that, for any U0 ∈ K, (8.11) and hence (8.1) possesses a unique global solution U (t; U0 ) in the function space (8.19). By S(t)U0 = U (t; U0 ), we define a nonlinear semigroup S(t) acting on K. Estimate (8.20) shows that (6.56) is fulfilled. So, (8.11) defines a dynamical system (S(t), K, L2 (Ω)). Recall that 1 D(A 2 ) = L2 (Ω).

4.2 Exponential Attractors We shall construct exponential attractors for the dynamical system (S(t), K, L2 (Ω)). To this end, however, it suffices to verify (6.59) and (6.60). The space condition (6.59) is clear. Meanwhile, (8.20) shows that the dissipative condition (6.60) is also fulfilled. Hence, (S(t), K, L2 (Ω)) enjoys a family of exponential attractors.

5 Three-Dimensional Problem In this section, we want to discuss the three-dimensional problem. Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary ∂Ω which is split into two parts ΓD and ΓN satisfying (2.41) and (2.42).

Notes and Further Researches

355

As a matter of fact, we can argue in a similar way as in the two-dimensional case. However, in order to extend definition of the drift term ∇ · [u∇χ] as (8.7), we have to assume the stronger shift property that Λu ∈ L2 (Ω) implies u ∈ Wp11 (Ω) with some p1 > 3 and the estimate u Wp1 ≤ C Λu L2 , 1

u ∈ D(Λ|L2 ).

(8.21)

We set the same underlying space X as (8.9). Three-dimensional problem (8.1) is then written as an abstract problem of the form (8.8), where A and F are the same linear and nonlinear operators, respectively, as before. The only difference is that F has the domain 3 3 u D(F ) = ; u ∈ H p1 (Ω) and v ∈ H p1 (Ω) , v where p1 > 3 is the number for which (8.21) is valid. Then, D(Aη ) ⊂ D(F ) with η = 12 + 2p3 1 < 1 due to (8.12) (which is independent of the dimensions), and F is

shown to fulfill (4.2) with β = 12 and η = 12 + 2p3 1 . We can then repeat the same arguments as in the preceding sections to obtain the local existence of solutions, nonnegativity of local solutions, and the a priori estimates of local solutions. Hence, under (8.21), we can deduce the uniqueness and global existence of solutions for any pair (u0 , v0 ) of initial functions in K (see (8.10)).

Notes and Further Researches The semiconductor model (8.1) was presented by Shockley to describe the flows of electrons and holes in a semiconductor in the middle of the last century. In his historic book [Sho50] published in 1950, we may find the origin of the macroscopic studies of physical systems which consist of many individuals possessing some uniform properties by the analytical methods. For the physical backgrounds of the semiconductor model and the details of modeling, we refer to the references Sze [Sze81], Selberherr [Sel83], and Gajewski–Gröger [GG90]. Many authors have already contributed to the study of semiconductor equations. The evolutional problems have been studied by many papers including [FI95a, FI95b, FI95c, Gaj85, GG86, Jer87, Moc74, Moc75, ST85]. In particular, the asymptotic behavior of solutions was studied in [FI95c, Gaj85, GG86, Moc74]. Mock [Moc74] first proved, in a simple case, that every solution converges to the stationary solution at an exponential rate. Gajewski [Gaj85] and Gajewski–Gröger [GG86] generalized this result, but they still assumed some conditions which guarantee the unique solvability of the stationary problem. In the general case where the stationary solutions are not unique, Fang–Ito [FI95c] constructed a global attractor for the dynamical system determined from (8.1). More precisely, they constructed a global attractor with finite Hausdorff dimension under the spectral gap condition for the operator Λ. The stationary problems have been studied, e.g., by [FN94, FN96, Jer85,

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8 Semiconductor Models

Moc72, Mar86] and [BF02, Chap. 6]. The existence of stationary solutions to (8.1) was shown in various situations. On the contrary, Bensoussan–Frehse [BF02, Theorem 6.2] gave some special conditions which imply the uniqueness of stationary solutions. The same results as in this chapter was already obtained in Favini–Lorenzi– Yagi [FLY06]. However, in [FLY06], the abstract result of [OY02] was used to construct local solutions. In addition, the result due to [EMZ00] was used to construct exponential attractors. As shown in Sect. 4.2, we were able to construct a finite-dimensional global attractor without any spectral gap condition as assumed by Fang–Ito [FI95c]. For constructing (local) solutions to (8.11), we indeed made an essential use of the shift property (8.6) which was established in Bensoussan– Frehse [BF02] (cf. also Lorenzi [Lor78]). In the three-dimensional case, however, no general conditions are known for ensuring such a shift property (8.21). Recently, mathematical models incorporating quantum effects are attracting interest of researchers. Ancona–Tiersten [AnT87] and Ancona–Iafrate [AI89] presented the following quantum-drift-diffusion model: √ ⎧ Δ u

⎨ ∂u √ + F, = aΔu − ∇ · [μu∇χ] − ∇ · αu∇ ∂t u √

⎩ ∂v = bΔv + ∇ · [νv∇χ] − ∇ · βv∇ Δ√ v + F ∂t v (cf. also Gardner [Gar94]). Here, u = u(x, t) and v = v(x, t) denote as before the densities√of electrons and holes, respectively, in a semiconductor device. The terms √ Δ u Δ v √ √ −αu∇ u and −βv v denote quantum corrections for the flows of u and v, respectively, α > 0 and β > 0 being some positive coefficients, and F denotes the rate of generation and recombination of electrons and holes. Problem of constructing the dynamical system determined from such a quantum model remains entirely to be studied.

Chapter 9

Activator–Inhibitor Models

Let us consider the activator–inhibitor system. The general model equation is given by the reaction–diffusion system ∂u ∂t = aΔu + γf (u, v) in Ω × (0, ∞), (9.1) ∂v ∂t = bΔv + γ g(u, v) in Ω × (0, ∞), in a domain Ω ⊂ Rd . Here, u denotes the concentration of the activator, and v the concentration of the inhibitor in Ω, respectively. The functions f (u, v) and g(u, v) denote the kinetics and are real smooth functions defined for u ≥ 0 and v ≥ 0. The activator and the inhibitor diffuse in Ω with diffusion rates a > 0 and b > 0, respectively. The constant γ > 0 denotes the reaction rate. Species of the activator or inhibitor must be specified in each activator–inhibitor system under consideration. The form of kinetic functions f (u, v) and g(u, v) must also be fixed in an adequate manner. Let the system possess a positive homogeneous stationary solution (u, v), i.e., f (u, v) = 0 and g(u, v) = 0. The partial derivatives of the kinetics at (u, v) are denoted by f u = fu (u, v), f v = fv (u, v), g u = gu (u, v), and g v = gv (u, v). We assume that f u > 0, that is, the activator is created autocatalystically, and that g v < 0 with the relation f u + g v < 0, that is, the inhibitor degrades faster than the activator grows. We assume also that f v < 0 and g u > 0; therefore, the inhibitor restrains growth of the activator, and oppositely, the activator produces the inhibitor, too. The relation f u g v > f v g u is also assumed. Hence, the partial derivatives are assumed to satisfy the following inequalities: f u > 0,

f u + g v < 0,

It then holds that f u + g v < 0,

f ugv > f v gu,

and f u g v > f v g u .

(9.2)

and ag v + bf u < 2 ab(f u g v − f v g u ) (9.3)

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_9, © Springer-Verlag Berlin Heidelberg 2010

357

358

9 Activator–Inhibitor Models

if the diffusion constants a and b coincide, i.e., a = b. According to the arguments in Chap. 6, Sect. 6.4 (since (6.82) is valid), this condition indicates that the homogeneous solution (u, v) is a stable stationary solution. Thus, under (9.2), the coincidence a = b provides the stability of the homogeneous solution (u, v). On the contrary, if the diffusion rate b of the inhibitor is greater than the a of the activator and their difference is large enough, then the last inequality in (9.3) becomes false, that is, (6.82) in Chap. 6, Sect. 6.4 is no longer valid. This means that the stationary solution (u, v) loses stability. Thus, under the same condition as (9.2), the relation b > a with sufficiently large difference can provide the instability of the homogeneous stationary solution. Such a principle is called the diffusion-driven instability and was originally proposed by Turing in 1952. This chapter is devoted to handling a typical model, choosing suitable functions f (u, v) and g(u, v), and to showing rigorously that the Turing instability principle actually takes place.

1 Prototype Reaction–Diffusion Model We take the kinetic functions f (u, v) = k1 − k2 u +

k3 u2 , v

g(u, v) = k4 u2 − k5 v,

presented by Gierer–Meinhardt [GM72] and Meinhardt [Mei82] (cf. also Meinhardt [Mei92] and Murray [Mur03, Chap. 2]), where ki are positive constants for i = 1, . . . , 5. After nondimensionalization (see [Mur03, p. 78]), the equations are written by ∂u u2 ∂t = aΔu + γ (d − cu + v ) in Ω × (0, ∞), ∂v ∂t

= bΔv + γ (u2 − v)

in Ω × (0, ∞).

So, we consider the initial-boundary-value problem ⎧ ∂u 2 = aΔu + γ (d − cu + uv ) ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂v 2 ∂t = bΔv + γ (u − v) ⎪ ∂u ⎪ = ∂v = 0 ⎪ ⎪ ⎩ ∂n ∂n v(x, 0) = v0 (x) u(x, 0) = u0 (x),

in Ω × (0, ∞), in Ω × (0, ∞), on ∂Ω × (0, ∞), in Ω,

(9.4)

where 0 < a ≤ b are diffusion constants of the activator and inhibitor, respectively, and where c, d, and γ are positive constants (> 0). We consider this problem in a three-dimensional C2 or convex, bounded domain Ω. Let from now on f (u, v) and g(u, v) denote the functions f (u, v) = d − cu +

u2 , v

g(u, v) = u2 − v,

(9.5)

2 Local Solutions

359

respectively. Then, fu (u, v) = −c + gu (u, v) = 2u,

2u u2 , fv (u, v) = − 2 , v v gv (u, v) − 1.

It is easy to verify that (9.4) possesses a unique homogeneous stationary solution (u, v), which is given by d +1 u= c

and v =

d +1 c

2 (9.6)

.

Obviously, f u = fu (u, v) =

c(1 − d) , d +1

f v = fv (u, v) = −

g u = gu (u, v) =

2(d + 1) , c

g v = gv (u, v) = −1.

c2 , (d + 1)2

We then easily verify that, at least under the assumptions that 0 < d < 1 and 0 < c

0 . (9.8) Ω v0 As the equation of (9.4) on u has a singularity at v = 0, we will impose the condition ess infΩ v0 > 0 on the initial functions v0 . We introduce a diagonal matrix operator A = diag{A1 , A2 } of X with domain D(A) = H2N (Ω), where A1 and A2 are realizations of the operators −aΔ + γ c

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9 Activator–Inhibitor Models

and −bΔ + γ in L2 (Ω), respectively, under the homogeneous Neumann boundary conditions on ∂Ω. On account of Theorem 1.25, A1 and A2 are positive definite self-adjoint operators of L2 (Ω) with domains HN2 (Ω) (see (2.34)), H2N (Ω) being the product space of HN2 (Ω). Consequently, A is a positive definite self-adjoint operator of X. Furthermore, on account of Theorems 16.7 and 16.9 in Chap. 16, it is verified that 3 D(Aθ ) = H2θ (Ω) if 0 ≤ θ < , 4 3 < θ ≤ 1. D(Aθ ) = H2θ N (Ω) if 4

(9.9) (9.10)

Let ε > 0 be an arbitrarily fixed positive number. We introduce a nonlinear operator defined as follows. Let η be a fixed exponent such that 34 < η < 1. Let Fε : D(Aη ) → X be given by γ d + γ u2 /χε (Re v) u (9.11) , U = ∈ D(Aη ). Fε (U ) = v γ u2 Here, χε (ξ ) is a cutoff function such that χε (ξ ) = ξ for ε ≤ ξ < ∞ and χε (ξ ) ≡ ε for −∞ < ξ ≤ ε; obviously, χε (ξ ) is a Lipschitz continuous function defined for ξ ∈ R. In view of (1.76) and (9.10), it holds true that D(Aη ) ⊂ H2η (Ω) ⊂ L∞ (Ω), which shows that Fε certainly maps D(Aη ) into X. Problem (9.4), at least for U0 ∈ K satisfying ess infΩ v0 > ε, is now formulated as the Cauchy problem for a semilinear abstract equation dU dt + AU = Fε (U ), 0 < t < ∞, (9.12) U (0) = U0 , in X. Let us apply the general results in Chap. 4 to construct local solutions. Our nonlinear operator Fε fulfills (4.2). In fact, we have γ u2

2 γ u˜ 2 ˜ 2 |χε (Re v) − χε (Re v)| ˜ + |u2 − u˜ 2 | ≤ Cε |u| + |u| χ (Re v) − χ (Re v) ˜ ε ε ≤ Cε |u|2 + |u| ˜ 2 + 1 (|v − v| ˜ + |u − u|). ˜ Therefore, ˜ L∞ + u − u ˜ L∞ ) Fε (U ) − Fε (U˜ ) X ≤ Cε ( u2 L2 + u˜ 2 L2 + 1)( v − v ≤ Cε u 2L4 + u ˜ 2L4 + 1 ( v − v ˜ L∞ + u − u ˜ L∞ ). 3

3

Noting that D(A 8 ) = H 4 (Ω) ⊂ L4 (Ω), we obtain that 3 2 3 2 Fε (U ) − Fε (U˜ ) X ≤ Cε A 8 U X + A 8 U˜ X + 1 × Aη (U − U˜ ) X ,

U, U˜ ∈ D(Aη ).

(9.13)

2 Local Solutions

361

This means that (4.2) is satisfied by Fε with β = 38 and the η fixed above. We can now apply Theorem 4.1. As a consequence, for any U0 ∈ K, (9.12) possesses a unique local solution in the function space: 3 U ∈ C((0, TU0 ]; D(A)) ∩ C [0, TU0 ]; D A 8 ∩ C1 ((0, TU0 ]; X),

(9.14)

3

where TU0 > 0 is determined by the norm A 8 U0 X only.

2.2 Nonnegativity of Local Solutions For U0 ∈ K, let U (t) = t (u(t), v(t)) be the local solution constructed above. We want to prove that u(t) ≥ 0 and v(t) > 0 for every 0 < t ≤ TU0 by using the truncation method. Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (9.12) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function given by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ϕ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative ϕ (t) = a

H (u)Δu dx + γ Ω

u2 H (u) d − cu + dx. χε (v) Ω

Property (1.96) provides that H (u)Δu dx = − ∇H (u) · ∇u dx = − |∇H (u)|2 dx ≤ 0. Ω

Ω

Ω

In addition, since H (u) ≤ 0 and H (u)u ≥ 0, it is seen that u2 H (u) d − cu + dx ≤ 0. χε (v) Ω Hence, ϕ (t) ≤ 0; consequently, ϕ(t) ≤ ϕ(0) for every 0 < t ≤ TU0 . Thus, ϕ(0) = 0 implies ϕ(t) ≡ 0, that is, u(t) ≥ 0 for 0 < t ≤ TU0 . As for v(t), we consider ψ(t) = Ω H (v(t) − e−γ t ε0 ) dx, where ε0 = ess infΩ v0 > 0. Then, from ψ (t) = b H (v − e−γ t ε0 )Δ[v − e−γ t ε0 ] dx Ω

−γ Ω

H (v − e−γ t ε0 )[v − e−γ t ε0 ] dx + γ

H (v − e−γ t ε0 )u2 dx

Ω

it follows that ψ (t) ≤ 0. Hence, v(t) ≥ e−γ t ε0 for 0 < t ≤ TU0 .

362

9 Activator–Inhibitor Models

This shows that, if ess infΩ v0 = ε0 > ε, then the local solution of (9.12) constructed above is a solution to (9.4), too, at least for t such that (v(t) ≥)e−γ t ε0 ≥ ε. The lower estimate for v(t) obtained in this subsection will be improved below by a uniform one.

3 Global Solutions 3.1 A Priori Estimates from Below Let U0 = t (u0 , v0 ) be an initial value in K with ess infx∈Ω v0 (x) = ε0 > 0. Let U = t (u, v) be any local solution of (9.12) on [0, T ] in the function space: U

3

0 ≤ u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; H 4 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 3

0 < v ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; H 4 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (9.15) Then, it holds that u(t) ≥ (d/c)(1 − e−γ ct ), v(t) ≥ ε0 e

−γ t

+ e(t),

0 ≤ t ≤ TU ,

0 ≤ t ≤ TU .

(9.16) (9.17)

t Here, e(t) denotes the function e(t) = γ (d/c)2 e−γ t 0 eγ s (1 − e−γ cs )2 ds; clearly, e(t) > 0 for t > 0 and limt→∞ e(t) = (d/c)2 . Note that these lower estimates are independent of the fixed number ε > 0 in (9.12). In fact, these estimates are verified by a quite analogous method to that used in Sect. 2.2. We consider the function u(t) ˜ = u(t) − (d/c)(1 − e−γ ct ), By (1.100), ϕ(t) ˜ =

Ω

H (u(t)) ˜ dx is continuously differentiable with the derivative

d u˜ H (u) ˜ dx = ϕ˜ (t) = dt Ω

0 ≤ t ≤ TU .

γ u2 −γ ct H (u) ˜ aΔu + γ (d − cu) + − γ de dx. χε (v) Ω

By the same arguments as before (note that ∇ u˜ = ∇u), we conclude that ϕ˜ (t) ≤ 0. Hence, ϕ(t) ˜ ≡ 0 for 0 ≤ t ≤ TU . This shows that (9.16) is valid. Consider next the function v(t) ˜ = v(t) − [ε0 e−γ t + e(t)], ˜ We put ψ(t) =

Ω

H (v(t)) ˜ dx. Then,

0 ≤ t ≤ TU .

3 Global Solutions

363

d v˜ dx dt Ω

= H (v) ˜ bΔv + γ (u2 − v) + ε0 γ e−γ t

ψ˜ (t) =

H (v) ˜

Ω

+ γ e(t) − γ (d/c)2 (1 − e−γ ct )2 dx. Since (9.16) is already known, it follows that 2 ˜ ψ (t) ≤ −b H (v)|∇v| ˜ dx − γ H (v) ˜ v˜ dx ≤ 0. Ω

Ω

˜ Hence, ψ(t) ≡ 0 for 0 ≤ t ≤ TU . We have thus verified (9.17).

3.2 A Priori Estimates for Sobolev Norms Let U0 ∈ K be an initial value satisfying ess infΩ v0 (x) = ε0 > 0. Let ε be such that 0 < ε ≤ min {e−γ t ε0 + e(t)}. 0≤t 0 being a constant independent of U0 and TU . Proof Let 2 < p < ∞ and 0 < q < ∞ be exponents to be fixed below. We put p w(t) = u(t) v(t)q . Then, d dt

qup ∂v pup−1 ∂u − dx v q ∂t v q+1 ∂t Ω p−1 qup pu aΔu − bΔv dx = vq v q+1 Ω

w dx = Ω

364

9 Activator–Inhibitor Models

+γ Ω

pup−1 u2 qup 2 d − cu + − (u − v) dx vq v v q+1

=I + II. By integration by parts, the first integral I is rewritten in the form up−2 up−1 ap(p − 1) q |∇u|2 − (a + b)pq q+1 ∇u · ∇v I =− v v Ω p u + bq(q + 1) q+2 |∇v|2 dx v up−2 =− ap(p − 1)|v∇u|2 − (a + b)pq(v∇u) · (u∇v) q+2 Ω v + bq(q + 1)|u∇v|2 dx. We here assume that p and q satisfy √ √ pq 2 ab . ≤ √ (p − 1)(q + 1) a + b

(9.20)

Then, it is easily observed that I ≤ 0. In the meantime, the second integral is equal to −q p−1 II = γ v du − (pc − q)up + v −(q+1) (pup+1 − qup+2 ) dx. Ω

Here, we notice that dup−1 ≤ ζ up + Cp,ζ p+1

pu

p+2

≤ζ u

for all u ≥ 0,

+ Cp,ζ

for all u ≥ 0,

with arbitrarily small numbers ζ > 0 and ζ > 0, provided that Cp,ζ > 0 and Cp,ζ > 0 are chosen in a suitable way. We assume in addition to (9.20) that q

0 and ζ > 0 sufficiently small and noting that v(t) ≥ ε (due to (9.18)), we obtain that γ cp w dx + Cε,p,q . II ≤ − 2 Ω γ cp d In this way, we have dt Ω w dx ≤ − 2 Ω w dx + Cε,p,q and conclude that w(t) L1 ≤ e−

γ cp 2 t

w(0) L1 + Cε,p,q ,

0 ≤ t ≤ TU .

(9.22)

3 Global Solutions

365

In view of (9.20) and (9.21), we can take p = 4, provided that q > 0 is taken sufficiently small. Let us indeed fix p = 4 and q such that 0 < q ≤ 2. Then, since 4 4 uv 2 L1 ≤ ε q−2 vuq L1 , (9.22) yields the desired estimate (9.19). The equation on u is written as the abstract equation du u(t)2 + A1 u = d + , dt v(t)

0 < t ≤ TU ,

in L2 (Ω). Therefore, u(t) = e−tA1 u0 +

0

t

u(s)2 ds, e−(t−s)A1 d + v(s)

where e−tA1 is the analytic semigroup on L2 (Ω) generated by −A1 and satisfies the estimate e−tA1 L2 ≤ e−γ ct . Furthermore, t 3 3 3 u(s)2 ds. A18 u(t) = e−tA1 A18 u0 + A18 e−(t−s)A1 d + v(s) 0 Using this expression, we immediately derive from (9.19) that 38 A u(t) 1

L2

3

≤ Cε e−δt A18 u0 L + 1 , 2

0 ≤ t ≤ TU ,

(9.23)

with some positive exponent δ > 0 and a constant Cε > 0 independent of U0 and TU . As well, 3

u(t) L4 ≤ Cε e−δt A18 u0 L + 1 , 0 ≤ t ≤ TU . 2

We can repeat the same argument for v(t) to conclude that 38 A v(t) 2

L2

3

3 ≤ Cε e−δt A18 u0 L + A28 v0 L + 1 , 2

2

0 ≤ t ≤ TU .

Hence, this, together with (9.23), provides the final estimate 3

A 8 U (t) ≤ Cε e−δt A 38 U0 + 1 , X X

0 ≤ t ≤ TU .

(9.24)

3.3 Global Solution Let U0 ∈ K be an initial value with ess infΩ v0 (x) = ε0 > 0. Taking ε > 0 small enough to satisfy min {ε0 e−γ t + e(t)} ≥ ε,

0≤t 0, this function U is not only a solution of (9.12) but also that of the original equation (9.4).

3.4 Global Norm Estimates For U0 ∈ K, let U (t) = U (t; U0 ) be the global of (9.12) with the initial value U0 in the function space (9.25). Of course, estimate (9.24) holds for the global solution, i.e., 3

A 8 U (t; U0 ) ≤ Cε e−δt A 38 U0 + 1 , 0 ≤ t < ∞. (9.26) X X

4 Dynamical System 4.1 Construction of Dynamical System For U0 ∈ K, let U (t; U0 ) denote the global solution of (9.4) with the initial value U0 . From (9.17) we notice that, if U (t; U0 ) = t (u(t), v(t)), then v(t) ≥ e(t) for every 0 ≤ t < ∞. This lower estimate is independent of the initial value. As we are concerned with asymptotic behavior of U (t; U0 ) for U0 ∈ K, without loss of generality we assume that all the initial functions v0 uniformly satisfy the estimate v0 (x) ≥ ε0 = e(1). Furthermore, v(t) ≥ ε for every 0 ≤ t < ∞, where ε > 0 is given by ε = min {e(1)e−γ t + e(t)}. 0≤t 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R ; U0 ∈ Kθ,R .

Meanwhile, by Proposition 6.1, 3 Aθ S(t)U0 X ≤ Cθ,R 1 + t −(θ− 8 ) ,

0 < t < ∞; U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞; U0 ∈ Kθ,R .

Hence, (6.56) is fulfilled with the exponent dynamical system in Dθ .

3 8

< θ < 1. So, (S(t), Kθ , Dθ ) defines a

4.2 Exponential Attractors Let us construct exponential attractors for (S(t), K, D 3 ). To this end, we can fol8 low the methods established in Chap. 6, Sect. 5, namely, it suffices to verify (6.59) and (6.60). It is clear that the space condition (6.59) is valid. Meanwhile, (9.26) shows that the dissipative condition (6.60) is also valid. So, it is concluded that (S(t), K, D 3 ) has exponential attractors. 8

For any exponent 38 ≤ θ < 1, we can equally construct exponential attractors of the dynamical system (S(t), Kθ , Dθ ), too.

4.3 Squeezing Property of S(t) As in Chap. 6, Sect. 5.2, we can construct a subset X of K which is a compact set of X, is a bounded set of D(A), and is an absorbing and invariant set of S(t). Then, (S(t), X, X) is a dynamical system in the space X. It is clear that (6.62) is valid. Then, S(t) enjoys the squeezing property (6.45)–(6.46). Hence, by Theorem 6.16, (S(t), X, X) has exponential attractors whose fractal dimensions are precisely estimated by (6.50).

5 Instability of Homogeneous Stationary Solution As mentioned in the beginning of this chapter, our next objective is to show that the homogeneous stationary solution U of (9.4) becomes unstable, provided that b is

368

9 Activator–Inhibitor Models

sufficiently large. For this purpose, we intend to construct a local unstable manifold of U employing the method described in Chap. 6, Sect. 6. Since we are concerned with only the local unstable manifold, we introduce a localized problem of (9.4) in a neighborhood of U .

5.1 Localized Problem We introduce the following initial-boundary-value problem ⎧ 2 ∂u ⎪ = aΔu + γ [d − cu + ϕ(u) ] in Ω × (0, ∞), ⎪ ⎪ ∂t ψ(v) ⎪ ⎪ ⎨ ∂v 2 in Ω × (0, ∞), ∂t = bΔv + γ [ϕ(u) − v] ⎪ ∂u ∂v ⎪ on ∂Ω × (0, ∞), ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v0 (x) in Ω. 0

(9.28)

Here, ϕ(u) is a cutoff function of u in a complex neighborhood of u such that u−u ϕ(u) ≡ u for |u − u| < 1 and ϕ(u) = |u−u| + u for |u − u| ≥ 1. Clearly, ϕ(u) is a bounded Lipschitz continuous function in C. Meanwhile, ψ(v) ≡ v for |v − v| < r v−v r + v for |v − v| ≥ r, where r > 0 is taken in such a way that and ψ(v) = |v−v|

r < v2 . Then, ψ(v) is a bounded Lipschitz continuous function in C with the estimate |ψ(v)| ≥ v2 for all v ∈ C. We can then repeat the same arguments as in Sects. 3 and 5 to construct local solutions and global solutions to (9.28) for all initial values U0 ∈ D(Aθ ) in the function space: U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); D(Aθ )) ∩ C1 ((0, ∞); X)

for any exponent θ such that 38 ≤ θ < 1. The homogeneous stationary solution U of (9.4) or of (9.12) is clearly a stationary solution to (9.28), too.

5.2 Complexified Dynamical System ˜ acting on The localized problem (9.28) introduced above defines a semigroup S(t) 3 θ ˜ Dθ = D(A ), 8 ≤ θ < 1, and defines a dynamical system (S(t), Dθ , Dθ ) which we call a comlexified dynamical system. We are concerned with the case 34 < θ < 1 because of the embedding Dθ = 2θ HN (Ω) ⊂ C(Ω). In this case, any solution of problem (9.12) is a solution of (9.28) in a small neighborhood of U in Dθ . Naturally, in such a neighborhood, any trajec˜ Dθ , Dθ ). In particular, U is an equilibrium tory of (S(t), Kθ , Dθ ) is that of (S(t), ˜ ˜ Dθ , Dθ ), then it is the of (S(t), Dθ , Dθ ). If U is stable as an equilibrium of (S(t),

5 Instability of Homogeneous Stationary Solution

369

same as that of (S(t), Kθ , Dθ ). On the other hand, even if U is unstable as an equi˜ librium of (S(t), Dθ , Dθ ), U is not automatically unstable as that of (S(t), Kθ , Dθ ). ˜ Nevertheless, as will be explained below, the instability of U in (S(t), Dθ , Dθ ) provides us an information which indicates its instability even in the original dynamical system (S(t), Kθ , Dθ ).

5.3 Unstable Manifold of U Problem (9.28) is now formulated in X as the Cauchy problem of the form (9.12) in which the nonlinear operator F˜ : D(Aη ) → X is given by u γ d + γ ϕ(u)2 /ψ(v) , U = ∈ D(Aη ), F˜ (U ) = v γ ϕ(u)2 where 34 < θ < η < 1. In the complex neighborhoods |u − u| < 1 of u and |v − v| < r of v, we have ϕ(u) ≡ u and ψ(v) ≡ v. So, in a small neighborhood of U in D(Aη ) ⊂ D(Aθ ) ⊂ C(Ω), F˜ is Fréchet differentiable with the derivative 2γ u γ u2 h1 − 2 ˜ v v F (U ) h = , h2 2γ u 0 η h u U= ∈ B D(A ) (U ; r), h = 1 ∈ D(Aη ). h2 v Then, it is possible to apply the results obtained in Chap. 6, Sect. 6. The derivative obviously fulfills conditions (6.68) and (6.69). In order to verify the spectrum separation condition (6.76), it suffices to check condition (6.83). To this end, we have to make the assumption that abμ2k − γ (ag v + bf u )μk + γ 2 (f u g v − f v g u ) = 0 for all μk ,

(9.29)

where 0 = μ0 < μ1 ≤ μ2 ≤ · · · are the eigenvalues of the Laplace operator −Δ under the homogeneous Neumann boundary conditions on ∂Ω. Since f u + g v < 0 (due to (9.2)), we have μk = (f u + g v )/(a + b) for all μk . Therefore, under (9.29), (6.83) and hence (6.76) is fulfilled. Under (9.7) (and hence under (9.2)), if a and b are such that ag v + bf u > 2 ab(f u g v − f v g u ), (9.30) then there may exist eigenvalues μk such that abμ2k − γ (ag v + bf u )μk + γ 2 (f u g v − f v g u ) < 0. Note that (9.30) is actually satisfied if a is fixed and b is sufficiently large because of ˜ + (U ; O) which f u > 0. In such a case, U has a nontrivial local unstable manifold W

370

9 Activator–Inhibitor Models

is tangential to the space U + X− at U , where X− is the sum of the eigenspaces of the linearized operator A = A − F˜ (U ) corresponding to the negative eigenvalues. ˜ + (U ; O) = dim X− = N (U ), where We already know that dim W (9.31) N(U ) = # μk ; abμ2k − γ (ag v + bf u )μk + γ 2 (f u g v − f v g u ) < 0 . ˜ + (U ; O) with dimenAs a consequence, U possesses a local unstable manifold W ˜ sion N(U ) in the dynamical system (S(t), Dθ , Dθ ). ˜ + (U ; O) by using (9.31). For Let us furthermore estimate the dimension of M simplicity, let Ω be a smooth bounded domain in R3 . According to [Tri78, Theorem 5.6.2], we have N (μ) ≡

0≤μk ≤μ

3 1 = O μ2

as μ → ∞.

In the meantime, let 0 < μ1 < μ2 < ∞ be the solution to the equation abμ2 − γ (ag v + bf u )μ + γ 2 (f u g v − f v g u ) = 0 for μ. Clearly, γ (ag v + bf u ) ∓ γ (ag v + bf u )2 − 4ab(f u g v − f v g u ) μ 1 , μ2 = . 2ab Then, we have N(U ) = N (μ2 ) − N (μ1 ). Remark 9.1 Let us consider some specific case where the diffusion constants are such that a = γ −1 a and b = γ b with a > 0 and b > 0. Fixing all other constants in (9.4), we let γ tend to infinity. It is easy to see that μ1 (γ ) ≤ C1 and μ2 (γ ) ≥ c2 γ 2 as γ → ∞ with some two positive constants C1 and c2 (> 0). Consequently, N(μ2 (γ )) − N (μ1 (γ )) ≥ Cγ 3 ; this means that, as γ → ∞, ˜ + (U ; O) = N (U ) ≥ Cγ 3 → ∞. dim W Let us consider finally the instability of U as an equilibrium of the original dynamical system (S(t), Kθ , Dθ ). We already know that U is an unstable equilib˜ + (U ; O) with dimen˜ rium of (S(t), Dθ , Dθ ) having a local unstable manifold W sion dim X− which is tangential to U + X− at U . Furthermore, the sum X− of the eigenspaces of A corresponding to the negative eigenvalues possesses an orthogonal basis which consists of real-valued functions (cf. Corollary 2.1). These facts then indicate that the local unstable manifold W+ (U ; O) in (S(t), Kθ , Dθ ) is neither trivial, that is, U is an unstable equilibrium of (S(t), Kθ , Dθ ).

Notes and Further Researches After the Turing’s pioneering work [Tur52], his instabilty principle was applied extensively. It is known that various pattern formations exhibited by physical, chemical, or biological systems can be reproduced by making use of activator–inhibitor

Notes and Further Researches

371

models. Accordingly, there are a certain number of forms for the kinetic functions on the activator and inhibitor, see [Mei92] and [Mur02, Chap. 6]. Gierer–Meinhardt [GM72] and Meihardt [Mei82] presented the functions k3 u2 , v

f (u, v) = k1 − k2 u +

g(u, v) = k4 u2 − k5 v.

(9.32)

The system (9.1) having these kinetic functions is called the Gierer–Meinhardt system. More generally, when the system has the kinetic functions f (u, v) = k1 − k2 u +

k3 up , vq

g(u, v) =

k4 ur − k5 v, vs

(9.33)

it is often called the generalized Gierer–Meinhardt system. When the kinetic functions are k2 up k3 ur g(u, v) = s − k4 v (9.34) f (u, v) = −k1 u + q , v v and when one imposes the Dirichlet boundary conditions on u and v, the problem is called the singular Gierer–Meinhardt system, see [CM00, CM03]. A system consisting of the diffusion equation on u and an ordinary differential equation on v which is obtained by letting b → ∞ and replacing g(u, v) by the integral Ω g(u, v) dx is called a shadow system of the Gierer–Meinhardt system. Murray [Mur88] (cf. also [Mur03, Chap. 3]) used the functions f (u, v) = k1 − k2 u − h(u, v), h(u, v) =

g(u, v) = k3 − k4 v − h(u, v),

k5 uv k6 + k7 u + k8 u2

in order to reproduce the animal coat patterns. A large number of papers have already been published for the Gierer–Meinhardt system. For the case of (9.32), the global existence was first obtained by Rothe [Rot84]. Masuda–Takahashi [MaT87] proved the global existence of solutions for the general case of (9.33), but they had to assume some restrictive condition on the exponents p, q, r, and s which excludes the case of (9.32). Afterward, Li–Chen– Qin [LCQ95] and Jiang [Jia06] got rid of such a restriction and proved the global q existence in the case where 0 < p − 1 < r and p−1 r < s+1 (which coves the case where p = 2, q = 1, r = 2, and s = 0). Wu–Huang [WH00] studied the asymptotic behavior of the global solutions. Evolutional problems for the shadow system were studied by Miyamoto [Miy06]. Exponential attractors are newly constructed for the Gierer–Meinhardt system. The results in Sect. 5 are also new. More articles have been published to contribute to the stationary problems of the Gierer–Meinhardt system than those to the evolutional problems. However, we here quote only pioneering papers Haken–Olbrich [HO78] and Takagi [Tak79] and recent papers Jiang–Ni [JN07] and Iron–Rumsey [IR07].

Chapter 10

Belousov–Zhabotinskii Reaction Models

This chapter is devoted to studying reaction–diffusion models for the Belousov– Zhabotinskii reaction. The reaction which consists of more than ten elementary chemical reactions arising simultaneously and which does not tend to any chemical equilibrium is known as one of typical self-organization phenomena in chemistry. In 1974, Field–Noyes presented a simple mathematical model for describing the complicated mechanism from a global point of view. Their reaction–diffusion model of dimensionless form is given by ⎧ ∂u −1 2 ⎪ ⎨ ∂t = aΔu + ε (qw − uw + u − u ) in Ω × (0, ∞), ∂v in Ω × (0, ∞), ∂t = bΔv + u − v ⎪ ⎩ ∂w −1 in Ω × (0, ∞). ∂t = dΔw + δ (−qw − uw + cv) Here, u denotes the concentration of HBrO2 , v the concentration of Ce4+ , and w the concentration of Br− , respectively, in a vessel represented by Ω. The positive constants δ, ε, q, and c are chemical parameters, where especially δ, ε, and q are considered to be small. In 1986, the model was simplified more by Keener–Tyson by assuming that δ cv is sufficiently small so that w is in quasi-equilibrium, i.e., w = u+q , and by using some scaling in the spatial variable so that the diffusions of u and v are εa and εb, respectively. Their model is then given by ∂u ∂t ∂v ∂t

= εaΔu +

1 ε

u(1 − u) − cv u−q in Ω × (0, ∞), u+q

= εbΔv + u − v

in Ω × (0, ∞).

(10.1)

In this chapter, we consider the initial-boundary-value problems of these equations under the Neumann boundary conditions. We show the global existence of solutions, construct a dynamical system, and construct exponential attractors. Especially, for the Keener–Tyson model, we show that, if ε and q are sufficiently small in a specific sense, then the problem has no stable homogeneous stationary solutions. In addition, in such a case, the fractal dimension of exponential attractor is estimated from below by some quantity which tends to infinity as ε → 0. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_10, © Springer-Verlag Berlin Heidelberg 2010

373

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10 Belousov–Zhabotinskii Reaction Models

1 Field–Noyes Model Let us first consider the Field–Noyes model. We consider the initial-boundary-value problem ⎧ ∂u −1 2 ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu + ε (qw − uw + u − u ) ⎪ ⎪ ⎪ ∂v ⎪ ⎪ = bΔv + u − v in Ω × (0, ∞), ⎪ ⎨ ∂t ∂w −1 in Ω × (0, ∞), ∂t = dΔw + δ (−qw − uw + cv) ⎪ ⎪ ⎪ ∂u ∂v ∂w ⎪ on ∂Ω × (0, ∞), ⎪ ∂n = ∂n = ∂n = 0 ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), w(x, 0) = w (x) in Ω, 0 0 0 (10.2) in a three-dimensional C2 or convex, bounded domain Ω. Here, δ, ε, q, and c are positive constants (>0), and a, b, and d are positive diffusion rates. The three unknown functions u, v and w are assumed to satisfy the homogeneous Neumann boundary conditions on ∂Ω.

2 Local Solutions 2.1 Construction of Local Solutions We construct local solutions by formulating (10.2) as the Cauchy problem for a semilinear abstract equation. We set the underlying space X as the product ⎧⎛ ⎞ ⎫ ⎨ f ⎬ (10.3) X = L2 (Ω) = ⎝ g ⎠ ; f ∈ L2 (Ω), g ∈ L2 (Ω), h ∈ L2 (Ω) . ⎩ ⎭ h We introduce a diagonal matrix operator A = diag {A1 , A2 , A3 } of X, where A1 , A2 and A3 are realizations of −aΔ + ε −1 , −bΔ + 1 and −dΔ + δ −1 q, respectively, in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. They are positive definite self-adjoint operators of L2 (Ω) with domains HN2 (Ω), see Theorems 2.6 and 2.7. Therefore, A is a positive definite self-adjoint operator of X. Furthermore, by Theorems 16.7 and 16.9 in Chap. 16, the domains of its fractional powers are given by 3 D(Aθ ) = H2θ (Ω) if 0 ≤ θ < , 4 3 < θ ≤ 1. D(Aθ ) = H2θ N (Ω) if 4

2 Local Solutions

375

We also define the nonlinear operator F : D(Aη ) → X by ⎛ −1 ⎞ ε (qw − uw + 2u − u2 ) ⎠ , U ∈ D(Aη ), u F (U ) = ⎝ δ −1 (−uw + cv)

(10.4) 2η

where the exponent η is fixed in such a way that 34 < η < 1. Since HN (Ω) ⊂ H 2η (Ω) ⊂ C(Ω) due to (1.76), it follows that D(Aη ) ⊂ C(Ω). We also set the space of initial values by ⎧⎛ ⎞ ⎫ ⎨ u0 ⎬ K = ⎝ v0 ⎠ ; 0 ≤ u0 ∈ L2 (Ω), 0 ≤ v0 ∈ L2 (Ω), 0 ≤ w0 ∈ L2 (Ω) . ⎩ ⎭ w0 We can then formulate (10.2) as the Cauchy problem dU 0 < t < ∞, dt + AU = F (U ),

(10.5)

U (0) = U0 ,

in X. Applications of the general results in Chap. 4 to (10.5) are quite easy. The structural assumption (4.21) on F is immediately verified. Indeed, from |u2 − u˜ 2 | ≤ (|u| + |u|)|u ˜ − u| ˜ it follows that ˜ L∞ )u − u ˜ L2 u2 − u˜ 2 L2 ≤ (uL∞ + u ≤ (uH 2η + u ˜ H 2η )u − u ˜ L2 ,

2η

u, u˜ ∈ HN (Ω).

Similarly, ˜ L2 + u ˜ H 2η w − w ˜ L2 ), uw − u˜ w ˜ L2 ≤ C(wH 2η u − u 2η

u, w, u, ˜ w˜ ∈ HN (Ω). η

η

η

2η

Therefore, in view of D(A1 ) = D(A2 ) = D(A3 ) = HN (Ω), we deduce that F (U ) − F (U˜ )X ≤ Cε (Aη U X + Aη U˜ X + 1)U − U˜ X ,

U, U˜ ∈ D(Aη ). (10.6)

Hence, F fulfills (4.21). Theorem 4.4 (with G(t) ≡ 0) thus provides that, for any initial value U0 ∈ X(⊃ K), (10.5) possesses a unique local solution in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; X) ∩ C1 ((0, TU0 ]; X),

(10.7)

where TU0 > 0 depends only on the norm U0 X . It also holds that tAU (t)X + U (t)X ≤ CU0 ,

0 < t ≤ TU 0 ,

(10.8)

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10 Belousov–Zhabotinskii Reaction Models

where CU0 > 0 also depends only on the norm U0 X .

2.2 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, v, w) be the local solution of (10.5) constructed above. We want to show that u(t) ≥ 0, v(t) ≥ 0, and w(t) ≥ 0 for every 0 < t ≤ TU0 . To this end, we have to use the modified nonlinear operator ⎞ ⎛ ⎞ ⎛ −1 u ε (qw − uw + 2u − u2 ) ⎠ , U = ⎝ v ⎠ ∈ D(Aη ), |u| F˜ (U ) = ⎝ w δ −1 (−uw + cv) to (10.4). And we consider an auxiliary problem ˜ dU ˜ ˜ ˜ dt + AU = F (U ), 0 ≤ t < ∞, U˜ (0) = U0 .

(10.9)

We can repeat the same arguments as in the preceding subsection to construct a unique local solution U˜ = t (u, ˜ v, ˜ w) ˜ to (10.9) on an interval [0, T˜U0 ] in the same ˜ solution space as (10.7), U also satisfying (10.8). The local solution U˜ (t) is then real valued, since the complex conjugate U˜ (t) of U˜ (t) is also a local solution of (10.9) with the same initial value, and the uniqueness of solution implies that U˜ (t) = U˜ (t). Furthermore, as is shown below, it holds that u(t) ˜ ≥ 0, v(t) ˜ ≥ 0 and w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . Let us verify first that v(t) ˜ ≥ 0 by the truncation method. Let H (v) ˜ be a C1,1 cut1 2 ˜ ≡0 off function for −∞ < v˜ < ∞ given by H (v) ˜ = 2v˜ for −∞ < v˜ < 0 and H (v) ˜ dx is continuously diffor 0 ≤ v˜ < ∞. By (1.100), the function ϕ(t) = Ω H (v(t)) ferentiable with the derivative H (v)Δ ˜ v˜ dx + H (v)(| ˜ u| ˜ − v) ˜ dx. ϕ (t) = b Ω

Ω

By property (1.96), H (v)Δ ˜ v˜ dx = − ∇H (v) ˜ · ∇ v˜ dx = − |∇H (v)| ˜ 2 dx ≤ 0. Ω

Ω

Ω

In addition, since H (v) ˜ ≤ 0 and H (v) ˜ v˜ ≥ 0, it follows that ϕ (t) ≤ 0. Conse˜ quently, ϕ(t) ≤ ϕ(0) for 0 < t ≤ TU0 . Thus, ϕ(0) = 0 implies ϕ(t) ≡ 0, that is, v(t) ˜ ≥ 0 for 0 ≤ t ≤ T˜U0 . ˜ By Let us next verify that w(t) ˜ ≥ 0. We use the function ψ(t) = Ω H (w(t))dx. the same reason as for ϕ(t), we have −1 H (w)Δ ˜ w˜ dx + δ H (w)(−q ˜ w˜ − u˜ w˜ + cv) ˜ dx. ψ (t) = d Ω

Ω

3 Global Solutions

377

Similarly, by property (1.96), Ω H (w)Δ ˜ w˜ dx ≤ 0. Since H (w) ˜ ≤ 0, 0 ≤ H (w) ˜ w˜ ≤ 2H (w) ˜ and v(t) ˜ ≥ 0, it is seen that H (w)(−q ˜ w˜ − u˜ w˜ + cv) ˜ dx ≤ 2u(t) ˜ H (w) ˜ dx. L∞ Ω

Ω

t ˜ ˜ Hence, ψ (t) ≤ 2u(t) L∞ ψ(t); consequently, ψ(t) ≤ ψ(0) exp[2 0 u(s) L∞ ds] η ˜ ˜ ˜ for 0 < t ≤ TU0 . We here know that u(s) L∞ ≤ CA U (s)X and, in view of (10.8), that Aη U (s)X ≤ CU0 s −η . Thus, ψ(0) = 0 implies ψ(t) ≡ 0, that is, w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . ˜ dx. Using the known reLet us finally consider the function φ(t) = Ω H (u(t)) −η , we can repeat the same ˜ sults that w(t) ˜ ≥ 0 and u(t) ˜ L∞ + w(t) L∞ ≤ CU0 t arguments as for w(t) ˜ to verify that u(t) ˜ ≥ 0. In this way, we have shown that u(t) ˜ ≥ 0, v(t) ˜ ≥ 0, and w(t) ˜ ≥ 0 for 0 ≤ t ≤ T˜U0 . The componentwise nonnegativity of U˜ (t) now implies that F˜ (U˜ (t)) = F (U˜ (t)) for 0 ≤ t ≤ T˜U0 ; this then means that U˜ (t) is a local solution of (10.5), too. By the uniqueness of a solution of (10.5), we must have U˜ (t) = U (t) for 0 ≤ t ≤ T˜U0 . Hence, if T˜U0 ≥ TU0 , we finish the proof. Otherwise, we put T0 = sup{0 < T ≤ TU0 ; u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for every 0 < t ≤ T }. Since Ω H (u(T0 )) dx = limtT0 Ω H (u(t)) dx = 0, we see that u(T0 ) ≥ 0. By a similar reason, we have v(T0 ) ≥ 0 and w(T0 ) ≥ 0. So, if T0 = TU0 , we finish the proof. If T0 < TU0 , we will again consider problem (10.9) but with initial time T0 and initial value U (T0 ). Then, we conclude that there is τ > 0 such that u(t) ≥ 0, v(t) ≥ 0, and w(t) ≥ 0 for every t such that T0 ≤ t ≤ T0 + τ . This is a contradiction; hence, T0 = TU0 .

3 Global Solutions 3.1 A Priori Estimates of Local Solutions For U0 ∈ K, let U = t (u, v, w) denote a local solution of (10.5) on [0, TU ] in the function space: 0 ≤ U ∈ C (0, TU ]; H2N (Ω) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (10.10) Consider the inner product of the equation of (10.5) and U in X. From the equation on u we have 1 d 2 2 −1 u dx + a |∇u| dx + ε u2 dx 2 dt Ω Ω Ω = ε −1 (quw − u2 w + 2u2 − u3 ) dx

Ω

≤ Ω

ε −1 (4−1 q 2 w + 2u2 − u3 ) dx,

(10.11)

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10 Belousov–Zhabotinskii Reaction Models

where we used the inequality qu ≤ 1 d 2 dt

q2 4

+ u2 . By the equation on v,

1 v dx + b |∇v| dx + 2 Ω Ω 2

(uv − 2−1 v 2 ) dx

v dx =

2

2

Ω

Ω

(u2 − 4−1 v 2 ) dx.

≤

(10.12)

Ω

Finally, by the equation on w, 1 d 2 dt

w 2 dx + d

Ω

=

|∇w|2 dx + Ω

δ −1 q 2

w 2 dx Ω

δ −1 (−2−1 qw 2 − uw 2 + cvw) dx

Ω

≤

δ −1 (−4−1 qw 2 + c2 q −1 v 2 ) dx.

(10.13)

Ω

After multiplying (10.12) by a parameter ξ > 0, we sum up (10.11), (10.12), and (10.13) to obtain that 1 1 d (u2 + ξ v 2 + w 2 ) dx + (2ε −1 u2 + ξ v 2 + δ −1 qw 2 ) dx 2 dt Ω 2 Ω −1 ≤ (2ε + ξ − ε−1 u)u2 + (c2 q −1 δ −1 − 4−1 ξ )v 2 Ω

+ 4−1 q(ε −1 q − δ −1 w)w dx. Let us fix the parameter ξ so that ξ = 4c2 q −1 δ −1 and let us notice the following two estimates (2ε−1 + ξ )u2 ≤ ε −1 u3 +

4 (2ε −1 + ξ )3 ε 2 27

ε −1 qw ≤ δ −1 w 2 + 4−1 δ 2 ε −2 q 2 Then, d dt

for all 0 ≤ u < ∞,

for all 0 ≤ w < ∞.

(u2 + ξ v 2 + w 2 ) dx + μ Ω

(u2 + ξ v 2 + w 2 ) dx ≤ C Ω

with μ = min{2ε −1 , 1, δ −1 q} > 0 and some constant C > 0. Solving this differential inequality (cf. (1.58)), we obtain that u(t)2L2 + ξ v(t)2L2 + w(t)2L2 ≤ e−μt u0 2L2 + ξ v0 2L2 + w0 2L2 + Cμ−1 ,

0 ≤ t ≤ TU . (10.14)

4 Dynamical System

379

3.2 Global Existence Let U0 ∈ K. For any local solution U of (10.5) in the function space (10.10), from (10.14) it follows that U (t)X ≤ C U0 X + 1 , 0 ≤ t ≤ TU . (10.15) Then, we can apply Corollary 4.3 to conclude the global existence of solution. Indeed, for any U0 ∈ K, (10.5) possesses a unique global solution in the function space: 0 ≤ U ∈ C (0, ∞); H2N (Ω) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (10.16)

3.3 Global Norm Estimate For U0 ∈ K, the global solution U (t; U0 ) of (10.5) in (10.16) with initial value U0 satisfies the estimate ˜ U (t; U0 )X ≤ C[e−μt U0 X + 1],

0 ≤ t < ∞, U0 ∈ K,

(10.17)

with some exponent μ˜ > 0 and constant C > 0. Indeed, this is verified directly from (10.14).

4 Dynamical System 4.1 Construction of Dynamical System We can now construct a dynamical system determined from (10.5). We take the universal space X given by (10.3) and follow the general methods described in Chap. 6, Sect. 5. We already know that, for any U0 ∈ K, (10.5) possesses a unique global solution U (t; U0 ) in the function space (10.16). By S(t)U0 = U (t; U0 ) we define a nonlinear semigroup S(t) acting on K. Estimate (10.17) shows that (6.56) is fulfilled. So, (10.5) determines a dynamical system (S(t), K, X).

4.2 Exponential Attractors Let us construct exponential attractors for the dynamical system (S(t), K, X). Thanks to the general results, it suffices to verify (6.59) and (6.60). But, the space condition (6.59) is clear. Meanwhile, (10.17) shows that the dissipative condition (6.60) is also fulfilled. Hence, there exists an absorbing and invariant set X of

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10 Belousov–Zhabotinskii Reaction Models

(S(t), K, X) such that X is a compact subset of X and is a bounded subset of D(A). Furthermore, the dynamical system (S(t), X, X) enjoys a family of exponential attractors M. It is clear in the present case that (6.62) is valid. Therefore, S(t) enjoys the squeezing property (6.45)–(6.46). Hence, (S(t), X, X) has exponential attractors whose fractal dimensions are estimated precisely by (6.50).

5 Keener–Tyson Model We now consider the initial-boundary-value problem ⎧ u−q

∂u 1 ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu + ε 2 u(1 − u) − cv u+q ⎪ ⎪ ⎪ ∂v 1 ⎨ = bΔv + (u − v) in Ω × (0, ∞), ∂t ε ∂v ⎪ on ∂Ω × (0, ∞), ⎪ ∂u ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩ v(x, 0) = v0 (x) in Ω, u(x, 0) = u0 (x),

(10.18)

in a three-dimensional C2 or convex, bounded domain Ω. Note that we have rewritten the equations for u and v in the original Keener–Tyson model (10.1) by scaling the temporal variable as τ = ε −1 t . The new temporal variable τ is again denoted by t as in (10.18). We fix the coefficients a > 0,

b > 0,

c > 0,

0 < q < 1.

(10.19)

In the meantime, 0 < ε ≤ 1 is considered as a control parameter of the system. We use the notation f (u, v) = u(1 − u) − cv u−q u+q , (10.20) g(u, v) = u − v. For u ≥ 0 and v ≥ 0, the equations f (u, v) = 0 and g(u, v) = 0 obviously have two solutions (0, 0) and (u, v). Here, u is a unique positive solution to the quadratic equation (u + q)(1 − u) = c(u − q),

(10.21)

and v = u. More precisely,

u = v = 1 − c − q + (c + q − 1)2 + 4q(c + 1) /2.

(10.22)

It is easy to see from (10.21) that q < u < 1. Thus, problem (10.18) has two homogeneous stationary solutions (u, v) ≡ (0, 0) and (u, v).

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381

We are then concerned with the stability and instability of these homogeneous stationary solutions. Clearly, we have fu (u, v) = 1 − 2u − gu (u, v) = 1,

2cqv , (u + q)2

fv (u, v) = −c

u−q , u+q

gv (u, v) = −1.

In particular, fu (0, 0) = 1,

fv (0, 0) = c,

gu (0, 0) = 1,

gv (0, 0) = −1.

These imply that the relation fu (0, 0)gv (0, 0) − fv (0, 0)gu (0, 0) = −(1 + c) < 0 always takes place, which shows on account of (6.84) that (0, 0) is always unstable. We denote the partial differential coefficients as f u = fu (u, v) = 1 − 2u − g u = gu (u, v) = 1,

2cqu , (u + q)2

f v = fv (u, v) = u − 1,

g v = gv (u, v) = −1.

It is clear that f u < 1. This estimate is actually optimal when the coefficients of equations are taken as in (10.19); indeed, when c = 1, it is seen from (10.22) that √ √ u = O( q), and hence f u = 1 − O( q) when q is small. Since f u − (1 − u) = 2cqu −u − (u+q) 2 < 0, it is seen that the differential coefficients always satisfy the relation f u g v > f v g u . According to (6.82), a sufficient condition in order that (u, v) satisfies the stability condition (6.67) is read as 1 f + g v < 0, ε u

f ugv > f v gu

and εag v + bf u < 2 εab(f u g v − f v g u ).

(10.23) When f u ≤ 0, all these inequalities are fulfilled automatically whatever ε > 0 is; hence, in this case, (u, v) is stable. On the other hand, when 0 < f u < 1, these inequalities may fail depending on the control parameter ε > 0. Indeed, if ε > max

2

b2 f u

, fu , ( ab(f u g v − f v g u ) + −abf v g u )2

then the inequalities in (10.23) are still valid; therefore, (u, v) is stable. But, if ε > 0 is small enough, then some of inequalities in (10.23) fail; so, (u, v) is no longer stable.

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10 Belousov–Zhabotinskii Reaction Models

6 Local Solutions 6.1 Construction of Local Solutions It is carried out in an analogous way to the Field–Noyes model by formulating (10.18) as the Cauchy problem of the form (10.5) in the product space X = L2 (Ω). We introduce a diagonal matrix operator A = diag{A1 , A2 } of X, where A1 and A2 are realizations of −aΔ + 1 and −bΔ + 1, respectively, in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. They are positive definite selfadjoint operators of L2 (Ω) with domains HN2 (Ω). Therefore, A is a positive definite self-adjoint operator of X. Furthermore, the domains of its fractional powers are 3 given by D(Aθ ) = H2θ (Ω) if 0 ≤ θ < 34 and D(Aθ ) = H2θ N (Ω) if 4 < θ ≤ 1. We η introduce also a nonlinear operator F : D(A ) → X given by u + ε −2 [u(1 − u) − cv(u − q)(|u| + q)−1 ] F (U ) = , U ∈ D(Aη ), (10.24) ε −1 u + (1 − ε−1 )v 2η

where the exponent η is fixed as 34 < η < 1. Since HN (Ω) ⊂ H 2η (Ω) ⊂ C(Ω) due to (1.76), it follows that D(Aη ) ⊂ C(Ω). The space of initial values is given by u0 K= (10.25) ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ v0 ∈ L2 (Ω) . v0 Problem (10.18) is then formulated as the Cauchy problem for an abstract evolution equation dU 0 < t < ∞, dt + AU = F (U ), (10.26) U (0) = U0 , of the same form as (10.5) in X. Applications of the general results in Chap. 4 are quite direct as before. The Lipschitz condition (4.21) on F is easily verified. Indeed, from uv u˜ v˜ ≤ C[|v − v| ˜ + (|v| + |v|)|u ˜ − u|] ˜ |u| + q − |u| ˜ +q it follows that uv u˜ v˜ ≤ C[v − v ˜ L2 + (vH 2η + v ˜ H 2η )u − u ˜ L2 ], |u| + q − |u| ˜ + q L2 2η

2η

u, v ∈ HN (Ω); u, ˜ v˜ ∈ HN (Ω). η

η

2η

Therefore, in view of D(A1 ) = D(A2 ) = HN (Ω), we deduce that F (U ) − F (U˜ )X ≤ Cε (Aη U X + Aη U˜ X + 1)U − U˜ X , U, U˜ ∈ D(Aη ),

(10.27)

7 Global Solutions

383

hence, F fulfills (4.21). We thus conclude that, for any initial value U0 ∈ K, (10.26) possesses a unique local solution in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; X) ∩ C1 ((0, TU0 ]; X),

(10.28)

where TU0 > 0 depends only on the norm U0 X .

6.2 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, v) be the local solution of (10.26) constructed above. It is then shown by the same methods as in Sect. 2.2 that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ T U0 . We introduce the modified nonlinear operator u u + ε−1 [|u|(1 − u) − c|v|(u − q)(|u| + q)−1 ] ˜ , U= ∈ D(Aη ), F (U ) = v ε −1 |u| + (1 − ε−1 )v to (10.24) and consider the same auxiliary problem as (10.9). Let U˜ (t) denote the local solution of the auxiliary problem with initial value U0 ∈ K on an interval [0, T˜U0 ]. Employing the truncation method as before, we conclude that U˜ (t) ≥ 0 componentwise for every 0 < t ≤ T˜U0 . Consequently, F˜ (U˜ (t)) = F (U˜ (t)) for 0 ≤ t ≤ T˜U0 . This means that U˜ (t) is a local solution of (10.26), too. By the uniqueness of local solution, we deduce that U˜ (t) = U (t) for 0 ≤ t ≤ T˜U0 . Therefore, U (t) ≥ 0 for 0 ≤ t ≤ T˜U0 . We can repeat the similar arguments until we obtain that U (t) ≥ 0 for every 0 < t ≤ TU0 . Now, since u(t) ≥ 0, we have |u(t)| = u(t). This means that the local solution U of (10.26) is a local solution of the original problem (10.18), too.

7 Global Solutions 7.1 A Priori Estimates of Local Solutions For U0 ∈ K, let U = t (u, v) denote a local solution of (10.26) on [0, TU ] in the function space: 0 ≤ U ∈ C (0, TU ]; H2N (Ω) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (10.29) Consider the inner product of the equation of (10.26) and U in X. From the equation on u we have 1 1 d 2 2 u dx + a |∇u| dx + u2 dx 2 dt Ω 2 Ω Ω

384

10 Belousov–Zhabotinskii Reaction Models

=

[(2

−1

+ε

−2

)u − ε 2

−2 3

u ] dx − ε

Ω

−2

c

uv Ω

u−q dx. u+q

u−q u−q If u > q, then −ε−2 cuv u+q ≤ 0. And, if 0 ≤ u ≤ q, then −ε −2 cuv u+q ≤ ε −2 cqv. So, 1 d 1 u2 dx + a |∇u|2 dx + u2 dx 2 dt Ω 2 Ω Ω −1 −2 2 −2 3 −2 ≤ [(2 + ε )u − ε u ] dx + ε cq v dx. Ω

Ω

In the meantime, from the equation on v, we have 1 d 1 v 2 dx + b |∇v|2 dx + v 2 dx = [ε −1 uv + (2−1 − ε −1 )v 2 ] dx 2 dt Ω 2 Ω Ω Ω 3 −2 2 −1 −ε ε u + v 2 dx. ≤ 4 Ω Therefore, it follows that 1 1 d (u2 + v 2 ) dx + (u2 + v 2 ) dx 2 dt Ω 2 Ω 3 −1 −2 2 −2 3 −2 −1 −ε ≤ [(2 + 2ε )u − ε u ] + ε cqv + v 2 dx. 4 Ω Here, we notice that 4 (2−1 + 2ε −2 )u2 ≤ ε −2 u3 + (2 + 2−1 ε 2 )3 ε −2 for all 0 ≤ u < ∞, 27 3 2 −2 −1 v + (4 − 3ε)−1 c2 q 2 ε −3 for all 0 ≤ v < ∞. ε cqv ≤ ε − 4 Hence, d dt

(u2 + v 2 ) dx + Ω

(u2 + v 2 ) dx ≤ Cε−3 .

Ω

Solving this differential inequality (cf. (1.58)), we obtain that u(t)2L2 + v(t)2L2 ≤ e−t u0 2L2 + v0 2L2 + Cε −3 , 0 ≤ t ≤ TU .

(10.30)

7.2 Global Existence Let U0 ∈ K. For any local solution U of (10.26) in (10.29), from (10.30) it follows that U (t)X ≤ U0 2X + Cε −3 , 0 ≤ t ≤ TU . (10.31)

8 Dynamical System

385

Then, we can use Corollary 4.3 to conclude the global existence of solution. Indeed, for any U0 ∈ K, (10.26) possesses a unique global solution in the function space: 0 ≤ U ∈ C (0, ∞); H2N (Ω) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (10.32)

7.3 Global Norm Estimate For U0 ∈ K, let U (t; U0 ) be the global solution of (10.26) in (10.32) with the initial value U0 . Then, (10.30) yields that U (t; U0 )2X ≤ e−t U0 2X + Cε −3 ,

0 ≤ t < ∞, U0 ∈ K.

(10.33)

8 Dynamical System 8.1 Construction of Dynamical System Let us now construct a dynamical system determined from (10.26). As before, we can follow the general methods described in Chap. 6, Sect. 5. We already know that, for any U0 ∈ K, (10.26) and hence (10.18) possesses a unique global solution U (t; U0 ) in the function space (10.32). By S(t)U0 = U (t; U0 ), we define a nonlinear semigroup S(t) acting on K. Estimate (10.33) shows that (6.56) is fulfilled. So, (10.26) determines a dynamical system (S(t), K, X). For any exponent 0 < θ < 1, (10.26) equally determines a dynamical system in the universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (10.24) fulfills (4.21) and a fortiori (4.2) with β = θ . D Let Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

Meanwhile, by Proposition 6.1, Aθ S(t)U0 X ≤ CAS(t)U0 θX S(t)U0 1−θ X ≤ Cθ,R (1 + t −θ ),

0 < t < ∞, U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that the global norm estimate (6.56) is valid in the space Dθ , too. Then, by the similar arguments as in the proof of Proposition 6.2, S(t)U0 is seen to be continuous from [0, ∞) × Kθ to Kθ . So, (S(t), Kθ , Dθ ) becomes a dynamical system in Dθ .

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10 Belousov–Zhabotinskii Reaction Models

8.2 Exponential Attractors We construct exponential attractors for (S(t), K, X). Thanks to the general results, it suffices to verify (6.59) and (6.60). However, the space condition (6.59) is clear. Meanwhile, (10.33) shows that the dissipative condition (6.60) is also fulfilled. Hence, there exists an absorbing and invariant set X of (S(t), K, X) such that X is a compact subset of X and is a bounded subset of D(A). In addition, (S(t), X, X) has a family of exponential attractors M. For any exponent 0 < θ < 1, these exponential attractors M keep their properties as exponential attractors in the universal space Dθ , too. Hence, the exponential attractors M of (S(t), K, X) are exponential attractors of (S(t), Kθ , Dθ ). As before, we can verify that S(t) enjoys the squeezing property (6.45)–(6.46). Hence, by Theorem 6.16, (S(t), X, X) has exponential attractors whose fractal dimensions are estimated by (6.50).

9 Instability of Homogeneous Stationary Solution 9.1 Localized Problem Let U = t (u, v) (actually, u = v) be the positive homogeneous stationary solution to (10.18) given by (10.22). We intend to investigate its stability and instability. For this purpose, we begin with introducing the localized problem of (10.18) in a neighborhood of U ⎧ ∂u ⎪ ∂t = aΔu − u + χ(u) ⎪ ⎪

⎪ ⎪ ⎪ + ε−2 χ(u)(1 − χ(u)) − cχ(v) χ(u)−q ⎪ χ(u)+q ⎨ ∂v −1 −1 ∂t = bΔv − v + ε χ(u) + (1 − ε )χ(v) ⎪ ⎪ ⎪ ⎪ ∂u = ∂v = 0 ⎪ ⎪ ⎪ ⎩ ∂n ∂n v(x, 0) = v0 (x) u(x, 0) = u0 (x),

in Ω × (0, ∞), in Ω × (0, ∞),

(10.34)

on ∂Ω × (0, ∞), in Ω.

Here, χ(u) is a cutoff function of u in a complex neighborhood of u such that u−u u + u for |u − u| ≥ u. It is clear χ(u) ≡ u for |u − u| < u and χ(u) = |u−u| that |χ(u) + q| ≥ q for u ∈ C, |χ(u)| ≤ 2u for u ∈ C, and |χ(u1 ) − χ(u2 )| ≤ |u1 − u2 | for u1 , u2 ∈ C. Since u = v, χ(v) also is a cutoff function of v in the same neighborhood of v. We can then repeat the same arguments as in Sects. 6 and 7 to construct local solutions and global solutions to (10.34) for any initial value U0 ∈ D(Aθ ), where 0 ≤ θ < 1, in the function space: U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); D(Aθ )) ∩ C1 ((0, ∞); X).

9 Instability of Homogeneous Stationary Solution

387

9.2 Complexified Dynamical System ˜ The localized problem (10.34) defines a semigroup S(t) acting on Dθ = D(Aθ ). ˜ Therefore, (10.34) defines a complexified dynamical system (S(t), Dθ , Dθ ) for any 0 ≤ θ < 1. Let us fix 34 < θ < 1. Then, Dθ ⊂ C(Ω). In a suitable neighborhood of U in Dθ , any solution to the original problem (10.18) is a solution of (10.34), too. In such ˜ a neighborhood, any trajectory of (S(t), Kθ , Dθ ) is that of (S(t), Dθ , Dθ ). In par˜ ticular, U is an equilibrium of (S(t), Dθ , Dθ ). If U is stable as an equilibrium of ˜ (S(t), Dθ , Dθ ), then it is equally stable as that of (S(t), Kθ , Dθ ). However, even if ˜ U is unstable as an equilibrium of (S(t), Dθ , Dθ ), we cannot say that U is automatically unstable as that of (S(t), Kθ , Dθ ). The situation is quite similar as for the ˜ activator–inhibitor model handled in Chap. 9. The instability of U in (S(t), Dθ , Dθ ) provides us an information which indicates the instability in the original dynamical system (S(t), Kθ , Dθ ), too.

9.3 Case where fu (u, v) ≤ 0 Problem (10.34) is formulated as the Cauchy problem of an abstract equation in X of the form (10.26) in which the nonlinear operator F˜ : D(Aη ) → X, where 34 < θ < η < 1, is given by χ(u) + ε −2 [χ(u)(1 − χ(u)) − cχ(v)(χ(u) − q)(χ(u) + q)−1 ] ˜ F (U ) = . ε −1 χ(u) + (1 − ε−1 )χ(v) In the complex neighborhood |u − u| < u and |v − v| < v of (u, v), we have χ(u) ≡ u and χ(v) ≡ v. So, in a suitable neighborhood of U in D(Aη ) ⊂ D(Aβ ) ⊂ C(Ω), F˜ is Fréchet differentiable with the derivative 1 + ε−2 [(1 − 2u) − 2cqv(u + q)−2 ] −ε −2 c(u − q)(u + q)−1 F˜ (U ) h = 1 1 − ε −1 η h h u × 1 , U= ∈ B D(A ) (U ; r), h = 1 ∈ D(Aη ). h2 h2 v It is now possible to apply the general results obtained in Chap. 6, Sect. 6. It is easy to check that F˜ fulfills conditions (6.68) and (6.69), and a fortiori (6.64). Let us consider in the present subsection the case where fu (u, v) ≤ 0.

(10.35)

Since the inequalities in (10.23) are all satisfied (whatever ε is), condition (6.82) and hence condition (6.67) is fulfilled. We already know that (6.64) and (6.67) provide the exponential stability. Hence, under (10.35), U is an exponentially stable ˜ equilibrium of (S(t), Dθ , Dθ ).

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10 Belousov–Zhabotinskii Reaction Models

9.4 Case where 0 < fu (u, v) < 1 Let us next consider the case where 0 < fu (u, v) < 1. We can still follow the arguments of Chap. 6, Sect. 6. If ε > 0 is large enough to satisfy ε > max

2

b2 f u

, fu , ( ab(f u g v − f v g u ) + −abf v g u )2

then the inequalities in (10.23) are all valid. Hence, by the same reason as before, ˜ U is an exponentially stable equilibrium of (S(t), Dθ , Dθ ). For the remaining ε, we make the assumption that

abμ2k − ε −2 (εag v + bf u )μk + ε −3 (f u g v − f v g u ) = 0 −εag v + bf u > (a + b) −εf v g u .

for all μk , and

(10.36) Here, 0 = μ0 < μ1 ≤ μ2 ≤ · · · are the eigenvalues of the Laplace operator −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. This assumption means that condition (6.83) is fulfilled. We already know that (6.83) implies the spectrum separation condition (6.76). Hence, (6.68), (6.69), and (6.76) are fulfilled. We have thus shown that, under (10.36), U has a smooth local un˜ + (U ; O) which is tangential to the space U + X− at U , where stable manifold W X− is the eigenspace of A = A − F˜ (U ) for the negative eigenvalues. Furthermore, ˜ + (U ; O) = dim X− = N (U ), where dim W N (U ) = # μk ; abμ2k − ε −2 (εag v + bf u )μk + ε −3 (f u g v − f v g u ) < 0 . (10.37) We remark that if ε is sufficiently small, then U has a nontrivial local unstable manifold. Indeed, if ε is sufficiently small, then we have εag v + bf u > 2 εab(f u g v − f v g u ), which actually implies N (U ) ≥ 1. We remark also that, if ε → 0 (the other parameters in (10.34) are fixed), then ˜ + (U , O) tends to infinity. For simplicity, assume that Ω is a smooth bounded dim W domain in R3 . Then, according to [Tri78, Theorem 5.6.2], N (μ) ≡ 0≤μk ≤μ 1 = 3

3

3

O(μ 2 ). Therefore, N (U ) = N (μ2 (ε)) − N (μ1 (ε)) = O(μ2 (ε) 2 ) − O(μ1 (ε) 2 ), where 0 < μ1 (ε) < μ2 (ε) < ∞ are the solutions to the equation abμ2 −ε −2 (εag v + bf u )μ + ε −3 (f u g v − f v g u ) = 0 on μ. From μ1 (ε), μ2 (ε) =

(εag v + bf u )2 ∓

(εag v + bf u )2 − 4εab(f u g v − f v g u ) 2ε 2 ab

Notes and Further Researches

389

we find two positive constants C1 and c2 such that μ1 (ε) ≤ C1 ε −1 and μ2 (ε) ≥ c2 ε −2 , respectively. Hence, as ε → 0, ˜ + (U ; O) = N (U ) ≥ Cε −3 → ∞. dim W Let us finally consider the instability of U as an equilibrium of (S(t), Kθ , Dθ ). ˜ As is proved above, U has in (S(t), Dθ , Dθ ) a smooth local unstable manifold ˜ W+ (U ; O) with dimension N (U ) which is tangential to U + X− at U . Furthermore, X− is the eigenspace which corresponds to negative eigenvalues of A and possesses an orthogonal basis which consists of real-valued functions (cf. Corollary 2.1). These facts indicate that U is an unstable equilibrium of (S(t), Kθ , Dθ ), too.

Notes and Further Researches The Field–Noyes model was introduced in the papers [FN74a, FN74b], cf. also [TF80]. The model has already attracted interest of many mathematicians. Traveling wave solutions for one-dimensional problem were first constructed by Murray [Mur76] and were succeeded by the papers [FT79, Gib80, GC95, LS96, Tro77, WXY94, YW87]. The stability of traveling wave solutions was discussed by KL90]. In the meantime, there are only a few papers on the asymptotic behavior of solutions of the Field and Noyes model. Wang–Wu WW95] constructed for the d-dimensional problem under the Dirichlet boundary conditions a finite-dimensional global attractor. (But they handle the case where the parameter q vanishes.) They succeeded also in constructing an inertial manifold when Ω is a square domain. Pao Pao88] proved in some specific cases that solutions satisfying the Robin boundary conditions converge to the homogeneous stationary solutions as t → ∞. The stationary problems were studied by BK93, PW04]. For the numerical study, we quote MM82]. The results obtained in Sect. 4 are new. The Keener–Tyson model (10.18) was presented in the paper KT86]. The results on the Keener–Tyson model in this chapter were announced in YOS] without full proof. In YOS], the continuous dependence of exponential attractors M with respect to the parameter ε is also proved by employing Theorem 6.17.

Chapter 11

Forest Kinematic Model

In this chapter, we handle a forest kinematic model. In 1994, Kuznetsov, Antonovsky, Biktashev, and Aponina presented a system of parabolic-ordinary differential equations for three unknown functions u, v, and w for describing kinetics of a simplified forest ecosystem. The unknown function w (density of seeds) satisfies a linear diffusion equation in a domain Ω ⊂ R2 , v (density of old age class trees) satisfies a linear ordinary differential equation at each point of Ω, and u (density of young age class trees) does a nonlinear ordinary differential equation at each point of Ω with a cubic kinetic function on u and v. We treat this system as a degenerate parabolic system; “degenerate” means that the ordinary differential equations for u and v are regarded as a diffusion equation with vanishing diffusion rate. We may lose of course spatial smoothing of solutions for u and v. What is worse, at least in the level of numerical simulation (see Sect. 4.5), u and v can tend to discontinuous functions of Ω as t → ∞ even if they start from continuous functions at t = 0. These evidences make us be careful for choosing underlying function spaces X1 and X2 for treating the equations on u and v, respectively. First, Xi (i = 1, 2) must be closed in the usual product of functions, because the equations contain a cubic kinetic function on u and v and the solution u, v has no smoothing. Secondly, Xi must contain discontinuous functions in Ω. We are thus led to choose L∞ (Ω) as the spaces Xi (i = 1, 2).

1 Model Equations Consider the initial-boundary-value problem ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = βδw − γ (v)u − f u ⎪ ⎪ ⎪ ∂v ⎪ ⎪ = f u − hv in Ω × (0, ∞), ⎪ ⎨ ∂t ∂w in Ω × (0, ∞), ∂t = dΔw − βw + αv ⎪ ⎪ ⎪ ⎪ w=0 on ∂Ω × (0, ∞), ⎪ ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), w(x, 0) = w (x) in Ω, 0 0 0 (11.1) A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_11, © Springer-Verlag Berlin Heidelberg 2010

391

392

11 Forest Kinematic Model

in a two-dimensional C2 or convex, bounded domain Ω. In Ω a prototype forest ecosystem which simply consists of mono-species is considered. We suppose that age generations of trees are divided only into two classes, namely, the young age class and old age class. The unknown functions u(x, t) and v(x, t) denote the tree densities of the young and old age classes in Ω at time t, respectively. The third unknown function w(x, t) denotes the density of seeds in the air over Ω at time t. The third equation describes the kinetics of seeds; d is a diffusion rate in the air, and α and β are the seed production and seed deposition rates, respectively. While the first and second equations describe the growth of the young and old age class trees, respectively, δ is the seed establishment rate (0 < δ ≤ 1), γ (v) is the mortality of young trees, which is allowed to depend on the old-tree density v, f is the aging rate, and h is the mortality of old generation trees. On w, we impose the homogeneous Dirichlet conditions on ∂Ω. The initial functions u0 (x), v0 (x), and w0 (x) are given nonnegative functions in Ω. We assume that the mortality γ (v) is given by a square function of the form γ (v) = a(v − b)2 + c,

(11.2)

which means that the mortality hits its minimum when the old-tree density is a specific optimal value b. We assume also that a, b, c, d, f, h, α, and β are positive constants (> 0) and that δ is a constant such that 0 < δ ≤ 1. We shall first prove the global existence of solutions and then construct a dynamical system determined from (11.1). Fortunately, our dynamical system has a Lyapunov function given by (11.31). Thanks to this Lyapunov function, we can investigate the asymptotic behavior of solutions, and indeed they are shown to tend to some stationary solutions in a certain weak sense. Furthermore, we see that the structure of stationary solutions changes drastically depending on the parameters in the system, especially on the parameter h.

2 Local Solutions 2.1 Abstract Formulation We will handle equations on u and v in L∞ (Ω) and that on w in L2 (Ω). The underlying space is therefore set by ⎛ u ⎞ X = ⎝ v ⎠ ; u, v ∈ L∞ (Ω), w ∈ L2 (Ω) . (11.3) w In addition, the space of initial functions is taken as ⎛ u ⎞ ⎝ ⎠ v ; 0 ≤ u, v ∈ L∞ (Ω), 0 ≤ w ∈ L2 (Ω) . K= w

(11.4)

2 Local Solutions

393

We formulate problem (11.1) as the Cauchy problem for an abstract semilinear equation dU dt + AU = F (U ), 0 < t < ∞, (11.5) U (0) = U0 , in X. Here, A = diag{f, h, Λ} is a diagonal matrix operator of X with domain ⎛ u ⎞ 2 D(A) = ⎝ v ⎠ ; u, v ∈ L∞ (Ω), w ∈ HD (Ω) , w where Λ is a realization of the operator −dΔ + β in L2 (Ω) under the homogeneous Dirichlet boundary conditions and is a positive definite self-adjoint operator of L2 (Ω) with D(Λ) = HD2 (Ω) (see (2.31)). By Theorems 16.12 and 16.13 in Chap. 16, the domains of Λθ , 0 < θ < 1, are characterized by D(Λθ ) = H 2θ (Ω) for 0 ≤ θ < 14 and D(Λθ ) = HD2θ (Ω) for 14 < θ ≤ 1, θ = 34 . Then, A is a sectorial operator of X. For 0 < θ < 1, its fractional powers are given by Aθ = diag{f θ , hθ , Λθ } with ⎛ u ⎞ D(Aθ ) = ⎝ v ⎠ ; u, v ∈ L∞ (Ω), w ∈ D(Λθ ) . w

The nonlinear operator F is given by ⎛ ⎞ βδw − γ (v)u ⎠, fu F (U ) = ⎝ αv

⎛ ⎞ u U = ⎝ v ⎠ ∈ D(Aη ), w

(11.6)

where η is an arbitrarily fixed exponent such that 12 < η < 1. As D(Λη ) ⊂ L∞ (Ω) due to (1.76) and consequently as D(Aη ) ⊂ L∞ (Ω), F is certainly an operator from D(Aη ) into X.

2.2 Construction of Local Solutions It is possible to apply Theorem 4.4 to (11.5). Indeed, after some direct calculations, we get

F (U ) − F (U˜ )X ≤ C U 2X + U˜ 2X + 1 Aη (U − U˜ )X , U, U˜ ∈ D(Aη ), (11.7) with some constant C > 0. This shows that (4.21) is fulfilled. Therefore, for any U0 ∈ K, (11.5) possesses a unique local solution in the function space: u, v ∈ C([0, TU0 ]; L∞ (Ω)) ∩ C1 ((0, TU0 ]; L∞ (Ω)), w ∈ C((0, TU0 ]; HD2 (Ω)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)). (11.8)

394

11 Forest Kinematic Model

Here, TU0 > 0 is determined by the norm U0 X only. Moreover, the local solution satisfies the estimate tAU (t)X + U (t)X ≤ CU0 ,

0 < t ≤ TU 0 ,

(11.9)

with some constant CU0 > 0 depending only on U0 X . We observe also that w(t)L∞ ≤ Cw(t)H 2η ≤ CU0 t −η ,

0 < t ≤ TU 0 .

(11.10)

Then, thanks to (1.104), we can verify the representation formulas u(t) = e

−

t

0 [γ (v(s))+f ] ds

v(t) = e−ht v0 + f

u0 + βδ

t

e−

t s

[γ (v(τ ))+f ] dτ

w(s) ds,

(11.11)

0

t

e−h(t−s) u(s) ds.

(11.12)

0

In verifying (11.11), (11.10) was used.

2.3 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, v, w) be the local solution of (11.5) constructed above. Our goal is to show that u(t) ≥ 0, v(t) ≥ 0, and w(t) ≥ 0 for every 0 < t ≤ TU0 . For this purpose, however, we have to introduce the modified nonlinear operator F˜ (U ) = t (βδw − γ (v)u, f u, αχ(Re v)),

U ∈ D(Aη ),

where χ(v) is a cutoff function given by χ(v) ≡ 0 for −∞ < v < 0 and χ(v) = v for 0 ≤ v < ∞. We also have to consider the auxiliary problem

d U˜ dt

+ AU˜ = F˜ (U˜ ), ˜ U (0) = U0 .

0 < t < ∞,

(11.13)

We can in fact repeat the same arguments as in the preceding subsection to de˜ v, ˜ w) ˜ duce that, for any U0 ∈ K, (11.13) possesses a unique local solution U˜ = t (u, on an interval [0, T˜U0 ] in the function space (11.8)–(11.9) (note that χ(Re v1 ) − χ(Re v2 )L2 ≤ v1 − v2 L2 for v1 , v2 ∈ L2 (Ω)). Before showing the nonnegativity of u, v, w, we will show that of u, ˜ v, ˜ w. ˜ Let us first notice that U˜ is real valued. Indeed, if U˜ is a local solution of (11.13), then its complex conjugate is also a local solution with the same initial value; so, by the uniqueness of solution, they must coincide; this implies that U˜ is real valued. Let us secondly verify the nonnegativity of w. ˜ We use another cutoff function H (w) ˜ which is the C1,1 function given by H (w) ˜ = 12 w˜ 2 for −∞ < w˜ < 0

3 Global Solutions

395

and H (w) ˜ ≡ 0 for 0 ≤ w˜ < ∞. According to (1.100), the function ψ(t) = H ( w(t)) ˜ dx is continuously differentiable with the derivative Ω ψ (t) = d H (w)Δ ˜ w˜ dx + H (w)[−β ˜ w˜ + αχ(v)] ˜ dx. Ω

Ω

By property (1.96), H (w)Δ ˜ w˜ dx = − ∇H (w) ˜ · ∇ w˜ dx = − |∇H (w)| ˜ 2 dx ≤ 0. Ω

Ω

Ω

H (w)[−β ˜ w˜

In addition, it is clear that Ω + αχ(v)] ˜ dx ≤ 0. Hence, ψ (t) ≤ 0; consequently, ψ(t) ≤ ψ(0) for 0 < t ≤ T˜U0 . Thus, ψ(0) = 0 implies ψ(t) ≡ 0, i.e., w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . As for u˜ and v, ˜ we use representations (11.11) and (11.12), which hold equally for U˜ . Therefore, we conclude that u(t) ˜ ≥ 0 and v(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . We now notice that v(t) ˜ ≥ 0 implies that χ(v(t)) ˜ = v(t); ˜ in other words, U˜ is a local solution of the original problem (11.5), too. The uniqueness of solution then yields that U (t) ≡ U˜ (t). Hence, u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for 0 < t ≤ T˜U0 . If T˜U0 ≥ TU0 , we finish the proof. If not, we put T 0 = sup{0 < T ≤ ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for every 0 < t ≤ T }. Since TU0 ; u(t) Ω H (u(T0 )) dx = limtT0 Ω H (u(t)) dx = 0, we see that u(T0 ) ≥ 0. By similar reasons, we have v(T0 ) ≥ 0, w(T0 ) ≥ 0. So, if T0 = TU0 , we finish the proof. If T0 < TU0 , we will consider again problem (11.13) but with the initial time T0 and the initial value U (T0 ). Repeating the same arguments above, we conclude that there is τ > 0 such that u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for every t such that TU0 ≤ t ≤ TU0 + τ . This is a contradiction; hence, T0 = TU0 .

3 Global Solutions 3.1 A Priori Estimates for Local Solutions For U0 ∈ K, let U = t (u, v, w) be a local solution of (11.5) on [0, TU ] in the function space: 0 ≤ u, v ∈ C([0, TU ]; L∞ (Ω)) ∩ C1 ((0, TU ]; L∞ (Ω)), 0 ≤ w ∈ C([0, TU ]; L2 (Ω)) ∩ C((0, TU ]; HD2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (11.14) We establish the following a priori estimates for local solutions. Proposition 11.1 There exist an exponent ρ > 0 and a constant C > 0 such that the estimate U (t)X ≤ C[e−ρt U0 X + 1],

0 ≤ t ≤ TU ,

holds for any local solution U in the function space (11.14).

(11.15)

396

11 Forest Kinematic Model

Proof In the proof, C, C1 , C2 , . . . , stand for some constants which are determined only by Ω and by the initial constants in the equations of (11.1). Similarly, ρ stands for some positive exponent determined in the same manner. Step 1. Let us first estimate the norm U (t)L2 . Multiply the first equation of (11.1) by u and integrate the product in Ω. Then, 1 d 2 2 u dx + f u dx = βδ wu dx − γ (v)u2 dx 2 dt Ω Ω Ω Ω f u2 dx + C1 w 2 dx − γ (v)u2 dx. ≤ 2 Ω Ω Ω (11.16) Multiply the third equation of (11.1) by w and integrate the product in Ω. Then, 1 d 2 2 w dx + d |∇w| dx + β w 2 dx 2 dt Ω Ω Ω β vw dx ≤ w 2 dx + C2 v 2 dx. =α 2 Ω Ω Ω Let C3 > 0 be a constant such that C1 C3 ≤ β4 . Multiply (11.16) by C3 and add the product to the above inequality. Then, we obtain that C3 d 1 d C3 f β 2 2 2 u dx + w dx + u dx + w 2 dx 2 dt Ω 2 dt Ω 2 Ω 4 Ω v 2 dx − C3 γ (v)u2 dx. (11.17) ≤ C2 Ω

Ω

Multiply next the second equation of (11.1) by v and integrate the product in Ω. Then, 1 d 2 2 v dx + h v dx = f uv dx. 2 dt Ω Ω Ω Let C4 > 0 be a constant such that C4 h ≥ 2C2 . Multiply the present equality by C4 and add the product to inequality (11.17) to obtain that C3 d C4 d 1 d C3 f u2 dx + v 2 dx + w 2 dx + u2 dx 2 dt Ω 2 dt Ω 2 dt Ω 2 Ω β 2 2 v dx + w dx ≤ C4 f uv dx − C3 γ (v)u2 dx. + C2 4 Ω Ω Ω Ω We here notice that C2f 2 C4 f uv − C3 γ (v)u2 = − C3 a(v − b)2 u2 − C4 f (v − b)u + 4 4C3 a

3 Global Solutions

397

C 2 f 2 b2 C 2 f 2 1 b2 − C3 cu2 − C4 f bu + 4 + 4 + 4C3 c 4C3 a c C 2 f 2 1 b2 ≤ 4 + . 4C3 a c Therefore, d dt

(C3 u + C4 v + w ) dx + ρ 2

2

(C3 u2 + C4 v 2 + w 2 ) dx ≤ C.

2

Ω

Ω

Hence (cf. (1.58)), we conclude that U (t)2L2 ≤ C e−ρt U0 2L2 + 1 ,

0 ≤ t ≤ TU .

(11.18)

Step 2. We will estimate the norm w(t)L∞ . We have t η − t−s Λ − t−s Λ βt βt Λη w(t) = Λη e− 2 Λ e− 2 Λ w0 + α Λ e 2 e 2 v(s) ds. 0

Then, βt

w(t)H 2η ≤ C(1 + t −η )e− 2 w0 L2 + C e−tΛ

t

0

β 1 + (t − s)−η e− 2 (t−s) v(s)L2 ds,

≤ e−βt

where we used the estimate for t ≥ 0 (recall that Λ ≥ β). FurtherL2 more, by (11.18), t β 1 + (t − s)−η e− 2 (t−s) v(s)L2 ds 0

t

β 1 + (t − s)−η e− 2 (t−s) ds

≤C 0

+ Ce

−ρ t

0

≤ C[e

−ρ t

t

β 1 + (t − s)−η e−( 2 −ρ )(t−s) e−(ρ−ρ )s dsU0 L2

U0 L2 + 1],

where 0 < ρ < min{ β2 , ρ}. Replacing ρ by new one, we thus conclude that w(t)L∞ ≤ Cw(t)H 2η ≤ C[(1 + t −η )e−ρt U0 L2 + 1],

0 < t ≤ TU . (11.19)

Step 3. Let us finally estimate the norms u(t)L∞ and v(t)L∞ by using (11.11) and (11.12). In view of (11.19), we have t −f t e−f (t−s) [(1 + s −η )e−ρs U0 L2 + 1] ds. u(t)L∞ ≤ e u0 L∞ + C 0

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Therefore, replacing ρ by new one, we conclude that u(t)L∞ ≤ C[e−ρt U0 X + 1],

0 ≤ t ≤ TU .

In a similarly way, by (11.12), v(t)L∞ ≤ C[e−ρt U0 X + 1],

0 ≤ t ≤ TU .

These, together with (11.18), finally yield the desired estimates (11.15).

3.2 Global Solutions As the a priori estimates have been established for local solutions of (11.5) with U0 ∈ K, we can apply Corollary 4.3 to conclude the global existence of solutions. Indeed, for any U0 ∈ K, there exists a unique global solution to (11.5) in the function space: 0 ≤ u, v ∈ C([0, ∞); L∞ (Ω)) ∩ C1 ((0, ∞); L∞ (Ω)), 0 ≤ w ∈ C((0, ∞); HD2 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (11.20)

3.3 Global Norm Estimates For U0 ∈ K, let U (t) = U (t; U0 ) be the global solution of (11.5) with the initial value U0 in the function space (11.20). From (11.15) it immediately follows that U (t; U0 )X ≤ C[e−ρt U0 X + 1],

0 ≤ t < ∞, U0 ∈ K.

(11.21)

In particular, U (t; U0 )X ≤ C[U0 X + 1],

0 ≤ t < ∞, U0 ∈ K.

This, together with (11.9), yields that AU (t; U0 ) ≤ (1 + t −1 )p0 (U0 X ),

0 < t < ∞, U0 ∈ K,

(11.22)

with a suitable continuous increasing function p0 (·). In particular, w(t)H 2 ≤ (1 + t −1 )p0 (U0 X ),

0 < t < ∞, U0 ∈ K.

As a consequence, w(t)L∞ ≤ (1 + t −η )p0 (U0 X ),

0 < t < ∞, U0 ∈ K.

We can further prove the following norm estimates.

(11.23)

3 Global Solutions

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Proposition 11.2 For the derivative U (t), it holds that u (t)L∞ ≤ (1 + t −η )p1 (U0 X ),

v (t)L∞ ≤ p1 (U0 X ),

0 < t < ∞, U0 ∈ K,

0 < t < ∞, U0 ∈ K,

w (t)L2 ≤ (1 + t −1 )p1 (U0 X ),

(11.24) (11.25)

0 < t < ∞, U0 ∈ K,

(11.26)

where p1 (·) is an appropriate continuous increasing function. Proof In view of (11.23), (11.24) is verified directly from the equation of u. In the meantime, (11.25) is obvious by (11.21). The last estimate (11.26) is an immediate consequence of (11.22). Proposition 11.3 For the second order derivative U (t), it holds that u (t)L∞ ≤ (1 + t −1−η )p2 (U0 X ),

v (t)L∞ ≤ (1 + t

−η

0 < t < ∞, U0 ∈ K,

(11.27)

)p2 (U0 X ),

0 < t < ∞, U0 ∈ K,

(11.28)

w (t)L2 ≤ (1 + t −2 )p2 (U0 X ),

0 < t < ∞, U0 ∈ K,

(11.29)

where p2 (·) is an appropriate continuous increasing function. Proof By the equation of v in (11.5), v (t) = f u (t) − hv (t),

0 < t < ∞.

Then, v ∈ C2 ((0, ∞); L∞ (Ω)), and estimate (11.28) is verified by (11.24) and (11.25). We can next use the regularity property of solutions of linear evolution equations, regarding the equation of w as a linear equation with the unknown function w and given force function v. Since v belongs to C2 ([τ, ∞); L2 (Ω)), we conclude by Theorem 3.6 that w ∈ C2 ((τ, ∞); L2 (Ω)) with the estimate w (t)L2 + w (t)H 2 ≤ C 1 + (t − τ )−1 w (τ )L2 ,

τ < t < ∞,

where τ > 0 is any positive time. So, taking τ = 2t , we verify (11.29) and the estimate w (t)H 2 ≤ (1 + t −2 )p2 (U0 X ),

0 < t < ∞.

As a consequence, we verify that w (t)L∞ ≤ Cw (t)H 2η ≤ (1 + t −1−η )p2 (U0 X ),

0 < t < ∞.

Then, (11.27) is observed by differentiating the equation of u in t .

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4 Dynamical System We now construct a dynamical system determined from (11.5). It is possible to follow the general strategy described in Chap. 6, Sect. 5.1 by taking β = 0. Estimate (11.21) indeed shows that (6.56) is fulfilled; therefore, it is immediate to conclude that (11.5) defines a dynamical system (S(t), K, X). For any exponent 0 < θ < 1, (11.5) equally defines a dynamical system in the universal space Dθ = D(Aθ ) with the phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (11.6) fulfills (4.21) and a fortiori (4.2) with β = θ . D Let Kθ,R = K ∩ B θ (0; R). Applying Theorem 3.12, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

This, together with (11.22), then yields that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that (6.56) is valid in the space Dθ , too. Therefore, for any 0 < θ < 1, (S(t), Kθ , Dθ ) defines a dynamical system. X Estimate (11.15) means that, for any KR = K ∩ B (0; R), there exists tR > 0 such that ˜ =K ˜ S(t)KR ⊂ K ∩ B X (0; 2C) 2C

for any tR ≤ t < ∞.

˜ Thus, we Moreover, let B = S(1)K2C˜ . Then, by (11.22), B ⊂ B D(A) (0; 2p(2C)). have observed that (S(t), K, X) possesses an absorbing set B (⊂ K) which is a bounded set of D(A).

4.1 Lyapunov Function We can build up a Lyapunov function Ψ (U ) for (S(t), K, X). Let U0 ∈ K, and let S(t)U0 = U (t). Put ϕ(t) = f u(t) − hv(t), 0 ≤ t < ∞. From the equations of u and v in (11.5) it is easily obtained that ∂ϕ = f {βδw − [γ (v) + f ]u} − h(f u − hv) ∂t = fβδw − [γ (v) + f + h]ϕ − h[γ (v) + f ]v. Multiply this by ϕ(t) = ∂v ∂t and integrate the product in Ω. Then, ∂v d 1 d ϕ 2 dx + h Γ (v) dx − fβδ w dx 2 dt Ω dt Ω Ω ∂t 2 ∂v = − [γ (v) + f + h] dx, ∂t Ω v where Γ (v) = 0 [γ (v)v + f v] dv.

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401

Meanwhile, multiply the equation of w in (11.5) by ∂w ∂t and integrate the product in Ω. Then, d d ∂w 2 β d ∂w 2 2 |∇w| dx + w dx − α v dx. dx = − 2 dt Ω 2 dt Ω Ω ∂t Ω ∂t From these two energy equalities we obtain that α 2 dfβδ d fβ 2 δ 2 2 ϕ + |∇w| + hαΓ (v) + w − (f αβδ)vw dx dt Ω 2 2 2 2 ∂v ∂w 2 α[γ (v) + f + h] dx ≤ 0. (11.30) =− + fβδ ∂t ∂t Ω Note that hαΓ (v) +

fβ 2 δ 2 w − (f αβδ)vw ≥ −C 2

for all v, w ≥ 0,

with some constant C > 0. So, the functional given by α dfβδ (f u − hv)2 + |∇w|2 + hαΓ (v) Ψ (U ) = 2 2 Ω 1 fβ 2 δ 2 w − (f αβδ)vw dx, U ∈ D A 2 , + 2

(11.31)

becomes a Lyapunov function for the dynamical system (S(t), K, X). The Lyapunov function provides the following two propositions. Proposition 11.4 For any trajectory S(t)U0 = U (t), it holds that ∞ dU 2 dt (t) dt < ∞. 1 L2 Proof Integrate both sides of (11.30) on an interval [1, T ]. Then, 2 T ∂v ∂w 2 α[γ (v) + f + h] dx dt + fβδ ∂t ∂t Ω 1 α dfβδ ≤ ϕ(1)2 + |∇w(1)|2 + hαΓ (v(1)) 2 Ω 2 fβ 2 δ 2 w(1) + f αβδv(T )w(T ) dx. + 2 By (11.21) it is observed that 2 ∞ ∂v ∂w 2 α[γ (v) + f + h] dx dt < ∞. + fβδ ∂t ∂t 1 Ω

(11.32)

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Differentiating the equation of u in (11.1) with respect to t, we have ∂u ∂w ∂v ∂ 2u − γ (v)u − [γ (v) + f ] . = βδ ∂t ∂t ∂t ∂t 2 ∂u ∂t

Multiply this by 1 d 2 dt

Ω

∂u ∂t

and integrate the product in Ω. Then,

2

2 ∂w ∂v ∂u ∂u − γ (v)u dx − (γ (v) + f ) dx ∂t ∂t ∂t ∂t Ω Ω 2 ∂v ∂w 2 dx ≤ C(U0 X + 1) + ∂t ∂t Ω 2 f ∂u − dx. 2 Ω ∂t

dx =

βδ

Integrating this inequality on [1, T ], we obtain that f 2

T 1

Ω

∂u ∂t

2 dx dt ≤

2 ∂u (1) dx Ω ∂t + C(U0 X + 1)

1 2

T

1

On account of (11.32), we conclude that sired integrability has been verified.

f 2

Ω

∞ 1

∂v ∂t

2

∂u 2 Ω ( ∂t ) dx dt

+

∂w ∂t

2 dx dt.

< ∞. Hence, the de

Proposition 11.5 For any trajectory S(t)U0 = U (t), the derivative 0 as t → ∞ in the L2 norm.

dU dt (t)

tends to

d dU dt (t)2L2 | ≤ CU0 for 1 ≤ t < ∞ Proof Propositions 11.2 and 11.3 yield that | dt with some constant CU0 > 0 depending on U0 X . This, together with integrability dU 2 of dU dt (t)L2 in (1, ∞), implies the desired convergence dt (t)L2 → 0.

We finally verify the following fact. 1

Theorem 11.1 The Lyapunov function Ψ : D(A 2 ) → C is Fréchet differentiable, and U ∈ D(A) is a stationary solution of (11.1) if and only if Ψ (U ) = 0. Proof By direct calculations it is observed that Ψ is Fréchet differentiable, and its derivative is given by Ψ (U )U = α(f u − hv)(f u − hv) + dfβδ∇w · ∇w + hαΓ (v)v Ω

+ fβ 2 δww − f αβδ(vw + wv) dx,

4 Dynamical System

403

⎛ ⎞ ⎛ ⎞ u u 1 U = ⎝v ⎠, U = ⎝v ⎠ ∈ D A2 . w w Consequently, Ψ (U )U =

f α(f u − hv)u + α[h2 v + hγ (v)v − fβδw]v

Ω

+ fβδ[d∇w · ∇w + (βw − αv)w] dx. Therefore, Ψ (U ) = 0 if and only if f u − hv = 0, h2 v + hγ (v)v − fβδw = 0 and [d∇w · ∇w + (βw − αv)w] dx = 0 for all w ∈ H˚ 1 (Ω). Ω

Furthermore, Ψ (U ) = 0 if and only if f u − hv = 0, hu + γ (v)u − βδw = 0, and −dΔw + βw − αv = 0 in H −1 (Ω). Hence, Ψ (U ) = 0 if and only if U is a stationary solution of (11.1).

4.2 ω-limit Sets For each U0 ∈ K, the ω-limit set of the trajectory S(t)U0 is defined by ω(U0 ) =

{S(τ )U0 ; t ≤ τ < ∞} (closure in X),

t≥0

see Chap. 6, Sect. 1.2. However, we have the following observation. Remark 11.1 As will be seen in Sect. 4.5, numerical examples suggest that there exists a trajectory which starts from an initial value U0 ∈ K consisting of a continuous function of Ω componentwise but, as t → ∞, converges to a discontinuous stationary solution U = t (u, v, w). If this is the case, then any sequence of the form S(tn )U0 cannot converge to U in the topology of X. (Indeed, on account of the topology of X, u and v must be continuous functions; in the meantime, since w ∈ H 2 (Ω), w is also a continuous function.) Therefore, for U0 ∈ K, it must hold that ω(U0 ) = ∅. This suggests also that the dynamical system (S(t), K, X) cannot have a global attractor. In view of this observation, we are led to introduce other types of ω-limit sets. A sequence Un = t (un , vn , wn ) in X is said to be L2 convergent to U = t (u, v, w) ∈ X as n → ∞ if u → u in L (Ω), v → v in L (Ω), and w → w n 2 n 2 n in L2 (Ω), respectively. Then, using this topology, we define the L2 -ω-limit set of

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S(t)U0 , U0 ∈ K, by L2 -ω(U0 ) =

{S(τ )U0 ; t ≤ τ < ∞}

(closure in the L2 topology of X).

t≥0

In the meantime, a sequence Un = t (un , vn , wn ) in X is said to be weak* convergent to U = t (u, v, w) ∈ X as n → ∞ if un → u weak* in L∞ (Ω), vn → v weak* in L∞ (Ω), and wn → w in L2 (Ω), respectively. Using this topology, we define the w*-ω-limit set of S(t)U0 , U0 ∈ K, by w*-ω(U0 ) =

{S(τ )U0 ; t ≤ τ < ∞} (closure in the weak* topology of X).

t≥0

Concerning these ω-limit sets, we have the following results. Theorem 11.2 For each U0 ∈ K, w*-ω(U0 ) is a nonempty set. Proof Let U0 ∈ K and S(t)U0 = U (t). On account of (11.22), we notice that the Banach–Alaoglu theorem of sequentially weak* compact version (see Brezis [Bre83, IV. 3. C]) and the compact embedding theorem, i.e., H 2 (Ω) ⊂ L2 (Ω), are available to conclude that there exists a time sequence {tn } tending to ∞ such that u(tn ) → u weak* in L∞ (Ω), v(tn ) → v weak* in L∞ (Ω), and w(tn ) → w in L2 (Ω), that is, U (tn ) → U in the weak* topology of X. Theorem 11.3 For each U0 ∈ K, ω(U0 ) ⊂ L2 -ω(U0 ) ⊂ w*-ω(U0 ). Proof The first relation ω(U0 ) ⊂ L2 -ω(U0 ) is obvious by the definition. Let U ∈ L2 -ω(U0 ). Then, there exists a sequence {tn } ∞ such that S(tn )U0 = Un (t) → U in the L2 topology of X. Let ϕ ∈ L1 (Ω). For any f ∈ L2 (Ω), ϕ[u(tn ) − u] dx ≤ ϕ − f L u(tn ) − uL + f [u(tn ) − u] dx . ∞ 1 Ω

Ω

Since L2 (Ω) is dense in L1 (Ω) and since (11.21) is true, we verify that, as tn → ∞, | Ω ϕ[u(tn ) − u] dx| → 0. Hence, u(tn ) → u in the weak* topology of L∞ (Ω). It is the same for the weak* convergence of v(tn ) to v. Thus, we have U ∈ w*-ω(U0 ). Theorem 11.4 For U0 ∈ K, let there exist a sequence {tn } tending to ∞ such that the sequence of functions S(tn )U0 converges to U ∈ X componentwise almost everywhere in Ω. Then, U ∈ L2 -ω(U0 ). Proof By (11.21) and (11.23), the almost everywhere convergence implies the L2 convergence of the sequences of u(tn ), v(tn ), and w(tn ). Hence, U ∈ L2 -ω(U0 ).

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405

4.3 Constituents of L2 -ω-limit Sets Generally speaking, if there exists a Lyapunov function, then one can prove that the ω-limit set consists of equilibria of a dynamical system. However, in the present case, we do not know that S(t) is continuous with respect to the weak* topology of X. We can merely prove the assertion only for the L2 -ω-limit set. Theorem 11.5 If L2 -ω(U0 ) is nonempty, then the set consists of equilibria of (S(t), K, X). Proof We first notice that S(t) : K → X is continuous with respect to the L2 topology of X. Indeed, by similar techniques as in Step 1 of the proof of Proposition 11.1, we can establish the local Lipschitz condition S(t)U0 − S(t)U˜ 0 L2 ≤ CT ,R U0 − U˜ 0 L2 , 0 ≤ t ≤ T ; U0 , U˜ 0 ∈ K with U0 L2 + U˜ 0 L2 ≤ R. The detailed proof is however left to the reader. Let U ∈ L2 -ω(U0 ). Then, there exists a sequence tn → ∞ such that S(tn )U0 = U (tn ) → U in the L2 topology. By (11.22), we can take a subsequence {w(tn )} of {w(tn )} such that w(tn ) → w in H 1 (Ω). It is then easy to see that w = w . Meanwhile, u(tn ) → u and v(tn ) → v in any Lp topology with any finite 2 ≤ p < ∞. By these facts we conclude that the value Ψ (U (tn )) of the Lyapunov function is convergent to Ψ (U ) as tn → ∞. That is, Ψ (U ) = lim Ψ (U (tn )) = inf Ψ (S(t)U0 ) ≡ Ψ∞ . n →∞

0≤t 0. 3f

We here notice that

2 (c + f )2 u2 + (αδv)2 + (βδw)2 − (c + f )αδuv − αβδ 2 vw − (c + f )βδwu + 3εv 2

2

(c + f )αδu 2 2 αβδ 2 w 2 − α δ +ε v + = √ α2 δ2 + ε v − √ α2δ2 + ε α2δ2 + ε (c + f )2 u2 (βδw)2 + v2 + 2 2 . + [βδw − (c + f )u]2 + ε 2 2 α δ +ε α δ +ε

4 Dynamical System

407

Therefore, with an appropriate exponent ρ > 0, d (C1 u2 + C2 v 2 + C3 w 2 ) dx + ρ (C1 u2 + C2 v 2 + C3 w 2 ) dx ≤ 0, dt Ω Ω where Ci > 0 (i = 1, 2, 3) are some positive constants. Thus, U (t)L2 ≤ Ce−ρt U0 L2 ,

0 < t < ∞, U0 ∈ K.

This shows that, as t → ∞, S(t)U0 converges to O exponentially in the L2 topology. We can then repeat similar calculations as in Steps 2 and 3 of the proof of Proposition 11.1 to derive the exponential decay of S(t)U0 in the X norm. In particular, the assertion of theorem is verified. We secondly consider the case where ab2 < 3(c + f ). We notice in this case that the cubic function Q(v) ≡

h [γ (v) + f ]v, fβδ

−∞ < v < ∞,

(11.36)

has the following properties. Lemma 11.1 When ab2 < 3(c + f ), Q (v) > 0 for all −∞ < v < ∞. Therefore, w = Q(v) is an increasing function, and its inverse v = Q−1 (w) is a single-valued smooth function for −∞ < w < ∞; moreover, the inverse function is uniformly Lipschitz continuous, i.e., |Q−1 (w) − Q−1 (w)| ˜ ≤ L|w − w|, ˜ −∞ < w, w˜ < ∞. On the contrary, when ab2 ≥ 3(c + f ), Q (v) has real roots 0 < v1 ≤ v2 < ∞ (when ab2 = 3(c + f ), v1 = v2 ). Proof We obviously see from (11.2) that h (3av 2 − 4abv + ab2 + c + f ) fβδ 2b 2 ab2 − 3(c + f ) h 3a v − . − = fβδ 3 3

Q (v) =

Therefore, all the assertions of lemma are clear.

Using these properties of Q(v), we prove the following theorem. Theorem 11.7 Let ab2 < 3(c + f ). Then, L2 -ω(U0 ) = w*-ω(U0 ) for every U0 ∈ K. Proof Let S(t)U0 = t (u(t), v(t), w(t)). Consider any time sequence {tn } which tends to ∞ as n → ∞. By (11.22), w(tn )H 2 is a bounded sequence; so, we can choose a subsequence {tn } for which {w(tn )} is convergent to w in H 1+ε (Ω) and

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11 Forest Kinematic Model

hence in L∞ (Ω). From the first and second equations of (11.1), it is easily observed that 1 du γ (v(tn )) + f dv Q(v(tn )) = w(tn ) − (tn ) − (tn ). βδ dt fβδ dt So, by (11.36), 1 du γ (v(tn )) + f dv v(tn ) = Q−1 w(tn ) − (tn ) − (tn ) . βδ dt fβδ dt Since w(tn ) → w in L∞ (Ω) and since Proposition 11.5 is valid, we conclude that v(tn ) converges to v = Q−1 (w) in L2 (Ω). Since Proposition 11.5 provides in particular that, as t → ∞, f u(t)−hv(t) → 0 in L2 (Ω), we conclude also that u(tn ) converges to fh v in L2 (Ω). Thus, we have shown that t (u(tn ), v(tn ), w(tn )) → t (u, v, w) in L (Ω). 2 We now know that any sequence t (u(tn ), v(tn ), w(tn )) has a subsequence which converges to some vector of X in the L2 topology. In particular, the relation w*-ω(U0 ) ⊂ L2 -ω(U0 ) must be true.

4.5 Numerical Examples We here present some numerical example. Let Ω = [0, 1] × [0, 1] be a quadratic domain. Set the parameters as α = β = δ = 1.0, f = g = 1.0, a = 1.0, b = 3.0, c = 0.2, and d = 0.05. We treat h as a control parameter and put h = 0.0488 in order that the parameters satisfy the relation 0 0 is the minimal eigenvalue of the operator −Δ in L2 (Ω) equipped with the homogeneous Dirichlet boundary conditions.

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11 Forest Kinematic Model

5.1 Localized Problem in a Neighborhood of O For investigating the stability and instability of O, we can follow the general strategy in Chap. 6, Sect. 6. Let χ(u) be a cutoff function defined on the complex plane C such that χ(u) = u for u : |u| < 1 and χ(u) = u/|u| for u : |u| ≥ 1. The localized problem in a neighborhood of O is given by ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = βδχ(w) − γ (χ(v))χ(u) − f u ⎪ ⎪ ⎪ ∂v ⎪ ⎪ in Ω × (0, ∞), ⎨ ∂t = f χ(u) − hv ∂w ⎪ ∂t = dΔw − βw + αχ(v) ⎪ ⎪ ⎪ w =0 ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), 0 0

in Ω × (0, ∞), on ∂Ω × (0, ∞), w(x, 0) = w0 (x) in Ω. (11.38) We can then repeat the same arguments as in Sect. 1 to construct local solutions and global solutions for all initial values from X in the function space: U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); X) ∩ C1 ((0, ∞); X). ˜ acting Therefore, the localized problem (11.38) defines a continuous semigroup S(t) ˜ on X. Furthermore, it is verified that (S(t), X, X) defines a dynamical system which is a complexified dynamical system of (S(t), K, X). It is also verified that, if U0 ∈ D(Aθ ) with 0 < θ < 1, then U belongs to: U ∈ C((0, ∞); D(A)) ∩ C([0, X); D(Aθ )) ∩ C1 ((0, ∞); D(Aθ )). ˜ From this we can equally verify that, for any 0 < θ < 1, (S(t), Dθ , Dθ ) is a dynamθ ical system in Dθ , where Dθ = D(A ). We now fix 12 < θ < 1 in order to have Dθ ⊂ L∞ (Ω). Then, in a suitable neighborhood of O in the space Dθ , any trajectory of (S(t), Kθ , Dθ ) is that of ˜ ˜ (S(t), Dθ , Dθ ). In particular, O is an equilibrium of (S(t), Dθ , Dθ ), too. Therefore, ˜ if O is stable as an equilibrium of (S(t), Dθ , Dθ ), then it is the same as that of (S(t), Kθ , Dθ ). However, we cannot say automatically that, even if O is unstable ˜ in (S(t), Dθ , Dθ ), it is the same as that of (S(t), Kθ , Dθ ). Nevertheless, the insta˜ bility of O in (S(t), Dθ , Dθ ) provides an information which indicates its instability in the original system (S(t), Kθ , Dθ ). In fact, that shows existence of a smooth lo˜ cal unstable manifold W˜ + (O; O) in (S(t), Dθ , Dθ ), which is tangential to the space X− at O, where X− is a finite- or infinite-dimensional eigenspace corresponding to the negative eigenvalues of the linearized operator A − F˜ (O) given by (11.39). Furthermore, X− has a basis consisting of vectors which are componentwise realvalued functions in Ω. ˜ Let us next verify the Fréchet differentiability of S(t) in a neighborhood of O. ˜ is determined by the Cauchy problem for a semilinear equation of We know that S(t)

5 Homogeneous Stationary Solution

411

the form (11.5) in which the nonlinear operator is replaced by the following operator F˜ (U ) : D(Aη ) → X, where 12 < θ < η < 1: ⎛ ⎞ ⎛ ⎞ βδχ(w) − γ (χ(v))χ(u) u ⎠ , U = ⎝ v ⎠ ∈ D(Aη ). f χ(u) F˜ (U ) = ⎝ αχ(v) w In a neighborhood of O in D(Aη ) ⊂ D(Aθ ) ⊂ L∞ (Ω), F˜ is Fréchet differentiable with the derivative ⎛ ⎞ βδr − γ (v)p − γ (v)uq ⎠, fp F˜ (U ) · H = ⎝ αq ⎛ ⎞ p η U ∈ B D(A ) (O; r), H = ⎝ q ⎠ ∈ D(Aη ). r Therefore, F˜ is of class C1,1 and satisfies conditions (6.68) and (6.69). So, the gen˜ : Dθ → eral results obtained in Chap. 6, Sect. 6 are available to conclude that S(t) Dθ is of class C1,1 in a neighborhood of O in Dθ for 0 ≤ t ≤ T , where T > 0. In ˜ is Fréchet differentiable at O for any 0 ≤ t < ∞ with the differential particular, S(t) ˜ O = e−tA , where e−tA is the analytic semigroup on X generated by S(t) ⎛ ⎞ m 0 −βδ h 0 ⎠ (11.39) −A = −(A − F˜ (O)) = ⎝−f 0 −α Λ with m = γ (0) + f = ab2 + c + f .

˜ O 5.2 Spectrum of S(t) Let us characterize in this subsection the spectrum of A in X, i.e., σ X (A). To this end, let us find λ ∈ C such that, for any H = t (p, q, r) ∈ X, the vector equation (λ − A)U = H has a unique solution U = t (u, v, w) ∈ D(A). Obviously, this problem is equivalent to ⎧ ⎪ ⎨(λ − m)u + βδw = p, f u + (λ − h)v = q, ⎪ ⎩ αv + (λ − Λ)w = r. It then follows that [(λ − m)(λ − h)(λ − Λ) + f αβδ]w = f αp − α(λ − m)q + (λ − m)(λ − h)r.

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11 Forest Kinematic Model

Therefore, if λ is a solution of the equation (λ − m)(λ − h) = 0, i.e., λ = m or λ = h, then w cannot belong to H 2 (Ω) in general. This means that m and h do not belong to ρ X (A), i.e., m, h ∈ σ X (A). So, let λ be neither m nor h. Then, we verify that λ ∈ ρ X (A)

⇔

λ+

f αβδ ∈ ρ L2 (Λ). (λ − m)(λ − h)

λ ∈ σ X (A)

⇔

λ+

f αβδ = dμn + β (λ − m)(λ − h)

In other words,

for some n = 0, 1, 2, . . . , where (0 0. The construction of a dynamical system for (11.1) in the underlying space X was done by Chuan–Yagi [CY06] and Shirai–Chuan–Yagi [SCY07a]. The structure of the ω-limit set of trajectories was studied by Chuan–Tsujikawa–Yagi [CTY06] and Shirai–Chuan–Yagi [SCY07b]. Although we know the existence of the Lyapunov function given by (11.31), there remain many essential questions unanswered for the ω-limit sets. For example, does the limit set w*-ω(U0 ) consist only of stationary solutions of (11.1)? In any case, is the limit set L2 -ω(U0 ) nonempty? When do the

Notes and Further Researches

415

two limit sets w*-ω(U0 ) and L2 -ω(U0 ) coincide? Can we utilize the Simon and Łojasiewicz method (cf. [Sim83] and [Loj63, Loj65]) in order to show that the ω-limit sets are singletons? The stability and instability of the homogeneous stationary solution O were studied by Shirai–Chuan–Yagi [SCY08]. In Chuan–Tsujikawa–Yagi [CTY09], the existence of an infinite number of discontinuous stationary solutions was indicated but under the homogeneous Neumann boundary conditions on w. In the Dirichlet case, however, there is no such a rigorous result. Mola–Yagi [MY09] treated a forest kinematic model with memory.

Chapter 12

Chemotaxis Models

Budrene–Berg discovered [BB91, BB95] a remarkable aggregation pattern generated by chemotactic bacteria. In 1995, Woodward–Tyson–Myerscough–Murray– Budrene–Berg presented the mathematical model ⎧ u ⎨ ∂u = aΔu − μ∇ · ∇ρ + cu 1 − us in Ω × (0, ∞), ∂t (1+αρ)2 (12.1) rρ p ∂ρ u su ⎩ = bΔρ + in Ω × (0, ∞), ∂t 1+βuq − 1+γρ for describing the process of pattern formation. Here, Ω is a two-dimensional domain where bacteria are bred. The function u(x, t) denotes the population density of bacteria in Ω at time t . The function ρ(x, t) denotes the concentration of chemical attractant in Ω at time t . The movement of biological individuals consists of two effects: one is a random walk, and the other is a directed movement in a sense that they have a tendency to move toward a higher concentration direction of the chemical substance. The latter directed movement is called chemotaxis. In 1970, Keller–Segel introduced a mathematical formulation for chemotaxis by the diffusion–advection equation ∂u = aΔu − μ∇ · [u∇χ(ρ)], ∂t where χ(ρ) is a sensitivity function of bacteria toward the chemoattractant, and μ > 0 is an attraction rate. Several forms are proposed as sensitivity functions. The ρ typical ones are such that χ(ρ) = ρ, ρ 2 , log(ρ + 1), log ρ, ρ+1 , etc. In (12.1), the ρ authors took the function χ(ρ) = 1+αρ . In (12.1), the term cu(1 − us ) denotes the proliferation of bacteria by the logistic law, where s denotes the concentration of the substrate. In this model, s is assumed sup to be spatially homogeneous and constant. The term 1+βu q denotes the production of chemoattractant by bacteria, where p, q are some nonnegative exponents and ur ρ denotes the consumption of chemoattractant β ≥ 0 is a constant. The term − 1+γρ by bacteria, where r is some nonnegative exponent, and γ ≥ 0 is a constant. The bacteria and the chemical substance diffuse with rates a > 0 and b > 0, respectively. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_12, © Springer-Verlag Berlin Heidelberg 2010

417

418

12 Chemotaxis Models

We will consider in this chapter the following simplified model: ∂u 2 in Ω × (0, ∞), ∂t = aΔu − μ∇ · [u∇χ(ρ)] + cu − γ u ∂ρ ∂t

= bΔρ − dρ + νu

in Ω × (0, ∞).

(12.2)

The sensitivity function will not be specified but will be assumed to satisfy condiρ tion (12.4). Actually, χ(ρ) = ρ, log(ρ +1), and ρ+1 are allowed. For the production term, we take p = 1 and β = 0. We ignore the consumption of chemical substance by bacteria, i.e., r = 0; instead we consider the unconstrained resolution which is denoted by −dρ with rate d > 0. First, we construct a dynamical system for the initial-boundary-value problem of (12.2) imposing the homogeneous Neumann boundary conditions for u and ρ and show that the dynamical system possesses exponential attractors. Secondly, we investigate stability and instability of the homogeneous stationary solution cνthe (u, ρ) = γc , dγ . In fact, we show that, when the product μν is small, (u, ρ) is stable and, on the other hand, that, when μν is large enough, (u, ρ) is no longer stable. Moreover, we show that the degree of instability tends to infinity as μν → ∞ and consequently the fractal dimensions of exponential attractors also tend to infinity.

1 Chemotaxis Model Without Proliferation Before studying (12.2), let us consider a chemotaxis model without proliferation. We are concerned with the following problem: ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu − μ∇ · [u∇χ(ρ)] ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu in Ω × (0, ∞), ∂t (12.3) ∂ρ ∂u ⎪ = ∂n = 0 on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ⎪ ⎪ ⎩u(x, 0) = u (x), ρ(x, 0) = ρ (x) in Ω. 0 0 As mentioned above, such a chemotaxis model was founded by Keller–Segel [KS70]. The form of the equations of (12.3) is of some particular case of the general ones presented in the paper. Here, Ω is a two-dimensional C2 or convex, bounded domain. We assume that the sensitivity function χ(ρ) is a real smooth function for 0 ≤ ρ < ∞ satisfying the condition i d χ sup i (ρ) < ∞ for i = 1, 2. (12.4) dρ 0≤ρ0). First, we construct local solutions for problem (12.3). Secondly, in the case where χ(ρ) is linear (= ρ), we establish a priori estimates of local solutions for the initial functions u0 (x) ≥ 0 and ρ0 (x) ≥ 0 with sufficiently small norm u0 L1 . Consequently, for such initial functions, we can show the global existence of solutions.

1 Chemotaxis Model Without Proliferation

419

However, we must notice the fact due to Herrero–Velázquez [HV96] that the local solutions to (12.3) blow up in finite time, provided that the norms u0 L1 of their initial functions are large.

1.1 Abstract Formulation Let us formulate problem (12.3) as the Cauchy problem for an abstract evolution equation. It will be spontaneous to try to handle the first equation of (12.3) on u in the space L2 (Ω). Then, we are naturally led to handle the second one in the Sobolev space H 2 (Ω). As will be observed in establishing a priori estimates for (12.3) (see Propositions 12.2 and 12.3), we have to couple the norms u(t)2L2 and ρ(t)2H 2 . This means that it is natural to handle (12.3) in the product space of u ∈ L2 (Ω) and ρ ∈ H 2 (Ω). By this reason, our underlying space is taken as u X= ; u ∈ L2 (Ω) and ρ ∈ HN2 (Ω) . (12.5) ρ Since ρ satisfies the homogeneous Neumann boundary conditions, the space for ρ is taken as HN2 (Ω). In the underlying space X, the chemotactic term ∇ · [u∇χ(ρ)] can be treated as a lower term. In other words, we are able to formulate problem (12.3) as a semilinear problem of the form (4.1), although the first diffusion equation of (12.3) on u is quasilinear in form. This is the merit of taking an underlying space of the form (12.5). In this way, we rewrite (12.3) by dU 0 < t < ∞, dt + AU = F (U ), (12.6) U (0) = U0 , in X, where U = t (u, ρ). Here, the linear operator A is given by

A1 0 . A= −ν A2

(12.7)

The two operators A1 = −aΔ + 1 and A2 = −bΔ + d are positive definite selfadjoint operators of L2 (Ω) with domains D(A1 ) = D(A2 ) = HN2 (Ω). However, A2 4 (Ω) = D(A2 ) = {ρ ∈ H 2 (Ω); Δρ ∈ H 2 (Ω)} is regarded as an operator from HN 2 N N 2 2 into D(A2 ) = HN (Ω). (Note that, as Ω is not assumed to be of C4 class, we cannot 4 (Ω) ⊂ H 4 (Ω).) So, the domain of A is given by expect that HN 2 u 2 4 D(A) = ; u ∈ HN (Ω) and ρ ∈ HN 2 (Ω) . ρ The nonlinear operator F is given by

−μ∇ · [u∇χ(Re ρ)] + u F (U ) = . 0

(12.8)

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12 Chemotaxis Models

Since ∇ · [u∇χ] = ∇u · ∇χ + uΔχ , it is possible to see that ∇ · [u∇χ]L2 ≤ CuH 1+ε χH 2 ,

u ∈ H 1+ε (Ω), χ ∈ HN2 (Ω)

(12.9)

with an arbitrary fixed exponent 0 < ε < 12 . Indeed, note that H 1+ε (Ω) ⊂ L∞ (Ω) due to (1.76) and H 1+ε (Ω) ⊂ Hp1 (Ω) for p such that 2 < p < 2/(1 − ε) due to (1.74). We then observe that F (U ) is an operator from D(Aη ) into X, where the exponent η is given by η = 1+ε 2 . Note also by (1.93) that ρ → χ(Re ρ) is a bounded operator from HN2 (Ω) into itself.

1.2 Construction of Local Solutions In order to show the existence of local solutions to (12.6), let us verify that A and F fulfill the structural assumptions made in Chap. 4, Sect. 1. Denote by B the multiplicative operator by the number −ν. Then, A is obviously written by

A1 0 A= . (12.10) B A2 We regard B as a linear operator from D(B) ⊂ L2 (Ω) to HN2 (Ω); clearly, D(B) = HN2 (Ω), and B is a bounded operator from D(A1 ) onto HN2 (Ω). Hence, it is an immediate consequence of Theorem 2.16 that the operator A is a sectorial operator of the product space X. Let us next investigate the domains of fractional powers Aθ for 0 ≤ θ ≤ 1. According to Theorems 16.7 and 16.7 in Chap. 16, the domains D(Aθ1 ) and D(Aθ2 ) are characterized by D Aθ1 = D Aθ2 = H 2θ (Ω), D Aθ1 = D Aθ2 = HN2θ (Ω),

3 0≤θ < , 4 3 < θ ≤ 1. 4

Consequently, we have 2θ (Ω) = ρ ∈ HN2 (Ω); Δρ ∈ H 2(θ−1) (Ω) , D Aθ2 = HN 2(θ−1) 2θ 2 (Ω) D(Aθ2 ) = HN 2 (Ω) = ρ ∈ HN (Ω); Δρ ∈ HN

7 1≤θ < , 4 7 < θ ≤ 2. 4

Consider a diagonal operator AD = diag {A1 , A2 } acting in X. Then, AD is a selfadjoint operator of X; therefore, AθD = diag {Aθ1 , Aθ2 }. More precisely, θ u 3 2(θ+1) 2θ ; u ∈ H (Ω) and ρ ∈ HN (Ω) , 0 < θ < , D AD = ρ 4 u 3 2(θ+1) D AθD = ; u ∈ HN2θ (Ω) and ρ ∈ HN 2 (Ω) , < θ ≤ 1. ρ 4

1 Chemotaxis Model Without Proliferation

421

We then verify the following property. Proposition 12.1 The domains of fractional powers of A are given by D(Aθ ) = D AθD , 0 ≤ θ ≤ 1, with the norm equivalence C −1 AθD ·X ≤ Aθ · X ≤ C AθD ·X ,

(12.11)

0 ≤ θ ≤ 1.

Proof We will apply Theorem 16.4 to A. It is clear that the analogous statement to Theorem 16.4 is true for matrix operators of the form (12.10). The multiplicative operator B is clearly an operator from L2 (Ω) to HN2 (Ω) = D(A2 ) with the domain D(B) = HN2 (Ω). Therefore, B is a bounded operator from D(A1 ) into HN2 (Ω). In the meantime, u ∈ HN2 (Ω) belongs to D(B ∗ ) if and only if the functional (u, B·)H 2 is continuous with respect to the L2 topology. Since N

(u, Bv)H 2 = (A2 u, −A2 νv)L2 = −ν(A2 u, A2 v)L2 , u ∈ D(A22 ) implies u ∈ D(B ∗ ). N

This indeed means that D(A22 ) ⊂ D(B ∗ ) and B ∗ is a bounded operator from D(A22 ) into L2 (Ω). Hence, we have verified that Theorem 16.4 is available to A. We now know that D(Aθ ) = [D(A), X]θ . From this we deduce that D(Aθ ) = [D(A), X]θ = [D(AD ), X]θ = D(AθD ) because of the coincidence D(A) = D(AD ). The norm equivalence is also verified by Theorem 16.4. Note that ρ → χ(Re ρ) is a continuous and bounded operator from HN2 (Ω) into itself due to (1.93). Furthermore, by (12.9) and (12.11) (θ = η), we observe that F satisfies the Lipschitz condition (4.21). We therefore know that Theorem 4.4 is available to problem (12.6). As (4.21) is valid in the present case, we are accordingly led to set the space of initial values by u0 (12.12) ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ ρ0 ∈ HN2 (Ω) . K= ρ0 We have thus proved that, for each U0 ∈ K, (12.6) (and hence (12.3)) possesses a unique local solution U = t (u, ρ) in the function space: u ∈ C((0, TU0 ]; HN2 (Ω)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)). ρ ∈ C((0, TU0 ]; HN U0 U0 2 N N (12.13) In addition, it holds for U that dU + tAU (t)X + U (t)X ≤ CU , 0 < t ≤ TU . t (12.14) (t) 0 0 dt X Here, the positive time TU0 > 0 and the constant CU0 > 0 are determined by the norm U0 X only.

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12 Chemotaxis Models

1.3 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, ρ) be the local solution of (12.6) constructed above. Our goal is to prove that u(t) ≥ 0 and ρ(t) ≥ 0 for every 0 < t ≤ TU0 by the truncation method. Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (12.6) with the same initial value. Therefore, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative ψ (t) = (H (u), aΔu − μ∇ · [u∇χ(ρ)])L2 . Applying (1.96) to H (u), we observe that

(H (u), aΔu)L2 = −a

∇H (u) · ∇u dx = −a Ω

|∇H (u)|2 dx.

Ω

In the meantime, by (1.96) again,

−(H (u), μ∇ · [u∇χ(ρ)])L2 = μ

u∇H (u) · ∇χ(ρ) dx

Ω

H (u)∇H (u) · ∇χ(ρ) dx

=μ Ω

=−

μ 2

H (u)2 Δχ(ρ) dx. Ω

Since Δχ(ρ) = χ (ρ)Δρ + χ (ρ)|∇ρ|2 ,

(12.15)

(12.13) shows that Δχ(ρ)L2 ≤ CU for 0 ≤ t ≤ TU0 . Therefore, −(H (u), μ∇ · [u∇χ(ρ)])L2 ≤

μ H (u)2L4 Δχ(ρ)L2 2

≤ CU H (u)H 1 H (u)L2 a ≤ H (u)2H 1 + CU H (u)2L2 . 2 Hence, ψ (t) ≤ CU ψ(t); consequently, ψ(t) ≤ ψ(0)eCU t . Then, ψ(0) = 0 implies ψ(t) ≡ 0, i.e., u(t) ≥ 0 for 0 < t ≤ TU0 . It is similar for the proof of ρ(t) ≥ 0 for 0 < t ≤ TU 0 .

2 Case where χ (ρ) = ρ

423

2 Case where χ(ρ) = ρ In this section, let us consider problem (12.6) in a particular case where the sensitivity function is given by χ(ρ) = ρ,

0 ≤ ρ < ∞.

(12.16)

We can establish a priori estimates for sufficiently small initial functions u0 , and consequently we obtain the global existence of solutions.

2.1 Global Solutions for Small Initial Functions Let U0 ∈ K, and let U be any local solution of (12.6) on [0, TU ] in the function space: 0 ≤ u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)). 0 ≤ ρ ∈ C((0, TU ]; HN U U 2 N N (12.17) Proposition 12.2 There exists a positive number r > 0 and a continuous increasing function p(·) such that the estimate u(t)L2 + ρ(t)H 2 ≤ p(u0 L2 + ρ0 H 2 ),

0 ≤ t ≤ TU ,

(12.18)

holds for any local solution U = t (u, ρ) of (12.6) in (12.17) with initial value U0 ∈ K satisfying the condition u0 L1 ≤ r. Proof In the proof, the notation p(·) stands for some continuous increasing functions which are determined only by Ω and by the initial constants in the equations of (12.3). Similarly, the notation C stands for some constants, which are determined in the same way as p(·). Step 1. Integrate the first equation of (12.6) in Ω. Then, obviously, d u dx = 0. dt Ω Since u(t) ≥ 0, it follows that u(t)L1 = u0 L1 ,

0 ≤ t ≤ TU .

Step 2. We here introduce the quantity u log(u + 1) dx, N1,log (u) =

0 ≤ u ∈ L2 (Ω).

Ω

The purpose of this step is to estimate N1,log (u(t)) for the local solution U .

(12.19)

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12 Chemotaxis Models

Multiply the first equation of (12.6) by log(u + 1) and integrate the product in Ω. Then, since

log(u + 1)Δu dx = − Ω

Ω

1 |∇(u + 1)|2 dx = −4 u+1

√ |∇ u + 1|2 dx

Ω

and

log(u + 1)∇ · [u∇ρ] dx = Ω

∇[(u + 1) log(u + 1) − u] · ∇ρ dx Ω

u log(u + 1)Δρ dx =

+

[u − log(u + 1)]Δρ dx,

Ω

Ω

it follows that d dt

[(u + 1) log(u + 1) − u] dx + 4a Ω

√ |∇ u + 1|2 dx

Ω

=μ Ω

[log(u + 1) − u]Δρ dx ≤ ζ Δρ2L2 + Cζ u2L2

(12.20)

with any ζ > 0. In the meantime, from the second equation of (12.6) we obtain that 1 d 2 dt

ρ 2 dx + b Ω

|∇ρ|2 dx + d Ω

ρ 2 dx = ν Ω

uρ dx ≤ Ω

d ρ2L2 + Cu2L2 2

and 1 d 2 dt

=ν

|∇ρ| dx + b Ω

|Δρ| dx + d

2

|∇ρ|2 dx

2

Ω

Ω

b uΔρ dx ≤ Δρ2L2 + Cu2L2 . 4 Ω

Fixing the parameter ζ as ζ = b4 , we sum up these three differential inequalities to obtain that √ d ψ(t) + dψ(t) + 4a |∇ u + 1|2 dx + b |Δρ|2 dx ≤ C u2L2 + 1 , dt Ω Ω where ψ(t) = (u + 1) log(u + 1) − uL1 + ρ2H 1 . We notice the fact that 1/3 2/3 1/3 √ 4/3 uL2 ≤ uL1 uL4 ≤ uL1 u + 1L8 √ √ 1/3 √ 1/3 ≤ CuL1 u + 1L2 (∇ u + 1L2 + u + 1L2 ),

0 ≤ u ∈ H 1 (Ω).

2 Case where χ (ρ) = ρ

425

Hence, on account of (12.19), d ψ(t) + dψ(t) + 4a dt

√

|∇ u + 1| dx + b

|Δρ|2 dx

2

Ω

Ω

2 √ ≤ u0 L3 1 p(u0 L1 ) ∇ u + 12L2 + 1 . 2

We now fix a radius r > 0 small enough to satisfy that r 3 p(r) ≤ 3a. Then, if u0 L1 ≤ r, it holds that √ d a|∇ u + 1|2 + b|Δρ|2 dx ≤ C. ψ(t) + dψ(t) + dt Ω As a consequence (cf. (1.58)), ψ(t) ≤ e−dt ψ(0) + C,

0 ≤ t ≤ TU .

In particular, N1,log (u(t)) + ρ(t)2H 1 ≤ C e−dt N1,log (u0 ) + ρ0 2H 1 + 1 ,

0 ≤ t ≤ TU . (12.21)

Proposition 1.5 is also available to conclude that t u(τ ) + 12H 1 + Δρ(τ )2L2 dτ s

≤ C (t − s) + N1,log (u0 ) + ρ0 2H 1 + 1 ,

0 ≤ s < t ≤ TU .

Step 3. The present step is devoted to showing the estimate u(t)2L2 + ρ(t)2H 2 ≤ p(N1,log (u0 ) + ρ0 H 1 ) × e−δt u0 2L2 + ρ0 2H 2 + 1 ,

0 ≤ t ≤ TU ,

with some exponent δ > 0. Now, when estimate (12.21) is established, the techniques of its proof are quite analogous to that of the proof (Step 4) of Proposition 12.3 below, which establishes a similar a priori estimate for a general sensitivity function including the case of (12.16). So, the continued proof will be given there. It is now possible to apply Corollary 4.3 to conclude the global existence of solutions. In fact, there is a number r > 0 such that, for any U0 ∈ K satisfying u0 L1 ≤ r, (12.3) possesses a unique global solution in the function space: 0 ≤ u ∈ C((0, ∞); HN2 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)), 4 (Ω)) ∩ C([0, ∞); H 2 (Ω)) ∩ C1 ((0, ∞); H 2 (Ω)). 0 ≤ ρ ∈ C((0, ∞); HN 2 N N (12.22)

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12 Chemotaxis Models

2.2 Lyapunov Function When (12.16) holds, we can construct a Lyapunov function. Let U = t (u, ρ) denote a local or global solution of (12.6) in the function space (12.17). Let ε > 0 be a ∂uε positive parameter and put uε = u + ε. It is clear that ∂u ∂t = ∂t and that aΔu − μ∇ · [u∇ρ] = aΔuε − μ∇ · [uε ∇ρ] + εμΔρ. Therefore, the first equation of (12.6) is written as ∂uε = ∇ · [uε ∇(a log uε − μρ)] + εμΔρ. ∂t Multiply this equation by (a log uε − μρ) and integrate the product in Ω. Then, ∂uε ∂u uε |∇(a log uε − μρ)|2 dx + Rε , a log uε dx − μ ρ dx = − Ω ∂t Ω ∂t Ω where Rε = εμ Ω (a log uε − μρ)Δρ dx. In the meantime, multiplying the equation of ρ by ∂ρ ∂t and integrating the product in Ω, we have Ω

∂ρ ∂t

2

b d dx = − 2 dt

d d |∇ρ| dx − 2 dt Ω

ρ dx + ν

2

2

Ω

u Ω

∂ρ dx. ∂t

Therefore, d dt

bμ dμ 2 |∇ρ|2 + ρ − μνuρ dx 2 2 Ω

2 ∂ρ νuε |∇(a log uε − μρ)|2 + μ dx + νRε ≤ νRε . =− ∂t Ω aν(uε log uε − uε ) +

Let 0 < s < t ≤ TU . Integrating this inequality in [s, t], we obtain that bμ dμ aν(uε (τ ) log uε (τ ) − uε (τ )) + |∇ρ(τ )|2 + ρ(τ )2 2 2 Ω τ =t t − μνu(τ )ρ(τ ) dx ≤ νRε (τ ) dτ. τ =s

It is not difficult to verify that limε→0

s

t s

Rε (τ ) dτ = 0. Hence, if we set

bμ 2 dμ 2 aν(u log u−u)+ |∇ρ| + ρ −μνuρ dx, Φ(U ) = 2 2 Ω

U ∈ X, (12.23)

then Φ(U (t)) ≤ Φ(U (s)). This means that Φ(U ) plays a role of the Lyapunov function for (12.3). Note that the function u log u can be extended for all 0 ≤ u < ∞ as a continuous function.

3 Chemotaxis Model with Proliferation

427

2.3 Blowup of Solutions Herrero–Velázquez [HV96, HV97] proved that, for suitable initial functions (at least, when u0 L1 are sufficiently large), the local solutions to (12.6) blow up in finite time. Theorem 12.1 There are initial values U0 ∈ K for which the local solutions to (12.6) satisfy lim supt→T U (t)X = ∞ with finite time T < ∞. Hence, (12.6) with the initial values U0 have no global solutions.

3 Chemotaxis Model with Proliferation Let us consider the initial-boundary-value problem for the chemotaxis model with proliferation ⎧ ∂u 2 in Ω × (0, ∞), ⎪ ⎪ ∂t = aΔu − μ∇ · [u∇χ(ρ)] + cu − γ u ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu in Ω × (0, ∞), ∂t (12.24) ∂ρ ∂u ⎪ = =0 on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ∂n ⎪ ⎪ ⎩u(x, 0) = u (x), ρ(x, 0) = ρ (x) in Ω, 0

0

in a two-dimensional C2 or convex, bounded domain Ω. We still assume that χ(ρ) satisfies (12.4) and assume that γ > 0 is a positive constant. As before, we handle problem (12.24) in the product space X given by (12.5), and, as a space of initial functions, we take the space K given by (12.12).

3.1 Local Solutions In order to construct local solutions to (12.24), we can argue in a quite analogous way as for problem (12.3). In fact, we rewrite (12.24) as an abstract problem of the form (12.6) in X. The linear operator A is given in a similar way by (12.7). However, now, the nonlinear operator F is given (instead of (12.8)) by

−μ∇ · [u∇χ(Re ρ)] + (c + 1)u − γ u2 . (12.25) F (U ) = 0 Let 0 < ε < 12 be arbitrarily fixed and put η = 1+ε 2 . Then, as before (by (12.9) and (12.11)), F is an operator from D(Aη ) into X. It is clear that u2 L2 ≤ CuL2 uH 1+ε ,

u ∈ H 1+ε (Ω).

Then, this, together with (12.9), implies that F fulfills the Lipschitz condition (4.21).

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For each U0 ∈ K, we can therefore construct a unique local solution to (12.24). Furthermore, by the truncation method, we can verify the nonnegativity of the local solution. In this way, for any initial value U0 ∈ K, problem (12.24) possesses a unique local solution in the function space: 0 ≤ u ∈ C((0, TU0 ]; HN2 (Ω)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)), 0 ≤ ρ ∈ C((0, TU0 ]; HN U0 U0 2 N N where the positive time TU0 > 0 is determined by the norm U0 X only. In addition, estimate (12.14) holds for the present local solution, too.

3.2 A Priori Estimates of Local Solutions For U0 ∈ K, let U = t (u, ρ) denote a local solution of (12.24) on [0, TU ] in the function space: 0 ≤ u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)). 0 ≤ ρ ∈ C((0, TU ]; HN U U 2 N N (12.26) We can establish the following a priori estimates for local solutions. Proposition 12.3 There exists a continuous increasing function p(·) such that the estimate U (t)X ≤ p(U0 X ),

0 ≤ t ≤ TU ,

(12.27)

holds for any local solution U of (12.24) in the function space (12.26). Proof We put f (u) = cu − γ u2 ,

0 ≤ u < ∞.

The notation p(·) stands for some continuous increasing functions which are determined only by χ(·) and Ω and by the initial constants in the equations of (12.24). Similarly, the notation C stands for some constants which are determined in the same way as p(·). Step 1. Integration of the first equation of (12.24) in Ω gives (c + 1)2 d 2 u dx = (cu − γ u ) dx ≤ dx. −u + dt Ω 4γ Ω Ω Solving this differential inequality (cf. (1.58)), we see that Ω

u dx ≤ e−t u0 L1 +

(c + 1)2 |Ω|. 4γ

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429

Since u(t) ≥ 0, we conclude that u(t)L1 ≤ e−t u0 L1 +

(c + 1)2 |Ω|, 4γ

0 ≤ t < TU .

(12.28)

Furthermore, for any 0 ≤ s < t ≤ TU , we have u(t)L1 − u(s)L1 = So,

γ s

t

u(τ )2L2

t s

t

dτ ≤ c s

(cu − γ u2 ) dx dτ.

Ω

u(τ )L1 dτ + u(s)L1 ;

hence, on account of (12.28),

t s

(c + 1)2 u(τ )2L2 dτ ≤ γ −1 u0 L1 + |Ω| [c(t − s) + 1], 4γ

0 ≤ s < t ≤ TU . (12.29)

Step 2. Multiply the second equation of (12.24) by Δρ and integrate the product in Ω. Then, 1 d 2 2 |∇ρ| dx + b |Δρ| dx + d |∇ρ|2 dx 2 dt Ω Ω Ω b ν2 uΔρ dx ≤ Δρ2L2 + u2L2 . = −ν 2 2b Ω Similarly, multiplying the second equation of (12.24) by ρ, we obtain that 1 d 2 2 2 ρ dx + b |∇ρ| dx + d ρ dx = −ν uρ dx 2 dt Ω Ω Ω Ω ≤

d ν2 ρ2L2 + u2L2 . 2 2d

Summing up these differential inequalities, we obtain that d |∇ρ|2 + ρ 2 dx + b|Δρ|2 + 2(d + b)|∇ρ|2 + dρ 2 dx dt Ω Ω ≤ ν 2 (b + d)(bd)−1 u(t)2L2 .

(12.30)

As (12.29) holds for u(t), it is possible to solve this by applying Proposition 1.4. Consequently, we deduce that ρ(t)2H 1 ≤ e−dt ρ0 2H 1 + C(u0 L1 + 1),

0 ≤ t ≤ TU .

(12.31)

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At the same time, t Δρ(τ )2L2 dτ ≤ C(u0 L1 + 1)[(t − s) + ρ0 H 1 + 1],

0 ≤ s < t ≤ TU .

s

Furthermore, since it holds that ρH 2 ≤ C(ΔρL2 + ρL2 ),

ρ ∈ HN2 (Ω),

due to (2.33), it is also verified that t ρ(τ )2H 2 dτ ≤ C(u0 L1 + 1)[(t − s) + ρ0 H 1 + 1],

(12.32)

0 ≤ s < t ≤ TU .

s

(12.33)

Step 3. In this step, we use the notation p1 (U0 ) = p(u0 L1 + ρ0 H 1 ), where p(·) is some continuous increasing function. Multiply the first equation of (12.24) by log(u + 1) and integrate the product in Ω. Then, by the same calculation as for (12.20), we have √ d [(u + 1) log(u + 1) − u] dx + 4a |∇ u + 1|2 dx dt Ω Ω = μ [log(u + 1) − u]Δχ(ρ) dx + f (u) log(u + 1) dx. Ω

Ω

In addition, we observe that Δχ(ρ) = χ (ρ)Δρ + χ (ρ)|∇ρ|2 . Then, the first integral in the right-hand side can be estimated as follows. By (12.4), [log(u + 1) − u]χ (ρ)Δρ dx ≤ CuL2 ΔρL2 ≤ C u2L2 + Δρ2L2 , Ω

while, by (12.4), (12.31), and (12.32), [log(u + 1) − u]χ (ρ)|∇ρ|2 dx ≤ CuL2 ∇ρ2L4 ≤ CuL2 ρH 2 ρH 1 Ω

≤ CρH 1 u2L2 + ρ2H 2 ≤ p1 (U0 ) u2L2 + ρ2H 2 .

As for the second integral, it is easy to see that f (u) log(u + 1) ≤ cu2

for all 0 ≤ u < ∞.

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431

Similarly, 1 (u + 1) log(u + 1) ≤ u2 + u for all 0 ≤ u < ∞. 2 These estimates thus yield that

1 [f (u) + u + 1] log(u + 1) dx ≤ c + u2L2 + uL1 . 2 Ω Our inequality is then reduced to √ d [(u + 1) log(u + 1) − u] dx + 4a |∇ u + 1|2 dx dt Ω Ω + [(u + 1) log(u + 1) − u] ≤ p1 (U0 ) u(t)2L2 + ρ(t)2H 2 + 1 . Ω

As we already know that (12.29) and (12.33) are valid, it is possible to solve this inequality by using Proposition 1.4. We then deduce that N1,log (u(t)) ≤ e−t N1,log (u0 ) + p1 (U0 ),

0 ≤ t ≤ TU .

(12.34)

Step 4. In this step, we use the notation p2 (U0 ) = p(N1,log (u0 ) + ρ0 H 1 ). Multiply the first equation of (12.24) by u and integrate the product in Ω. Then, 1 d u2 dx + a |∇u|2 dx = μ u∇u · ∇χ(ρ) dx + f (u)u dx. 2 dt Ω Ω Ω Ω Here, u∇u · ∇χ(ρ) dx = − Ω

1 2

u2 Δχ(ρ) dx, Ω

Δχ(ρ) = χ (ρ)Δρ + χ (ρ)|∇ρ|2 . Furthermore, with the aid of (1.98) (p = 2, q = 1, and r = 3), we verify that 2

1

uL3 ≤ ζ uH3 1 N1,log (u) 3 + Cζ uL1 with any ζ > 0, and, by the moment inequality (2.117), we observe that 1 2 1 2 3 2 1 1 ΔρL3 ≤ CΔρH3 1 ΔρL3 2 ≤ CA2 ρH3 1 A2 ρL3 2 ≤ C A22 ρ L3 2 A22 ρ L3 2 .

Therefore, on account of (12.4), (12.31), and (12.34), 3 2 χ (ρ)u2 Δρ dx ≤ Cu2L3 ΔρL3 ≤ p2 (U0 ) ζ u2H 1 + A22 ρ L + Cζ . − Ω

2

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12 Chemotaxis Models

Similarly, − Ω

4

2

χ (ρ)u2 |∇ρ|2 dx ≤ Cu2L3 ∇ρ2L6 ≤ Cu2L3 ρH3 2 ρH3 1 3 2 ≤ p2 (U0 ) ζ u2H 1 + A22 ρ L + Cζ 2

with any ζ > 0. Thus, we obtain that 1 d 2 2 u dx + a |∇u| dx + γ u3 dx 2 dt Ω Ω Ω 3 2 ≤ p2 (U0 ) ζ u2H 1 + A22 ρ L + Cζ . 2

(12.35)

In this step, it is necessary to combine two inequalities for u and ρ. Multiply the second equation of (12.24) by A22 ρ and integrate the product in Ω. Then, 1 3 3 2 1 d |A2 ρ|2 dx + A22 ρ dx = ν A22 u · A22 ρ dx 2 dt Ω Ω Ω 3 1 2 ≤ A22 ρ L + Cu2H 1 , 2 2 i.e., d dt

|A2 ρ|2 dx + Ω

Ω

32 2 A ρ dx ≤ Cu(t)2 1 . 2 H

After multiplying a parameter 2ξ > 0 to (12.35), we add the product to this inequality to obtain that 3 2 2 d ξ u + |A2 ρ|2 dx + (2aξ − C)|∇u|2 + 2γ ξ u3 dx + A22 ρ dx dt Ω Ω Ω 3 2 ≤ p2 (U0 )ξ ζ u2H 1 + A22 ρ L + Cζ . 2

Now, fix the parameters ξ and ζ so that 2aξ − C ≥ 1 and p2 (U0 )ξ ζ ≤ 12 . Then, we arrive at the differential inequality 3 2 1 dψ + δψ + u(t)2H 1 + A22 ρ(t)L ≤ p2 (U0 ) 2 dt 2 for ψ(t) = ξ u(t)2L2 +A2 ρ(t)2L2 with some exponent δ > 0. We can now employ Proposition 1.4 to conclude that ξ u(t)2L2 + A2 ρ(t)2L2 ≤ e−δt ξ u0 2L2 + A2 ρ0 2H 2 + p2 (U0 ), 0 ≤ t ≤ TU , i.e., u(t)2L2 + ρ(t)2H 2 ≤ p2 (U0 ) e−δt u0 2L2 + ρ0 2H 2 + 1 ,

0 ≤ t ≤ TU . (12.36)

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433

As well, t 3 2 u(τ )2H 1 + A22 ρ(τ )L dτ ≤ p2 (U0 ) (t − s) + u0 2L2 + ρ0 2H 2 + 1 , 2

s

0 ≤ s < t ≤ TU . We have in this way established the desired a priori estimate (12.27).

3.3 Global Solutions We can now apply Corollary 4.3 to conclude the global existence of solutions. As a result, for any U0 ∈ K, (12.24) possesses a unique global solution in the function space: 0 ≤ u ∈ C((0, ∞); HN2 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)), 4 (Ω)) ∩ C([0, ∞); H 2 (Ω)) ∩ C1 ((0, ∞); H 2 (Ω)). 0 ≤ ρ ∈ C((0, ∞); HN 2 N N (12.37) For U0 ∈ K, let U (t; U0 ) be the global solution of (12.24) with the initial value U0 in the function space (12.37). Recalling that Proposition 12.3 was established by four steps, let us apply (12.28) to U (t; U0 ) in the interval 0, 4t , (12.31) in the interval 4t , 2t4 , (12.34) in the interval 2t4 , 3t4 , and (12.36) in the interval 3t4 , t , respectively. Then, we obtain the (more refined than (12.27)) estimate U (t; U0 )X ≤ p(e−δt p(U0 X ) + 1),

0 ≤ t < ∞, U0 ∈ K,

(12.38)

with some suitable exponent δ > 0.

3.4 Dynamical System Let us construct a dynamical system determined from problem (12.24) in the universal space X by following the general methods in Chap. 6, Sect. 5. Indeed, we can take β = 0. Furthermore, (12.38) shows that (6.56) is fulfilled. Consequently, it is deduced that problem (12.24) determines a dynamical system (S(t), K, X). For any exponent 0 < θ < 1, (12.24) equally determines a dynamical system in a universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (12.25) fulfills (4.21) and a fortiori (4.2) with β = θ . Let D Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

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12 Chemotaxis Models

Meanwhile, by Proposition 6.1, Aθ S(t)U0 X ≤ Cθ,R t −θ ,

0 < t < ∞, U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that (6.56) is valid in the space Dθ , too. So, (S(t), Kθ , Dθ ) defines a dynamical system.

3.5 Exponential Attractors Let us next construct exponential attractors for (S(t), K, X). To this end, it suffices to verify (6.59) and (6.60). However, the space condition (6.59) is fulfilled in the present case. Meanwhile, (12.38) shows that the dissipative condition (6.60) is also fulfilled. Hence, (S(t), K, X) possesses a family of exponential attractors. For any exponent 0 < θ < 1, these exponential attractors are seen to keep their properties as exponential attractors in the universal space Dθ , too. Remark 12.1 As the linear operator A given by (12.7) is not symmetric, A does not satisfy (6.62). So, it is unclear whether S(t) enjoys the squeezing property (6.45)– (6.46) or not. However, if Ω is of C3 class, then one can use the shift property (Δu ∈ H 1 (Ω), together with u ∈ HN2 (Ω), implies u ∈ H 3 (Ω)) to rewrite (12.24) into the form (12.6) in an underlying space different from (12.5) in which the linear operator is a self-adjoint operator satisfying (6.62). For details, see Osaki– Tsujikawa–Yagi–Mimura [OTYM02]. In such an abstract formulation, the semigroup enjoys the squeezing property, and therefore the dimension of exponential attractor is estimated precisely by (6.50).

3.6 Numerical Examples In this subsection, we present some numerical examples. Let Ω = [−8, 8] × [−8, 8] be a quadratic domain. We consider the linear sensitivity function χ(ρ) = ρ and the growth function f (u) = u2 (1 − u). More precisely, we consider the problem ⎧ ∂u 2 ⎪ ∂t = 0.0625Δu − μ∇ · [u∇ρ] + u (1 − u) in Ω × (0, ∞), ⎪ ⎪ ⎪ ⎨ ∂ρ = Δρ − 32ρ + u in Ω × (0, ∞), ∂t ∂ρ ⎪ ∂u ⎪ = =0 on ∂Ω × (0, ∞), ⎪ ⎪ ⎩ ∂n ∂n ρ(x, 0) = ρ0 (x) in Ω. u(x, 0) = u0 (x),

(12.39)

3 Chemotaxis Model with Proliferation

435

Fig. 1 Stationary patterns for μ = 6.2, μ = 7.2, and μ = 8.5

We treat μ > 0 as a control parameter. As the growth function in (12.39) is a cubic function, we cannot apply the results obtained in this section to (12.39) directly, but, as a matter of fact, we can repeat the same arguments in order to conclude for any 0 < μ < ∞ that (12.39) equally defines a dynamical system (S(t), K, X) which possesses exponential attractors, cf. Aida–Efendiev–Yagi [AEY05]. Note 1 ) whatever also that (12.39) has a homogeneous stationary solution (u, ρ) = (1, 32 0 < μ < ∞ is. 1 ), where ε(x) is a The initial functions (u0 (x), ρ0 (x)) are taken as (1 + ε(x), 32 small perturbation which vanishes outside of a small disk centered at the origin. So, (u0 , ρ0 ) is taken in a neighborhood of the stationary solution (u, ρ). When μ is small enough, the homogeneous stationary solution (u, ρ) is asymptotically stable (cf. (12.48)). The numerical solution is also observed to tend to the stationary solution (u, ρ). When μ = 6.2, the homogeneous stationary solution is no longer stable (cf. (12.49)), and the numerical solution is observed to tend to an inhomogeneous stationary solution with honeycomb structure as in Fig. 1(a). (In the figure, white indicates high concentration of biological individuals, and black oppositely shows the graph of u.) As μ increases from 6.2 to 7.2 and 8.5, the inhomogeneous stationary solution changes its types from honeycomb to stripe and perforated stripe as Fig. 1 shows. When μ still increases, such an inhomogeneous and ordered stationary solution is no longer found. That may lose its stability or completely vanish. Instead, the numerical solution is observed to tend to some moving pattern. When μ = 9.0, a moving perforated labyrinthine pattern is observed as in Fig. 2. When μ = 11.0, a chaotic spot pattern is observed as in Fig. 3. Each spot continues to move in a chaotic manner, a few spots are combined here and there in Ω; on the other hand, some new spots are generated to conserve the total number of spots at every moment. When μ becomes larger, moving patterns as in Figs. 2 and 3 disappear. Instead, the numerical solution is observed to tend to a stationary pattern again. A number of steady spots are regularly located in Ω as in Fig. 4, but these stationary patterns seem to be quite different from those in Fig. 1.

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Fig. 2 Moving perforated labyrinthine pattern for μ = 9.0

Fig. 3 Chaotic spot pattern for μ = 11.0

Fig. 4 Stationary patterns for μ = 25.0, μ = 40.0, and μ = 70.0

These numerical results seem to be very interesting when we interpret them in accordance with the analytical result that the dimension of exponential attractors of the dynamical system for (12.39) increases as the parameter μ increases (cf. (12.50)).

4 Instability of Homogeneous Stationary Solution of (12.24)

437

4 Instability of Homogeneous Stationary Solution of (12.24) We are concerned with homogeneous stationary solutions to (12.24). The homogeneous solution is given as a solution to cu − γ u2 = 0, −dρ + νu = 0, cν . We want to investigate the stability and instanamely, (u, ρ) = (0, 0) and γc , dγ cν . bility of the nonzero homogeneous stationary solution (u, ρ) = γc , dγ In this section, we make the new assumption on χ(ρ) that χ(ρ) is real analytic in a neighborhood of ρ.

(12.40)

4.1 Localized Problem From (12.40) we can extend the sensitivity function as an analytic function in a complex neighborhood |ρ − ρ| < R; furthermore, this analytic function can be extended over the whole complex plane in such a way that the extended function, denoted by χ˜ (ρ), is a C2 function with respect to the real variables (ρ , ρ ) ∈ R2 such that ρ = ρ + iρ ∈ C with uniformly bounded partial derivatives in C up to the second order. Similarly, let ϕ(u) be an extension of the function u such that ϕ(u) ≡ u in a complex neighborhood |u − u| < R and ϕ(u) is a C1 function with respect to the real variable (u , u ) ∈ R2 such that u = u + iu ∈ C with uniformly bounded partial derivatives up to the first order. Using these functions, we introduce the localized problem ⎧ ∂u 2 ⎪ ∂t = aΔu − u − μ∇ · [ϕ(u)∇ χ˜ (ρ)] + (c + 1)ϕ(u) − γ ϕ(u) ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu ∂t ∂ρ ⎪ ∂u ⎪ ∂n = ∂n =0 ⎪ ⎪ ⎩ ρ(x, 0) = ρ0 (x) u(x, 0) = u0 (x),

in Ω × (0, ∞), in Ω × (0, ∞),

on ∂Ω × (0, ∞), in Ω. (12.41) We can handle this localized problem in a quite analogous way as for (12.24). In fact, the problem is formulated as the Cauchy problem for an abstract equation in X of the form (12.6) in which the nonlinear operator F˜ : D(Aη ) → X, where 1+ε 1 2 ≤ η < 1 and 0 < ε < 2 , is given by

−μ∇ · [ϕ(u)∇ χ˜ (ρ)] + (c + 1)ϕ(u) − γ ϕ(u)2 ˜ , F (U ) = 0

u U= ∈ D(Aη ). ρ

Obviously, F˜ is a nonlinear operator from D(Aη ) into X. Let 0 ≤ θ < η. It is then possible to construct local solutions and global solutions to (12.41) for any initial

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value U0 ∈ D(Aθ ) in the function space: U ∈ C([0, ∞); D(Aθ )) ∩ C((0, ∞); D(A)) ∩ C 1 ((0, ∞); X).

4.2 Complexified Dynamical System ˜ Problem (12.41) then defines a nonlinear semigroup S(t) acting on Dθ = D(Aθ ), ˜ 0 ≤ θ < η. Furthermore, (12.41) defines a complexified dynamical system (S(t), Dθ , Dθ ) in Dθ . We are concerned with the case where 12 < θ < η. Since Dθ ⊂ C(Ω) for θ > 12 , any solution to (12.24) is a solution of (12.41) in a suitable neighborhood of U in Dθ . Naturally, in such a neighborhood, any trajectory of (S(t), Kθ , Dθ ) is that of ˜ ˜ Dθ , Dθ ). If U is stable (S(t), Dθ , Dθ ). In particular, U is an equilibrium of (S(t), ˜ as an equilibrium of (S(t), Dθ , Dθ ), then it is the same as that of (S(t), Kθ , Dθ ). ˜ Dθ , Dθ ), we On the other hand, even if U is unstable as an equilibrium of (S(t), cannot say that U is automatically unstable as that of (S(t), Kθ , Dθ ). However, as ˜ Dθ , Dθ ) provides us an important information before, the instability of U in (S(t), which indicates the instability even in the original system (S(t), Kθ , Dθ ), too. We can follow the general methods described in Chap. 6, Sect. 6. It is indeed sufficient to verify the differentiability conditions (6.68) and (6.69) for the nonlinear operator F˜ (U ) and the spectrum separation condition (6.76) for the linearized operator A − F˜ (U ).

4.3 Differentiability of F˜ (U ) If r > 0 is sufficiently small so that Aθ (U − U ) < r implies U − U C < R, then, θ in the open ball B D(A ) (U ; r), F˜ : D(Aη ) → X is seen to be Fréchet differentiable with the derivative

−μ∇ · [h∇χ(ρ)] − μ∇ · {u∇[χ (ρ)]} + (c + 1)h − 2γ uh ˜ , F (U )H = 0

u h η D(Aβ ) ∈ D(A ) ∩ B U= (U ; r), H = ∈ D(Aη ). (12.42) ρ Furthermore, it is immediate to check that F˜ (U ) fulfills conditions (6.68) and (6.69).

4.4 Spectrum Separation Condition of A For A = A − F˜ (U ), let us verify condition (6.76).

4 Instability of Homogeneous Stationary Solution of (12.24)

439

Let Λ be the realization of −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions. Let 0 = μ 0 < μ1 ≤ μ 2 ≤ · · · → ∞ 1

be the eigenvalues of Λ in L2 (Ω), and let φ0 = |Ω|− 2 , φ1 , φ2 , . . . , be the corresponding real-valued eigenfunctions which constitute an orthonormal basis of L2 (Ω) (cf. Corollary 2.1). Using Λ, we can characterize the space HN2 (Ω) by ∞ ∞ 2 2 2 HN (Ω) = D(Λ) = ρ = ηk φk ; (μk + 1) |ηk | < ∞ k=0

k=0

with the inner product (ρ, ρ )H 2 = N

∞ (μk + 1)2 ηk ηk ,

ρ=

k=0

∞

ηk φk , ρ =

k=0

∞

ηk φk .

k=0

Then, we notice that our underlying space X given by (12.5) is obtained as the infinite sum of two-dimensional orthogonal subspaces

φk 0 + ηk ; ξk , ηk ∈ C , k = 0, 1, 2, . . . , Xk = U = ξ k 0 φk equipped with the norm U 2Xk = |ξk |2 + (μk + 1)2 |ηk |2 . It is then obvious that Φk =

φk 0

and Ψk = (μk + 1)−1

0 φk

are orthonormal bases of Xk . According to (12.42), we have

c + 1 − 2γ u −μuχ (ρ)Δ . F˜ (U ) = 0 0 Note that u =

c γ

and put χ1 = χ (ρ) > 0. Since

aΛ + 1 0 A= −ν bΛ + d

−c + 1 ˜ and F (U ) = 0

μuχ1 Λ , 0

we observe that A = A − F˜ (U ) maps the subspace Xk into itself, that is, Xk is invariantunder A for every k. Consequently, the operator A can also be decomposed as A = ∞ k=0 Ak , where Ak is the part of A in Xk . Obviously, Ak Φk = (aμk + c)Φk − ν(μk + 1)Ψk ,

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12 Chemotaxis Models

Ak Ψk = −μuχ1 μk (μk + 1)−1 Φk + (bμk + d)Ψk . The transformation matrix M k of Ak with respect to the basis Φk , Ψk is then given by

aμk + c −ν(μk + 1) . Mk = bμk + d −μuχ1 μk (μk + 1)−1 Since |λ − M k | = [λ − (aμk + c)][λ − (bμk + d)] − μνuχ1 μk , the characteristic equation of M k is written by λ2 − [(a + b)μk + c + d]λ + (aμk + c)(bμk + d) − μνuχ1 μk = 0. So, the discriminant satisfies Dk = [(a − b)μk + c − d]2 + 4μνuχ1 μk ≥ 0. That is, for every k = 0, 1, 2, . . . , the matrix M k has two real eigenvalues λk ≤ λk . Furthermore, for λ = λk , λk , (λ − M k )−1 =

1 λ − (bμk + d) (λ − λk )(λ − λk ) −μuχ1 μk (μk + 1)−1

−ν(μk + 1) . λ − (aμk + c) (12.43)

For verifying (6.76), we here make the assumption that abμ2k + (ad + bc − μνuχ1 )μk + cd = 0 for all k = 0, 1, 2, . . . .

(12.44)

If (aμk + c)(bμk + d) − μνuχ1 μk = abμ2k + (ad + bc − μνuχ1 )μk + cd > 0, (12.45) then the eigenvalues of M k are both positive, i.e., 0 < λk ≤ λk . Furthermore, the following estimate abμ2k + (ad + bc − μνuχ1 )μk + cd < λk ≤ λk (a + b)μk + c + d < (a + b)μk + c + d −

abμ2k + (ad + bc − μνuχ1 )μk + cd (12.46) (a + b)μk + c + d

holds for the eigenvalues. In the meantime, if abμ2k + (ad + bc − μνuχ1 )μk + cd < 0,

(12.47)

then one of the eigenvalues is negative, and the other is positive, i.e., λk < 0 < λk .

4 Instability of Homogeneous Stationary Solution of (12.24)

441

Let λ ∈ iR. Consider the parts Ak on Xk . Let ψk : Xk → C2 be a natural isomorphism such that ψk (ξk Φk + ηk Ψk ) = t (ξk , ηk ) (ψk = ψk−1 = 1); then Ak is written as Ak = ψk−1 M k ψk . Consequently, (λ − Ak )−1 = ψk−1 (λ − M k )−1 ψk . Therefore, if k are sufficiently large so that (12.45) take place, then (12.43) provides, with the aid of (12.46) and the obvious inequalities |λ − λk | ≥ λk and |λ − λk | ≥ λk , that (λ − Ak )−1 ≤ Cλ , L(X ) k

Cλ > 0 being independent of k. This fact means that λ ∈ iR belongs to ρ(A) if and / σ (A) if and only only if λ ∈ ρ(Ak ) for every k = 0, 1, 2, . . . . In other words, λ ∈ λ∈ / σ (Ak ) = {λk , λk } for every k. We already know that, under (12.44), iR ∩ σ (Ak ) = ∅ for every k = 0, 1, 2, . . . . Hence, iR ∩ σ (A) = ∅, that is, the separation condition (6.76) is fulfilled.

4.5 Stability or Instability Conditions Assume the condition

μνuχ1

0) and that the parameters c, d, f, g, μ, ν, ζ , and K are positive (> 0) constants. It is easy to see that (13.1) has the unique homogeneous stationary solution Kf ν Kf νζ f , , . (13.2) U= c Kcd + f ν (Kcd + f ν)g We shall first prove the global existence of solutions to (13.1) and construct a dynamical system. We shall next show that the dynamical system possesses exponential attractors and that the homogeneous stationary solution U becomes unstable if μ and ζ are sufficiently large.

2 Local Solutions 2.1 Abstract Formulation Let us formulate problem (13.1) as the Cauchy problem for an abstract evolution equation. We want to handle the first equation of (13.1) on u in the L2 space. In order to rewrite (13.1) into a semilinear abstract equation, we will choose the H 1 Sobolev space for handling the third equation on ρ. It is then natural to handle the second equation on v in the same H 1 space, because the equation has no smoothing effect. So, our underlying space is set by ⎧⎛ ⎞ ⎫ ⎨ u ⎬ X = ⎝ v ⎠ ; u ∈ L2 (Ω), v ∈ H 1 (Ω), ρ ∈ H 1 (Ω) . (13.3) ⎩ ⎭ ρ We are then led to formulate (13.1) as the problem dU 0 < t < ∞, dt + AU = F (U ), U (0) = U0 ,

(13.4)

in X. Here, A = diag {A1 , d, A2 } is a diagonal matrix operator of X, where A1 = −aΔ + c and A2 = −bΔ + g are positive definite self-adjoint operators of L2 (Ω)

2 Local Solutions

447

with domains D(A1 ) = D(A2 ) = HN2 (Ω) (see (2.34)). But, A2 must be regarded 3

1

as an operator from D(A22 ) = {ρ ∈ HN2 (Ω); Δρ ∈ H 1 (Ω)} into D(A22 ) = H 1 (Ω). Note that, as Ω is of C3 class, Theorem 2.9 ensures the shift property that Δρ ∈ ∂ρ = 0 implies that ρ ∈ H 3 (Ω). Therefore, H 1 (Ω) with ∂n 32 ∂ρ 3 3 D A2 = HN (Ω) = ρ ∈ H (Ω); = 0 on ∂Ω . ∂n Consequently, the domain of A is given by ⎧⎛ ⎞ ⎫ ⎨ u ⎬ D(A) = ⎝ v ⎠ ; u ∈ HN2 (Ω), v ∈ H 1 (Ω), ρ ∈ HN3 (Ω) . ⎩ ⎭ ρ

(13.5)

Obviously, A is a positive definite self-adjoint operator of X. The nonlinear operator F is given by ⎛ ⎞ ⎞ ⎛ u −μ∇ · [u∇ρ] + f (13.6) F (U ) = ⎝ ν(1 − Kv )u ⎠ , U = ⎝ v ⎠ ∈ D(Aη ), ζv ρ 3 4

where η is a fixed exponent such that

< η < 1.

2.2 Construction of Local Solutions Let us first characterize the domains of fractional powers Aθ for 0 < θ < 1. Since A is diagonal, it is clear that Aθ = diag {Aθ1 , d θ , Aθ2 } for 0 < θ < 1, where Aθ2 is θ+ 1

1

an operator from D(A2 2 ) to D(A22 ). According to Theorems 16.7 and 16.9 in Chap. 16, it is known that D(Aθ1 ) = D(Aθ2 ) = H 2θ (Ω), 0 ≤ θ < 34 , D(Aθ1 ) = D(Aθ2 ) = HN2θ (Ω), 34 < θ ≤ 1. In addition, as Ω is of C3 class, it is seen that ∂ρ D(Aθ2 ) = HN2θ (Ω) = ρ ∈ H 2θ (Ω); = 0 on ∂Ω , ∂n

3 1≤θ ≤ . 2

As a consequence, it is deduced that ⎛u⎞ D(Aθ ) = ⎝ v ⎠ ; u ∈ H 2θ (Ω), v ∈ H 1 (Ω), ρ ∈ H 2θ+1 (Ω) , ρ

1 0≤θ < , 4 (13.7)

448

13 Termite Mound Building Model

⎛u⎞

D(Aθ ) = ⎝ v ⎠ ; u ∈ H 2θ (Ω), v ∈ H 1 (Ω), ρ ρ

∈ HN2θ+1 (Ω)

,

⎛u⎞ D(Aθ ) = ⎝ v ⎠ ; u ∈ HN2θ (Ω), v ∈ H 1 (Ω), ρ ∈ HN2θ+1 (Ω) , ρ

1 3 0 is determined by the norm A 2 U0 X only. In addition, 1 1 t 2 AU (t)X + A 2 U (t)X ≤ CU0 , 1

where CU0 > 0 is also determined by A 2 U0 X only.

0 < t ≤ TU 0 ,

(13.12)

2 Local Solutions

449

2.3 Nonnegativity of Local Solutions Let U (t) be a unique local solution of (13.4) in (13.11) for U0 ∈ K. We want to show that U (t) ∈ K for every 0 < t ≤ TU0 . The local solution U is real valued. Indeed, the complex conjugate U of U is clearly a solution of (13.4) with the same initial value; then, by the uniqueness of solution, it must hold that U (t) = U (t) for every 0 < t ≤ TU0 ; hence, U (t) is a real-valued function. Let us denote U = t (u, v, ρ). We first prove that u ≥ 0 by the truncation method. 2 Let H (u) be a C1,1 cutoff function such that H (u) = u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. Consider the function ϕ(t) = Ω H (u(t)) dx. By (1.100), ϕ is a continuously differentiable function with the derivative ϕ (t) = H (u){aΔu − μ∇ · [u∇ρ] − cu + f } dx. Ω

Applying (1.96) to H (u), it is observed that H (u)aΔu = −a ∇H (u) · ∇u dx = −a |∇H (u)|2 dx. Ω

Ω

Ω

In the meantime, − H (u)μ∇ · [u∇ρ] dx = μ u∇H (u) · ∇ρ dx = μ H (u)∇H (u) · ∇ρ dx Ω

Ω

μ = 2

Ω

μ ∇H (u)2 · ∇ρ dx = − 2 Ω

H (u)2 Δρ dx.

Ω

Since Δρ(t)L2 ≤ CU for 0 ≤ t ≤ TU0 , we have μ H (u)2 Δρ dx ≤ C H (u)2 L ΔρL2 ≤ CH (u)2L4 ΔρL2 − 2 2 Ω 3

1

≤ CU H (u)H2 1 H (u)L2 2 a ≤ H (u)2H 1 + CU H (u)2L2 . 2

Finally, it is clear that Ω H (u)[−cu + f ] dx ≤ 0. Hence, we proved that ϕ (t) ≤ CU ϕ(t) and that ϕ(t) ≤ eCU t ϕ(0). Consequently, ϕ(0) = 0 implies ϕ(t) ≡ 0, i.e., u(t) ≥ 0 for 0 < t ≤ TU0 . We next verify that 0 ≤ v(t) ≤ K for 0 < t ≤ TU0 . Since u ≥ 0 is already seen, the proof for v(t) ≥ 0 is similar or even easier. In order to prove that v(t) ≤ K, we rewrite the equation of v in the form ∂w νu =− d+ w + dK, ∂t K

450

13 Termite Mound Building Model

where w = K − v. Then, the property w0 = K − v0 ≥ 0 at t = 0 implies w(t) ≥ 0 for every 0 < t ≤ TU0 . In this way, we observe that 0 ≤ v(t) ≤ K for 0 < t ≤ TU0 . We finally have to prove that ρ(t) ≥ 0 for 0 < t ≤ TU0 . Since v ≥ 0 is already known, this is also seen by the truncation method employed above. We have thus proved that U0 ∈ K implies U (t) ∈ K for each 0 < t ≤ TU0 .

3 Global Solutions 3.1 A Priori Estimates for Local Solutions For U0 ∈ K ∩ D(A), let U (t) denote any local solution of (13.4) on [0, TU ] in the function space: 1 U ∈ C((0, TU ]; D(A)) ∩ C([0, TU ]; D(A 2 ) ∩ C1 ((0, TU ]; X), (13.13) U (t) ∈ K, 0 ≤ t ≤ TU . We intend to establish the a priori estimate for such a local solution. For this purpose, we have to employ a realization of −bΔ + g under the homogeneous Neumann boundary conditions in the space L6 (Ω). We know by Theorem 2.12 that the realization, denoted by A2,6 , is a sectorial operator of L6 (Ω) with angle < π2 and generates an analytic semigroup e−tA2,6 on L6 (Ω) which satisfies the estimate e−tA2,6 L6 ≤ Ce−δ1 t , 0 ≤ t < ∞, with some exponent δ1 > 0. Moreover, ∂ρ = 0 on ∂Ω}. Since we know by Theorem 2.15 that D(A2,6 ) = {ρ ∈ H62 (Ω); ∂n 3

HN3 (Ω) ⊂ D(A2,6 ), we see that ρ0 ∈ D(A22 ) implies ρ0 ∈ D(A2,6 ). Furthermore, for 0 < θ < 1, consider the fractional power Aθ2,6 . On account of Remark 16.6 in

Chap. 16, we verify that D(Aθ2,6 ) ⊂ H62θ (Ω) for any 0 < θ < θ . In particular, we 9

8

9 8 10 1 (Ω) due to (1.76). and θ = 10 ; then, D(A2,6 ) ⊂ H65 (Ω) ⊂ H∞ take θ = 10 We can state what we want to prove now.

Proposition 13.1 There exists a continuous increasing function p(·) such that the estimate 9 9 1 A 2 U (t) + A 10 ρ(t) ≤ p A 12 U0 + A 10 ρ0 , 2,6 2,6 X L X L

0 ≤ t ≤ TU , (13.14) holds for any local solution U of (13.4) with U0 ∈ K ∩ D(A) in the function space (13.13). 6

6

Proof Throughout the proof, the notation p(·) stands for some continuous increasing functions which are determined only by Ω and by the initial constants in the equations of (13.1). Similarly, the notation C stands for some constants which are determined only by Ω and by the initial constants in (13.1).

3 Global Solutions

451

Step 1. Integrate the first equation of (13.4) in Ω. Then, d u dx = −c u dx + f |Ω|. dt Ω Ω So,

Ω

u dx = e−ct u0 L1 + f |Ω|

t

e−c(t−s) ds.

0

Since u(t) ≥ 0, it follows that u(t)L1 ≤ e−ct u0 L1 +

f |Ω|, c

0 ≤ t ≤ TU .

(13.15)

Step 2. In this step, let us treat the third equation of (13.4) in L6 (Ω). From (13.13) v we see that dv dt L2 = − dv + ν(1 − K )uL2 ≤ C, so that v(t) − v(s)L2 ≤ 1

C|t − s|. This then implies v ∈ C 3 ([0, TU ]; L6 (Ω)) because of 2

1

v(t) − v(s)L6 ≤ v(t) − v(s)L3 ∞ v(t) − v(s)L3 2 ,

0 ≤ s < t ≤ TU .

Then, thanks to Theorem 3.4, ρ(t) is written by t e−(t−s)A2,6 ζ v(s) ds, ρ(t) = e−tA2,6 ρ0 + 0 9

10 where A2,6 is the realization of −bΔ + g in L6 (Ω) mentioned above. Operate A2,6 to the equality. Then, t 9 9 9 10 10 10 −(t−s)A2,6 A2,6 ρ(t) = e−A2,6 t A2,6 ρ0 + A2,6 e ζ v(s) ds.

0

On account of 0 ≤ v(s) ≤ K, we have 9 109 A ρ(t) ≤ Ce−δ1 t A 10 ρ0 + C 2,6 2,6 L L 6

6

t 0

≤ C e−δ1 t A2,6 ρ0 L + 1 , 9 10

9

δ1

(t − s)− 10 e− 2 (t−s) v(s)L6 ds 0 ≤ t ≤ TU .

6

In particular, −δ t 109 1 A ρ(t)H∞ 1 ≤C e 2,6 ρ0 L + 1 , 6

0 ≤ t ≤ TU ,

with some exponent δ1 > 0. Step 3. In this step, let p1 (U0 ) denote the quantity 109 p1 (U0 ) = p u0 L1 + A2,6 ρ0 L . 6

(13.16)

452

13 Termite Mound Building Model

Multiply the first equation of (13.4) by u and integrate the product in Ω. Then, 1 d u2 dx + a |∇u|2 dx + c u2 dx 2 dt Ω Ω Ω =μ u∇u · ∇ρ dx + f u dx Ω

Ω

7 a ≤ ∇u2L2 + C uL3 7 + ∇ρ14 L14 + f uL1 . 4 3 Here, by (1.77), we have 28

7

uL 7 ≤ CuH351 uL351 . 3

So, 7

28

7

uL3 7 ≤ CuH151 uL151 ≤ ζ˜ u2H 1 + Cζ˜ u7L1 3

with an arbitrary number ζ˜ > 0. In the meantime, it is clear that ∇ρL14 ≤ ρH∞ 1 . We take ζ˜ > 0 small enough to ensure that a c 1 d u2 dx + |∇u|2 dx + u2 dx ≤ p1 (U0 ). 2 dt Ω 2 Ω 2 Ω Hence, solving this inequality (see (1.57)), we conclude that u(t)2L2 ≤ e−ct u0 2L2 + p1 (U0 ),

0 < t ≤ TU .

(13.17)

In addition, by Proposition 1.5, we conclude that

t s

u(τ )2H 1 dτ ≤ C p1 (U0 )(t − s) + u0 2L2 ,

0 ≤ s < t ≤ TU .

(13.18)

Step 4. Differentiate the second equation of (13.4) by xi (i = 1, 2, 3). Then, νu v ∂ Di v = − d + Di v + ν 1 − Di u. ∂t K K Multiply this equation by Di v and integrate the product in Ω. Then, 1 d νu v 2 2 |Di v| dx + |Di v| dx = (Di u)(Di v) dx d+ ν− 2 dt Ω K K Ω Ω ≤

d Di v2L2 + CDi u2L2 . 2

3 Global Solutions

453

So, it follows that d dt

|Di v|2 dx + d Ω

Ω

|Di v|2 dx ≤ CDi u(t)2L2 .

Employing Proposition 1.4, in view of (13.18), we conclude that Di v(t)2L2 ≤ e−dt Di v0 2L2 + p1 (U0 ) + u0 2L2 ,

0 ≤ t ≤ TU .

We sum up these estimates for i = 1, 2, 3 to conclude that v(t)2H 1 ≤ C e−dt v0 2H 1 + p1 (U0 ) + u0 2L2 , 0 ≤ t ≤ TU .

(13.19)

Step 5. In view of (13.19), let p2 (U0 ) denote p2 (U0 ) = p(v0 H 1 + u0 L2 + p1 (U0 )). Let us use the triplet H 1 (Ω) ⊂ L2 (Ω) ⊂ H 1 (Ω)∗ . We remember that A2 can be extended as an operator from H 1 (Ω) to H 1 (Ω)∗ , see Theorem 2.4. On the other hand, A2 is an operator from HN3 (Ω) to H 1 (Ω). So, A22 maps HN3 (Ω) to H 1 (Ω)∗ . We consider the duality product of the third equation of (13.4), which has a meaning in H 1 (Ω), and the vector A22 ρ(t) ∈ H 1 (Ω)∗ . Then, since ρ1 , A22 ρ2 H 1 ×H 1∗ = A2 ρ1 , A2 ρ2 H 1 ×H 1∗ = (A2 ρ1 , A2 ρ2 )L2 we have 1 d 2 dt

32 2 A ρ dx = ζ

|A2 ρ|2 dx + Ω

for ρ1 ∈ HN2 (Ω), ρ2 ∈ HN3 (Ω),

Ω

2

1

Ω

3

A22 v · A22 ρ dx

1 3 2 ≤ A22 ρ L + Cv2H 1 . 2 2 On account of (13.19), we obtain that 3 2 d 2 |A2 ρ| dx + A22 ρ dx ≤ p2 (U0 ). dt Ω Ω −1

(13.20)

3

Since A2 ρL2 ≤ A2 2 L(L2 ) A22 ρL2 , it follows that 2 d 2 |A2 ρ| dx + δ2 A2 ρ dx ≤ p2 (U0 ), dt Ω Ω −1

where δ2 = A2 2 −1 L(L2 ) > 0. Solving this (cf. (1.58)), we conclude that ρ(t)2H 2 ≤ Ce−δ2 t ρ0 2H 2 + p2 (U0 ),

0 ≤ t ≤ TU .

(13.21)

454

13 Termite Mound Building Model

As well, by Proposition 1.5, we conclude that t 3 2 A ρ(τ )2 dt ≤ C[p2 (U0 )(t − s) + ρ0 2 ], H 2 L

0 ≤ s < t ≤ TU .

2

s

In other words, t ρ(τ )2 3 dτ ≤ C[p2 (U0 )(t − s) + ρ0 2 ], H H

0 ≤ s < t ≤ TU .

s

Step 6. On account of (13.21), let us denote p3 (U0 ) = p(ρ0 H 2 + p2 (U0 )). Multiply the first equation of (13.4) by Δu and integrate the product in Ω. Then, 1 d |∇u|2 dx + a |Δu|2 dx + c |∇u|2 dx = μ ∇ · [u∇ρ]Δu dx 2 dt Ω Ω Ω Ω a 2 ≤ |Δu| dx + C (|∇u · ∇ρ| + u|Δρ|)2 dx. 4 Ω Ω Here, Ω

|∇u|2 |∇ρ|2 dx ≤ C∇u2L2 ∇ρ2L∞ ≤ CuH 2 uL2 ρ2H 1

∞

≤ ζ˜ u2H 2 + Cζ˜ u2L2 ρ4H 1 ∞ 2 2 ≤ ζ˜ ΔuL2 + Cζ˜ uL2 ρ4H 1 + 1 , ∞

where ζ˜ > 0 is an arbitrary number. Similarly, using H (Ω) ⊂ L∞ (Ω) (due to (1.76)), we have 5 1 u2 |Δρ|2 dx ≤ u2L∞ Δρ2L2 ≤ u2 5 ρ2H 2 ≤ uH3 2 uL3 2 ρ2H 2 5 3

H3

Ω

≤ ζ˜ u2H 2 + Cζ˜ u2L2 ρ12 H2 ≤ ζ˜ Δu2L2 + Cζ˜ u2L2 ρ12 +1 , H2 where ζ˜ > 0 is an arbitrary number. Therefore, by (13.17) and (13.21), we obtain that d |∇u|2 dx + a |Δu|2 dx + c |∇u|2 dx ≤ p3 (U0 ). (13.22) dt Ω Ω Ω Solving this (cf. (1.58)), we conclude that ∇u(t)2L2 ≤ e−ct ∇u0 2L2 + p3 (U0 ),

0 ≤ t ≤ TU .

3 Global Solutions

455

This, together with (13.17), yields that u(t)2H 1 ≤ e−ct u0 2H 1 + p3 (U0 ),

0 ≤ t ≤ TU .

As well, by Proposition 1.5, t Δu(τ )2L2 dτ ≤ C p3 (U0 )(t − s) + u0 2H 1 ,

(13.23)

0 ≤ s ≤ t ≤ TU .

s

By (13.16), (13.19), (13.21), and (13.23), we have obtained the desired estimate.

3.2 Construction of Global Solutions Let U0 ∈ K. As shown in Sect. 2.2, there exists a unique local solution to (13.4) with the initial value U0 on an interval [0, TU0 ]. Let 0 < t1 < TU0 . Then, U1 = U (t1 ) ∈ D(A) ∩ K. We consider problem (13.4) but substituting t1 for the initial time 0 and U1 for the initial value U0 . Of course, the problem also possesses a unique local solution in (13.11). Since U0 ∈ D(A) ∩ K, we can apply Proposition 13.1 to any local solution of the problem with initial time t1 and initial value U1 . Then, Corollary 4.1 provides the existence of the global solution. In this way, we verify that, for any U0 ∈ K, (13.4) possesses a unique global solution in the function space: 1 U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); D(A 2 )) ∩ C1 ((0, ∞); X), (13.24) U (t) ∈ K, 0 < t < ∞.

3.3 Global Norm Estimates For U0 ∈ K, let U (t; U0 ) be the global solution of (13.4) with the initial value U0 in (13.24). We observe that 1 A 2 U (t; U0 ) ≤ p e−δt p A 12 U0 + 1 , 0 ≤ t < ∞, U0 ∈ K, (13.25) X X where p(·)’s are some continuous increasing functions, and δ is a positive exponent. In fact, let us first assume that U0 ∈ D(A) ∩ K. We will apply estimates (13.15), (13.16), (13.17), (13.19), (13.21), and (13.23) established above. Applying (13.23) on [ 4t5 , t] and using (13.14), we have u(t)2H 1

2 4t 4t ≤ e u + p3 U 5 H 1 5 c 4t ≤ e− 5 t p(AU0 X ) + p3 U . 5 − 5c t

456

13 Termite Mound Building Model

By the definition of p3 (·), we observe that 2 4t 4t 4t = p ρ . p3 U + p2 U 5 5 H 2 5 Applying (13.21) on

3t 5

, 4t5 and using (13.14), we have

2 4t δ ≤ Ce− 52 t ρ 3t + p2 U 3t ρ 5 H 2 5 H 2 5 δ2 3t . ≤ e− 5 t p(AU0 X ) + p2 U 5 Let us estimate p2 (U ( 3t5 )). It is the same for p2 (U ( 4t5 )). By the definition of p2 (·), we observe that 2 2 3t 3t 3t + p1 U 3t v p2 U = p u . + 5 5 L2 5 H 1 5 Applying (13.19) on

2t 5

, 3t5 and using (13.14), we have

2 2 2 3t v ≤ e− d5 t v 2t + u 2t + p1 U 2t 5 1 5 1 5 5 H

H

L2

2 2t 2t − d5 t . p(AU0 X ) + u + p1 U ≤e 5 5 L2

In addition, applying (13.17) on

t

2t

5, 5

, we have

2 2 2t u ≤ e− 5c t u t + p1 U t 5 5 L2 5 L2 c t ≤ e− 5 t p(AU0 X ) + p1 U . 5 (It is the same for the estimate u( 3t5 )2L2 .) Let us next estimate p1 (U ( 5t )). (It is the same for p1 (U ( 2t5 )).) By the definition of p1 (·), we observe that 9 t 10 t t A2,6 ρ p1 U + . = p u 5 5 L1 5 L6 Finally, we apply (13.15) and (13.16) on [0, 5t ] to obtain that t −ct u 5 ≤ C e 5 AU0 X + 1 , L1

4 Dynamical System

457

9 10 A ρ t 2,6 5

δ1 ≤ C e− 5 t AU0 X + 1 .

L6

We have thus verified that u(t)H 1 ≤ p(e−δt p(AU0 X ) + 1),

0 ≤ t < ∞,

with some exponent δ > 0. It is possible to estimate v(t)H 1 and ρ(t)H 2 in an analogous way. We therefore observe that 1 A 2 U (t) ≤ p(e−δt p(AU0 X ) + 1), 0 ≤ t < ∞, (13.26) X with some exponent δ > 0. Let now U0 ∈ K, and let us verify (13.25). By (13.12), there exists a time τ = 1 1 τ (A 2 U0 X ) > 0 depending only on the norm A 2 U0 X such that 1 1 1 t 2 AU (t)X + A 2 U (t)X ≤ p A 2 U0 X ,

0 < t ≤ τ.

(13.27)

For τ ≤ t < ∞, (13.26) yields that 1 A 2 U (t) ≤ p(e−δ(t−τ ) p(AU (τ )X ) + 1), X

τ ≤ t < ∞.

Hence, 1 A 2 U (t) ≤ p e−δt eδτ p τ − 12 p A 12 U0 +1 , X X

τ ≤ t < ∞.

This, together with (13.27), immediately implies (13.25).

4 Dynamical System 4.1 Construction of Dynamical System Let us construct a dynamical system determined from problem (13.4). We can entirely follow the general methods in Chap. 6, Sect. 5 by taking β = 12 . Indeed, (13.25) shows that (6.56) is fulfilled. Hence, problem (13.4) determines a dynamical 1 system (S(t), K, D 1 ), where D 1 = D(A 2 ). 2

2

For any exponent 12 < θ < 1, (13.4) equally determines a dynamical system in the universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (13.6) fulfills (4.2) with β = 12 and a fortiori with β = θ . D

Let Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

458

13 Termite Mound Building Model

Meanwhile, by Proposition 6.1, 1

Aθ S(t)U0 X ≤ Cθ,R t 2 −θ ,

0 < t < ∞, U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that (6.56) is valid in the space Dθ , too. So, (S(t), Kθ , Dθ ) is a dynamical system. Let us finally construct a dynamicalsystem in the universal space X. For any 0 < R < ∞, let KR = K 1 ,R . Let XR = 0≤t 0 for which the following assertion is valid. For any bounded subset B of K, there exists a time tB > 0 such that ˜ sup sup S(t)U0 D 1 ≤ C.

U0 ∈B t≥tB

(13.28)

2

D1

˜ is an absorbing set. This means that the closed ball B 2 (0; C) We can then follow the same arguments as in Chap. 6, Sect. 5.2 to construct an absorbing set X ⊂ K which is invariant under S(t), is a closed subset of X, and is a bounded subset of D(A). Let R > 0 be a radius large enough to satisfy X ⊂ KR ⊂ XR . As (S(t), XR , X) is already known to be a dynamical system in X, we conclude that (S(t), X, X) also defines a dynamical system in X. Furthermore, since X is an absorbing set of (S(t), K, D 1 ), the asymptotic behavior of trajectories 2 of (S(t), K, D 1 ) is reduced to that of (S(t), X, X). 2

4.3 Global Attractor We are concerned with constructing a global attractor for (S(t), X, X). We must however notice that the general methods described in Chap. 6, Sect. 5.3 are not available, because the space condition (6.59) fails now. The domain D(A) given by (13.3) is not compactly embedded in X. So, we have to construct it in a direct way.

4 Dynamical System

459

We set !

A=

(closure in X).

S(s)X

(13.29)

0≤t

For further volumes: www.springer.com/series/3733

Atsushi Yagi

Abstract Parabolic Evolution Equations and their Applications

Atsushi Yagi Department of Applied Physics Graduate School of Engineering Osaka University Suita, Osaka 565-0871 Japan [email protected]

ISSN 1439-7382 ISBN 978-3-642-04630-8 e-ISBN 978-3-642-04631-5 DOI 10.1007/978-3-642-04631-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009939692 Mathematics Subject Classification (2000): 35K90, 35K57, 37L30, 92D25, 92D40 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Contents

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 1 Banach Spaces . . . . . . . . . . . . . . . . . . . . . 1.1 Distance of Subsets . . . . . . . . . . . . . . 1.2 Fractal Dimension . . . . . . . . . . . . . . . 2 Function Spaces with Values in a Banach Space . . . 2.1 Uniformly Bounded Function Spaces . . . . . 2.2 Continuously Differentiable Function Spaces . 2.3 Hölder Continuous Function Spaces . . . . . 2.4 Weighted Hölder Continuous Function Spaces 2.5 Space of Analytic Functions . . . . . . . . . . 3 Linear Operators . . . . . . . . . . . . . . . . . . . . 3.1 Bounded Linear Operators . . . . . . . . . . . 3.2 Part of Linear Operator . . . . . . . . . . . . 3.3 Multivalued Linear Operators . . . . . . . . . 3.4 Resolvent Equation and Pseudo-Resolvent . . 3.5 Spectra of Operators . . . . . . . . . . . . . . 4 Nonlinear Operators . . . . . . . . . . . . . . . . . . 4.1 Contraction Mappings . . . . . . . . . . . . . 4.2 Fréchet Differentiation . . . . . . . . . . . . . 5 Interpolation of Banach Spaces . . . . . . . . . . . . 5.1 Interpolation Spaces . . . . . . . . . . . . . . 5.2 Interpolation Theorem . . . . . . . . . . . . . 6 Adjoint Spaces and Operators . . . . . . . . . . . . . 6.1 Dual Spaces . . . . . . . . . . . . . . . . . . 6.2 Adjoint Spaces . . . . . . . . . . . . . . . . . 6.3 Adjoint Operators . . . . . . . . . . . . . . . 7 Extrapolation of Hilbert Spaces . . . . . . . . . . . . 7.1 Triplets of Spaces . . . . . . . . . . . . . . . 7.2 Extrapolation Theorem . . . . . . . . . . . . 8 Linear Operators Associated with Sesquilinear Forms 8.1 Sesquilinear Forms and Associated Operators 8.2 Adjoint Forms and Adjoint Operators . . . . .

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Integral Equations and Inequalities of Volterra Type . . . . . . 9.1 Generalized Inequality of Gronwall Type . . . . . . . . 9.2 Integral Inequality of Volterra Type . . . . . . . . . . . 9.3 Integral Equations of Volterra Type . . . . . . . . . . . 9.4 Dominate Convergence Theorem for Integral Equations Differential Inequalities . . . . . . . . . . . . . . . . . . . . . Sobolev–Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . 11.1 Boundaries of Domains . . . . . . . . . . . . . . . . . 11.2 Sobolev Spaces of Integral Order . . . . . . . . . . . . 11.3 Sobolev–Lebesgue Spaces in Rn . . . . . . . . . . . . 11.4 Sobolev–Lebesgue Spaces in Rn+ or Bounded Domains 11.5 Interpolation Property . . . . . . . . . . . . . . . . . . 11.6 Embedding Theorems . . . . . . . . . . . . . . . . . . 11.7 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Spaces H˚ ps (Ω) and Hp−s (Ω) . . . . . . . . . . . . . . 11.9 Product Spaces . . . . . . . . . . . . . . . . . . . . . . 11.10 Miscellaneous Results . . . . . . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . . . . . . . . . .

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27 27 29 31 34 35 38 39 40 41 42 43 44 45 46 48 48 53

Sectorial Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Sectorial Operators in Hilbert Spaces . . . . . . . . . . . . . . 1.1 Sectorial Operators Associated with Sesquilinear Form 1.2 Sectorial Operators in L2 Spaces . . . . . . . . . . . . 1.3 Shift Property in L2 . . . . . . . . . . . . . . . . . . . 1.4 Sectorial Operators in Product Spaces of L2 . . . . . . 2 Sectorial Operators in Banach Spaces . . . . . . . . . . . . . . 2.1 Sectorial Operators in Lp (1 < p < ∞) Spaces . . . . . 2.2 Extreme Cases p = 1 and p = ∞ . . . . . . . . . . . . 2.3 Adjoint Operators . . . . . . . . . . . . . . . . . . . . 2.4 Shift Property in Lp Spaces . . . . . . . . . . . . . . . 3 Sectorial Operators in Product Spaces . . . . . . . . . . . . . . 4 Yosida Approximation . . . . . . . . . . . . . . . . . . . . . . 4.1 Uniform Estimates . . . . . . . . . . . . . . . . . . . . 4.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . 5 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 5.1 Analytic Semigroups . . . . . . . . . . . . . . . . . . 5.2 Contraction Semigroups . . . . . . . . . . . . . . . . . 5.3 Adjoint Semigroups . . . . . . . . . . . . . . . . . . . 5.4 Approximate Semigroups . . . . . . . . . . . . . . . . 6 Generation of Analytic Semigroups . . . . . . . . . . . . . . . 6.1 Generation Results in Hilbert Spaces . . . . . . . . . . 6.2 Generation Results in Lp Spaces . . . . . . . . . . . . 7 Fractional Powers of Linear Operators . . . . . . . . . . . . . 7.1 Integral Formula . . . . . . . . . . . . . . . . . . . . . 7.2 Law of Exponent . . . . . . . . . . . . . . . . . . . . . 7.3 Law of Exponent, II . . . . . . . . . . . . . . . . . . .

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7.4 Moment Inequality . . . . . . . . . . . . . . . . . . 7.5 Comparison of Domains of Fractional Powers . . . 7.6 Fractional Powers of Adjoint Operators . . . . . . . 7.7 Fractional Powers and Analytic Semigroups . . . . 7.8 Generation Theorem of Continuous Semigroups . . Purely Imaginary Powers and Square Root Problem . . . . 8.1 Bounded Purely Imaginary Powers . . . . . . . . . 8.2 Imaginary Powers for Maximal Accretive Operators 8.3 Heinz and Kato Inequality . . . . . . . . . . . . . . 8.4 Square Root Problem . . . . . . . . . . . . . . . . 8.5 Symmetric Forms and Square Root Problem . . . . Notes and Further Researches . . . . . . . . . . . . . . . .

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97 98 101 101 104 105 105 106 109 110 112 114

Linear Evolution Equations . . . . . . . . . . . . . . . . . . . Part I. Autonomous Equations . . . . . . . . . . . . . . . . 1 Analytic Semigroups . . . . . . . . . . . . . . . . . . . . . 1.1 Generation of Analytic Semigroups . . . . . . . . . 1.2 Generator of Analytic Semigroup . . . . . . . . . . 1.3 Perturbation for Generators of Analytic Semigroups 2 Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . 2.1 Construction of Solutions . . . . . . . . . . . . . . 2.2 Maximal Regularity . . . . . . . . . . . . . . . . . 2.3 Higher-Order Maximal Regularity . . . . . . . . . 2.4 Spatial Regularity . . . . . . . . . . . . . . . . . . 3 Autonomous Parabolic Equations . . . . . . . . . . . . . . 3.1 Problems in Hilbert Spaces . . . . . . . . . . . . . 3.2 Problems in Lp Spaces . . . . . . . . . . . . . . . Part II. Nonautonomous Equations . . . . . . . . . . . . . 4 Nonautonomous Abstract Evolution Equations . . . . . . . 4.1 Structural Assumptions . . . . . . . . . . . . . . . 4.2 Some Consequences from (3.30) . . . . . . . . . . 4.3 Structural Assumptions and Yosida Approximation . 5 Evolution Operators . . . . . . . . . . . . . . . . . . . . . 5.1 Integral Equations . . . . . . . . . . . . . . . . . . 5.2 Convergence of Un (t, s) . . . . . . . . . . . . . . . 5.3 Basic Properties of U (t, s) . . . . . . . . . . . . . 6 Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . 7 Conditions (3.29) and (3.30) . . . . . . . . . . . . . . . . . 7.1 Case of Hilbert Spaces . . . . . . . . . . . . . . . . 7.2 Case of Lp Spaces . . . . . . . . . . . . . . . . . . 8 Maximal Regularity . . . . . . . . . . . . . . . . . . . . . 8.1 Refined Properties of U (t, s) . . . . . . . . . . . . 8.2 Maximal Regularity . . . . . . . . . . . . . . . . . 8.3 Spatial Regularity . . . . . . . . . . . . . . . . . . 9 Variational Methods . . . . . . . . . . . . . . . . . . . . .

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Semilinear Evolution Equations . . . . . . . . . . . . . . . . 1 Semilinear Abstract Evolution Equations . . . . . . . . . 1.1 Cauchy Problems . . . . . . . . . . . . . . . . . 1.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . 1.3 Regularity for More Regular Initial Data . . . . . 1.4 Global Existence . . . . . . . . . . . . . . . . . . 1.5 Lipschitz Continuity of Solutions in Initial Data . 1.6 Case β = 0 . . . . . . . . . . . . . . . . . . . . . 2 Variational Methods . . . . . . . . . . . . . . . . . . . . 3 Semilinear Parabolic Equations . . . . . . . . . . . . . . 4 Competition System with Diffusions . . . . . . . . . . . 4.1 Construction of Local Solutions . . . . . . . . . . 4.2 Nonnegativity of Local Solutions . . . . . . . . . 4.3 Global Solutions . . . . . . . . . . . . . . . . . . 5 Some Model in Immunology . . . . . . . . . . . . . . . 5.1 Construction of Local Solutions . . . . . . . . . . 5.2 Nonnegativity of Local Solutions . . . . . . . . . 5.3 Global Solutions . . . . . . . . . . . . . . . . . . 6 Nonautonomous Semilinear Abstract Evolution Equations Notes and Further Researches . . . . . . . . . . . . . . .

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Quasilinear Evolution Equations . . . . . . . . . . 1 Quasilinear Abstract Evolution Equations . . . . 1.1 Structural Assumptions . . . . . . . . . 1.2 Proof of Theorem 5.1 . . . . . . . . . . 1.3 Stronger Results for More Regular Data 1.4 Case ν ≤ β . . . . . . . . . . . . . . . . 2 Lipschitz Continuity of Solutions in Initial Data 3 Equations Including Semilinear Terms . . . . . 3.1 Existence Theorem . . . . . . . . . . . 3.2 Proof of Theorem 5.5 . . . . . . . . . . 3.3 Stronger Results for More Regular Data 3.4 Case ν ≤ β . . . . . . . . . . . . . . . . 4 Lipschitz Continuity of Solutions in Initial Data 5 Variational Methods . . . . . . . . . . . . . . . 6 Quasilinear Parabolic Equations . . . . . . . . . 6.1 Abstract Formulation . . . . . . . . . . 6.2 Construction of Local Solutions . . . . .

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Nonautonomous Parabolic Equations . . . . . . 10.1 Problems in Hilbert Spaces . . . . . . . 10.2 Problems in Lp Spaces . . . . . . . . . 10.3 Some Example of the Critical Case ν = 0 Perturbed Problems . . . . . . . . . . . . . . . 11.1 Evolution Operator . . . . . . . . . . . 11.2 Cauchy Problem . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . .

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Some Model for Honeybee Colonies . . . . . . . . . . . . 7.1 Abstract Formulation . . . . . . . . . . . . . . . . 7.2 Construction of Local Solutions . . . . . . . . . . . 7.3 Nonnegativity of Local Solutions . . . . . . . . . . 7.4 A Priori Estimates . . . . . . . . . . . . . . . . . . 7.5 Global Solutions . . . . . . . . . . . . . . . . . . . Some Chemotaxis Model . . . . . . . . . . . . . . . . . . 8.1 Abstract Formulation . . . . . . . . . . . . . . . . 8.2 Basic Properties of Operators A(U ) . . . . . . . . . 8.3 Construction of Local Solutions . . . . . . . . . . . 8.4 Nonnegativity of Local Solutions . . . . . . . . . . 8.5 Global Solutions . . . . . . . . . . . . . . . . . . . Nonautonomous Quasilinear Abstract Evolution Equations Notes and Further Researches . . . . . . . . . . . . . . . .

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235 236 238 239 239 242 243 243 244 245 246 247 247 249

Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . 1 Basics of Dynamical Systems . . . . . . . . . . . . . . . . 1.1 Equilibria and Attractors . . . . . . . . . . . . . . . 1.2 ω-limit Sets and Global Attractor . . . . . . . . . . 1.3 Stable and Unstable Manifolds . . . . . . . . . . . 1.4 Exponential Attraction . . . . . . . . . . . . . . . . 2 Representation Theorem of Stable and Unstable Manifolds 2.1 Discrete Case . . . . . . . . . . . . . . . . . . . . 2.2 Continuous Case . . . . . . . . . . . . . . . . . . . 3 Exponentially Stable Equilibria . . . . . . . . . . . . . . . 4 Exponential Attractors . . . . . . . . . . . . . . . . . . . . 4.1 Contraction Semigroups . . . . . . . . . . . . . . . 4.2 Compact Perturbation of Contraction Operator, I . . 4.3 Compact Perturbation of Contraction Operator, II . 4.4 Squeezing Property . . . . . . . . . . . . . . . . . 4.5 Continuous Dynamical System . . . . . . . . . . . 4.6 Continuous Dependence on Parameter . . . . . . . 5 Dynamical Systems for Semilinear Evolution Equations . . 5.1 Dynamical System . . . . . . . . . . . . . . . . . . 5.2 Absorbing and Invariant Compact Set . . . . . . . . 5.3 Exponential Attractors . . . . . . . . . . . . . . . . 5.4 Squeezing Property of S(t) . . . . . . . . . . . . . 6 Stationary Solutions to Semilinear Equations . . . . . . . . 6.1 Equilibria of Dynamical System . . . . . . . . . . . 6.2 Exponential Stability of U . . . . . . . . . . . . . . 6.3 Unstable Manifold of U . . . . . . . . . . . . . . . 6.4 Reaction–Diffusion Systems . . . . . . . . . . . . . 7 Dynamical Systems for Quasilinear Evolution Equations . . 7.1 Structural Assumptions . . . . . . . . . . . . . . . 7.2 Dynamical System . . . . . . . . . . . . . . . . . .

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7.3 Absorbing and Invariant Compact Set . 7.4 Exponential Attractors . . . . . . . . . Stationary Solutions to Quasilinear Equations 8.1 Equilibria of Dynamical System . . . . 8.2 Exponential Stability of U . . . . . . . 8.3 Unstable Manifold of U . . . . . . . . 8.4 Proof of Proposition 6.10 . . . . . . . Notes and Further Researches . . . . . . . . .

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Numerical Analysis . . . . . . . . . . . . . . . . 1 Semilinear Evolution Equations . . . . . . . 1.1 Approximate Problems . . . . . . . 1.2 Local Estimates of Convergence . . 2 Quasilinear Evolution Equations . . . . . . 2.1 Approximate Problems . . . . . . . 2.2 Local Estimates of Convergence . . 2.3 Proof of Lemma 7.1 . . . . . . . . . 3 Two-Dimensional Finite Element Methods . 3.1 Approximate Operators Aξ . . . . . 3.2 Some properties of Cξ (Ω) and Πξ . 3.3 Projection Operator Pξ . . . . . . . 3.4 Ritz Operator Rξ . . . . . . . . . . . 3.5 Domains of Fractional Powers of Aξ 4 Convergence of Exponential Attractors . . . 4.1 Discrete Case . . . . . . . . . . . . 4.2 Continuous Case . . . . . . . . . . . Notes and Further Researches . . . . . . . .

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Semiconductor Models . . . . . . . . . . . . . . . 1 Prototype Semiconductor Equations . . . . . . 1.1 Model Equations . . . . . . . . . . . . 1.2 Some Preliminary Results . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . 2.1 Construction of Local Solutions . . . . 2.2 Nonnegativity of Solutions . . . . . . 3 Global Solutions . . . . . . . . . . . . . . . . 3.1 A Priori Estimates for Local Solutions 3.2 Global Solutions . . . . . . . . . . . . 3.3 Global Norm Estimates . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . 4.2 Exponential Attractors . . . . . . . . . 5 Three-Dimensional Problem . . . . . . . . . . Notes and Further Researches . . . . . . . . .

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Activator–Inhibitor Models . . . . . . . . . . . . . 1 Prototype Reaction–Diffusion Model . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . 2.1 Construction of Local Solutions . . . . . 2.2 Nonnegativity of Local Solutions . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates from Below . . . . . 3.2 A Priori Estimates for Sobolev Norms . 3.3 Global Solution . . . . . . . . . . . . . 3.4 Global Norm Estimates . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . 4.2 Exponential Attractors . . . . . . . . . . 4.3 Squeezing Property of S(t) . . . . . . . 5 Instability of Homogeneous Stationary Solution 5.1 Localized Problem . . . . . . . . . . . . 5.2 Complexified Dynamical System . . . . 5.3 Unstable Manifold of U . . . . . . . . . Notes and Further Researches . . . . . . . . . .

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357 358 359 359 361 362 362 363 365 366 366 366 367 367 367 368 368 369 370

10 Belousov–Zhabotinskii Reaction Models . . . . . . 1 Field–Noyes Model . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . 2.1 Construction of Local Solutions . . . . . 2.2 Nonnegativity of Local Solutions . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates of Local Solutions . . 3.2 Global Existence . . . . . . . . . . . . . 3.3 Global Norm Estimate . . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . 4.2 Exponential Attractors . . . . . . . . . . 5 Keener–Tyson Model . . . . . . . . . . . . . . 6 Local Solutions . . . . . . . . . . . . . . . . . 6.1 Construction of Local Solutions . . . . . 6.2 Nonnegativity of Local Solutions . . . . 7 Global Solutions . . . . . . . . . . . . . . . . . 7.1 A Priori Estimates of Local Solutions . . 7.2 Global Existence . . . . . . . . . . . . . 7.3 Global Norm Estimate . . . . . . . . . . 8 Dynamical System . . . . . . . . . . . . . . . . 8.1 Construction of Dynamical System . . . 8.2 Exponential Attractors . . . . . . . . . . 9 Instability of Homogeneous Stationary Solution 9.1 Localized Problem . . . . . . . . . . . . 9.2 Complexified Dynamical System . . . .

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373 374 374 374 376 377 377 379 379 379 379 379 380 382 382 383 383 383 384 385 385 385 386 386 386 387

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9.3 Case where fu (u, v) ≤ 0 . . . . . . . . . . . . . . . . . . . 387 9.4 Case where 0 < fu (u, v) < 1 . . . . . . . . . . . . . . . . 388 Notes and Further Researches . . . . . . . . . . . . . . . . . . . . 389 11 Forest Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . 1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . . . . . . . . . 2.2 Construction of Local Solutions . . . . . . . . . . . . . 2.3 Nonnegativity of Local Solutions . . . . . . . . . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates for Local Solutions . . . . . . . . . 3.2 Global Solutions . . . . . . . . . . . . . . . . . . . . . 3.3 Global Norm Estimates . . . . . . . . . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Lyapunov Function . . . . . . . . . . . . . . . . . . . 4.2 ω-limit Sets . . . . . . . . . . . . . . . . . . . . . . . 4.3 Constituents of L2 -ω-limit Sets . . . . . . . . . . . . . f αδ < h < ∞ and where ab2 < 3(c + f ) . 4.4 Cases where c+f 4.5 Numerical Examples . . . . . . . . . . . . . . . . . . . 5 Homogeneous Stationary Solution . . . . . . . . . . . . . . . 5.1 Localized Problem in a Neighborhood of O . . . . . . ˜ O . . . . . . . . . . . . . . . . . . . 5.2 Spectrum of S(t) 5.3 Stability and Instability of O . . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . . . . . . . . . .

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391 391 392 392 393 394 395 395 398 398 400 400 403 405 405 408 409 410 411 412 414

12 Chemotaxis Models . . . . . . . . . . . . . . . . . . . . . . . 1 Chemotaxis Model Without Proliferation . . . . . . . . . 1.1 Abstract Formulation . . . . . . . . . . . . . . . 1.2 Construction of Local Solutions . . . . . . . . . . 1.3 Nonnegativity of Local Solutions . . . . . . . . . 2 Case where χ(ρ) = ρ . . . . . . . . . . . . . . . . . . . 2.1 Global Solutions for Small Initial Functions . . . 2.2 Lyapunov Function . . . . . . . . . . . . . . . . 2.3 Blowup of Solutions . . . . . . . . . . . . . . . . 3 Chemotaxis Model with Proliferation . . . . . . . . . . . 3.1 Local Solutions . . . . . . . . . . . . . . . . . . 3.2 A Priori Estimates of Local Solutions . . . . . . . 3.3 Global Solutions . . . . . . . . . . . . . . . . . . 3.4 Dynamical System . . . . . . . . . . . . . . . . . 3.5 Exponential Attractors . . . . . . . . . . . . . . . 3.6 Numerical Examples . . . . . . . . . . . . . . . . 4 Instability of Homogeneous Stationary Solution of (12.24) 4.1 Localized Problem . . . . . . . . . . . . . . . . . 4.2 Complexified Dynamical System . . . . . . . . . 4.3 Differentiability of F˜ (U ) . . . . . . . . . . . . .

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417 418 419 420 422 423 423 426 427 427 427 428 433 433 434 434 437 437 438 438

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4.4 Spectrum Separation Condition of A . . . . . . . . . . . . 438 4.5 Stability or Instability Conditions . . . . . . . . . . . . . . 441 Notes and Further Researches . . . . . . . . . . . . . . . . . . . . 442 13 Termite Mound Building Model . . . . . . . . . . . 1 Model equations . . . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . 2.2 Construction of Local Solutions . . . . . 2.3 Nonnegativity of Local Solutions . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates for Local Solutions . 3.2 Construction of Global Solutions . . . . 3.3 Global Norm Estimates . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . 4.2 Bounded absorbing set . . . . . . . . . . 4.3 Global Attractor . . . . . . . . . . . . . 4.4 Exponential Attractors . . . . . . . . . . 5 Instability of Homogeneous Stationary Solution 5.1 Localized Problem . . . . . . . . . . . . 5.2 Complexified Dynamical System . . . . 5.3 Differentiability of F˜ (U ) . . . . . . . . 5.4 Spectrum Separation Condition of A . . 5.5 Stability and Instability Conditions . . . Notes and Further Researches . . . . . . . . . .

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445 445 446 446 447 449 450 450 455 455 457 457 458 458 460 463 464 464 465 465 469 470

14 Adsorbate-Induced Phase Transition Model . . . . . . 1 Model Equations . . . . . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . . . 2.2 Basic Properties of A(U ) . . . . . . . . . . 2.3 Construction of Local Solutions . . . . . . . 2.4 Lower and Upper Bound of Local Solutions 3 Global Solutions . . . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates . . . . . . . . . . . . . . 3.2 Global Solutions . . . . . . . . . . . . . . . 3.3 Global Norm Estimates . . . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . . . 4.2 Exponential Attractors . . . . . . . . . . . . 5 Homogeneous Stationary Solutions . . . . . . . . . Notes and Further Researches . . . . . . . . . . . .

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471 472 472 472 474 475 477 479 479 483 483 483 483 484 484 485

15 Lotka–Volterra Competition Model with Cross-Diffusion . . . . . . 487 1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

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489 489 490 492 493 494 495 495 505 511 511 512 513 513 514 514 514 517 521 521 524 524

16 Characterization of Domains of Fractional Powers . . . . . . . . . 1 Domains of Fractional Powers in Hilbert Spaces . . . . . . . . . 1.1 Case of Self-Adjoint Operators . . . . . . . . . . . . . . 1.2 Bounded H∞ Functional Calculus . . . . . . . . . . . . 1.3 Integrable Condition Along Contour . . . . . . . . . . . 1.4 Equivalence Theorem . . . . . . . . . . . . . . . . . . . 1.5 Counterexample . . . . . . . . . . . . . . . . . . . . . . 2 Some Result on Matrix Operators . . . . . . . . . . . . . . . . . 3 Domains of Fractional Powers in Banach Spaces . . . . . . . . . 3.1 Equivalence Theorem in Banach Spaces . . . . . . . . . 4 Domains of Fractional Powers of Elliptic Operators in L2 Spaces 4.1 Case of Domains with Lipschitz Boundary . . . . . . . . 4.2 Case of Domains with C2 Boundary . . . . . . . . . . . . 4.3 Case of Convex Domains . . . . . . . . . . . . . . . . . 5 Domains of Fractional Powers of Elliptic Operators in Lp Spaces 6 Domains of Fractional Powers Under Dirichlet Conditions . . . . 6.1 Domains in L2 Spaces . . . . . . . . . . . . . . . . . . . 6.2 Domains in Lp Spaces . . . . . . . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . . . . . . . . . . .

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527 527 527 529 532 532 537 540 542 542 545 545 546 552 552 558 558 560 561

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Local Solutions . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . 2.2 Basic Properties of Operators A(U ) . . . 2.3 Construction of Local Solutions . . . . . 2.4 Some Regularity Properties . . . . . . . 2.5 Nonnegativity of Local Solutions . . . . A Priori Estimates for Local Solutions . . . . . 3.1 Case where (15.16) is Valid . . . . . . . 3.2 Case where (15.17) is Valid . . . . . . . Global Solutions . . . . . . . . . . . . . . . . . 4.1 Global Solutions . . . . . . . . . . . . . 4.2 Global Norm Estimates . . . . . . . . . Dynamical System . . . . . . . . . . . . . . . . 5.1 Construction of Dynamical System . . . 5.2 Exponential Attractors . . . . . . . . . . Instability of Homogeneous Stationary Solution 6.1 Localized Problem . . . . . . . . . . . . 6.2 Dynamical System of Localized Problem ˜ ) . . . . . . . . 6.3 Differentiability of A(U 6.4 Spectrum Separation Condition of A . . 6.5 Instability of U . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . .

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

Preface

This monograph is intended to present the fundamentals of the theory of abstract parabolic evolution equations and to show how to apply to various nonlinear diffusion equations and systems arising in science. The theory gives us a unified and systematic treatment for concrete nonlinear diffusion models. Three main approaches are known to the abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. In order to keep the volume of the monograph in reasonable length, we will focus on the semigroup methods. For other two approaches, see the related references in Bibliography. The semigroup methods, which go back to the invention of the analytic semigroups in the middle of the last century, are characterized by precise formulas representing the solutions of the Cauchy problem for evolution equations. The analytic semigroup e−tA generated by a linear operator −A provides directly a fundamental solution to the Cauchy problem for an autonomous linear evolution equation, dU dt + AU = F (t), 0 < t ≤ T , U (0) = U0 , and the solution is t given by the formula U (t) = e−tA U0 + 0 e−(t−s)A F (s) ds. The evolution opU (t, s) for linear operators A(t) gives the formula U (t) = U (t, 0)U0 + erator t U (t, s)F (s) ds representing a solution to a nonautonomous linear evolution 0 equation, dU dt + A(t)U = F (t), 0 < t ≤ T , U (0) = U0 , and the evolution operator U (t, s) can be obtained from the analytic semigroups e−sA(t) through an operator-valued integral equation of Volterra type. Furthermore, a solution to the Cauchy problem for semilinear evolution equation, dU dt + AU = F (U ), 0 < , can be found as a solution of the integral equation U (t) = t ≤ T , U (0) = U 0 t e−tA U0 + 0 e−(t−s)A F (U (s)) ds. Similarly, a solution to the quasilinear problem, dU = U , can be obtained as a solution of the dt + A(U )U = F (U ), 0 < t ≤ T , U (0) t 0 integral equation U (t) = UU (t, 0)U0 + 0 UU (t, s)F (U (s)) ds, where UU (t, s) denotes the evolution operator for linear operators AU (t) = A(U (t)). For these problems, the solution formulas provide us important information on solutions such as the uniqueness, maximal regularity, smoothing effect, and so forth. Especially, for nonlinear problems, one can derive the Lipschitz continuity of solutions with respect to the initial values, even their Fréchet differentiability. Such information on xv

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solutions then enable us develop the study to next stages. In fact, we can construct dynamical systems determined from the Cauchy problems, investigate asymptotic behavior of solutions, show the existence of attractors, investigate the stability and instability of stationary solutions, construct smooth stable and unstable manifolds, and even can analyze solutions numerically in a general framework. So far, a great number of nonlinear diffusion models have already been presented in order to describe diverse interesting phenomena in the real world. Our abstract results are available to these models. Among others, the second half of the present monograph is devoted to handling various diffusion systems in the science of self-organizations. The self-organizing process is usually described by a nonlinear dynamical system. When it is well abstractly formulated, the basic rules of selforganization, which are locally uniform in general, can be described by diffusions of constituents and suitable interactions or reactions among them, and consequently the process can be written down into a nonlinear diffusion system. We will pick up prototype diffusion systems arising in physics, chemistry, biology, and ecology. After constructing dynamical systems, we shall further study the asymptotic behavior of trajectories, existence of attractors, stability and instability of equilibria, construction of smooth stable and unstable manifolds. These results then give a clear meaning or a reasonable interpretation for many notions and ideas suggested in the science of self-organizations such as the emergent property, robustness of process, fluctuations, instabilization of homogeneous states, reduction of active degree of freedom, and so forth. They also seem to give us a certain standpoint for studying the self-organizations theoretically in the framework of infinitedimensional dynamical systems. Chapter 1 prepares all basic matters which are used as terminology or fundamental facts such as Banach and Hilbert spaces, interpolation of Banach spaces, extrapolation of Hilbert spaces, closed linear operators, contraction operators, linear operators associated with sesquilinear forms, integral equations and inequalities of Volterra type, Sobolev–Lebesgue spaces, etc. Chapter 2 reviews the basic properties and matters of sectorial operators in Banach spaces generally with their proof. Sectorial operators indeed play the principal role in the theory of abstract parabolic evolution equations. Especially, we shed light upon the two subject notions, the fractional powers of sectorial operators which are a complex interpolation of operator powers of integral order with the law of exponent and the analytic semigroups which are exponential functions generated by sectorial operators of angles less than π2 . In both we utilize the techniques for analytic functions in complex domains with values in a complex Banach space. We also present many concrete sectorial operators determined from strongly elliptic operators in Hilbert and Banach spaces. In applications, complete characterization or sharp estimation of the domains of fractional powers have critical importance. However, they require some more sophisticated techniques called H∞ functional calculus for sectorial operators. So, they are reviewed separately in Chap. 16. Chapter 3 is devoted to studying the Cauchy problems for linear evolution equations. We are concerned with construction of fundamental solutions which give us formulas representing the solutions to the Cauchy problem. For the autonomous problems, the fundamental solutions are given simply by analytic semi-

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groups. Meanwhile, for the nonautonomous problems, the fundamental solutions are given by evolution operators which are constructed by using operator-valued integral equations of Volterra type. Chapters 4 and 5 then handle the Cauchy problems for nonlinear evolution equations. In Chap. 4, we construct local solutions for the semilinear evolution equations and derive various properties of the local solutions from the solution formulas, uniqueness, maximal regularity, smoothing effect, Lipschitz continuity with respect to the initial values, etc. A sufficient condition for the global existence is also given. Chapter 5 is devoted to handling the quasilinear evolution equations. We similarly construct the local solutions and derive their significant properties from the solution formula as for the semilinear equations. Chapters 6 and 7 continue the study of the Cauchy problem for nonlinear evolution equations. In Chap. 6, first, we review basic matters and results of the theory of infinite-dimensional dynamical systems, generally with their proofs. Secondly, we consider applications of the theory to the dynamical systems determined from semilinear and quasilinear evolution equations; more precisely, we show general methods how to construct ω-limit sets, global attractors, exponential attractors, stable and unstable manifolds of equilibria, etc. In Chap. 7, we develop numerical analysis for semilinear and quasilinear evolution equations. Setting suitable approximate evolution equations in finite-dimensional spaces, we present general methods how to estimate the order of convergence of approximate solutions. All the subsequent Chaps. 8, 9, 10, 11, 12, 13, 14, and 15 are devoted to studying prototype nonlinear diffusion models which describe the kinetics of self-organizing processes. We are concerned with constructing dynamical systems, constructing attractors, investigating stability and instability of stationary solutions, and constructing the stable and unstable manifolds. Those chapters can be read independently. Chapter 8 handles Semiconductor models with drift-diffusions. The Semiconductor models seem to be the first nonlinear diffusion models presented for describing the kinetics of systems made of a number of constituents which interact each other from the macroscopic view. Chapters 9 and 10 handle Activator–Inhibitor models and Belousov–Zhabotinskii reaction models, respectively. They are both well known as classical examples of pattern formations. Their model equations are written down by reaction–diffusion systems. Chapter 11 treats some diffusion model describing the kinetics of a forest. The model equations are written down by a system of one linear diffusion equation and two ordinary differential equations, one is nonlinear, and the other linear. By the kinematic model, we can find a natural forest boundary. Chapters 12 and 13 are concerned with chemotaxis models. Chemotaxis is known as one of indispensable machinery which sustain the life. In Chap. 12, we handle a bacterium aggregating model by interaction between bacteria and chemoattractant in the manner of positive feedback. The model equations are written down by diffusion–advection–reaction equations. In Chap. 13, a termite mound building model is handled as well. Chapter 14 treats a phase transition model of a metallic surface adsorbing CO molecules. Remarkable adsorption patterns are self-organized on the surface by in-

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teraction between adsorbed molecules and a phase of the surface. The model equations are written down by diffusion–advection–reaction equations. Chapter 15 is finally devoted to handling a segregation model of competing biological species. We consider the species segregating process by using Lotka– Volterra competition model with self-diffusions and cross-diffusions. I would like to express my profound gratitude to the numerous collaborators and colleagues around the world who have made, for the long period, valuable comments and suggestions. In particular, I would like to thank Professors Alan McIntosh, Angelo Favini, Masayasu Mimura, Messoud Efendiev, Tohru Tsujikawa, and Shin-ichi Nakagiri. I would also like to express my hearty thanks to my past students, Drs. Etsushi Nakaguchi, Koichi Osaki, Sang-Uk Ryu, Masashi Aida, Yasuhiro Takei, Le Huy Chuan, Takanori Shirai, and Hideaki Fujimura, my present postgraduate student Doan Duy Hai and postdoctoral Gianluca Mola. The joint works with them were extremely important for presenting this monograph. They also kindly read the draft very carefully. Finally, I would like to express my appreciation to Springer for their efficient handling of the publication. Osaka, July 2009

Atsushi Yagi

Chapter 1

Preliminaries

1 Banach Spaces 1.1 Distance of Subsets Let X be a Banach space with norm · . Throughout book, unless others are notified, Banach and Hilbert spaces are always defined over the complex field C. As usual, X is a metric space with the distance d(U, V ) = U − V determined from · . For U ∈ X and 0 < r < ∞, the set B(U ; r) = {U ∈ X; d(U , U ) < r} is called the ball with center U and radius r. Similarly, B(U ; r) = {U ∈ X; d(U , U ) ≤ r} is called the closed ball with center U and radius r. For a point U ∈ X and a subset A of X, their distance is defined by d(U, A) = inf d(U, V ). V ∈A

For a subset A ⊂ X and 0 < r < ∞, the r-neighborhood of A is defined by Wr (A) = {U ∈ X; d(U, A) < r}. For two subsets A and B of X, the Hausdorff pseudo-distance is defined by h(A, B) = sup d(U, B) = sup inf d(U, V ). U ∈A

U ∈A V ∈B

(1.1)

Unless A is a bounded subset, h(A, B) may possibly be infinity. It is seen by definition that h(A1 , A3 ) ≤ h(A1 , A2 ) + h(A2 , A3 ). A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_1, © Springer-Verlag Berlin Heidelberg 2010

(1.2) 1

2

1 Preliminaries

For two subsets A and B, their distance is defined by d(A, B) = max{h(A, B), h(B, A)}.

(1.3)

Unless A and B are bounded subsets, d(A, B) may possibly be infinity.

1.2 Fractal Dimension Let M be a compact subset of a Banach space X. For any ε > 0, M can be covered by a finite number of closed balls having a radius ε. Let N (ε) be the minimal number of balls with radius ε which cover the set M. We then define d(M) = lim sup ε→0

log N (ε) log 1ε

.

This superior limit, 0 ≤ d(M) ≤ ∞, is called the fractal dimension of M. In particular, when d(M) < ∞, M is said to have a finite fractal dimension. The following result is easily verified from the definition. Theorem 1.1 Let M be a compact subset of X with finite fractal dimension. Let Φ be a Lipschitz-continuous operator from M into another Banach space Y , namely, ΦU − ΦV Y ≤ LU − V X ,

U, V ∈ M,

with some constant L. Then, Φ(M) also has a finite fractal dimension with the relation dY (Φ(M)) ≤ dX (M). If M has a finite fractal dimension, then M can be embedded into a finitedimensional space by Mañé–Hölder-type projection as follows. Theorem 1.2 Let M be a compact subset of a real Banach space X with fractal dimension d < ∞. Let N > 2d be an integer, and let α0 = NN−2d (1+d) . Then, for any 0 < α < α0 , we have that for almost every bounded linear operator Π : X → RN , CΠ(U − V )α ≥ U − V ,

U, V ∈ M,

(1.4)

with some constant C > 0. For details, see [BEFN93, FO96, HY99]. Condition (1.4) shows in particular that Π is injective on the subset M and the inverse of Π on Π(M) is Hölder continuous with exponent α. As N → ∞, α0 → 1; so, α can also be taken close to 1; we get by Theorem 1.1 that, if (1.4) is valid with α = 1, then d(M) = d(Π(M)). The term “almost every” is in the sense of prevalence. Prevalence is a generalized notion of “almost every” for the points in Lebesgue measurable sets in Euclidean spaces for the vectors in infinite-dimensional spaces.

2 Function Spaces with Values in a Banach Space

3

Consider two Banach spaces X and Y such that Y ⊂ X continuously. Let M be a compact subset of Y with dY (M) < ∞. By Theorem 1.1, we verify that M has a finite fractal dimension in X, too, with dX (M) ≤ dY (M). However, the following theorem gives some converse estimation. Theorem 1.3 Let three Banach spaces X, Y , and Z be such that Z ⊂ Y ⊂ X with 1−θ continuous embeddings, and let · Y ≤ D · θZ · X on Z with some exponent 0 < θ < 1. If M is a compact set of Y with dY (M) < ∞ and is a bounded set of Z, then dX (M) ≤ dY (M) ≤ (1 − θ )−1 dX (M). Proof Let ε > 0, and let N X (ε) be the minimal number of balls of X with radius ε which cover M. There exist N X (ε) points Ui ∈ X such that M⊂

X (ε) N

X

B (Ui ; ε).

i=1 X

X

For each i, since M ∩ B (Ui ; ε) = ∅, there is a point Vi ∈ M ∩ B (Ui ; ε). Then, 1−θ provides the relation · Y ≤ D · θZ · X X

X

M ∩ B (Ui ; ε) ⊂ M ∩ B (Vi ; 2ε) ⊂ B

Y

Vi ; D[δZ (M)]θ (2ε)1−θ ,

where δZ (M) denotes the diameter of M in Z. Therefore, M⊂

X (ε) N

B

Y

Vi ; D[δZ (M)]θ (2ε)1−θ .

i=0

This shows that M can be covered by at least N X (ε)-balls of Y with radius ε˜ = D[δZ (M)]θ (2ε)1−θ , that is, N Y (˜ε ) ≤ N X (ε). We hence obtain that dY (M) = lim sup ε˜ →0

log N Y (˜ε ) log

1 ε˜

= (1 − θ )−1 lim sup ε→0

≤ lim sup ε→0

log N X (ε) log

1 ε

log N X (ε) log D[δ

1 θ 1−θ Z (M)] 2

1 + log ε1−θ

= (1 − θ )−1 dX (M).

It is already known that dX (M) ≤ dY (M).

2 Function Spaces with Values in a Banach Space Let X be a Banach space with norm · . We introduce various function spaces consisting of functions defined on an interval of R or on a domain of C with values in X.

4

1 Preliminaries

2.1 Uniformly Bounded Function Spaces Let [a, b] be a bounded closed interval. By B([a, b]; X) we denote the space of uniformly bounded functions on [a, b] (not necessarily smooth or measurable). The space is a Banach space with the supremum norm F B = sup F (t). a≤t≤b

−η

By B{a} ((a, b]; X), where η > 0, we denote the space of X-valued functions which are uniformly bounded on (a, b] with weight function (t − a)η , namely, F ∈ −η B{a} ((a, b]; X) if and only if (t − a)η F ∈ B((a, b]; X). We equip the space with the norm F B−η = sup (t − a)η F (t) = (t − a)η F B . {a}

a 0 be fixed exponents. If a continuous function ϕ(t, s) ≥ 0 defined for 0 ≤ s < t ≤ T has a weak singularity at t = s and satisfies the integral inequality t (1.39) ϕ(t, s) ≤ a(t − s)μ−1 + b (t − τ )ν−1 ϕ(τ, s) dτ, 0 ≤ s < t ≤ T , s

then 1 ϕ(t, s) ≤ a(μ)(t − s)μ−1 Eμ,ν [b(ν)] ν (t − s) ,

0 ≤ s < t ≤ T,

(1.40)

where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35). Proof By induction, we will show the estimate ϕ(t, s) ≤

n

abk

k=0

+ bn+1

(μ)(ν)k (t − s)μ+kν−1 (μ + kν) (ν)n+1 ((n + 1)ν)

t

(t − τ )(n+1)ν−1 ϕ(τ, s) ds.

(1.41)

s

Indeed, it is obvious for n = 0. Assume that this is true for n. Then, ϕ(t, s) ≤

n

(μ)(ν)k (ν)n+1 (t − s)μ+kν−1 + bn+1 (μ + kν) ((n + 1)ν) k=0 t τ (n+1)ν−1 μ−1 ν−1 a(τ − s) × (t − τ ) +b (τ − σ ) ϕ(σ, s) dσ dτ. abk

s

s

Hence, by (1.38), t (μ)((n + 1)ν) . (t − τ )(n+1)ν−1 (τ − s)μ−1 dτ = (t − s)μ+(n+1)ν−1 (μ + (n + 1)ν) s In addition, by (1.38), t τ s

(t − τ )(n+1)ν−1 (τ − σ )ν−1 ϕ(σ, s) dσ dτ

s

=

t s

=

t

(t − τ )(n+1)ν−1 (τ − σ )ν−1 ϕ(σ, s) dτ dσ

σ

((n + 1)ν)(ν) ((n + 2)ν)

t

(t − σ )(n+2)ν−1 ϕ(σ, s) dσ.

s

Hence, (1.41) is obtained for n + 1, too. Letting n → ∞ in (1.41), we observe that ϕ(t, s) ≤

∞ n=0

abn

(μ)(ν)n (t − s)μ+nν−1 , (μ + nν)

0 ≤ s < t ≤ T.

9 Integral Equations and Inequalities of Volterra Type

31

By the definition (1.35), the desired estimate (1.40) is concluded.

In a quite analogous way, we can show the following result. The proof will be left to the reader. Theorem 1.28 Let a ≥ 0 and b > 0 be constants, and let μ > 0 and ν > 0 be fixed exponents. If a continuous function ψ(t, s) ≥ 0 defined for 0 ≤ s < t ≤ T has a weak singularity at t = s and satisfies the integral inequality t ψ(t, τ )(τ − s)ν−1 dτ, 0 ≤ s < t ≤ T , (1.42) ψ(t, s) ≤ a(t − s)μ−1 + b s

then 1 ψ(t, s) ≤ a(μ)(t − s)μ−1 Eμ,ν [b(ν)] ν (t − s) ,

0 ≤ s < t ≤ T,

where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35).

9.3 Integral Equations of Volterra Type In this subsection, we consider integral equations of Volterra type. Let X be a Banach space, and let L(X) be the Banach space of all bounded linear operators on X equipped with the uniform operator norm · . Let 0 < T < ∞ be a fixed number and put = {(t, s) ∈ R2 ; 0 ≤ s < t ≤ T }. We consider the integral equation of Volterra type t Φ(t, s) = P (t, s) + Q(t, τ )Φ(τ, s) dτ, 0 ≤ s < t ≤ T ,

(1.43)

s

in L(X). Here, P and Q are in C(; L(X)) and are assumed to satisfy the estimates P (t, s) ≤ a(t − s)μ−1 ,

0 ≤ s < t ≤ T,

(1.44)

Q(t, s) ≤ b(t − s)ν−1 ,

0 ≤ s < t ≤ T,

(1.45)

with some fixed exponents μ > 0 and ν > 0 and some constants a > 0 and b > 0, respectively. These mean that P and Q have weak singularities at t = s. We can construct a unique solution to (1.43). Theorem 1.29 Under (1.44) and (1.45), there exists a unique function Φ ∈ C(; L(X)) such that Φ has a weak singularity at t = s and satisfies (1.43). Moreover, Φ is estimated by 1 Φ(t, s) ≤ a(μ)(t − s)μ−1 Eμ.ν [b(ν)] ν (t − s) , 0 ≤ s < t ≤ T , (1.46) where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35).

32

1 Preliminaries

Proof Before proving the assertion of theorem, we present a lemma. ˜ ∈ C(; L(X)) satisfy Lemma 1.3 Let P˜ , Q ˜ P˜ (t, s) ≤ a(t ˜ − s)μ−1 ,

0 ≤ s < t ≤ T,

(1.47)

˜ s) ≤ b(t ˜ − s)ν˜ −1 , Q(t,

0 ≤ s < t ≤ T,

(1.48)

with some exponents μ˜ > 0 and ν˜ > 0 and some constants a˜ > 0 and b˜ > 0, respectively. Put t ˜ τ )P˜ (τ, s) dτ, 0 ≤ s < t ≤ T . ˜ s) = Q(t, R(t, s

Then, R˜ belongs to C(; L(X)) and satisfies the estimate ˜ ˜ ν) ˜ ν −1 ˜ s) ≤ a˜ b(μ)(˜ , R(t, (t − s)μ+˜ (μ˜ + ν˜ )

0 ≤ s < t ≤ T.

(1.49)

Proof We observe in view of (1.38) that R˜ is well defined and satisfies (1.49). ˜ But this is not obvious because P˜ So, it suffices to verify the continuity of R. ˜ and Q have singularities. We here introduce a sequence of cutoff functions P˜k defined for all integers k > T1 such that P˜k ’s are defined for all 0 ≤ s ≤ t ≤ T and are continuous functions with values in L(X), that P˜k (t, s) = P˜ (t, s) for 0 ≤ s ≤ T − k1 , s + k1 ≤ t ≤ T , and that P˜k ’s satisfy the same estimate as (1.47). We also introduce a sequence Q˜ k satisfying (1.48), which is defined in a quite analogous way to P˜k . ˜ k ∈ C(; L(X)), it is easy to see that the function As P˜k , Q t 1 R˜ k (t, s) = Q˜ k (t, τ )P˜k (τ, s) dτ, 0 ≤ s ≤ t ≤ T , k > , T s is also in C(; L(X)). Moreover, for 0 ≤ s ≤ T − k1 , s + R(t, ˜ s) − R˜ k (t, s) ≤

≤t ≤T,

Q(t, ˜ τ ) − Q˜ k (t, τ )P˜ (τ, s) dτ

t

s

Q ˜ k (t, τ )P˜ (τ, s) − P˜k (τ, s) dτ

t

+ s

≤C

1 k

t t− k1

+

s+ k1

˜ (t − τ )μ−1 (τ − s)ν˜ −1 dτ

s

˜ 1 ν˜ −1 1 μ−1 −1 −μ˜ −1 −˜ν ≤ C μ˜ k + ν˜ k t −s− t −s− . k k ˜ s) in L(X) as k → ∞ This estimate indeed shows that R˜ k (t, s) converges to R(t, and, for arbitrarily fixed 0 < δ < T , the convergence is uniform for t and s such that

9 Integral Equations and Inequalities of Volterra Type

33

˜ s) for 0 ≤ s ≤ T − δ, s + δ ≤ t ≤ T . As a result, we observe the continuity of R(t, 0≤s 0, then u(t) ≤ e−δt u(0) +

t

e−δ(t−s) f (s) ds,

0 < t ≤ T.

(1.57)

0

In addition, if f (t) ≡ f > 0, then it follows that u(t) ≤ e−δt u(0) + f δ −1 ,

0 < t ≤ T.

(1.58)

We next consider the differential inequality du + δu ≤ f (t), dt

0 < t ≤ T,

(1.59)

with δ > 0 and f ∈ C([0, T ]; R). We assume that f (t) satisfies the condition t f (τ ) dτ ≤ α(t − s) + β, 0 ≤ s < t ≤ T , (1.60) s

with some constants α ≥ 0 and β ≥ 0. Then, as seen above, if u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfies (1.59), then u is estimated by (1.57). By integration by parts, t t t d −δ(t−s) −δ(t−s) − e f (s) ds = e f (τ ) dτ ds ds s 0 0 t t t = e−δt f (τ ) dτ + δ e−δ(t−s) f (τ ) dτ ds. 0

0

s

Therefore, (1.60) yields that t t e−δ(t−s) f (s) ds ≤ e−δt (αt + β) + δ e−δ(t−s) [α(t − s) + β] ds 0

0

= αδ

−1

(1 − e

−δt

) + β.

(1.61)

Hence, we have proved the following proposition. Proposition 1.4 Let δ > 0, and let f ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0. If u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfies the differential inequality (1.59), then u is estimated by u(t) ≤ e−δt u(0) + αδ −1 + β,

0 < t ≤ T.

(1.62)

On the basis of these results, we can prove some other propositions which will be often utilized in showing a dissipative condition for nonlinear diffusion equations. Proposition 1.5 Let δ > 0, γ > 0, and let f ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0. Assume that 0 ≤ u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) and 0 ≤ v ∈

10 Differential Inequalities

37

C([0, T ]; R) satisfy the differential inequality du + δu + γ v ≤ f (t), dt

0 < t ≤ T.

Then, it holds for v that t v(τ ) dτ ≤ αγ −1 (t − s) + [u(0) + αδ −1 + 2β]γ −1 ,

(1.63)

0 ≤ s < t ≤ T . (1.64)

s

Proof Since γ v ≥ 0, u satisfies (1.59), too. So, u is estimated by (1.62). Fix 0 ≤ s < t ≤ T . Integrate both hand sides of (1.63) for τ in the interval [s, t]. Then, t t t u(t) − u(s) + δ u(τ ) dτ + γ v(τ ) dτ ≤ f (τ ) dτ. s

Since u ≥ 0,

γ

t

s

s

v(τ ) dτ ≤ u(s) +

s

t

f (τ ) dτ. s

By (1.60) and (1.65), we see that t γ v(τ ) dτ ≤ e−δs u(0) + αδ −1 + β + α(t − s) + β. s

Hence, (1.64) is obtained.

Proposition 1.6 Let δ > 0, let f ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0, and let 0 ≤ p ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0, where δ > α ≥ 0. If 0 ≤ u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfies the differential inequality du + δu ≤ p(t)u + f (t), dt

0 < t ≤ T,

(1.65)

then u is estimated by u(t) ≤ eβ e−(δ−α)t u(0) + eβ α (δ − α)−1 + β ,

0 < t ≤ T.

(1.66)

Proof Obviously, u satisfies (1.55) with a(t) = δ − p(t). So, the assumption on p yields that t t t − 0 [δ−p(τ )]dτ u(t) ≤ e u(0) + e− s [δ−p(τ )]dτ f (s) ds ≤ eβ e−(δ−α)t u(0) + eβ

0 t

e−(δ−α)(t−s) f (s) ds.

0

Therefore, (1.66) is verified by (1.61).

38

1 Preliminaries

Proposition 1.7 Let δ > 0, γ > 0, and let f ∈ C([0, T ]; R) (resp. g ∈ C([0, T ]; R)) satisfy (1.60) with α ≥ 0 and β ≥ 0 (resp. α ≥ 0 and β ≥ 0). Let 0 ≤ p ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0. Assume that 0 ≤ u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) and 0 ≤ v ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfy the two differential inequalities du 0 < t ≤ T, dt + δu + γ v ≤ f (t), (1.67) dv + δv ≤ p(t)v + g(t), 0 < t ≤ T. dt Then, v is estimated by v(t) ≤ eβ e−δt v(0) + eβ αγ −1 [u(0) + α γ −1 + α δ −1 + 2β ] + eβ (α δ −1 + β ),

0 ≤ t ≤ T.

(1.68)

Proof Let ξ > 0 be a parameter. From the second inequality of (1.67) we have dv + [δ + ξ − p(t)]v ≤ ξ v(t) + g(t). dt Therefore, using (1.56) with a(t) = δ + ξ − p(t), we obtain that t t − 0t [δ+ξ −p(τ )]dτ v(t) ≤ e v(0) + e− s [δ+ξ −p(τ )]dτ [ξ v(s) + g(s)] ds. 0

The assumption on p then yields that v(t) ≤ eβ e−(δ+ξ −α)t v(0) + eβ

t

e−(δ+ξ −α)(t−s) [ξ v(s) + g(s)] ds.

0

We here fix the ξ such as ξ = α. Then, since v satisfies (1.64), we observe by (1.61) that t e−δ(t−s) αv(s) ds ≤ αγ −1 [u(0) + α γ −1 + α δ −1 + 2β ]. 0

Similarly, by (1.61),

t

e−δ(t−s) g(s) ds ≤ α δ −1 + β .

0

Hence, the desired estimate has been shown.

11 Sobolev–Lebesgue Spaces An open and connected subset Ω of Rn is called a domain in Rn . Throughout the book, unless otherwise notified, the functions in Ω are always complex valued.

11 Sobolev–Lebesgue Spaces

39

Let Ω be a domain in Rn . For 1 ≤ p ≤ ∞, the set of all Lp functions in Ω is denoted by Lp (Ω). With the usual Lp norm · Lp , Lp (Ω) is a Banach space. When Ω is a bounded domain, C(Ω) (resp. Ck (Ω), k = 1, 2, . . .) is the set of all continuous (resp. k-times continuously differentiable)functions on Ω. With the norm uC = maxx∈Ω |u(x)| (resp. uCk = uC + |α|≤k D α uC ), C(Ω) (resp. Ck (Ω)) becomes a Banach space. For 0 < σ < 1, Cσ (Ω) is the set of all Hölder continuous functions on Ω with exponent σ . Cσ (Ω) is a Banach space k+σ (Ω), k = with the norm uCσ = uC + supx,y∈Ω, x =y |u(x)−u(y)| |x−y|σ . Similarly, C 1, 2, . . . , are the sets of all Ck functions on Ω with kth-order derivatives in Cσ (Ω). Ck+σ (Ω)’s are also Banach spaces with the norms uCk+σ = uCk + α |α|=k D uCσ . Meanwhile, the space of Lipschitz continuous functions on Ω is denoted by C0,1 (Ω). C0,1 (Ω) is a Banach space with the norm uC0,1 = uC + k,1 supx,y∈Ω, x =y |u(x)−u(y)| |x−y| . In an analogous way, the spaces C (Ω), k = 1, 2, . . . , and their norms are defined. By D(Ω) we denote the space of all infinitely differentiable functions in Ω with compact supports.

11.1 Boundaries of Domains Following [Gri85], we review basic things concerning the boundary of domain in Rn . In this subsection, we are concerned with a bounded domain Ω in Rn . We say that Ω has a continuous (resp. Lipschitz, Cm class (m = 1, 2, 3, . . .)) boundary ∂Ω if for every x ∈ ∂Ω, there exists a neighborhood V of x in Rn and a new orthogonal coordinate system (y1 , . . . , yn ) such that (1) V is a cube in the new coordinates: V = {(y1 , . . . , yn ); −ai < yi < ai , i = 1, . . . , n}. (2) There exists a continuous (resp. Lipschitz, Cm class) function ϕ which is defined in V = {(y1 , . . . , yn−1 ); −ai < yi < ai , i = 1, . . . , n − 1} and satisfies |ϕ(y )| ≤ an /2 for every y = (y1 , . . . , yn−1 ) ∈ V , Ω ∩ V = {y = (y , yn ) ∈ V ; yn > ϕ(y )}, ∂Ω ∩ V = {y = (y , yn ) ∈ V ; yn = ϕ(y )}. (3) ϕC(V ) ≤ c (resp. ϕLip(V ) ≤ c, ϕCm (V ) ≤ c) with some constant c > 0. If Ω is an n-dimensional manifold of Cm class (m = 1, 2, . . .) with boundary, then its boundary ∂Ω is automatically of Cm class. However, the same is not necessarily true for a Lipschitz manifold. Even if Φ(y1 , . . . , yn ) is a Lipschitz continuous function for which the equation Φ(y1 , . . . , yn ) = 0 defines a hypersurface

40

1 Preliminaries

yn = ϕ(y1 , . . . , yn−1 ), the function ϕ may be non-Lipschitz. A counterexample is given in [Gri85, Lemma 1.2.1.4]. Let Ω be a bounded domain with Lipschitz boundary. Then, on the boundary ∂Ω, a measure dS can be defined in a similar manner as in the case of smooth boundary, and an integral with respect to dS is defined for the functions on ∂Ω. If a function f has a support in ∂Ω ∩ V , where V is one of the neighborhoods mentioned above, then its integral is given in the form f dS = f 1 + |∇ ϕ(y )|2 dy , V

∂Ω

where ∇ ϕ = (D1 ϕ, . . . , Dn−1 ϕ). Furthermore, for almost all boundary points x ∈ ∂Ω, the outer normal vectors ν(x) are defined. Indeed, if (y , yn ) ∈ ∂Ω ∩ V , then ν is given by ν(y , yn ) = (D1 ϕ(y ), . . . , Dn−1 ϕ(y ), −1) 1 + |∇ ϕ(y )|2 . Then, the formula of integration by parts Di u · v dx + u · Di v dx = [uv]νi (x) dS, Ω

Ω

i = 1, 2, . . . , n,

(1.69)

∂Ω

is valid for u, v ∈ C1 (Ω), where ν(x) = (ν1 (x), . . . , νn (x)), x ∈ ∂Ω.

11.2 Sobolev Spaces of Integral Order Let Ω be a domain in Rn . For 1 ≤ p ≤ ∞ and k = 0, 1, 2, . . . , the space Hpk (Ω) is defined as a set of all complex-valued Lp functions u whose derivatives D α u in the distribution sense up to order k belong to Lp (Ω), where α = (α1 , α2 , . . . , αn ) denotes a multiindex, and |α| = α1 + α2 + · · · + αn . The space Hpk (Ω) becomes a Banach space with the norm uHpk =

|α|≤k

1 D

α

p uLp

p

,

u ∈ Hpk (Ω).

When p = 2, H2k (Ω) is obviously a Hilbert space with the inner product (D α u, D α v)L2 , u, v ∈ H2k (Ω). (u, v)H k = 2

|α|≤k

Let Ω be the half-space Rn+ = {x = (x , xn ); x ∈ Rn−1 , xn > 0} or a bounded domain in Rn with Lipschitz boundary. According to [Ste70, Chap. VI, Theorems 5 and 5 ], we can construct an extension operator of functions in Ω to functions in Rn universally in 1 ≤ p ≤ ∞ and k = 0, 1, 2, . . . .

11 Sobolev–Lebesgue Spaces

41

Theorem 1.33 Let Ω be Rn+ or a bounded domain with Lipschitz boundary. There exists a linear operator E mapping functions in Ω to functions in Rn with the following properties: (1) (Eu)|Ω = u. (2) E is a continuous operator from Hpk (Ω) to Hpk (Rn ) for any 1 ≤ p ≤ ∞ and any k = 0, 1, 2, . . . , namely, EuHpk (Rn ) ≤ Ap,k uHpk (Ω) ,

u ∈ Hpk (Ω),

with some constant Ap,k > 0 depending on p and k.

11.3 Sobolev–Lebesgue Spaces in Rn When 1 < p < ∞, the Sobolev spaces Hpk (Ω) can be extended for nonintegral orders. In this subsection, we consider the case where Ω = Rn . Let s ≥ 0 be a nonnegative number. The space Hps (Rn ) is defined by s Hps (Rn ) = u ∈ S(Rn ) ; F−1 1 + |ξ |2 2 Fu ∈ Lp (Rn ) , where S(Rn ) denotes the set of tempered distributions in Rn , and F and F−1 denote the Fourier transform and the inverse Fourier transform on S(Rn ) , respectively. The space Hps (Rn ) is a Banach space with the norm s uHps = F−1 1 + |ξ |2 2 Fu L , p

u ∈ Hps (Rn ).

By the theory of Fourier multipliers, it is possible to verify that the two definitions of Hpk (Rn ) and Hps (Rn ) are equivalent for integral orders s = 0, 1, 2, . . . , that is, Hpk (Rn ) = Hps (Rn ) with norm equivalence for nonnegative integers s = k. When p = 2, the space H2s (Rn ) is seen to be a Hilbert space with the inner product (u, v)H2s =

1 + |ξ |2

s

2

s Fu, 1 + |ξ |2 2 Fv L , 2

u, v ∈ H2s (Rn ).

Furthermore, for s = k + σ , where k = [s] is an integer and 0 < σ < 1, the norm of H2s (Rn ) is equivalent to u H s 2

= uL2 +

|α|≤k

see [Tri78, 2.5.1, (15)].

Rn ×Rn

|D α u(x) − D α u(y)|2 dx dy |x − y|n+2σ

1 2

,

u ∈ H2s (Rn ),

42

1 Preliminaries

11.4 Sobolev–Lebesgue Spaces in Rn+ or Bounded Domains We intend to extend the definition of Hpk (Ω) for nonintegral orders. Let Ω denote the half-space Rn+ or a bounded domain in Rn with Lipschitz boundary. For 1 < p < ∞ and s ≥ 0, Hps (Ω) is defined as the set of all restrictions u of the functions in Hps (Rn ) to Ω. That is, a function u ∈ Lp (Ω) is in Hps (Ω) if and only if there exists a function U ∈ Hps (Rn ) such that U|Ω = u almost everywhere in Ω. For u ∈ Hps (Ω), its Hps norm is defined by uHps (Ω) =

inf

U ∈Hps (Rn ),U|Ω =u

U Hps (Rn ) .

With this norm, Hps (Ω) is a Banach space. Indeed, since the space K = {U ∈ Hps (Rn ); U = 0 in Ω} is a closed subspace of Hps (Rn ), Hps (Ω) is nothing more than the quotient space Hps (Rn ) K. By virtue of Theorem 1.33, we can easily verify that such a definition is consistent with the former one when s is an integer. Let p = 2, and let s = k + σ , where k = [s] is an integer and 0 < σ < 1. Then, the norm of H2s (Ω) is as before equivalent to u H s 2

= uL2 +

|α|≤k

Ω×Ω

|D α u(x) − D α u(y)|2 dx dy |x − y|n+2σ

1

u ∈ H2s (Ω),

2

, (1.70)

see [Tri78, Remark 4.4.2/2]. One sees that, for 0 < s0 < s1 < ∞, Hps1 (Ω) ⊂ Hps0 (Ω) ⊂ Lp (Ω) with continuous embeddings. Theorem 1.34 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. Let 1 < p < ∞ and s ≥ 0. Then, u ∈ Hps (Ω) if and only if Re u ∈ Hps (Ω) and Im u ∈ Hps (Ω). It holds that max{ Re uHps , Im uHps } ≤ uHps ≤ Re uHps + Im uHps ,

u ∈ Hps (Ω).

Proof We will prove that u ∈ Hps (Ω) implies u ∈ Hps (Ω) with uHps = uHps , where u(x) = u(x) is the complex conjugate of u(x). Clearly, all the assertions of the theorem are easily verified by this property. Consider first the case where Ω = Rn . By definition, u ∈ Hps (Rn ) is equivalent to s ei(y−x)·ξ 1 + |ξ |2 2 u(x) dx dξ ∈ Lp Rny . Rn ×Rn

11 Sobolev–Lebesgue Spaces

43

Taking the complex conjugate of this function and changing integral variables by ξ = −ξ , we have s n (−1) ei(y−x)·ξ 1 + |ξ |2 2 u(x) dx dξ ∈ Lp Rny . Rn ×Rn

This then means that u ∈ Hps (Rn ) with uHps = uHps . Consider next the general case of Ω. By definition, u ∈ Hps (Ω) implies that there exists a sequence of functions Uk ∈ Hps (Rn ) such that u = Uk|Ω and Uk Hps (Rn ) → uHps (Ω) as k → ∞. Since u = U k|Ω , we conclude that u ∈ Hps (Ω) with uHps ≤ infk U k Hps = infk Uk Hps = uHps . Similarly, uHps = uHps ≤ uHps . These spaces Hps (Ω) are called by different names. When the order is integral, Hpk (Ω) is called the Sobolev space. When p = 2, H2s (Ω) is also called the Sobolev space. In the meantime, when 1 < p < ∞, p = 2, Hps (Ω) is called the Lebesgue space. When p = 2, H2s (Ω) is often abbreviated by H s (Ω). The Sobolev–Lebesgue spaces consisting of real-valued functions are denoted by Hps (Ω; R), 1 < p < ∞, s ≥ 0.

11.5 Interpolation Property In this subsection, we present the interpolation property of the Sobolev–Lebesgue spaces. This property is extremely useful for deriving various properties for the Sobolev–Lebesgue spaces. According to [Tri78, Theorems 2.4.2/1, 2.10.1 and 4.3.1/2], we have the following theorem. Theorem 1.35 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. Let 1 < p < ∞ and 0 ≤ s0 < s1 < ∞. Then s (1.71) Hp0 (Ω), Hps1 (Ω) θ = Hps (Ω) with norm equivalence, where 0 ≤ θ ≤ 1 and s = (1 − θ )s0 + θ s1 . Let us use here this theorem to verify the boundedness of the extension operator E introduced above from Hps (Ω) to Hps (Rn ). Let s > 0 be nonintegral, and let Ω = Rn . Let k = [s] + 1 > s be an integer. Since E is a bounded operator from Lp (Ω) to Lp (Rn ) and from Hpk (Ω) to Hpk (Rn ) simultaneously, the interpolation theorem (Theorem 1.15) is applied to E to conclude that E is a bounded operator from Hps (Ω) to Hps (Rn ). In other words, if u ∈ Hps (Ω), then Eu ∈ Hps (Rn ) with the estimate EuHps (Rn ) ≤ Ap,s uHps (Ω) . We thus observe the following estimates: A−1 p,s EuHps (Rn ) ≤ uHps (Ω) ≤ EuHps (Rn ) ,

u ∈ Hps (Ω).

(1.72)

44

1 Preliminaries

As an immediate consequence, we get that a function u ∈ Lp (Ω) belongs to Hps (Ω) if and only if its extension Eu is in Hps (Rn ). Estimate (1.72) shows also that the norm of Hps (Ω) is equivalent to the graph norm E · Hps (Rn ) of E. When p = 2, since H2s (Rn ) is a Hilbert space, this fact yields that H s (Ω) = H2s (Ω) also enjoys a Hilbert structure with the inner product (u, v)H s (Ω) = (Eu, Ev)H s (Rn ) ,

u, v ∈ H s (Ω).

(1.73)

11.6 Embedding Theorems According to [Tri78, Theorem 2.8.1/Remark 2 and Theorem 4.6.1], we get the following embedding theorem. Theorem 1.36 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. Let 1 < p < ∞ and 0 ≤ s < ∞. (1) If 0 ≤ s < pn , then Hps (Ω) ⊂ Lr (Ω) where p ≤ r ≤ (2) If s = pn , then

with continuous embedding,

(1.74)

Hpp (Ω) ⊂ Lr (Ω) with continuous embedding,

(1.75)

pn n−ps .

n

where r is any finite number such that p ≤ r < ∞. (3) If s > pn , then Hps (Ω) ⊂

C(Rn ) (resp. C(Rn+ )) when Ω = Rn (resp. Rn+ ), when Ω is bounded. C(Ω)

(1.76)

When Ω is bounded, the embedding is continuous. The following estimates are known as Gagliardo–Nirenberg’s inequality. When Ω = Rn , the proof is given in [Tan97, Theorems 3.3, 3.4, and 3.5]. Thanks to the extension theorem, Theorem 1.33, the proofs for other cases are immediately reduced to this case. Theorem 1.37 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary, and let 1 ≤ q ≤ p ≤ ∞. (1) If 1 ≤ p < n, then Hp1 (Ω) ∩ Lq (Ω) ⊂ Lr (Ω)

11 Sobolev–Lebesgue Spaces

45

with the estimate uLr ≤ Cp,q,r uaH 1 u1−a Lq , p

where q ≤ r ≤

np n−p ,

u ∈ Hp1 (Ω) ∩ Lq (Ω),

and a is given by 1 1 1 1−a =a − + . r p n q

(1.77)

(2) If p = n, then Hn1 (Ω) ∩ Lq (Ω) ⊂ Lr (Ω) with the estimate uLr ≤ Cp,q,r uaH 1 u1−a Lq , n

u ∈ Hn1 (Ω) ∩ Lq (Ω),

where r is any finite number such that q ≤ r < ∞, and a is given by 1 1−a = . r q

(1.78)

(3) If n < p ≤ ∞, then Hp1 (Ω) ∩ Lq (Ω) ⊂ Lr (Ω) with the estimate uLr ≤ Cp,q,r uaH 1 u1−a Lq , p

u ∈ Hp1 (Ω) ∩ Lq (Ω),

where q ≤ r ≤ ∞, and a is given by 1 1−a 1 1 =a − + . r p n q

(1.79)

We finally present the compact embedding theorem. The proof is given in [Tri78, Theorem 4.10.1]. Theorem 1.38 Let Ω be a bounded domain in Rn with Lipschitz boundary. For 1 < p < ∞ and s > 0, the embedding i : Hps (Ω) → Lp (Ω) is compact.

11.7 Traces We are concerned with defining the values of f (x) on the boundary ∂Ω for the functions f (x) in Ω. Let us first consider the case where Ω = Rn+ . If 1 < p < ∞ and s > pn , we see from (1.76) that H s (Rn+ ) ⊂ C(Rn+ ). Therefore, the trace operator γ : f → f|∂Rn+ is

46

1 Preliminaries

defined from H s (Rn+ ) to C(∂Rn+ ), where ∂Rn+ = {x = (x , 0); x ∈ Rn−1 }. According to [Tri78, Theorem 2.9.3], if s > p1 , the operator γ can be extended as a bounded operator from Hps (Rn+ ) to Lp (∂Rn+ ). We want to extend this result to bounded domains with Lipschitz boundaries. In [Tri78], only the case where the boundary ∂Ω is sufficiently smooth is treated (see [Tri78, Theorem 4.7.1]). In [Gri85], the main interest is only in the Sobolev spaces Wps (Ω). So, it may be necessary to write up the full proof for Lebesgue spaces. However, we are able to carry out the proof in a quite analogous way to the case of Wps (Ω). So, we will omit the proof of the following theorem. It is the same for the proofs of Theorems 1.40, 1.41, and 1.42 below. Theorem 1.39 Let Ω be Rn+ or a bounded domain with Lipschitz boundary. Let 1 < p < ∞. If s > p1 , then the trace γ : f → f|∂Ω is a bounded operator from Hps (Ω) to Lp (∂Ω). On the contrary, the following two theorems show that the similar trace is no longer meaningful for s such that 0 ≤ s ≤ p1 . Theorem 1.40 Let Ω be as above, and let 1 < p < ∞. For 0 ≤ s ≤ p1 , the space D(Ω) is dense in Hps (Ω). Theorem 1.41 Let Ω be as above, and let 1 < p < ∞. For 0 ≤ s < p1 , the zero extension operator f → f ∼ , where f ∼ = f in Ω and f ∼ = 0 in Rn − Ω, is a bounded operator from Hps (Ω) to Hps (Rn ). For

1 p

< s ≤ 1, we can verify the following fact.

Theorem 1.42 Let Ω be as above, and let 1 < p < ∞. For p1 < s ≤ 1, a function u ∈ Hps (Ω) belongs to the closure of D(Ω) in Hps (Ω) if and only if u|∂Ω = 0.

11.8 Spaces H˚ ps (Ω) and Hp−s (Ω) For 1 < p < ∞ and s ≥ 0, the space H˚ ps (Ω) is defined as the closure of the set D(Ω) in the space Hps (Ω). Therefore, u ∈ H˚ ps (Ω) if and only if there exists a sequence un ∈ D(Ω) such that un → u in Hps (Ω). When Ω = Rn , H˚ ps (Rn ) = Hps (Rn ) for all 0 ≤ s < ∞. When Ω is Rn+ or a bounded domain with Lipschitz boundary, Theorem 1.40 implies that H˚ ps (Ω) = Hps (Ω)

for all 0 ≤ s ≤

1 , p

(1.80)

11 Sobolev–Lebesgue Spaces

47

but H˚ ps (Ω) = Hps (Ω)

for any

1 < s < ∞. p

In addition, Theorem 1.42 implies that 1 H˚ ps (Ω) = u ∈ Hps (Ω); u|∂Ω = 0 for < s ≤ 1. p

(1.81)

When p = 2, the space H˚ 2s (Ω) is abbreviated by H˚ s (Ω). For s ≥ 0, the space Hp−s (Ω) is defined as the dual space of H˚ ps (Ω), where 1 < p < ∞, 1 + 1 = 1. Therefore, {H˚ s (Ω), Hp−s (Ω)} is an adjoint pair with p

p

p

duality product ·, ·H˚ s

×Hp−s p

on H˚ ps (Ω) × Hp−s (Ω). Since D(Ω) ⊂ H˚ ps (Ω) with

dense embedding, we have Hp−s (Ω) = H˚ ps (Ω) ⊂ D(Ω) , namely, ϕ ∈ Hp−s (Ω) is characterized as a distribution in Ω which is continuous in D(Ω) with respect to the Hps (Ω) topology. As an immediate consequence, Lp (Ω) ⊂ Hp−s (Ω) with the relation u, f H˚ s

p

×Hp−s

= u, f Lp ×Lp ,

u ∈ H˚ ps (Ω), f ∈ Lp (Ω).

(1.82)

When p = p = 2, H2−s (Ω) is abbreviated as H −s (Ω). We easily verify that the three spaces H˚ s (Ω) ⊂ L2 (Ω) ⊂ H −s (Ω),

0 < s < ∞,

(1.83)

form a triplet. When Ω = Rn , Hp−s (Rn ) is given by − s Hp−s (Rn ) = f ∈ S(Rn ) ; F−1 1 + |ξ |2 2 Ff ∈ Lp (Rn ) . According to [Tri78, Theorem 2.6.1], for any −∞ < s < ∞, s n 1 1 −s n Hp (R ) = Hp (R ) =1 . + p p This in particular shows that, for any s ≥ 0 and 1 < p < ∞, Hps (Rn ) is a reflexive Banach space. Estimate (1.72) then yields that Hps (Ω) is also a reflexive Banach space. In fact, (1.72) shows that Hps (Ω) is isomorphic to the subspace E(Hps (Ω)) of H s (Rn ). But it is clear that the image E(Hps (Ω)) is a closed subspace of H s (Rn ). It is then known that any closed subspace of a reflexive Banach space is reflexive. Therefore, Hps (Ω) must be reflexive. Furthermore, by the similar reason, H˚ ps (Ω) is also reflexive. We remember that H s (Ω), s ≥ 0, is a Hilbert space with the inner product given by (1.73). It is the same for H˚ s (Ω). We can finally show the following fundamental fact. Since the proof is quite analogous to the case of the Sobolev spaces Wps (Ω) (cf. [Gri85, Theorem 1.4.4.6]), we will omit the proof.

48

1 Preliminaries

Theorem 1.43 Let Ω be Rn+ or a bounded domain with Lipschitz boundary. Let 1 < p < ∞. For any −∞ < s < ∞, s = p1 , the partial derivation Di (i = 1, . . . , n) is a bounded operator from Hps (Ω) to Hps−1 (Ω).

11.9 Product Spaces Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. For 1 ≤ p ≤ ∞, the product space Lp (Ω) is defined by ⎧ ⎫ ⎛ ⎞ f1 ⎪ ⎪ ⎨ ⎬ ⎜ .. ⎟ Lp (Ω) = F = ⎝ . ⎠ ; fj ∈ Lp (Ω) for j = 1, . . . , ⎪ ⎪ ⎩ ⎭ f

(1.84)

p 1 and is equipped with the product norm F Lp = ( j =1 fj Lp ) p if 1 ≤ p < ∞ and F L∞ = max{f1 L∞ , . . . , f L∞ } if p = ∞. Similarly, for 1 < p < ∞ and s ≥ 0, the product space Hsp (Ω) is defined by ⎧ ⎫ ⎛ ⎞ u1 ⎪ ⎪ ⎨ ⎬ ⎜ ⎟ Hsp (Ω) = U = ⎝ ... ⎠ ; uj ∈ Hps (Ω) for j = 1, . . . , ⎪ ⎪ ⎩ ⎭ v

(1.85)

1 p and is equipped with the product norm U Hsp = ( j =1 uj H s ) p . p All the results concerning Lp (Ω) and Hps (Ω) can be naturally generalized to Lp (Ω) and Hsp (Ω). For example, when p = 2, L2 (Ω) and Hs (Ω) = Hs2 (Ω) are Hilbert spaces. We shall also use the notation t (f1 , . . . , f ) (resp. t (u1 , . . . , u )), in order to denote the vectors F ∈ Lp (Ω) (resp. U ∈ Hsp (Ω)). Here, t (· · · ) means the transposition of matrices.

11.10 Miscellaneous Results Throughout this subsection, Ω denotes a bounded domain with Lipschitz boundary. (1) For 1 ≤ p, q, r ≤ ∞ such that

1 p

+

1 q

uvLr ≤ uLp vLq ,

= 1r , it holds that u ∈ Lp (Ω), v ∈ Lq (Ω).

(1.86)

Indeed, this estimate is immediately observed by using the usual Hölder inequality for p1 + q1 = 1, where p = pr and q = qr .

11 Sobolev–Lebesgue Spaces

49

(2) Let k be a positive integer. If a ∈ Ck (Ω), then the multiplication u → au by a is a bounded operator on Hps (Ω) for any 1 < p < ∞ and any 0 ≤ s ≤ k with the estimate s

1− s

auHps ≤ Ck aCk k aC k uHps ,

u ∈ Hps (Ω).

(1.87)

It is clear that auLp ≤ aC uLp and that auHpk ≤ Ck aCk uHpk . Then, the estimate is immediately verified by applying the interpolation theorem and property, namely, Theorems 1.15 and 1.35. (3) Let n < p < ∞. Then, u, v ∈ Hp1 (Ω) implies uv ∈ Hp1 (Ω) with the estimate uvHp1 ≤ Cp uHp1 vHp1 ,

u, v ∈ Hp1 (Ω).

(1.88)

Indeed, due to (1.76), Hp1 (Ω) ⊂ L∞ (Ω). Then, the estimate is verified directly from ∇(uv) = v∇u + u∇v. (4) Let s ≥ 1 and s > n2 . Then, similarly, u, v ∈ H s (Ω) implies uv ∈ H s (Ω) with the estimate uvH s ≤ Cs uH s vH s ,

u, v ∈ H s (Ω).

(1.89)

In order to show this result, let us prove first that for any a ∈ H s (Ω), the multiplication u → au by a is a bounded operator from H s−1 (Ω) into itself and the estimate auH s−1 ≤ Cs aH s uH s−1 ,

u ∈ H s−1 (Ω)

(1.90)

holds. Indeed, let s = k + σ with an integer k ≥ 1 and 0 ≤ σ < 1. Then, u → au is a bounded operator from H k−1 (Ω) into itself, for we have D α (av) =

Cα ,α (D α a)(D α u),

|α |+|α |=k−1

where α denotes any multiindex such that |α| = k − 1. Here, due to (1.74), (1.75), and (1.76), we observe that (D α a)(D α u) ∈ L2 (Ω); hence, au ∈ H k−1 (Ω) with auH k−1 ≤ Ck−1 aH s uH k−1 . In a similar way, we can observe that u → au is a bounded operator from H k (Ω) into itself with auH k ≤ Ck aH s uH k . Then, the interpolation theorem and property (Theorems 1.15 and 1.35) provide that u → au is a bounded operator from H k−1+σ (Ω) into itself with auH k−1+σ ≤ Ck−1+σ aH s uH k−1−σ , namely, (1.90) is proved. Now, the proof of (1.89) is immediate. Let u, v ∈ H s (Ω); obviously, ∇(uv) = v∇u + u∇v and ∇u, ∇v ∈ H s−1 (Ω); (1.90) then ensures ∇(uv) ∈ H s−1 (Ω); consequently, for any multiindex α such that |α| = k, it holds that D α (uv) ∈ H σ (Ω). Therefore, on account of (1.70), we conclude that uv ∈ H k+σ (Ω). At the same time, estimate (1.90) is also verified.

50

1 Preliminaries

(5) Let F : R → R be a C1 function. Let n < p < ∞. Then, the correspondence u → F (u) is an operator from Hp1 (Ω; R) into itself. Moreover, there is an increasing function p(·) determined from F such that F (u)Hp1 ≤ p(uHp1 ),

u ∈ Hp1 (Ω; R).

(1.91)

We have ∇F (u) = F (u)∇u. So, (1.91) is verified directly from (1.76). Let, more strongly, F : R → R be a C2 function, and n < p < ∞. Then, the operator u → F (u) from Hp1 (Ω; R) into itself satisfies the Lipschitz condition F (u) − F (v)Hp1 ≤ sup p1 (θ u + (1 − θ )vHp1 )u − vHp1 , 0≤θ≤1

u, v ∈ Hp1 (Ω; R),

(1.92)

where p1 (·) is an increasing function determined from F . Indeed, since 1 F (u) − F (v) = 0 F (θ u + (1 − θ )v) dθ (u − v), this estimate is verified by (1.88) and (1.91). (6) Let s ≥ 1 and s > n2 . Let F : R → R be a Cm function, where m is an integer such that m ≥ s. Then, the correspondence u → F (u) is an operator from H s (Ω; R) into itself. Moreover, there is an increasing function p(·) determined from F such that F (u)H s ≤ p(uH s ),

u ∈ H s (Ω; R).

(1.93)

Let s = k + σ , where 0 ≤ σ < 1, and k ≥ 1 is an integer. For any u ∈ H s (Ω; R), consider the linear operator Tu w = F (u)∇w. Due to (1.74), (1.75), and (1.76), we can observe that Tu is a bounded operator from H k (Ω) into H k−1 (Ω) with Tu wH k−1 ≤ p(uH s )wH k and is a bounded operator from H k+1 (Ω) into H k (Ω) with Tu wH k ≤ p(uH s )uH k+1 . The interpolation theorem and property (Theorems 1.15 and 1.35) then provide that Tu is a bounded operator from H k+σ (Ω) into H k−1+σ (Ω) with Tu wH k−1+σ ≤ p(uH s )wH k−1+σ . We now apply Tu to u itself to verify that ∇F (u) = F (u)∇u = Tu u ∈ H k−1+σ (Ω). Then, for any multiindex α such that |α| = k, we have D α F (u) ∈ H σ (Ω). Hence, on account of (1.70), we conclude that F (u) ∈ H k+σ (Ω). Estimate (1.93) is also verified. Let s ≥ 1 and s > n2 . Let, more strongly, F : R → R be a Cm+1 function with m ≥ s. Then, the operator u → F (u) from H s (Ω; R) into itself satisfies the Lipschitz condition F (u) − F (v)H s ≤ sup p1 (θ u + (1 − θ )vH s )u − vH s , 0≤θ≤1

u, v ∈ H s (Ω; R),

(1.94)

where p1 (·) is an increasing function determined from F . This estimate is verified in an analogous way for (1.92) by using (1.89) and (1.93).

11 Sobolev–Lebesgue Spaces

51

(7) Let F : R → R be a piecewise smooth function with F ∈ L∞ (R) and with F (0) = 0 (hence, |F (u)| ≤ F L∞ |u| for u ∈ R). Then, according to [GT83, Theorem 7.8], the correspondence u → F (u) is an operator from Hp1 (Ω; R) into itself for 1 ≤ p ≤ ∞ with the estimate F (u)Hp1 ≤ CF L∞ uHp1 ,

u ∈ Hp1 (Ω; R).

(1.95)

i = 1, . . . , n,

(1.96)

In addition, F (u) has the property / Fs , F (u)Di u if u(x) ∈ Di F (u) = 0 if u(x) ∈ Fs ,

where Fs denotes the set of singular points of F (u). (8) The correspondence f → ef is an operator from L∞ (Ω) into itself. The operator is Fréchet differentiable in L∞ (Ω). Its Fréchet differential at f ∈ L∞ (Ω) is the multiplication operator by the function ef , i.e., [Def ]h = ef h,

f, h ∈ L∞ (Ω).

(1.97)

This fact is easily verified by a direct calculation. (9) For 1 ≤ q < ∞, let us introduce the quantity Nq,log (u) = |u|q log(|u| + 1) dx. Ω

All the results announced in Theorem 1.37 can be modified in the following form. For any ζ > 0, estimates in (1), (2), and (3) can be rewritten as q uLr ≤ Cp,q,r ζ 1−a uaH 1 Nq,log (u)1−a + Cζ uLr q , p

u ∈ Hp1 (Ω; R) ∩ Lq (Ω),

(1.98)

with some constant Cζ > 0 dependent on ζ . The relations among p, q, and r np ; if p = n, then are the same as before, that is, if 1 ≤ p < n, then q ≤ r ≤ n−p q ≤ r < ∞; and if n < p ≤ ∞, then q ≤ r ≤ ∞. The exponent a is also given in each case as before, that is, by (1.77), (1.78), and (1.79), respectively. To prove the estimate, we use a cutoff function χR (u) such that χR (u) = −R for u ≤ −R, χR (u) = u for |u| ≤ R, and χR (u) = R for R ≤ u. Then, by Theorem 1.37 and (1.95), we observe in each case that u − χR (u)Lr ≤ Cp,q,r u − χR (u)aH 1 u − χR (u)1−a Lq p

1−a

≤

Cp,q,r uaH 1 p

|u| dx

≤

Cp,q,r uaH 1 Nq,log (u)1−a . p [log(R + 1)]1−a

q

|u|≥R

52

1 Preliminaries

Meanwhile, q

q

χR (u)Lr ≤ R 1− r uLr q . These two estimates readily yield the desired estimate (1.98). (10) Let H (u) be a C1,1 function defined by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. For any function u ∈ C1 ((a, b); L2 (Ω; R)), if we put ψ(t) = H (u(t)) dx, a < t < b, (1.99) Ω

then ψ

∈ C1 (a, b)

with the derivative H (u(t))u (t) dx, ψ (t) =

(1.100)

a < t < b.

Ω

In fact, we notice that, since H : R → R is Lipschitz continuous, the mapping u → H (u) is also Lipschitz continuous from L2 (Ω; R) into itself. From 1 H (θ u(t + h, x) + (1 − θ )u(t, x)) dθ H (u(t + h, x)) − H (u(t, x)) = 0

× [u(t + h, x) − u(t, x)],

a.e. x ∈ Ω,

we have ψ(t + h) − ψ(t) =

1

H (θ u(t + h) + (1 − θ )u(t)) dθ, u(t + h) − u(t) .

0

L2

Dividing this equality by h and letting h → 0, we obtain (1.100). (11) As a variation of (1.100), for any u ∈ C((a, b); H 1 (Ω; R)) ∩ C1 ((a, b); H 1 (Ω)∗ ), the function ψ(t) defined similarly by (1.99) is in C1 (a, b) with the derivative ψ (t) = H (u(t)), u (t)H 1 ×H 1∗ ,

(1.101)

a < t < b.

In fact, due to (1.95), H (u) ∈ H 1 (Ω; R) and H (u)H 1 ≤ uH 1 for any u ∈ H 1 (Ω; R). Furthermore, as v → 0 in H 1 (Ω; R), H (u + v)H 1 ≤ u + vH 1 and H (u + v) → H (u) in L2 (Ω; R); therefore, H (u + v) → H (u) in the weak topology of H 1 (Ω). Hence, u → H (u) is continuous in H 1 (Ω; R) from strong topology to weak topology. In the meantime, we have ψ(t + h) − ψ(t) 1 = H (θ u(t + h) + (1 − θ )u(t)) dθ, u(t + h) − u(t) (

0

= 0

) 1 H (θ u(t + h) + (1 − θ )u(t)) dθ, u(t + h) − u(t)

L2

. H 1 ×H 1∗

Notes and Further Researches

53

Hence, (1.101) is obtained. The continuity of ψ (t) is also verified by the continuity of the mapping u → H (u) from strong topology to weak topology. (12) Consider the Cauchy problem du dt + p(t)u = q(t), 0 < t ≤ T , (1.102) u(0) = u0 , in the space L∞ (Ω). Here, p and q are given functions such that p ∈ C((0, T ]; L∞ (Ω)) ∩ B((0, T ]; L∞ (Ω)), t 1−ν q ∈ B((0, T ]; L∞ (Ω)), q ∈ C((0, T ]; L∞ (Ω)), with some exponent 0 < ν ≤ 1. Then, for any initial value u0 ∈ L∞ (Ω), (1.102) has a unique solution u in the function space C([0, T ]; L∞ (Ω)) ∩ C1 ((0, T ]; L∞ (Ω)), which is given by the formula t t t e− s p(τ )dτ q(s) ds, u(t) = e− 0 p(s)ds u0 +

0 ≤ t ≤ T.

(1.103)

(1.104)

0

Indeed, by (1.97), the operator g → eg from L∞ (Ω) into itself is Fréchet differentiable with the derivative [Deg ]h = eg h. Therefore, we verify that the function u given by (1.104) certainly belongs to (1.103) and satisfies the equation in (1.102). Conversely, let u be any solution of (1.102). Then, for 0 < s ≤ t , we have t t d − t p(τ )dτ u(s) = e− s p(τ )dτ p(s)u(s) + e− s p(τ )dτ u (s) e s ds

= e−

t s

p(τ )dτ

q(s).

Integrating this equality in [0, t], we obtain (1.104).

Notes and Further Researches For the basic matters of the theory of functional analysis, we quote some of the classics, e.g., Hille–Phillips [HP57], Yosida [Yos80], and Brezis [Bre83]. For the basic matters of fractal dimension for the compact sets, see Eden–Foias– Nicolaenko–Temam [EFNT94]. Especially, for Mañé–Hölder-type projection, see [EFNT94, Chap. 10] and also Ben Artzi–Eden–Foias–Nicolaenko [BEFN93], Foias–Olson [FO96], and Hunt–Kaloshin [HY99]. Theorem 1.2 was obtained in [HY99]. For Dunford integrals, see Dunford–Schwartz [DS58, DS63]. Basics of multivalued linear operators are given in Favini–Yagi [FY99, Chap. I]. Basics of

54

1 Preliminaries

nonlinear operators are given in Deimling [Dei85] and Zeidler [Zei86]. Basics of interpolation spaces are given in Bergh–Löfström [BL76] and Tribel [Tri78, Chap. 1]. The triplets of spaces were presented and systematically used in Lions–Magenes [LM72], cf. also Dautray–Lions [DL88]. In these literatures, the triplet was defined by identifying X and its dual X , i.e., Z ⊂ X = X ⊂ Z . Such an identification is possible, since Z ⊂ X implies X ⊂ Z with dense and continuous embedding and since X and X are canonically isomorphic by the Riesz Theorem. By the uniqueness of triplet Z ⊂ X ⊂ Z ∗ , such a definition is of course essentially equivalent to ours presented in Sect. 7. For Sobolev–Lebesgue spaces, we quote Adams [Ada75], Grisvard [Gri85, Chap. 1], and Tribel [Tri78, Chap. 4]. The fundamental extension theorem, Theorem 1.33, was presented by Stein [Ste70].

Chapter 2

Sectorial Operators

Let X be a Banach space with norm · . Let A be a densely defined, closed linear operator in X. We assume that the spectrum of A is contained in an open sectorial domain such that σ (A) ⊂ Σω = {λ ∈ C; |arg λ| < ω},

0 < ω ≤ π,

(2.1)

and its resolvent satisfies the estimate (λ − A)−1 ≤ M , |λ|

λ∈ / Σω ,

(2.2)

with some constant M ≥ 1. We call such an operator A a sectorial operator of X. The condition (2.1) implicitly means that the origin is not in σ (A), namely, A has a bounded inverse A−1 on X. By (1.11) it then follows that λ ∈ ρ(A) if |λ| < A−1 −1 with the estimate (λ − A)−1 ≤

A−1 , 1 − A−1 |λ|

|λ| < A−1 −1 .

(2.3)

By the similar reason, for λ0 = r0 e±iω , r0 > 0, we see that r0 ⊂ ρ(A) λ ∈ C; |λ − λ0 | < M with the estimate (λ − A)−1 ≤

M , r0 − M|λ − λ0 |

|λ − λ0 |

δ > 0. Let A be the linear operator associated with the form, and let A|X and A|Z be its parts in the spaces X and Z, respectively. As already shown in Chap. 1, Sect. 8, A, A|X , and A|Z are densely defined, closed linear operators acting in Z ∗ , X, and Z, respectively. For each Re λ ≤ 0, consider the sesquilinear form a(U, V ) − λ(U, V ),

U, V ∈ Z.

Obviously this form is also continuous and coercive on Z. Therefore, Theorem 1.24 shows that the associated operator, which is clearly nothing more than A − λ, is an isomorphism from Z onto Z ∗ . In particular, λ ∈ ρ(A) if A is considered as an operator of Z ∗ . Let us next establish various estimates concerning the resolvent (λ − A)−1 for Re λ ≤ 0. Let U ∈ Z. Then, δU 2 ≤ Re a(U, U ) − Re λ|U |2 = Re (A − λ)U, U ≤ (A − λ)U ∗ U . Putting Φ = (λ − A)U , we obtain that (λ − A)−1 Φ ≤ δ −1 Φ∗ .

(2.8)

Since λ(λ − A)−1 Φ = A(λ − A)−1 Φ + Φ, the estimate then yields that |λ|(λ − A)−1 Φ ∗ ≤ AL(Z,Z ∗ ) (λ − A)−1 Φ + Φ∗ ≤ (Mδ −1 + 1)Φ∗ . Consequently, |λ|(λ − A)−1 Φ ∗ ≤ (Mδ −1 + 1)Φ∗ ,

Φ ∈ Z∗.

Let now U ∈ D(A|X ). For Re λ ≤ 0, δU 2 ≤ Re a(U, U ) − Re λ|U |2 = Re((A − λ)U, U ) ≤ |(A − λ)U ||U |. From λ|U |2 = a(U, U ) + ((λ − A)U, U ) it follows that |λ||U |2 ≤ MU 2 + |(λ − A)U | |U | ≤ (Mδ −1 + 1)|(λ − A)U | |U |. Putting F = (λ − A)U , we obtain that |λ|(λ − A)−1 F ≤ (Mδ −1 + 1)|F |,

F ∈ X.

58

2 Sectorial Operators

Let again U ∈ Z. From λ(λ − A)−1 U − (λ − A)−1 AU = U it follows that |λ|(λ − A)−1 U ≤ (λ − A)−1 L(Z ∗ ,Z) AL(Z,Z ∗ ) U + U . Therefore, by (2.8), |λ|(λ − A)−1 U ≤ (Mδ −1 + 1)U ,

U ∈ Z.

We have therefore shown for Re λ ≤ 0 the estimates |λ|(λ − A)−1 Φ ∗ ≤ (Mδ −1 + 1)Φ∗ , Φ ∈ Z ∗ , |λ|(λ − A)−1 F ≤ (Mδ −1 + 1)|F |, F ∈ X, |λ|(λ − A)−1 U ≤ (Mδ −1 + 1)U , U ∈ Z.

(2.9) (2.10) (2.11)

These estimates mean that all the operators A, A|X , and A|Z satisfy conditions (2.1) and (2.2) with angle ω = π2 and constant Mδ −1 + 1 in the spaces Z ∗ , X, and Z, respectively. In this way, we have proved that they are sectorial operators. Theorem 2.1 Let a(U, V ) be a sesquilinear form on Z satisfying (2.6) and (2.7), and let A, A|X , and A|Z be the linear operators of Z ∗ , X, and Z, respectively, which are determined from the form. Then, they satisfy (2.1) and (2.2) with angle ∗ ω = π2 and constant M+δ δ . Hence, they are sectorial operators of Z , X, and Z, π respectively, with angles < 2 . Remark 2.1 For λ ≤ 0 and U ∈ D(A|X ), we have δU 2 + |λ||U |2 ≤ Re a(U, U ) − λ|U |2 = Re((A − λ)U, U ) ≤ |(λ − A)U | |U |, (c−2 δ + |λ|)|U |2 ≤ |(λ − A)U ||U |, where c is the constant of embedding: |U | ≤ cU , U ∈ Z. Putting F = (λ − A)U , we obtain that (|λ| + c−2 δ)(λ − A)−1 F ≤ |F |. Hence, A|X satisfies the estimate (λ − A|X )−1 ≤ 1/(|λ| + c−2 δ),

λ ≤ 0.

(2.12)

1.2 Sectorial Operators in L2 Spaces Let us utilize Theorem 2.1 for constructing various sectorial operators in L2 spaces.

1 Sectorial Operators in Hilbert Spaces

59

Consider the sesquilinear form a(u, v) =

n n i,j =1 R

aij (x)Di uDj v dx +

Rn

c(x)uv dx,

u, v ∈ H 1 (Rn ), (2.13)

defined on H 1 (Rn ). Here, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Rn satisfying the conditions aij ∈ L∞ (Rn ), n

1 ≤ i, j ≤ n,

aij (x)ξi ξj ≥ δ|ξ |2 ,

(2.14)

ξ = (ξ1 , . . . , ξn ) ∈ Rn , a.e. x ∈ Rn ,

(2.15)

i,j =1

with some constant δ > 0, and c(x) is a real-valued function satisfying c ∈ L∞ (Rn )

and c(x) ≥ c0 > 0,

Clearly, a(u, v) fulfills (2.6) with M = max{ tion, (2.15) implies that Re a(u, u) =

n n i,j =1 R

i,j

a.e. x ∈ Rn .

(2.16)

ai,j L∞ , cL∞ }. In addi-

aij (x)[(Re Di u)(Re Dj u) + (Im Di u)(Im Dj u)] dx

+

Rn

c(x)|u|2 dx ≥ δ∇u2L2 + c0 u2L2 .

Therefore, a(u, v) fulfills (2.7), too. We consider a triplet H 1 (Rn ) ⊂ L2 (Rn ) ⊂ H 1 (Rn )∗ = H −1 (Rn ) and an operator A associated with (2.13). Since

n n aij (x)Di uDj v dx = − Dj [aij (x)Di u], v n i,j =1 R

i,j =1

H −1 ×H 1

in the distribution sense, it naturally holds that Au = −

n

Dj [aij (x)Di u] + c(x)u

in H −1 (Rn ).

i,j =1

Theorem 2.1 then yields the following results. Theorem 2.2 Let (2.14), (2.15), and (2.16) be satisfied. Then, the operator A associated with the form (2.13) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H −1 (Rn ), L2 (Rn ), and H 1 (Rn ), respectively, with angles < π2 .

60

2 Sectorial Operators

Let now Ω be any domain in Rn , and let Z be any closed subspace of H 1 (Ω) such that H˚ 1 (Ω) ⊂ Z ⊂ H 1 (Ω). We consider the sesquilinear form n

a(u, v) =

aij (x)Di uDj v dx +

i,j =1 Ω

c(x)uv dx,

u, v ∈ Z,

(2.17)

Ω

on Z. As before, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Ω satisfying the conditions aij ∈ L∞ (Ω), n

1 ≤ i, j ≤ n,

aij (x)ξi ξj ≥ δ|ξ |2 ,

(2.18)

ξ = (ξ1 , . . . , ξn ) ∈ Rn , a.e. x ∈ Ω,

(2.19)

i,j =1

with some constant δ > 0, and c(x) is a real-valued function in Ω satisfying c ∈ L∞ (Ω)

and c(x) ≥ c0 > 0,

a.e. x ∈ Ω.

(2.20)

By the same reason as above, these conditions readily imply that a(u, v) fulfills (2.6) and (2.7) on Z. So, if we consider a triplet Z ⊂ L2 (Ω) ⊂ Z ∗ (note that, due to D(Ω) ⊂ H˚ 1 (Ω) ⊂ Z, Z is always dense in L2 (Ω)), then the operator A associated with the form (2.17) and its parts are all sectorial operators of Z ∗ , L2 (Ω), and Z, respectively, with angles < π2 . Let us investigate below more specific cases. Case where Z = H˚ 1 (Ω). In this case, Z ∗ coincides with H −1 (Ω). Since the space D(Ω) is dense in H˚ 1 (Ω), H −1 (Ω) is contained in the space of distributions D(Ω) . Therefore, it is possible to represent Au as Au = −

n

Dj [aij (x)Di u] + c(x)u

in distribution sense.

(2.21)

i,j =1

This shows us that A and its parts are regarded as realizations of the differential operator (2.21) in H −1 (Ω), L2 (Ω), and H˚ 1 (Ω), respectively. We also know that, if Ω is a bounded domain with Lipschitz boundary, then D(A) = H˚ 1 (Ω) is given by (1.81), that is, u ∈ D(A) satisfies the Dirichlet boundary conditions γ u = 0 on ∂Ω.

(2.22)

Theorem 2.3 Let Ω be any domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied. Then, the operator A associated with the form (2.17) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H −1 (Ω), L2 (Ω), and H˚ 1 (Ω), respectively, with angles < π2 . In addition, if Ω is a bounded domain with Lipschitz boundary, then the functions in D(A) satisfy (2.22).

1 Sectorial Operators in Hilbert Spaces

61

Case where Z = H 1 (Ω). In this case, Z ∗ coincides no longer with any subspace of distributions. So, the associated operator A is not represented as a usual differential operator. But, let u ∈ D(A|L2 ), i.e., Au ∈ L2 (Ω), and assume that all are regular enough to justify the following arguments. Then, by Green’s formula, (Au, v)L2 = a(u, v) = −

+

n

Dj [aij (x)Di u]v dx

i,j =1 Ω

n

[νj (x)aij (x)Di u]v dS +

i,j =1 ∂Ω

c(x)uv dx,

v ∈ H 1 (Ω),

Ω

where ν(x) = (ν1 (x), . . . , νn (x)) denotes the outer normal vector at x ∈ ∂Ω. Since a(u, v) must be continuous in v with respect to the L2 (Ω) topology, the integral on ∂Ω must vanish. In other words, u must satisfy the boundary conditions n ∂u ≡ νj (x)aij (x)Di u = 0 ∂νA

on ∂Ω.

(2.23)

i,j =1

Such conditions are called the Neumann-type boundary conditions on ∂Ω. We then arrive at n Dj [aij (x)Di u] + c(x)u, v , v ∈ H 1 (Ω); (Au, v)L2 = − i,j =1

L2

therefore, Au = −

n

Dj [aij (x)Di u] + c(x)u

in L2 (Ω).

(2.24)

i,j =1

In this way, u satisfies the boundary conditions (2.23), and Au is given by (2.24). We also notice that, if Au = ϕ with u ∈ H 1 (Ω) and ϕ ∈ H 1 (Ω)∗ , then there exist sequences uk ∈ D(A|L2 ) and fk ∈ L2 (Ω) such that A|L2 uk = fk and that uk → u in H 1 (Ω) and fk → ϕ in H 1 (Ω)∗ . Indeed, it is sufficient to choose any sequence fk → ϕ in H 1 (Ω)∗ with fk ∈ L2 (Ω) and to put uk = (A|L2 )−1 fk . In this sense, A and its parts are regarded as realizations of the differential operator (2.21) under the Neumann-type boundary conditions (2.23) in H 1 (Ω)∗ , L2 (Ω), and H 1 (Ω), respectively. Theorem 2.4 Let Ω be any domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied. Then, the operator A associated with the form (2.17) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H 1 (Ω)∗ , L2 (Ω), and H 1 (Ω), respectively, with angles < π2 . Case where Z = H˚ D1 (Ω). Let Ω be a bounded domain with Lipschitz boundary. We now consider a splitting of the boundary ∂Ω into ΓD and ΓN , namely, ∂Ω =

62

2 Sectorial Operators

ΓD ∪ ΓN , ΓD ∩ ΓN = ∅, and ΓD is a nonempty open set of ∂Ω. We set Z = H˚ D1 (Ω) = {u ∈ H 1 (Ω); γ u = 0 on ΓD }.

(2.25)

The operator A associated with the form (2.17) and its parts are regarded as realizations of the differential operator (2.21) under the splitting boundary conditions γu=0

on ΓD

and

∂u = 0 on ΓN . ∂νA

(2.26)

Theorem 2.5 Let Ω be a bounded domain with Lipschitz boundary. Let (2.18), (2.19), and (2.20) be satisfied. Then, the operator A associated with the form (2.17) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H˚ D1 (Ω)∗ , L2 (Ω), and H˚ D1 (Ω), respectively, with angles < π2 . Remark 2.2 When aij (x) ≡ δij (Kronecker’s delta), the differential operator n ∂u 2 Δ n= i=1 Di is called the Laplace operator and the boundary conditions ∂ν ≡ i=1 νi (x)Di u = 0 are called the Neumann boundary conditions. Following the procedure described above, we can define a realization of the Laplace operator in Ω under the Dirichlet, Neumann, or splitting boundary conditions on ∂Ω. Remark 2.3 Let Ω be a bounded domain with Lipschitz boundary. When Z = H˚ 1 (Ω) or H˚ D1 (Ω), it is possible to show the coercivity (2.7) without using the positivity of the function c(x). In fact, according to the Poincaré inequality (e.g., see [BF02, 1.1.3.b]), uL2 ≤ C∇uL2 for all u ∈ H˚ D1 (Ω), a fortiori for all u ∈ H˚ 1 (Ω). Therefore, (2.19), together with the condition c(x) ≥ 0, can imply (2.7).

1.3 Shift Property in L2 When condition (2.19) is satisfied, the differential operator A(D)u = −

n

Dj [aij (x)Di u] + c(x)u

(2.27)

i,j =1

is said to be strongly elliptic. In general, strongly elliptic operators are expected to enjoy the regularity property that A(D)u ∈ L2 (Ω) implies u ∈ H 2 (Ω). Such a property is called the shift property. However, our operators introduced in the preceding subsection cannot enjoy this property, because aij (x) are merely measurable functions. In order to have the shift property, we assume that Ω is a bounded domain in Rn with C2 boundary and that aij (x) satisfy aij ∈ C1 (Ω),

1 ≤ i, j ≤ n.

(2.28)

1 Sectorial Operators in Hilbert Spaces

63

The shift property is verified by the widely developed theory of elliptic operators. Let A(D) be the elliptic operator in Ω given by (2.27). Let the conditions (2.19), (2.20), and (2.28) be satisfied. We will first consider the case where A(D) is equipped with the Dirichlet conditions. According to the theory of elliptic operators (see [Gri85, Theorem 2.2.2.3 and Corollary 2.2.2.4]), the mapping u → (A(D) + k)u is an isomorphism from the space {u ∈ H 2 (Ω); γ u = 0 on ∂Ω} onto L2 (Ω) if k is sufficiently large. In addition, it holds that uH 2 ≤ C[A(D) + k]uL2 ,

u ∈ H 2 (Ω), γ u = 0.

Let then A be the operator associated with the form (2.17) on H˚ 1 (Ω), and let A|L2 be its part in L2 (Ω). If A|L2 u = f ∈ L2 (Ω), then there exists a unique function u˜ ∈ ˜ v) = (A(D)u, ˜ v) = H 2 (Ω) such that [A(D) + k]u˜ = f + ku and γ u˜ = 0; since a(u, (k(u − u) ˜ + f, v), v ∈ H˚ 1 (Ω), it follows that A|L2 u˜ = k(u − u) ˜ + f ; therefore, (A|L2 +k)(u− ˜ u) = 0, i.e., u˜ = u. In this way, u ∈ D(A|L2 ) implies that u ∈ H 2 (Ω). We have therefore verified that

D(A|L2 ) = {u ∈ H 2 (Ω); γ u = 0 on ∂Ω}, (2.29) A|L2 u = A(D)u, as well as uH 2 ≤ C(A|L2 uL2 + uL2 ),

u ∈ D(A|L2 ).

(2.30)

In what follows, the domain will often be denoted by HD2 (Ω) = {u ∈ H 2 (Ω); γ u = 0 on ∂Ω}.

(2.31)

We next consider the case where A(D) is equipped with the Neumann-type boundary conditions. Similarly, according to the theory of elliptic operators (see [Gri85, Theorem 2.2.2.5 and Corollary 2.2.2.6]), the mapping u → (A(D) + k)u is ∂u an isomorphism from the space {u ∈ H 2 (Ω); ∂ν = 0 on ∂Ω} onto L2 (Ω) if k is A sufficiently large. In addition, it holds that uH 2 ≤ C[A(D) + k]uL2 ,

u ∈ H 2 (Ω),

∂u = 0. ∂νA

Let A be the operator associated with the form (2.17) on H 1 (Ω). Let A|L2 be its part in L2 (Ω). If A|L2 u = f ∈ L2 (Ω), then there exists a unique function u˜ ∈ H 2 (Ω) ∂ u˜ = 0 on ∂Ω. As before, it is easy to see such that [A(D) + k]u˜ = f + ku and ∂ν A that u˜ = u. Therefore, u ∈ D(A|L2 ) implies that u ∈ H 2 (Ω). We thus conclude that

∂u D(A|L2 ) = {u ∈ H 2 (Ω); ∂ν = 0 on ∂Ω}, A (2.32) A|L2 u = A(D)u as well as uH 2 ≤ C(A|L2 uL2 + uL2 ),

u ∈ D(A|L2 ).

(2.33)

64

2 Sectorial Operators

In what follows, the domain will often be denoted by ∂u HN2 (Ω) = u ∈ H 2 (Ω); = 0 on ∂Ω . ∂νA

(2.34)

The following theorem has therefore been verified. Theorem 2.6 Let Ω be a bounded domain with C2 boundary, and let (2.19), (2.20), and (2.28) be satisfied. Then, the parts A|L2 of the operators associated with the form (2.17) on H˚ 1 (Ω) and H 1 (Ω) are characterized by (2.29) and (2.32), respectively. In the case where Ω is convex and where aij (x) are symmetric in the sense that aij (x) = aj i (x),

1 ≤ i, j ≤ n,

(2.35)

we can weaken the C2 regularity of ∂Ω to Lipschitz. Indeed, according to Grisvard [Gri85, Theorems 3.1.3.1 and 3.1.3.3], the constants C appearing in the energy estimates (2.30) and (2.33) depend (roughly speaking) only on the negative part of the curvature of ∂Ω and are independent of the positive part. On the basis of this fact, the same shift properties are established for bounded convex domains. Note that any bounded convex domain has a Lipschitz boundary. Theorem 2.7 Let Ω be a bounded convex domain, and let (2.19), (2.20), (2.28), and (2.35) be satisfied. Then, the parts A|L2 of the operators associated with the form (2.17) on H˚ 1 (Ω) and H 1 (Ω) are again characterized by (2.29) and (2.32), respectively. Estimates (2.30) and (2.33) also hold true. For the proofs, see [Gri85, Theorems 3.2.1.2 and 3.2.1.3]. Let us now consider a very particular case where aij (x) = a(x)δij but a(x) is less regular than (2.28). Our elliptic operator is then written as A(D)u = −

n

Di [a(x)Di u] + c(x)u.

(2.36)

i=1

We assume (2.19) for a(x)δij , which reads as a(x) ≥ δ > 0 in Ω. We assume also that a(x) belongs to either a ∈ Hp1 (Ω; R)

with n < p < ∞

(2.37)

n with s ≥ 1 and s > . 2

(2.38)

or a ∈ H s (Ω; R)

Note that, due to (1.76), Hp1 (Ω; R) ⊂ C(Ω) and H s (Ω; R) ⊂ C(Ω).

1 Sectorial Operators in Hilbert Spaces

65

Theorem 2.8 Let Ω be a C2 or convex, bounded domain in Rn . Let (2.19) and (2.20) be satisfied, and let a(x) belong to either (2.37) or (2.38). Let A(D) be the elliptic operator given by (2.36). Then, realizations A|L2 of A(D) in L2 (Ω) under the boundary conditions (2.22) and (2.23) (cf. Remark 2.2) have their domains given by (2.31) and (2.34), respectively. Estimates (2.30) and (2.33) also hold true. Proof Let us describe here the proof only for the Dirichlet boundary conditions. For the other case, it is quite similar. Let u ∈ D(A|L2 ). Then, Au = f ∈ L2 (Ω), and −

n

Di [a(x)Di u] + c(x)u = f

in distribution sense.

(2.39)

i=1

We rewrite −a(x)Δu −

n [Di a(x)]Di u + c(x)u = f i=1

and (−Δ + 1)u −

n

a(x)−1 [Di a(x)]Di u + a(x)−1 c(x) − 1 u = a(x)−1 f.

i=1

In view of this formula, let us introduce a realization Λ of elliptic operator −Δ + 1 under (2.22) in L2 (Ω). We know by Theorems 2.6 and 2.7 that Λ is a positive definite self-adjoint operator of L2 (Ω) with domain (2.31). In addition, by Theorems 16.12 and 16.13 in Chap. 16, we know also characterization of the domains of fractional powers Λθ for 0 < θ < 1, in particular D(Λθ ) ⊂ H 2θ (Ω). Keeping in mind this fact, we consider a realization B of the differential operator B(D)v = −

n

a(x)−1 [Di a(x)]Di v + a(x)−1 c(x) − 1 v

i=1 1+ n

in L2 (Ω) whose domain is given by D(B) = H p (Ω) (resp. D(B) = n H 2−s+ 2 (Ω)) if (2.37) (resp. (2.38)) is assumed. Since 1 + pn < 2 (resp. 2 − s + n 2 < 2), the operator B is dominated by Λ in the sense of (3.8). As a consequence, if β > 0 is a sufficiently large number, then −β ∈ ρ(Λ + B), see Theorem 3.3. We recall that Au = f implies that (Λ + B)u = a(x)−1 f . Then, let u˜ be the solution of (Λ + B + β)u˜ = βu + a(x)−1 f . As shown above, u˜ ∈ HD2 (Ω) with the estimate u ˜ H 2 ≤ Cβu + a(x)−1 f L2 . We can follow the calculations above inversely to obtain the equality −

n i=1

Di [a(x)Di u˜ ] + c(x)u˜ + βa(x)u˜ = βa(x)u + f

in L2 (Ω).

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2 Sectorial Operators

Subtract (2.39) from this. Then, −

n

Di [a(x)Di (u˜ − u)] + c(x)(u˜ − u) + βa(x)(u˜ − u) = 0.

i=1

Hence, u˜ = u. In addition, uH 2 ≤ Cβu + a(x)−1 f L2 ≤ Cδ −1 f L2 .

We often need a higher shift property of strongly elliptic operators. For this, we assume that aij ∈ Cm+1 (Ω),

1 ≤ i, j ≤ n; m = 1, 2, 3, . . . .

(2.40)

Theorem 2.9 Let Ω be a bounded domain with Cm+2 boundary, and let (2.19), (2.20), and (2.40) be satisfied. Then, the parts A|L2 of the operators associated with the form (2.17) on H˚ 1 (Ω) and H 1 (Ω), respectively, have the property that A|L2 u ∈ H m (Ω) implies u ∈ H m+2 (Ω). The estimate uH m+2 ≤ C(A|L2 uH m + uH m ) also holds true. For the proof, see [Gri85, Theorem 2.5.1.1]. Let us consider finally the splitting boundary conditions. Let Ω be a bounded domain of Rn with Lipschitz boundary. Consider a splitting of ∂Ω, ∂Ω = ΓD ∪ ΓN and ΓD ∩ ΓN = ∅, ΓD being a nonempty open subset. We assume the two conditions |B(x0 ; R) ∩ ΓD | ≥ γ R n−1 |B(x0 ; R) ∩ Ω| ≥ γ R

n

for any x0 ∈ ΓD ,

for any x0 ∈ ΓN , B(x0 ; R) ∩ ΓD = ∅

(2.41) (2.42)

with some constant γ > 0 (cf. [BF02, (1.22)]), where B(x0 ; R) denotes the open ball with center x0 ∈ ∂Ω and radius R > 0. Theorem 2.10 Let Ω be a bounded domain with Lipschitz boundary which is split into two parts ΓD and ΓN as above with conditions (2.41) and (2.42). Let (2.18), (2.19), and (2.20) be satisfied. Then, the part A|L2 of the operator associated with the form (2.17) on H˚ D1 (Ω) satisfies D(A|L2 ) ⊂ Wp1 (Ω) and uWp1 ≤ C(A|L2 uL2 + uL2 ),

u ∈ D(A|L2 ),

(2.43)

for a certain exponent p > 2. This property was proved by Bensoussan–Frehse [BF02, Theorem 2.2].

1.4 Sectorial Operators in Product Spaces of L2 Let Ω be a bounded domain in Rn . We introduce two product Hilbert spaces L2 (Ω) and H1 (Ω) with = 2 (see (1.84) and (1.85)). Consider the triplet of spaces

1 Sectorial Operators in Hilbert Spaces

67

H1 (Ω) ⊂ L2 (Ω) ⊂ H1 (Ω)∗ . By definition, it is easy to see that H1 (Ω)∗ is also given by the product space ϕ H1 (Ω)∗ = ; ϕ ∈ H 1 (Ω)∗ and ψ ∈ H 1 (Ω)∗ (2.44) ψ with duality product ϕ u , = ϕ, u + ψ, v , ψ v

ϕ u 1 ∗ ∈ H (Ω) , ∈ H1 (Ω). ψ v

Consider the sesquilinear form ˜ a(U, U ) = [a(x)∇u · ∇ u˜ + b(x)∇v · ∇ u˜ + c(x)∇u · ∇ v˜ Ω

˜ dx + k + d(x)∇v · ∇ v] u u˜ ˜ U= ,U= ∈ H1 (Ω), v v˜

(uu˜ + v v) ˜ dx, Ω

(2.45)

defined on H1 (Ω). Here, the functions a(x), b(x), c(x), d(x) are real-valued continuous functions on Ω. These functions are assumed to satisfy inf a(x) = a0 > 0,

x∈Ω

inf d(x) = d0 > 0,

x∈Ω

(2.46)

and to satisfy either b(x)c(x) ≤ 0

on Ω

(2.47)

or b(x)c(x) ≥ 0 on Ω

and

inf [a(x)d(x) − b(x)c(x)] = η0 > 0.

x∈Ω

(2.48)

The constant k is a positive constant which will be fixed below. Our goal is to show that the associated operator A and its parts in L2 (Ω) and H1 (Ω) are sectorial operators in H1 (Ω)∗ , L2 (Ω), and H1 (Ω), respectively. It seems, however, difficult to apply Theorem 2.1 to the form (2.45) directly. Our way to arrive at the goal will be indirect. We begin with verifying the following lemma. ϕ(x) 0 b(x) Lemma 2.1 Let P (x) = a(x) c(x) d(x) . There exist a diagonal matrix D(x) = 0 1 with ϕ ∈ C1 (Ω) and infx∈Ω ϕ(x) = ϕ0 > 0 and a constant δ > 0 for which it holds that D(x)P (x)D(x)−1 U, U ≥ δU 2 , x ∈ Ω, U ∈ R2 . (2.49) In addition, when (a, b, c, d) varies in a compact subset of C(Ω; R)4 satisfying (2.46) and either (2.47) or (2.48), the quantities ϕC1 , ϕ0 , and δ can be uniform.

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2 Sectorial Operators

Proof (I) Let us first consider a case where b(x) and c(x) are C1 functions on Ω. We have D(x)P (x)D(x)−1 U, U = a(x)u2 + b(x)ϕ(x) + c(x)ϕ(x)−1 uv + d(x)v 2 for U = t (u, v) ∈ R2 . Let condition (2.47) be satisfied. In this case, we put −b(x)c(x) + ε ϕε (x) = b(x)2 + ε 2 with parameter ε > 0. Then, b(x)ϕε (x) + c(x)ϕε (x)−1 =

|b(x)ε + c(x)ε 2 |

√ ≤ (1 + cC ) ε.

(b(x)2 + ε 2 )(−b(x)c(x) + ε) √ √ If ε > 0 is fixed in such a way that (1 + cC ) ε ≤ a0 d0 , then a(x)u2 + b(x)ϕε (x) + c(x)ϕε (x)−1 uv + d(x)v 2 ≥ a0 u2 −

1 a0 d0 |u||v| + d0 v 2 ≥ (a0 u2 + d0 v 2 ). 2

Let next condition (2.48) be satisfied. In this case, we put b(x)c(x) + ε ϕε (x) = , ε > 0. b(x)2 + ε 2 By (2.48), 2 c(x) + b(x)ε + c(x)ε 2 | b(x)ϕε (x) + c(x)ϕε (x)−1 = |2b(x) (b(x)2 + ε 2 )(b(x)c(x) + ε) √ ≤ 2 b(x)c(x) + (1 + cC ) ε √ ≤ 2 a(x)d(x) − η0 + (1 + cC ) ε.

Therefore, a(x)u2 + b(x)ϕε (x) + c(x)ϕε (x)−1 uv + d(x)v 2 ≥ [a(x) − μ]u2 − 2 a(x)d(x) − η √ + (1 + cC ) ε |u||v| + [d(x) − μ]v 2 + μ(u2 + v 2 ). Let μ =

η0 a(x)+d(x) .

Then, since

[a(x) − μ]u2 + [d(x) − μ]v 2 ≥ 2 (a(x) − μ)(d(x) − μ)|uv| > 2 a(x)d(x) − η0 |uv|,

1 Sectorial Operators in Hilbert Spaces

69

it follows that a(x)u2 + b(x)ϕε (x) + c(x)ϕε (x)−1 uv + d(x)v 2 √ η0 ≥ (u2 + v 2 ) − (1 + cC ) ε|uv|. a + dC Hence, if ε > 0 is sufficiently small, then the desired estimate (2.49) is satisfied. Since b(x) and c(x) are C1 functions, it is the same for ϕε (x). (II) Secondly, let us consider the general case. We notice that the C1 regularity of b(x) and c(x) is necessary only for ensuring that ϕε (x) is a C1 function and not for yielding estimate (2.49). So, if a new function ϕ(x), ˜ which is smooth on Ω, is taken sufficiently close to ϕε (x) in the topology of C(Ω), then (2.49) can be valid with ϕ(x). ˜ We also notice that, if (2.49) is satisfied for P (x) with some matrix D(x) = diag{ϕ(x), 1}, ϕ(x) being smooth, then it is satisfied for every P˜ (x), provided that ˜ ˜ a(x), ˜ b(x), c(x), ˜ and d(x) are all sufficiently close to a(x), b(x), c(x), and d(x), respectively, in the topology C(Ω). This shows that, when a(x), b(x), c(x), and d(x) vary in a compact subset of C(Ω), one can choose a finite number of matrices so that (2.49) is valid for every P (x) with at least one of these matrices. We notice that multiplication of D(x) is an isomorphism of H1 (Ω). In view of this fact, we here introduce the auxiliary sesquilinear form aD (U, V ) = a D(x)−1 U, D(x)V , U, V ∈ H1 (Ω), on H1 (Ω). On account of Lemma 2.1, aD (U, V ) is seen to satisfy the coercivity (2.7). In fact, we see that a D(x)−1 U, D(x)U = a(x)|∇u|2 + b(x)ϕ(x)∇v · ∇u Ω

+ c(x)ϕ(x)−1 ∇u · ∇v + d(x)|∇v|2 dx 2 +k |u| + |v|2 dx + b(U, U ), u U= ∈ H1 (Ω), v

Ω

where b(U, V ) denotes a sesquilinear form containing first derivatives of ϕ in its coefficients. Obviously, b(U, V ) satisfies |b(U, U )| ≤ Cϕ U |U |. Therefore, Re aD (U, U ) ≥ δ |∇u|2 + |∇v|2 + k |u|2 + |v|2 − Cϕ U |U | δ ≥ U 2 + (k − Cϕ )|U |2 . 2

70

2 Sectorial Operators

Fix k large enough to satisfy k ≥ Cϕ . Then, aD (U, V ) satisfies (2.7) and (2.6). Hence, Theorem 2.1 is available to conclude that the operator AD associated with the form aD (U, V ) and its parts satisfy (λ − AD )−1

H1∗

C , + (λ − AD|L2 )−1 L + (λ − AD|H1 )−1 H1 ≤ 2 |λ|

λ∈ / Σω ,

(2.50)

with some 0 ≤ ω < π2 . We now notice that AU, V = a(U, V ) = aD D(x)U, D(x)−1 V = AD D(x)U, D(x)−1 V . In addition, we observe by the lemma below that multiplications of D(x) and D(x)−1 are isomorphisms of H1 (Ω)∗ as well as of L2 (Ω) and H1 (Ω) and that the duality product has the property D(x)−1 Φ, V = Φ, D(x)−1 V , Φ ∈ H1 (Ω)∗ , V ∈ H1 (Ω). Then, it is deduced that A = D(x)−1 AD D(x) and the resolvent satisfies the relation (λ − A)−1 = D(x)(λ − AD )−1 D(x)−1 ,

λ∈ / Σω .

(2.51)

As a consequence of (2.50), (λ − A)−1

H1∗

C , + (λ − A|L2 )−1 L + (λ − A|H1 )−1 H1 ≤ 2 |λ|

λ∈ / Σω . (2.52)

Lemma 2.2 The operation of multiplication by D(x) can be extended as a bounded operator on H1 (Ω)∗ with the property D(x)Φ, V = Φ, D(x)V ,

Φ ∈ H1 (Ω)∗ , V ∈ H1 (Ω).

(2.53)

It is the same for D(x)−1 . Proof For F ∈ X, we have D(x)F ∗ = sup |(D(x)F, V )| = sup |(F, D(x)V )| ≤ CϕC1 F ∗ . V ≤1

V ≤1

Since L2 (Ω) is dense in H1 (Ω)∗ , this shows that multiplication by D(x) can be extended to H1 (Ω)∗ uniquely as a bounded operator. Since D(x)Φ, V = Φ, D(x)V holds for Φ ∈ L2 (Ω) and V ∈ H1 (Ω), (2.53) is also verified by the usual argument. We are now able to announce the following result.

2 Sectorial Operators in Banach Spaces

71

Theorem 2.11 Let Ω be any domain in Rn . Let a, b, c, d ∈ C(Ω; R) satisfy (2.46) and either (2.47) or (2.48). Then, for sufficiently large k, the operator A associated with the form (2.45) and its parts in L2 (Ω) and H1 (Ω) satisfy (2.1) and (2.2) in H1 (Ω)∗ , L2 (Ω), and H1 (Ω), respectively, with ω = π2 and some constant M. When (a, b, c, d) varies in a compact subset of C(Ω; R)4 satisfying (2.46) and either (2.47) or (2.48), the constants k and M can be uniform. We finally investigate a representation of A. Let us consider a case where a(x), b(x), c(x), and d(x) are all C1 functions. Let Λ be the operator associated with the sesquilinear form ∇u · ∇v dx + uv dx, u, v ∈ H 1 (Ω), a(u, v) = Ω

Ω

H 1 (Ω)∗

i.e., Λ is a realization of −Δ + 1 in under the Neumann boundary conditions. Then the first integral in the right-hand side of (2.45) is written as a(x)∇u · ∇ u˜ dx = ∇u · ∇(a u) ˜ dx − (∇u · ∇a)u˜ dx Ω

Ω

Ω

= a(x)Λu, u

˜ − ∇a · ∇u + a(x)u, u . ˜ It is the same for other integrals. Therefore, we have AU, V = a(U, V ) = a(x)Λu + b(x)Λv, u

˜ + c(x)Λu + d(x)Λv, v

˜ − ∇a · ∇u + ∇b · ∇v, u

˜ − ∇c · ∇u + ∇d · ∇v, v

˜ + (k − a(x))u − b(x)v, u

˜ + −c(x)u + (k − d(x))v, v . ˜ This then means that AU is given explicitly by the formula a(x) b(x) ∇a · ∇ ∇b · ∇ AU = (Λ − 1)U − U + kU, c(x) d(x) ∇c · ∇ ∇d · ∇

U ∈ H1 (Ω). (2.54)

Furthermore, for A|L2 , we can see that ∇ · (a∇·) ∇ · (b∇·) A|L2 U = − U + kU, ∇ · (c∇·) ∇ · (d∇·)

U ∈ H2N (Ω).

(2.55)

2 Sectorial Operators in Banach Spaces 2.1 Sectorial Operators in Lp (1 < p < ∞) Spaces Let Ω be a bounded domain in Rn . Consider the sesquilinear form a(u, v) =

n i,j =1 Ω

aij (x)Di uDj v dx +

c(x)uv dx, Ω

u, v ∈ Z,

(2.56)

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2 Sectorial Operators

on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Here, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Ω satisfying conditions (2.18) and (2.19), and c(x) is a real-valued function in Ω satisfying (2.20). We consider the triplet Z ⊂ L2 (Ω) ⊂ Z ∗ and the operator A associated with the form (2.56). We already know that its part A|L2 in L2 (Ω) is a sectorial operator of L2 (Ω). In this section, we shall study A in the space Lp (Ω), where 1 < p < ∞. It is shown for 2 < p < ∞ that its part in Lp (Ω) becomes a sectorial operator of Lp (Ω). On the other hand, it is shown for 1 < p < 2 that a natural extension of A in Lp (Ω) defines a sectorial operator of Lp (Ω). Consider first the case where 2 < p < ∞. We begin with showing the following lemma. Lemma 2.3 Let 2 < p < ∞. For u ∈ Z (= H˚ 1 (Ω) or H 1 (Ω)), put u∗ε = |u|2 p−2 2 u with a parameter ε > 0. Then, 1+ε|u|2 Re Au, u∗ε ≥ c0 u, u∗ε

(2.57)

and Im Au, u∗ ≤

Cnp Re Au, u∗ε √ 2δ p − 1

ε

(2.58)

with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20). Proof It is easily seen that u∗ε ∈ H 1 (Ω) with ∇u∗ε =

|u|2 1 + ε|u|2

|u|2 1 + ε|u|2

=

p−2 2

∇u +

p − 2 |u|p−4 u 2 p ∇|u| 2 (1 + ε|u|2 ) 2

∇u +

p − 2 |u|p−4 2 2 p |u| ∇u + u ∇u . 2 2 (1 + ε|u| ) 2

p−2 2

We therefore have Au, u∗ε =

Ω

|u|2 1 + ε|u|2

p−2 + 2 Ω

n

|u|p−4

Ω (1 + ε|u|2 )

(1 + ε|u|2 )

p−2 2

n

|u|p−2 (1 + ε|u|2 ) 2

p 2

aij (x)Di u u2 Dj u + |u|2 Dj u dx

i,j =1

dx

p

Ω

aij (x)Di uDj u dx

i,j =1

c(x)|u|p

+ =ε

p−2 n 2

i,j =1

aij (x)|u|2 Di uDj u dx + Ω

c(x)|u|p (1 + ε|u|2 )

p−2 2

dx

2 Sectorial Operators in Banach Spaces

1 + 2

73 n

|u|p−4

Ω (1 + ε|u|2 )

p 2

aij (x) p|u|2 Di uDj u

i,j =1

+ (p − 2)u2 Di uDj u dx.

Here, we put uDj u = αj + iβj , 1 ≤ j ≤ n. Then, it turns out that Au, u∗ε = ε

n

|u|p−2

aij (x)[(αi αj + βi βj ) + i(βi αj − αi βj )] dx

p

Ω

(1 + ε|u|2 ) 2

i,j =1 n

|u|p−4

+

aij (x){[(p − 1)αi αj + βi βj ]

p

Ω

(1 + ε|u|2 ) 2

i,j =1

c(x)|u|p

+ i[(p − 1)βi αj − αi βj ]} dx + Ω

(1 + ε|u|2 )

p−2 2

dx.

Then, Re(Au, u∗ε ) can be estimated from below by using (2.19) and (2.20). In particular, we verify the desired estimate (2.57). In the meantime, | Im(Au, u∗ε )| is estimated as follows. We have n

n

aij |βi αj − αi βj | ≤ 2C

i,j =1

|αi βj | ≤ C

i,j =1

= Cn

n 2 αi + βj2 i,j =1

n n 2 Cn αj + βj2 ≤ aij (αi αj + βi βj ). δ j =1

i,j =1

In a similar way, n

aij |(p − 1)βi αj − αi βj | ≤ Cp

i,j =1

≤√

≤

Noting that

√p 2 p−1

n

n Cp |αi βj | = √ p − 1|αi βj | p − 1 i,j =1 i,j =1

n Cp (p − 1)αi2 + βj2 p − 1 i,j =1

n Cnp aij [(p − 1)αi αj + βi βj ]. √ 2δ p − 1 i,j =1

≥ 1, we obtain from these the desired estimate (2.58).

Let Re λ ≤ 0. For the solution u to the equation (A − λ)u = f , where f ∈ Lp (Ω) ⊂ L2 (Ω), we verify the following fact.

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2 Sectorial Operators

Proposition 2.1 Let 2 < p < ∞, and let Re λ ≤ 0. If (A − λ)u = f with f ∈ Lp (Ω), then u belongs to Lp (Ω), and the estimates (| Re λ| + c0 )uLp ≤ f Lp , Cnp + 2 f Lp |λ|uLp ≤ √ 2δ p − 1

(2.59) (2.60)

hold with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20). Proof Taking the real part of the inner product of (A − λ)u = f and u∗ε in L2 (Ω), we have Re Au, u∗ε − Re λ u, u∗ε = Re f, u∗ε . Since Re(Au, u∗ε ) ≥ 0 and (u, u∗ε ) ≥ 0, it follows that Re Au, u∗ε + | Re λ|(u, u∗ ) = Re f, u∗ε ≤ f Lp u∗ε Lq , where

1 p

+

1 q

(2.61)

= 1. This, together with (2.57), yields that

(c0 + | Re λ|) Ω

|u|p (1 + ε|u|2 )

p−2 2

dx ≤ f Lp

Ω

(1 + ε|u|2 )

≤ f Lp

1

|u|(p−1)q

(1 + ε|u|2 )

dx 1

|u|p

Ω

q

(p−2)q 2

q

p−2 2

dx

.

Hence, (c0 + | Re λ|)

1

|u|p Ω

(1 + ε|u|2 )

p

p−2 2

dx

≤ f Lp .

Letting ε → 0, we deduce by Fatou’s lemma that u ∈ Lp (Ω) with (2.59). In order to prove (2.60), we notice that |λ| u, u∗ε ≤ Au, u∗ε + f, u∗ε ≤ Re Au, u∗ε + Im Au, u∗ε + f, u∗ε . Then, it follows from (2.58) and (2.61) that Cnp |λ| u, u∗ε ≤ + 2 f Lp u∗ε L . √ q 2δ p − 1 Letting ε → 0, we obtain that p |λ|uLp

≤

Cnp p−1 + 2 f Lp uLp . √ 2δ p − 1

Hence, estimate (2.60) is deduced.

2 Sectorial Operators in Banach Spaces

75

Let us next consider the case where 1 < p < 2. We verify the following analogous lemma. Lemma 2.4 Let 1 < p < 2. For u ∈ Z, put u∗ε = (|u|2 + ε) ε > 0. Then, Re Au, u∗ε ≥ c0 u, u∗ε

p−2 2

u with parameter (2.62)

and Im Au, u∗ ≤ ε

Cnp Re Au, u∗ε √ 2δ p − 1

(2.63)

with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20). Proof We can argue in an analogous way as before. It is observed that u∗ε ∈ H 1 (Ω) with p−2 p−4 p − 2 2 |u| + ε 2 |u|2 ∇u + u2 ∇u . ∇u∗ε = |u|2 + ε 2 ∇u + 2 From this we can write Au, u∗ε = ε

+

n 2 p−4 −2 2 |u| + ε |u| aij (x)|u|2 Di uDj u dx

Ω

1 2

i,j =1

Ω

n 2 p−4 |u| + ε 2 aij (x) p|u|2 Di uDj u i,j =1

+ (p − 2)u2 Di uDj u dx +

p−2 c(x) |u|2 + ε 2 |u|2 dx.

Ω

By putting uDj u = αj +iβj , 1 ≤ j ≤ n, we can repeat the same argument as before. Then, the desired estimates (2.62) and (2.63) are obtained. Let Re λ ≤ 0. For the solution u to the equation (A−λ)u = f , where f ∈ L2 (Ω), we verify the following proposition. Proposition 2.2 Let 1 < p < 2, and let Re λ ≤ 0. If (A − λ)u = f with f ∈ L2 (Ω), then u ∈ H 1 (Ω) satisfies the estimates (| Re λ| + c0 )uLp ≤ f Lp , Cnp + 2 f Lp |λ|uLp ≤ √ 2δ p − 1

(2.64) (2.65)

with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20).

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2 Sectorial Operators

Proof We consider the real part of the inner product (A − λ)u = f and u∗ε in L2 (Ω). Then, since Re(Au, u∗ε ) ≥ 0 and (u, u∗ε ) ≥ 0, it follows that Re Au, u∗ε + | Re λ| u, u∗ε ≤ f Lp u∗ε L . (2.66) q

By (2.62),

|u|2 + ε

(c0 + | Re λ|)

p−2 2

Ω

p−1

|u|2 dx ≤ f Lp uLp .

We here used the fact that u∗ε Lq ≤ uLp . Hence, letting ε → 0, we deduce by the Lebesgue dominate convergence theorem the desired estimate (2.64). Estimate (2.65) is obtained in a similar way from |λ| u, u∗ε ≤ Re Au, u∗ε + Im Au, u∗ε + f, u∗ε . p−1

Indeed, by (2.63) and (2.66), 2 p−2 2 2 |λ| |u| + ε |u| dx ≤ Ω

Cnp + 2 f Lp u∗ε L . √ q 2δ p − 1

Hence, letting ε → 0, (2.65) is also deduced.

We are now ready to define from A a linear operator Ap for each 1 < p < ∞. Let us first consider the case where 2 < p < ∞. In this case, the operator Ap is defined as the part of A in the space Lp (Ω). As Proposition 2.1 (λ = 0) shows that Au ∈ Lp (Ω) implies u ∈ Lp (Ω), we have in fact

D(Ap ) = {u ∈ Z; Au ∈ Lp (Ω)}, Ap u = Au. Proposition 2.1 shows also that, for every Re λ ≤ 0, the resolvent (λ − Ap )−1 exists and that the following estimates (λ − Ap )−1 ≤ 1/(| Re λ| + c0 ), Re λ ≤ 0, (2.67) Lp Cnp (λ − Ap )−1 ≤ + 2 |λ|, Re λ ≤ 0, λ = 0 (2.68) √ Lp 2δ p − 1 hold due to (2.59) and (2.60). Consider next the case where 1 < p < 2. Proposition 2.2 shows that, for every Re λ ≤ 0, the resolvent (λ − A)−1 acting on L2 (Ω) is continuous with respect to the Lp (Ω) norm. This means that the resolvent can be extended on Lp (Ω) as a bounded operator. Denote the extended operator by Rp (λ) for Re λ ≤ 0. Then the family of operators Rp (λ) defines a pseudo-resolvent in L(Lp (Ω)). According to Theorem 1.8, there exists a multivalued linear operator Ap of Lp (Ω) such that

2 Sectorial Operators in Banach Spaces

77

Rp (λ) = (λ − Ap )−1 for every Re λ ≤ 0. In this way, A−1 p is a continuous extension on L (Ω). Therefore, D(A ) ⊂ D(A ) and f = A2 u implies that f ∈ Ap u; of A−1 p 2 p 2 in particular, Ap is densely defined in Lp (Ω) (Indeed, D(A2 ) is dense in L2 (Ω), a fortiori in Lp (Ω)). Moreover, it is readily verified that u ∈ D(Ap ) and f ∈ Ap u if and only if there exist sequences un ∈ D(A2 ) and fn = A2 un such that un → u and fn → f in Lp (Ω). Finally, (2.64) and (2.65) yield that Ap also satisfies the same estimates as (2.67) and (2.68). To conclude for 1 < p < ∞ that Ap is a sectorial operator of Lp (Ω), it therefore remains to verify that Ap is densely defined in Lp (Ω) in the case where 2 < p < ∞ and that Ap is single valued in the case where 1 < p < 2. For these purposes, we notice the following proposition. Proposition 2.3 Let 1 < p < ∞ and f ∈ Lp (Ω). As λ → −∞, λ(λ − Ap )−1 f converges weakly to f in Lp (Ω). Proof When p = 2, the result is already known. Indeed, λ(λ − A2 )−1 f − f = A2 (λ − A2 )−1 f ; therefore, if f ∈ D(A2 ), then λ(λ − A2 )−1 f − f → 0 in L2 (Ω). By the denseness of D(A2 ) in L2 (Ω), λ(λ − A2 )−1 indeed converges strongly to the identity operator on L2 (Ω). Let 2 < p < ∞. Since the norms λ(λ − Ap )−1 f Lp are uniformly bounded for all λ ≤ 0, we can extract a sequence λn → −∞ such that λn (λn − Ap )−1 f → g weakly in Lp (Ω). However, since the sequence is strongly convergent to f in L2 (Ω), it must hold that g = f . Therefore, without extracting a sequence for λ, λ(λ − Ap )−1 f is weakly convergent to f in Lp (Ω). Let 1 < p < 2. If f ∈ L2 (Ω), then λ(λ − Ap )−1 f is strongly convergent to f in Lp (Ω). Then, by the uniform boundedness of λ(λ − Ap )−1 Lp for λ ≤ 0 and the denseness of L2 (Ω) in Lp (Ω), we conclude that λ(λ − Ap )−1 f → f in Lp (Ω) as λ → −∞. We can now state the final result and complete its proof. Theorem 2.12 Let Ω be a bounded domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied, and let A be an operator associated with the form (2.56) on Z = H˚ 1 (Ω) or H 1 (Ω). Then, for any 1 < p < ∞, the operator Ap defined from A in the way √ + 2, where specified above satisfies (2.1) and (2.2) with ω = π2 and constant 2δCnp p−1 C = maxi,j aij L∞ , δ, and c0 being the constants in (2.19) and (2.20). Therefore, Ap is a sectorial operator of Lp (Ω) with angle ωAp < π2 . Proof Let 2 < p < ∞. Since R((λ − Ap )−1 ) ⊂ D(Ap ), Proposition 2.3 shows that D(Ap ) is dense in the weak topology of Lp (Ω). But since D(Ap ) is a linear subspace, its denseness is valid even in the usual topology of Lp (Ω). Let 1 < p < 2. Assume that Ap 0 f ; then, (λ − Ap )0 f for any λ ≤ 0; that is, 0 = (λ − Ap )−1 f ; then Proposition 2.3 yields that f = 0. Therefore, Ap is a single-valued operator.

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2 Sectorial Operators

The operator Ap is called a realization of the elliptic operator (2.21) in Lp (Ω), 1 < p < ∞, under the Dirichlet- or Neumann-type boundary conditions ((2.22) or (2.23)) if Z = H˚ 1 (Ω) or H 1 (Ω), respectively. Remark 2.4 We notice that the resolvent of Ap satisfies the optimal estimate along the negative real axis (−∞, 0]. In fact, (2.67) shows that (λ − Ap )−1 ≤ 1/(|λ| + c0 ), λ ≤ 0, (2.69) L p

holds for every 1 < p < ∞.

2.2 Extreme Cases p = 1 and p = ∞ In this subsection, we consider the extreme cases p = 1 and p = ∞. It is possible to define analogously sectorial operators A1 and A∞ acting in the spaces L1 (Ω) and L∞ (Ω), respectively. However, (2.60) and (2.65) have no more meaning in these spaces. Let Ω be a bounded domain in Rn . Let A be a linear operator associated with the form (2.56) on Z = H˚ 1 (Ω) or H 1 (Ω). Conditions (2.18), (2.19), and (2.20) are assumed as before. First, let us consider the case p = ∞. For Re λ ≤ 0, consider the equation (A − λ)u = f , where f ∈ L∞ (Ω), and let u ∈ H 1 (Ω) be its solution. By 1

(2.59), we see that c0 uLp ≤ f Lp ≤ |Ω| p f L∞ for any finite p > 2. Using the fact that uL∞ = limp→∞ uLp , we can conclude that u ∈ L∞ (Ω). Furthermore, due to (2.59), (| Re λ| + c0 )uL∞ ≤ f L∞ .

(2.70)

The operator A∞ is then defined as the part of A in the space L∞ (Ω). That is, D(A∞ ) = {u ∈ Z; Au ∈ L∞ (Ω)} and A∞ u = Au. Due to (2.70), λ − A∞ has a bounded inverse on L∞ (Ω) with the estimates (λ − A∞ )−1 ≤ 1/(| Re λ| + c0 ), Re λ ≤ 0. (2.71) L ∞

In this way, we have observed that A∞ satisfies (2.1) and (2.2) with any ω > π2 . However, it is no longer possible to verify that D(A∞ ) is dense in L∞ (Ω), because Proposition 2.3 cannot be verified for p = ∞. We observe that λ(λ − A∞ )−1 f → f in L∞ (Ω) as λ → −∞ for f ∈ D(A∞ ) only. So, we have to consider again a part of A∞ in such a way that D(A˜ ∞ ) = {u ∈ D(A∞ ); A∞ u ∈ D(A∞ )} and A˜ ∞ u = A∞ u, where the closure is taken in L∞ (Ω). Then, the operator A˜ ∞ is densely defined in D(A∞ ). In fact, for any u ∈ D(A∞ ), uλ = λ(λ − A∞ )−1 u ∈ D(A˜ ∞ ) for λ < 0, and uλ → u in L∞ (Ω) as λ → −∞. Hence, D(A˜ ∞ ) is dense in D(A∞ ). Of course, A˜ ∞ also satisfies the same condition as (2.71). Secondly, let us consider the case p = 1. For Re λ ≤ 0, consider the equation (A − λ)u = f with f ∈ L2 (Ω). It is verified from (2.64) that (| Re λ| + c0 )uL1 ≤ f L1 .

(2.72)

2 Sectorial Operators in Banach Spaces

79

Here, we used the fact that uL1 = limp→1 uLp . This means that the resolvent (λ − A2 )−1 is continuous with respect to the L1 topology. As a consequence, each operator (λ − A2 )−1 is extended to a bounded operator R1 (λ) on L1 (Ω), and the family R1 (λ), Re λ ≤ 0, defines a pseudo-resolvent in L(L1 (Ω)). By a multivalued linear operator A1 of L1 (Ω), we have (λ − A1 )−1 = R1 (λ). Since D(A2 ) ⊂ D(A1 ), A1 is densely defined. Furthermore, as Proposition 2.3 is valid for p = 1, too, A1 is a single-valued linear operator. Finally, it is clear from (2.72) that (λ − A1 )−1

L1

≤ 1/(| Re λ| + c0 ),

Re λ ≤ 0.

(2.73)

In the cases where p = 1 and p = ∞, we have therefore obtained the following results. Theorem 2.13 Let Ω be a bounded domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied, and let A be an operator associated with the form (2.56) on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Then, the linear operators A1 and A˜ ∞ defined from A as above satisfy (2.1) and (2.2) in L1 (Ω) and D(A∞ ) ⊂ L∞ (Ω), respectively, with any angle ω > π2 and constant M = | cos1 ω| , and are sectorial operators of L1 (Ω) and D(A∞ ), respectively, with angles ωA1 ≤

π 2

and ωA˜ ∞ ≤ π2 .

2.3 Adjoint Operators We return to the study for the case where 1 < p < ∞. In Sect. 2.1 above, we defined the realization Ap of strongly elliptic operator (2.27) in Lp (Ω) under the Dirichletor Neumann type-boundary conditions. We are then concerned with the adjoint operator A∗p with respect to the adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1. Let Ω be a bounded domain in Rn . Under conditions (2.18), (2.19), and (2.20), consider the sesquilinear form (2.56) on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Let A be the operator associated with the from. Furthermore, let Ap be the sectorial operator of Lp (Ω) determined from A, where 1 < p < ∞. We consider an adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1, and denote by A∗p the adjoint of A∗p with respect to this adjoint pair. In order to characterize A∗p , we use the adjoint form whose notion was already presented in Chap. 1, Sect. 8.2. Actually, the adjoint form to (2.56) is given by a ∗ (u, v) = a(v, u) =

n i,j =1 Ω

aj i (x)Di uDj v dx +

c(x)uv dx,

u, v ∈ Z,

Ω

(2.74) on Z. Let A∗ be the operator associated with this form. It is of course possible to determine a sectorial operator Bp of Lp (Ω) from A∗ , where 1 < p < ∞. Our goal is to show that A∗p = Bp if p1 + p1 = 1.

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Theorem 2.14 Let Ω be a bounded domain in Rn , and let (2.18), (2.19), and (2.20) be satisfied. For 1 < p, p < ∞, let Ap and Bp be sectorial operators of Lp (Ω) and Lp (Ω) determined from the operators associated with the form (2.56) and the adjoint form (2.74), respectively, on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Then, A∗p = Bp if p1 + p1 = 1. Proof When p = 2, L2 (Ω) is a Hilbert space. So, A∗2 = B2 is observed directly by applying Theorem 1.25 to the form (2.56) and its adjoint form (2.74). Let 2 < p < ∞ and p1 + p1 = 1. Let u ∈ D(Ap ), and let v ∈ D(Bp ). Since D(Ap ) ⊂ D(A2 ), we have Ap u = A2 u. In the meantime, we recall that Bp v = g if and only if there exist sequences vn ∈ D(B2 ) and gn = B2 vn such that vn → v and gn → g in Lp (Ω). Then, since Ap u, vn Lp ×Lp = (A2 u, vn )L2 = (u, B2 vn )L2 = u, Bp vn Lp ×Lp , it follows that Ap u, v Lp ×Lp = u, Bp v Lp ×Lp ,

u ∈ D(Ap ), v ∈ D(Bp ).

This then shows that, if v ∈ D(Bp ), then v ∈ D(A∗p ) with A∗p v = Bp v. On the other hand, we know that both Bp and A∗p have their bounded inverses on Lp (Ω). Hence, it must hold that D(A∗p ) = D(Bp ) and A∗p = Bp . Let next 1 < p < 2 and p1 + p1 = 1. Since 2 < p < ∞, we can apply the result just obtained to Bp to conclude that Bp∗ = Ap . By virtue of Theorem 1.19, it then follows that A∗p = Bp∗∗ = Bp . As a consequence of Theorem 2.14, we know that, if aij (x) are symmetric in the sense of (2.35), then A∗2 = B2 = A2 . Therefore, A2 = A is a positive definite selfadjoint operator of L2 (Ω). Furthermore, A has a domain D(A) ⊂ Z (= H˚ 1 (Ω) or H 1 (Ω)). By Theorem 1.38, A−1 is a compact operator of L2 (Ω). Then, the theory of spectral resolutions (see [Yos80, Sect. XI.9]) provides that A possesses an infinite number of positive eigenvalues 0 < λ1 < λ2 < · · · and is written by

∞

A= 0

λ dE(λ) =

∞

λk Pk ,

k=1

where Pk = E(λk ) − E(λk − 0) denotes the orthogonal projection to the eigenspace Nk corresponding to the eigenvalue λk . Each Nk is a finite-dimensional subspace of L2 (Ω), and L2 (Ω) is the direct sum of Nk . Since ai,j (x) and c(x) are real-valued functions, we see, for each k, that u ∈ Nk implies u ∈ Nk , that is, u ∈ Nk implies Re u ∈ Nk and Im u ∈ Nk . This means that Nk can be generated by real functions alone. By the usual manner, we can then construct an orthonormal basis of Nk which is composed by real-valued functions alone. Corollary 2.1 Let Ω be a bounded domain in Rn . Let (2.18), (2.19), (2.20), and (2.35) be satisfied. Then, A2 is a positive definite self-adjoint operator of L2 (Ω).

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81

There exists an orthonormal basis of L2 (Ω) which is composed by real-valued eigenfunctions of A2 .

2.4 Shift Property in Lp Spaces For 1 < p < ∞, let Ap be the realization of strongly elliptic operator (2.27) in Lp (Ω) under the Dirichlet- or Neumann-type boundary conditions. In the case where p = 2, we already know that regularity of ∂Ω and aij (x) imply the shift property D(A2 ) ⊂ H 2 (Ω). This subsection is devoted to remarking that a similar property is valid for the general Ap , 1 < p < ∞, too. Let Ω be a bounded domain in Rn with C2 boundary. Let (2.19), (2.20), and (2.28) be satisfied. Let A(D) be the elliptic operator given by (2.27). Let us first consider the realization Ap which is equipped with the Dirichlet boundary conditions. According to Grisvard [Gri85, Theorem 2.4.1.3], the mapping u → (A(D) + k)u is an isomorphism from the space {u ∈ Hp2 (Ω); γ u = 0} onto Lp (Ω) if k is sufficiently large. For u ∈ D(Ap ), put Ap u = f ∈ Lp (Ω); then, there exists a function u˜ ∈ Hp2 (Ω) satisfying γ u˜ = 0 such that (A(D) + k)u˜ = f + ku, i.e., (Ap + k)u˜ = f + ku. Therefore, u˜ = u. In this way, we verify that

D(Ap ) = {u ∈ Hp2 (Ω); γ u = 0 on ∂Ω}, (2.75) Ap u = A(D)u, with uHp2 ≤ Cp (Ap uLp + uLp ),

u ∈ D(Ap ).

(2.76)

It is quite similar for the realization Ap equipped with the Neumann-type boundary conditions. According to Grisvard [Gri85, Theorem 2.4.1.3], the mapping u → ∂u = 0} onto Lp (Ω) (A(D) + k)u is an isomorphism from the space {u ∈ Hp2 (Ω); ∂ν A if k is sufficiently large. Then, by the same reason, we verify that

∂u = 0 on ∂Ω}, D(Ap ) = {u ∈ Hp2 (Ω); ∂ν A (2.77) Ap u = A(D)u, with uHp2 ≤ Cp (Ap uLp + uLp ),

u ∈ D(Ap ).

(2.78)

Theorem 2.15 Let Ω be a bounded domain with C2 boundary. Let (2.19), (2.20), and (2.28) be satisfied. Then, for each 1 < p < ∞, the realizations Ap of A(D) in Lp (Ω) under the Dirichlet- and Neumann-type boundary conditions are characterized by (2.75) and (2.77), respectively. Remark 2.5 A similar shift property is verified even in the extreme cases p = 1 and p = ∞. But the optimal shift property (i.e., D(A1 ) ⊂ H12 (Ω) and D(A∞ ) ⊂ 2 (Ω)) fails except the one-dimensional case. For characterization of these operH∞ ators, we refer the reader to Tanabe [Tan97, Chap. 5].

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2 Sectorial Operators

3 Sectorial Operators in Product Spaces Let X1 and X2 be Banach spaces with norms · 1 and · 2 , respectively, and let X be the product Banach space of X1 and X2 with norm F = f1 1 + f2 2 , F = t (f , f ) ∈ X. 1 2 Let A1 and A2 be sectorial operators of X1 and X2 , respectively, with angles 0 ≤ ω1 < π and 0 ≤ ω2 < π . We consider a matrix operator of the form A1 B , 0 A2 u1 ; u1 ∈ D(A1 ), u2 ∈ D(A2 ) ∩ D(B) , D(A) = u2

A=

(2.79)

where B is a third linear operator from D(B) ⊂ X2 into X1 . If B is a bounded operator from D(A2 ) into X1 , then A is a sectorial operator of X. Theorem 2.16 Let Ak be two sectorial operators of Xk with angles 0 ≤ ωk < π , respectively, where k = 1, 2. Let B ∈ L(D(A2 ), X1 ). Then, the matrix operator A defined by (2.79) satisfies (2.1) and (2.2) for any ω such that ωA < ω ≤ π with constant Mω = max{M1,ω , M2,ω + M1,ω (M2,ω + 1)BA−1 2 }, where ωA = max{ω1 , ω2 }. Therefore, A is a sectorial operator of X with angle ≤ ωA . Proof The equality f u (λ − A) 1 = 1 u2 f2 is obviously equivalent to

(λ − A1 )u1 − Bu2 = f1 , (λ − A2 )u2 = f2 .

/ Σω , then λ ∈ ρ(A), and the resolvent is written by Let ωA < ω < π . If λ ∈ −1

(λ − A)

(λ − A1 )−1 = 0

−1 (λ − A1 )−1 BA−1 2 A2 (λ − A2 ) . (λ − A2 )−1

The resolvent is estimated by (λ − A)−1 ≤ max M1,ω , M2,ω + M1,ω (M2,ω + 1)BA−1 |λ|−1 , 2

This shows that the assertion of theorem is true.

λ∈ / Σω .

4 Yosida Approximation

83

4 Yosida Approximation 4.1 Uniform Estimates Let A be a sectorial operator of X with angle ωA . We define the sequence An of bounded linear operators on X by An = A(1 + n−1 A)−1 = n − n2 (n + A)−1 ,

n = 1, 2, 3, . . . .

(2.80)

This sequence {An }n=1,2,3,... is called the Yosida approximation for A. It is observed by some calculations that nλ λ − An = n2 − (n − λ)(n + A) (n + A)−1 = (n − λ) − A (n + A)−1 . n−λ Therefore, if

nλ n−λ

∈ ρ(A), then λ ∈ ρ(An ), and the following representation is valid: −1

(λ − An )

−1 nλ 1 (n + A) −A = n−λ n−λ 2 −1 n nλ 1 + −A = . λ−n n−λ n−λ

(2.81) (2.82)

Let ωA < ω < π . By definition, C − Σω ⊂ ρ(A). Note that λ ∈ C − Σω implies ∈ C − Σω . Indeed, consider the transform

nλ n−λ

λ →

nλ = n−λ

1 1 λ

−

1 n

(2.83)

in C which is composed with two elementary transforms λ → λ−1 and λ → λ − n1 . nλ These two transforms map C − Σω into itself. Consequently, the transform λ → n−λ also maps C − Σω into itself. In view of (2.81), we have C − Σω ⊂ ρ(An ), i.e., σ (An ) ⊂ Σω ,

n = 1, 2, 3, . . . .

(2.84)

Let us estimate next (λ − An )−1 . By the cosine theorem, we have |λ − n|2 = |λ|2 + n2 − 2|λ|n cos(arg λ) ≥ |λ|2 + n2 − 2|λ|n cos ω

(1 − cos ω)(|λ|2 + n2 ) if 0 < ω ≤ π2 , λ ∈ / Σω , ≥ π 2 2 if 2 < ω < π, λ ∈ / Σω . |λ| + n

(2.85)

We then obtain from (2.82) that ˜ (λ − An )−1 ≤ Mω , |λ|

λ∈ / Σω , n = 1, 2, 3, . . . ,

with some constant M˜ ω ≥ Mω independent of n.

(2.86)

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2 Sectorial Operators

We have thus verified that the Yosida approximation An satisfies (2.1) and (2.2) for any ω such that ωA < ω < π with some constant M˜ ω independent of n. In particular, An are sectorial operators of X with angles ωAn ≤ ωA .

4.2 Convergence As remarked in Chap. 1, Sect. 3.4, (λ − A)−1 is an L(X)-valued analytic function for λ ∈ ρ(A). It then follows from (2.82) that, for each λ ∈ / Σω , as n → ∞, (λ − An )−1 → (λ − A)−1

in L(X).

(2.87)

Let us see the convergence of An itself. To this end, we introduce the sequence of bounded operators Jn = (1 + n−1 A)−1 = 1 − A(n + A)−1 ,

n = 1, 2, 3, . . . .

(2.88)

It is clear that Jn ≤ M. Meanwhile, (1 − Jn )u = (n + A)−1 Au → 0 for u ∈ D(A). The denseness of D(A) in X then yields that Jn converges to 1 strongly on X

(2.89)

(see Chap. 1, Sect. 3). We therefore observe for U ∈ D(A) that, as n → ∞, An U = Jn AU → AU

in X.

(2.90)

Let k be any positive integer. Then, as n → ∞, Jnk also converges to 1 strongly on X. On the other hand, R(Jnk ) is contained in D(Ak ). This then shows that D(Ak ) is dense in X for any integer k = 1, 2, 3, . . . .

(2.91)

5 Exponential Functions In this section, we consider a sectorial operator A of a Banach space X with angle 0 ≤ ωA < π2 . By ω, we denote an angle such that ωA < ω < π2 . By definition, σ (A) ⊂ Σω = {λ ∈ C; | arg λ| < ω},

ωA < ω

0 such that e−zA ≤ Cφ e−δφ |z| ,

π 2

− ω, there exist a positive exponent

z ∈ Σφ − {0}.

(2.97)

Proof First, let us consider variables such that 0 < |z| ≤ 1 and | arg z| ≤ φ. We shift the integral contour in (2.95) to C1 ∪ Γ1 , where C1 : λ = |z|−1 e−iϕ ,

ω ≤ ϕ ≤ 2π − ω,

and

Γ1 : λ = ρe±iω ,

|z|−1 ≤ ρ < ∞.

5 Exponential Functions

87

Then, since |e−zλ | ≤ e for λ ∈ C1 , −zλ −1 e (λ − A) dλ ≤ C1

2π−ω

ω

eMω −1 |z| dϕ ≤ 2πeMω . |z|−1

Since |e−zλ | ≤ e−|z||λ| cos(φ+ω) for λ ∈ Γ1 , ∞ ∞ dρ −zλ −1 −ρ|z| cos(φ+ω) dρ ≤ Mω = M . e (λ−A) dλ e e−ρ cos(φ+ω) ω −1 ρ ρ |z| 1 Γ1 Therefore, e−zA ≤ Cφ if 0 < |z| ≤ 1 and | arg z| ≤ φ. Next, let us consider variables such that |z| ≥ 1 and | arg z| ≤ φ. Let 0 < δ < A−1 −1 . On account of (2.3), we can shift the integral contour to C2 ∪ Γ2 , where C2 : λ = δeiϕ ,

and Γ2 : λ = ρe±iω ,

−ω ≤ ϕ ≤ ω,

δ ≤ ρ < ∞.

Then, since |e−zλ | ≤ e−δ|z| cos(φ+ω) for λ ∈ C2 , ω −zλ −1 ≤ Mω e (λ − A) dλ e−δ|z| cos(φ+ω) dϕ ≤ 2ωMω e−δ|z| cos(φ+ω) . C2

−ω

In addition, −zλ −1 e (λ − A) dλ ≤ Mω Γ2

∞

e−ρ|z| cos(φ+ω) dρ =

δ

Mω e−δ|z| cos(φ+ω) . |z| cos(φ + ω)

Therefore, e−zA ≤ Cφ e−δ|z| cos(φ+ω) if |z| ≥ 1 and | arg z| ≤ φ. We have thus obtained the desired estimate (2.97).

We are now concerned with the limit of e−zA as z → 0. Proposition 2.6 For any φ such that 0 < φ < on X as z → 0 with z ∈ Σφ − {0}.

π 2

− ω, e−zA converges to 1 strongly

Proof First, let U ∈ D(A). Then, by the same argument as in the proof of Proposition 2.4, 1 −zA e U= e−zλ (λ − A)−1 (1 + A)−1 dλ(1 + A)U 2πi Γ 1 e−zλ (λ − A)−1 + (1 + A)−1 dλ(1 + A)U = 2πi Γ λ + 1 1 e−zλ = (λ − A)−1 dλ(1 + A)U. 2πi Γ λ + 1 Then, by the Lebesgue dominated convergence theorem, we see that

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2 Sectorial Operators

e−zA U →

1 2πi

1 =− 2πi

Γ

1 (λ − A)−1 dλ(1 + A)U λ+1

|λ+1|=1

1 (λ − A)−1 dλ(1 + A)U = U λ+1

in X.

Next, let F ∈ X be a general vector. The denseness of D(A) in X and uniform boundedness of the norms e−zA verified by (2.97) readily yield that e−zA F → F in X. In view of this convergence, we are naturally led to define the value of e−zA at z = 0 by e−0A = 1 (the identity mapping on X). We finally show the differentiability of e−zA U at z = 0 for U ∈ D(A). Proposition 2.7 Let U ∈ D(A). For any φ such that 0 < φ < z−1 [e−zA − 1]U converges to −AU in X as z → 0 with z ∈ Σφ − {0}.

π 2

− ω,

Proof Let z ∈ Σφ − {0} and ε > 0. Then, [e−zA − e−εA ]U = (ε − z)

1

e−[θz+(1−θ)ε]A AU dθ.

0

Letting ε → 0, we have [e−zA − 1]U = −z z−1 [e−zA − 1]U → −AU .

1 0

e−θzA AU dθ . Therefore, as z → 0,

In this subsection, we have proved for a sectorial operator A satisfying (2.92) and (2.93) that its exponential function e−zA is an analytic function in Σ π2 −ω enjoying the properties described in Propositions 2.4, 2.5, 2.6, and 2.7. It is also verified that e−zA satisfies the semigroup property

e−zA e−z A = e−(z+z )A ,

z, z ∈ Σ π2 −ω .

(2.98)

Such an analytic function is called an analytic semigroup on X generated by −A. The precise definitions of analytic semigroups and their generators will be presented in Chap. 3.

5.2 Contraction Semigroups Let A be a sectorial operator of X satisfying (2.92) and (2.93). We assume that A satisfies the optimal estimate on the negative real axis, i.e., (λ − A)−1 ≤ 1/|λ|, λ < 0. (2.99) This condition implies that e−tA ≤ 1 for all t ≥ 0. Such a semigroup is called a contraction semigroup.

5 Exponential Functions

89

Theorem 2.17 Let A be a sectorial operator of X satisfying (2.92), (2.93), and (2.99). Then, it holds that e−tA ≤ 1 for any 0 ≤ t < ∞. Proof Let U ∈ D(A). By Proposition 2.7, limh0 h−1 [e−hA − 1]U = −AU . In the meantime, by (1.12), h−1 [(1+hA)−1 −1]U = −(1+hA)−1 AU → −AU as h 0. Therefore, we observe that limh0 h−1 [e−hA − (1 + hA)−1 ]U = 0. Furthermore, we verify from (2.99) that e−hA U − U [e−hA − (1 + hA)−1 ]U ≤ , h h

h > 0.

Hence, lim suph0 h−1 [e−hA U − U ] ≤ 0. By the same argument, we verify that D + e−tA U ≤ 0 for every t ≥ 0, where D + denotes Dini’s derivative. We then have e−tA U ≤ U . Indeed, suppose the contrary; then there exists some τ > 0 such that e−τ A U > U . Let α = (e−τ A U − U )/τ , and let the continuous function e−tA U − αt on [0, τ ] hit its minimum at t0 . Since e−τ A U − ατ = U , it is possible to assume that 0 ≤ t0 < τ . Then, e−tA U − e−t0 A U ≥ α(t − t0 ) for any t0 < t ≤ τ , and we conclude that (D + e−tA U )|t=t0 ≥ α > 0, which is a contradiction. Estimate (2.99) thus implies that, for U ∈ D(A), e−tA U ≤ U for every t ≥ 0. But, since D(A) is dense in X, the same estimate is valid for general F ∈ X, too.

5.3 Adjoint Semigroups Let X be a reflexive Banach space, and let X ∗ be an adjoint space of X. Let {X, X ∗ } be an adjoint pair with duality product ·, · . Let A be a sectorial operator of X satisfying (2.92) and (2.93). So, −A generates an analytic semigroup e−zA on X which is given by (2.95). Let A∗ be the adjoint operator of A with respect to {X, X ∗ }. By Theorem 1.19, ∗ A is a sectorial operator of X ∗ with the same angle ωA as A. Therefore, −A∗ also generates an analytic semigroup. We observe that, for any F ∈ X and any G ∈ X ∗ , −zλ 1 e (λ − A)−1 F, G dλ e−zA F, G = 2πi Γ 1 F, e−zλ (λ − A∗ )−1 G dλ. = 2πi Γ Here, let us choose an integral contour as Γω : λ = ρe±iω , 0 ≤ ρ < ∞, where ωA < ω < π2 . If the integral variable λ varies on Γω counterclockwise, namely, from ∞eiω through 0 to ∞e−iω , then the conjugate λ varies on Γω in the opposite direction. Therefore, 1 ∗ e−zA F, G = F, − e−zλ (λ − A∗ )−1 G dλ = F, e−zA G , 2πi −Γω

90

2 Sectorial Operators

i.e., ∗

e−zA = [e−zA ]∗ .

(2.100)

∗

We call the exponential function e−zA the adjoint semigroup of e−zA .

5.4 Approximate Semigroups Let A be a sectorial operator of X satisfying (2.92) and (2.93). Let An be the Yosida approximation of A defined by (2.80). As verified by (2.84) and (2.86), An are also sectorial operators of X with angles ≤ ωA . So, each −An generates an analytic semigroup e−zAn given by the integral (2.95). By (2.97), for any 0 < φ < π2 − ω, e−zAn ≤ Cφ e−δφ |z| ,

z ∈ Σφ ,

(2.101)

with some exponent δφ > 0 and constant Cφ which are independent of n. The sequence of analytic semigroups e−zAn is then called approximate semigroups of e−zA . As shown by (2.87), as n → ∞, (λ − An )−1 converges to (λ − A)−1 in L(X). Therefore, by the Lebesgue dominate convergence theorem, we verify for any z ∈ Σ π2 −ωA that e−zAn → e−zA

in L(X).

(2.102)

6 Generation of Analytic Semigroups 6.1 Generation Results in Hilbert Spaces Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Consider a continuous and coercive sesquilinear form a(U, V ) defined on Z, that is, (2.6) and (2.7) are satisfied. Let A be a linear operator associated with a(U, V ), and let A|X and A|Z be its parts in X and Z, respectively. By Theorem 2.1, we already know that A, A|X , and A|Z are all sectorial operators with angles < π2 . Therefore, they are negative generators of analytic semigroups on Z ∗ , X, and Z, respectively. Summarizing known results for A and its parts, we can state the following generation theorem. Theorem 2.18 Let a(U, V ) satisfy (2.6) and (2.7). Let A be the associated operator, and let A|X and A|Z be its parts. Then, A, A|X , and A|Z satisfy (2.92) and (2.93) δ < ω < π2 with a in Z ∗ , X, and Z, respectively, for any ω such that π2 − sin−1 M+δ (M+δ) sin ω constant δ−(M+δ) cos ω , where M and δ are the constants in (2.6) and (2.7).

6 Generation of Analytic Semigroups

91

Proof By (2.8), (2.9), (2.10), and (2.11), A, A|X , and A|Z satisfy (2.1) and (2.2) in Z ∗ , X, and Z, respectively, with angle π2 and constant M+δ δ . Then the results are obtained by applying (2.4) and (2.5). We know by (2.12) that the part A|X in X satisfies (2.99). Consequently, Theorem 2.17 yields that |e−tA|X | ≤ 1,

0 ≤ t < ∞.

We also remark that the operator A given by (2.54) and its parts are negative generators of analytic semigroups on the product Hilbert spaces. See Theorem 2.11.

6.2 Generation Results in Lp Spaces Let Ω be a bounded domain in Rn . For 1 < p < ∞, let Ap be the realization of the elliptic operator A(D) given by (2.27) in Lp (Ω) under the Dirichlet- or Neumanntype boundary conditions. By Theorem 2.12, we already know that Ap satisfies (2.1) and (2.2) with angle π2 and constant Mp given by (2.68). If we apply (2.4) and (2.5), then Ap is seen to satisfy (2.92) and (2.93). On the other hand, due to (2.69), Ap also satisfies (2.99). We thus obtain the following generation theorem. Theorem 2.19 Let Ω be a bounded domain in Rn . For 1 < p < ∞, let Ap be the realization of A(D) given by (2.27) in Lp (Ω) under the Dirichlet- and Neumanntype boundary conditions. Then, Ap satisfies (2.92) and (2.93) for any ω such M sin ω that π2 − sin−1 M1p < ω < π2 with constant 1−Mp p cos ω , where Mp is the constant in (2.68). The analytic semigroup e−tAp generated by −Ap satisfies the estimate e−tAp L(Lp ) ≤ 1 for 0 ≤ t < ∞.

For p = 1 and p = ∞, let A1 and A˜ ∞ be the realizations of A(D) in L1 (Ω) and L∞ (Ω), respectively. As seen by Theorem 2.13, there is no guarantee which ensures generation of analytic semigroup for A1 and A˜ ∞ . But (2.71) and (2.73) mean that they satisfy (2.99). As will be mentioned in Sect. 7.8, such a sectorial operator can generate a continuous semigroup. The following result is an immediate consequence of Theorem 2.28 announced below. Theorem 2.20 The realizations A1 and A˜ ∞ of A(D) are negative generators of continuous and contraction semigroups on L1 (Ω) and D(A∞ ), respectively. Remark 2.6 It is known in the regular case (see Sect. 2.4) that −A1 and −A˜ ∞ can generate analytic semigroups. Especially, let AC be a realization of A in the continuous function space C(Ω), then −AC generates an analytic semigroup in C(Ω). For details, see Tanabe [Tan97, Chap. 5] and Taira [Tai03].

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2 Sectorial Operators

7 Fractional Powers of Linear Operators Let X be a Banach space with norm · . Let A be a sectorial operator of X with angle 0 ≤ ωA < π . We know that, for any integer n ∈ Z, the operator An is defined; indeed, when n > 0, An is a densely defined, closed operator of X (cf. (2.91)); when n < 0, An = (A−1 )−n = (A−n )−1 is a bounded operator of X; and, when n = 0, A0 = 1 (the identity on X). We are then concerned with extending the definition for all real exponents x ∈ R in a natural manner. By ω we denote an angle such that ωA < ω < π . By definition, σ (A) ⊂ Σω = {λ ∈ C; | arg λ| < ω},

ωA < ω < π,

(2.103)

and (λ − A)−1 ≤ Mω , |λ|

λ∈ / Σω , ωA < ω < π,

(2.104)

with some constant Mω ≥ 1. As noticed in the beginning of the chapter, (2.103) implicitly means that 0 ∈ ρ(A). We may remark again that {λ ∈ C; |λ| ≤ δ} ⊂ ρ(A),

(2.105)

provided that 0 < δ < A−1 −1 .

7.1 Integral Formula We define, for each complex number z such that Re z > 0, the bounded linear operator 1 A−z = λ−z (λ − A)−1 dλ, (2.106) 2πi Γ using the Dunford integral in L(X). As a branch of the analytic function λ−z , we take the principal branch {λ ∈ C; | arg λ| < π} obtained by excluding the negative real axis from the complex plane. So, Γ is an integral contour surrounding σ (A) counterclockwise in C − (∞, 0] ∩ ρ(A). Here, we take Γ = Γ− ∪ Γ0 ∪ Γ+ such that Γ± : λ = ρe±iω ,

δ ≤ ρ < ∞,

and Γ0 : λ = δeiϕ ,

−ω ≤ ϕ ≤ ω,

(2.107)

where ω and δ are as in (2.103), (2.104), and (2.105). In addition, Γ is oriented from ∞eiω to δeiω , from δeiω to δe−iω , and from δe−iω to ∞e−iω . Since |λ−z | = |e−z log λ | = |e−z(log ρ±iω) | = e±(Im z)ω ρ − Re z ,

λ ∈ Γ± ,

we observe that the integral (2.106) is convergent in L(X). When z = n is a positive integer, we can shift the contour from Γ to a circle Cδ : |λ| = δ. Consequently, by

7 Fractional Powers of Linear Operators

93

the Cauchy integral formula, 1 1 λ−n (λ − A)−1 dλ = − λ−n (λ − A)−1 dλ 2πi Γ 2πi Cδ n−1 d 1 =− (λ − A)−1 (n − 1)! dλn−1 λ=0 n −n −n = (−1) (λ − A) λ=0 = A . This formula shows that the new definition (2.106) is consistent with the former one for any positive integer. By a direct calculation, we can verify that A−z is an analytic function for Re z > 0 with values in L(X). Indeed, we have 1 d −z A =− (log λ)λ−z (λ − A)−1 dλ. dz 2πi Γ However, since the proof is direct, we may omit it. We are now concerned with convergence of A−z as z → 0. Theorem 2.21 For any 0 < φ < to 1 strongly on X.

π 2,

as z → 0 with z ∈ Σφ − {0}, A−z converges

Proof We shall first verify the uniform boundedness of A−z when z varies in the set {z ∈ C; | arg z| ≤ φ, 0 < |z| < 1}. We notice that for such a z, the contour can be shifted from Γ to the reduplicated half lines (∞eiπ , 0] ∪ [0, ∞e−iπ ) by letting ±ω → ±π and δ → 0. From (2.107) it indeed follows that ∞ πi −z 1 A−z = (ρe ) − (ρe−πi )−z (ρ + A)−1 dρ 2πi 0 e−πiz − eπiz ∞ −z = ρ (ρ + A)−1 dρ 2πi 0 sin(πz) ∞ −z =− ρ (ρ + A)−1 dρ. (2.108) π 0 By (2.104) and (2.105), there exists some constant C ≥ 1 such that (ρ + A)−1 ≤

C , ρ +1

0 ≤ ρ < ∞.

Therefore, C| sin(πz)| A ≤ π −z

0

∞

C| sin(πz)| ρ − Re z dρ = . ρ +1 sin(π Re z)

Noting that (cos φ)|z| ≤ Re z ≤ |z|, we conclude that sup

| arg z|≤φ, 0 0, ∞ 1 1 A−z = λ−x (λ − A)−1 dλ = λ−z (λ − μ)−1 dE(μ) dλ 2πi Γ 2πi Γ δ ∞ ∞ 1 −z −1 λ (λ − μ) dλ dE(μ) = μ−z dE(μ). (2.110) = 2πi Γ δ δ

7.2 Law of Exponent We shall show that the fractional power A−z of sectorial operator satisfies the law of exponent, i.e.,

A−z A−z = A−(z+z ) ,

Re z > 0, Re z > 0.

(2.111)

By (1.12) we have −z

−z

A A

= =

1 2πi 1 2πi

2 Γ

2

Γ

Γ

Γ

λ−z (λ )−z (λ − A)−1 (λ − A)−1 dλ dλ λ−z (λ )−z (λ − A)−1 − (λ − A)−1 dλ dλ, λ −λ

where Γ is the integral contour given by (2.107), and Γ is another integral contour obtained by substituting ω and δ for ω and δ in such a way that ω < ω < π and 0 < δ < δ, respectively. Then, by the Cauchy integral formula, 1 2πi

Γ

(λ )−z 1 dλ = λ −λ 2πi

|λ −λ|= 2δ

(λ )−z dλ = λ−z , λ −λ

7 Fractional Powers of Linear Operators

1 2πi

Γ

95

λ−z dλ = 0. λ − λ

Therefore, we obtain the desired law 1 A−z A−z = λ−(z+z ) (λ − A)−1 dλ = A−(z+z ) . 2πi Γ We have thus verified the following fact. Theorem 2.22 The L(X)-valued function A−z is an analytic semigroup defined in the half-plane {z ∈ C; Re z > 0}. Using this fact, let us next observe that A−z is invertible for every Re z > 0. Indeed, if A−z0 F = 0 for some z0 , then, for any x > 0, A−(z0 +x) F = A−x A−z0 F = 0; since A−z F is analytic, A−z F must vanish identically. Then, by Theorem 2.21, F is necessarily equal to 0. Hence, for Re z > 0, A−z is one-to-one, namely, its inverse Az is a single-valued linear operator of X. The following definition is meaningful: Az = (A−z )−1

for Re z > 0.

If 0 < Re z1 < Re z2 , then A−z2 = A−z1 A−(z2 −z1 ) ; therefore, R(A−z2 ) ⊂ R(A−z1 ); this then means that D(Az2 ) ⊂ D(Az1 ). In particular, if 0 < Re z < k, where k is an integer, then D(Ak ) ⊂ D(Az ); therefore, D(Az ) is dense in X due to (2.91). Hence, Az is a densely defined, closed linear operator of X. In view of Theorem 2.21 it is natural to define A0 = 1. Then, for every real number −∞ < x < ∞, the fractional power Ax of A has been defined. The following properties are already established: (1) Ax are bounded operators on X for −∞ < x < 0, A0 = 1, and Ax are densely defined, closed linear operators of X for 0 < x < ∞. (2) D(Ax2 ) ⊂ D(Ax1 ) for 0 ≤ x1 < x2 < ∞. (3) Ax Ax = Ax Ax = Ax+x for any −∞ < x, x < ∞.

7.3 Law of Exponent, II Let α be a fixed exponent such that 0 < α < 1. We shall show that the fractional power Aα of a sectorial operator A is a sectorial operator with angle ≤ αωA and satisfies the second law of exponent, i.e., (Aα )z = Aαz ,

z ∈ C, Re z = 0.

We consider the family of bounded operators defined by 1 1 Rα (λ) = (μ − A)−1 dμ, 2πi Γ λ − μα

(2.112)

(2.113)

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2 Sectorial Operators

where Γ is the same integral contour given in (2.107). Let ω be another angle satisfying ω < ω < π . Then, if λ ∈ / Σαω , then λ = μα for any μ ∈ Γ . So, Rα (λ) is defined for λ ∈ / Σαω . By a direct calculation, Rα (λ) is seen to be an analytic function with values in L(X). Furthermore, by similar techniques as used to verify (2.111), we can verify that 1 1 / Σαω . (μ − A)−1 dμ, λ, λ ∈ Rα (λ)Rα (λ ) = 2πi Γ (λ − μα )(λ − μα ) 1 1 1 1 Since (λ−μα )(λ −μα ) = − λ−λ λ−μα − λ −μα , it follows that Rα (λ)Rα (λ ) = −

1 [Rα (λ) − Rα (λ )], λ − λ

λ, λ ∈ / Σαω ,

that is, Rα (λ) define a pseudo-resolvent. Since 0 − R(0)−1 = Aα , we see by Theorem 1.8 that this pseudo-resolvent is generated by Aα , i.e., Rα (λ) = (λ − Aα )−1 . Hence, we conclude that σ (Aα ) ⊂ Σαω . In order to verify (2.2), however, we need more careful calculations in which the integral contour in (2.113) must be shifted suitably depending on φ = | arg λ|, π−ωA (φ − αωA ) + ωA . Clearly, where αω ≤ φ ≤ π . We introduce the angle ω˜ = π−αω A ω˜ satisfies

π(1−α) (φ − αωA ), π > φ − α ω˜ = π−αω A (2.114) π(1−α) 2π(1 − α) > α(ω˜ − 2π) − (φ − 2π) > π−αω (φ − αωA ). A As a branch of the function μα , we retake the domain {μ; ω˜ − 2π < arg μ < ω}. ˜ Then, in view of the relation {arg μα ; ω˜ − 2π < arg μ < ω} ˜ ⊂ [α(ω˜ − 2π), α ω] ˜ ⊂ (φ − 2π, φ) (due to (2.114)), the equation λ − μα = 0 (λ being fixed) has no solution in this domain. Noting this fact, we then shift the integral contour Γ to the replicated ˜ ). After some calculations, it turns out that half-lines (∞ei ω˜ , 0] ∪ [0, ∞ei(ω−2π) ˜ ω−π)} ˜ sin πα ei{ω+α( π ∞ ρα × (ρei ω˜ − A)−1 dρ. ˜ (λ − ρ α eiα ω˜ )(λ − ρ α eiα(ω−2π) ) 0

Rα (λ) = −

From this representation we have

∞

Mω˜ ρ α−1 dρ ˜ |λ − ρ α eiα ω˜ ||λ − ρ α eiα(ω−2π) | 0 ∞ α−1 Mω˜ sin πα ρ dρ ≤ iφ α iα ω ˜ i(φ−2π) ˜ π|λ| |e − ρ e ||e − ρ α eiα(ω−2π) | 0 ∞ α−1 ρ Mω˜ sin πα = 2α α π|λ| 0 ρ − 2ρ cos(φ − α ω) ˜ +1

Rα (λ) ≤

sin πα π

7 Fractional Powers of Linear Operators

×

97

dρ ρ 2α

− 2ρ α cos[α(ω˜ − 2π) − (φ

− 2π)] + 1

.

Due to (2.114), cos(φ − α ω) ˜ < 1 and cos[α(ω˜ − 2π) − (φ − 2π)] < 1. Therefore, C for λ ∈ / Σαω . we conclude that Rα (λ) ≤ |λ| Theorem 2.23 For 0 < α < 1, Aα is a sectorial operator of X with angle ≤ αωA . Proof To complete the proof, it suffices to notice that ω , together with ω, can be taken arbitrarily close to ωA . We now know that Aα is a sectorial operator and its resolvent is given by (2.113). Then, verification of (2.112) is immediate. The proof may be left to the reader. Remark 2.8 For λ < 0, we can obtain more precise estimates. Indeed, if φ = ω˜ = π , then ∞ ρ α−1 dρ (λ − Aα )−1 ≤ Mπ sin πα π|λ| ρ 2α − 2ρ α cos π(1 − α) + 1 0 ∞ dρ Mπ Mπ sin πα = = , λ < 0, 2 2 πα|λ| |λ| (ρ + cos πα) + sin πα 0 (2.115) where Mπ is the constant appearing in (2.2) with ω = π .

7.4 Moment Inequality This subsection is devoted to showing the moment inequality of the fractional powers. For 0 < θ < 1, let Aθ be the fractional powers of a sectorial operator A. Let Mπ π be the constant in (2.2) with ω = π , i.e., (λ − A)−1 ≤ M |λ| for λ < 0. Applying θ−1 (2.108) to A , we have for U ∈ D(A), sin(πθ ) ∞ θ−1 Aθ U = Aθ−1 AU = − ρ (ρ + A)−1 AU dρ. (2.116) π 0 Here,

0

R

ρ θ−1 A(ρ + A)−1 U dρ ≤

R

ρ θ−1 A(ρ + A)−1 U dρ

0

≤ (Mπ + 1) 0

R

ρ θ−1 dρ U =

Mπ + 1 θ R U , θ

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2 Sectorial Operators

where 0 < R < ∞ is an arbitrary number. Similarly, ∞ ∞ θ−1 −1 ρ (ρ + A) AU dρ ρ θ−1 (ρ + A)−1 AU dρ ≤ R

R

≤ Mπ

∞

Mπ θ−1 R AU . 1−θ

ρ θ−2 dρAU =

R

Therefore,

sin(πθ ) U θ AU θ−1 A U ≤ , (Mπ + 1) R + R π θ 1−θ θ

0 < R < ∞.

The right-hand side hits the minimum at R = AU /U . Hence, Aθ U ≤

sin(πθ ) (Mπ + 1)AU θ U 1−θ , πθ(1 − θ )

U ∈ D(A).

(2.117)

This inequality is called a moment inequality for Aθ . If we put U = A−1 F , then we have another moment inequality of the form Aθ−1 F ≤

sin(πθ ) (Mπ + 1)F θ A−1 F 1−θ , πθ(1 − θ )

F ∈ X.

(2.118)

Let 0 ≤ α < β < γ ≤ 1. We know that Aγ −α is also a sectorial operator. So, applying the moment inequality to Aγ −α with θ = γβ−α −α , we obtain that Aβ−α V ≤

2 (sin π(β−α) γ −α )(γ − α)

π(γ − β)(β − α)

β−α

γ −β

(Mπ + 1)Aγ −α V γ −α V γ −α .

Putting V = Aα U , we conclude that Aβ U ≤

2 (sin π(β−α) γ −α )(γ − α)

π(γ − β)(β − α) β−α

(Mπ + 1) γ −β

× Aγ U γ −α Aα U γ −α , Remark 2.9 We remark that the function θ=

1 2

and hence

sin(πθ) πθ(1−θ)

≤

sin(πθ) πθ(1−θ)

U ∈ D(Aγ ).

(2.119)

for 0 < θ < 1 hits its maximum at

4 π.

Remark 2.10 The generalized moment inequality (2.119) can be shown for any triplet of exponents 0 ≤ α < β < γ < ∞.

7.5 Comparison of Domains of Fractional Powers Let A be a sectorial operator of a Banach space X. Let Aθ , 0 < θ < 1, be the fractional powers of A. We are concerned with comparing D(Aθ ) with some other spaces.

7 Fractional Powers of Linear Operators

99

Let us introduce the spaces " ! Dθ (A) = U ∈ X; sup ρ θ A(ρ + A)−1 U < ∞ ,

0 ≤ θ ≤ 1.

(2.120)

0 θ ), we observe that ∞ θ −1 −ν ν−θ −1 ρ (ρ + B) [A − B]A A (ρ + A) U dρ 1 ∞ ρ θ−1 ρ max{ν−θ−1,−1} dρU . ≤ (Mπ + 1)2 (1 + N ) 1

Therefore, [Aθ − B θ ]A−θ U ≤ C(A−θ U + U ),

U ∈ D(A).

The denseness of D(A) in X and closedness of B θ yield that this estimate holds true for any F ∈ X. It then follows that D(B θ ) ⊃ R(A−θ ) = D(Aθ ).

7.6 Fractional Powers of Adjoint Operators Let X be a reflexive Banach space, and let X ∗ be the adjoint space of X. Let {X, X ∗ } be an adjoint pair. Let A be a sectorial operator of X, and let A∗ be its adjoint operator. By Theorem 1.19, A∗ is seen to be a sectorial operator of X ∗ . So, their fractional powers A−z and A∗(−z) are defined for Re z > 0 by the integral formula (2.106). We can then easily prove that A∗(−z) = A(−z)∗ for Re z > 0 (by the same argument as for (2.100)). Considering these inverses, we prove that A∗z = Az∗ for Re z > 0. In particular, for real powers, we see that A∗x = Ax∗ ,

−∞ < x < ∞.

(2.126)

7.7 Fractional Powers and Analytic Semigroups Let A be a sectorial operator of X satisfying (2.92) and (2.93). As verified in Sect. 5, −A generates an analytic semigroup e−zA on X. By Proposition 2.4, R(e−tA ) ⊂ D(Aθ ) for any t > 0 and any 0 < θ < ∞. In this subsection, we shall investigate various properties concerning Aθ e−tA . We first estimate the norm of Aθ e−tA . Consider the operator defined by 1 θ E (t) = λθ e−tλ (λ − A)−1 dλ, 0 < t < ∞, 0 < θ < ∞, 2πi Γ

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2 Sectorial Operators

where Γ is the integral contour given by (2.107) but with ωA < ω < π2 . Then, by arguing in a similar way as for (2.111), it is easily verified that A−θ E θ (t) = E θ (t)A−θ = e−tA . Hence, E θ (t) = Aθ e−tA = e−tA Aθ , i.e., 1 λθ e−tλ (λ − A)−1 dλ, 0 < t < ∞, 0 < θ < ∞. (2.127) Aθ e−tA = 2πi Γ We here shift Γ to Γω : λ = ρe±iω , 0 ≤ ρ < ∞ (i.e., δ → 0). Then, we obtain that ∞ |λ|θ−1 e−t Re λ |dλ| = Mω ρ θ−1 e−tρ cos ω dρ Aθ e−tA ≤ Mω 0

Γw

= Mω (cos ω)−θ (θ )t −θ ,

0 < t < ∞, 0 < θ < ∞,

where (·) denotes the gamma function. Let now 0 < θ ≤ 1. We have t t 1−θ −τ A ≤ Cθ [e−tA − 1]A−θ ≤ A e dτ τ θ−1 dτ ≤ Cθ t θ , 0

0

(2.128)

0 < t < ∞. (2.129)

Meanwhile, we can verify for each 0 < θ ≤ 1 that, as t → 0, t θ Aθ e−tA converges to 0 strongly on X.

(2.130)

Indeed, if U ∈ D(Aθ ), it is clear that t θ e−tA Aθ U → 0 in X as t → 0. Then, for general F ∈ X, the convergence is verified by the uniform boundedness of t θ Aθ e−tA due to (2.128) and the denseness of D(Aθ ) in X. Similarly, for each 0 ≤ θ < 1, as t → 0, t −θ [e−tA − 1]A−θ converges to 0 strongly on X.

(2.131)

Indeed, if U ∈ D(A1−θ ), then t −θ [e−tA − 1]A−θ U = t −θ [e−tA − 1]A−1 A1−θ U → 0 in X as t → 0 due to (2.129). Then, for general F ∈ X, the convergence is verified by the uniform boundedness of t −θ [e−tA − 1]A−θ due to (2.129) and the denseness of D(A1−θ ) in X. Using the properties obtained above, we will show the following result. Theorem 2.27 Let 0 < β ≤ 1 and U0 ∈ D(Aβ ). Then, for any σ such that 0 < σ < β ≤ 1 and any 0 < T < ∞, Ae−tA U0 ∈ Fβ,σ ((0, T ]; X) with the estimate Ae−tA U0 Fβ,σ ((0,T ];X) ≤ CT Aβ U0 X . Proof In fact, since t 1−β Ae−tA U0 = t 1−β A1−β e−tA Aβ U0 ,

(2.132)

7 Fractional Powers of Linear Operators

103

(2.130) yields that lim t 1−β Ae−tA U0 = 0 when 0 < β < 1,

t→0

lim t 1−β Ae−tA U0 = AU0

t→0

when β = 1.

Therefore, t 1−β Ae−tA U0 has a limit as t → 0. By (2.128) and (2.129), s 1−β+σ A[e−tA − e−sA ]U0 [e−(t−s)A − 1]A−σ ≤ (t − s)σ (t − s)σ × s 1−β+σ A1−β+σ e−sA (Aβ U0 ) ≤ CT Aβ U0 ,

0 < s < t ≤ T.

Therefore, (1.6) is fulfilled. In a similar way, s 1−β+σ A[e−tA − e−sA ]U0 ≤ CT s 1−β+σ A1−β+σ e−sA (Aβ U0 ). (t − s)σ Hence, by (2.130), lim sup

t→0 0<s 0. Hence, by (1.10), e−tAU = U (t).

1.3 Perturbation for Generators of Analytic Semigroups We shall consider some perturbation of generators of analytic semigroup. Let A be a sectorial operator of a Banach space X with angle ωA < π2 . Let B a linear operator of X. For some exponent 0 < ν ≤ 1, we assume that B is dominated by A1−ν , where A1−ν is the fractional power of A, in the sense that D(A1−ν ) ⊂ D(B), and the estimate BAν−1 ≤ D

(3.8)

holds with some constant D > 0. Let ω > ωA . Since D(A) ⊂ D(B), we have

λ − (A + B) = 1 − B(λ − A)−1 (λ − A)

for λ ∈ / Σω .

Here, B(λ − A)−1 ≤ BAν−1 A1−ν (λ − A)−1 . In addition, by the moment inequality (2.117) (with θ = 1 − ν), 1−ν A (λ − A)−1 ≤ C A(λ − A)−1 1−ν (λ − A)−1 ν ≤ C|λ|−ν . Therefore, if λ ∈ / Σω and |λ| ≥ R and if R > 0 is fixed sufficiently large, then λ ∈ ρ(A + B), and the resolvent is given by

−1 (λ − (A + B))−1 = (λ − A)−1 1 − B(λ − A)−1 . This then means that A + B satisfies (3.2) and (3.3) with the ω fixed above and β = −(sin ω)−1 R. We thus obtain the following theorem. Theorem 3.3 Let A be a sectorial operator with angle ωA < π2 . If a linear operator B satisfies (3.8), then −(A + B) satisfies (3.2) and (3.3) with any angle ω > ωA . So, A + B generates an analytic semigroup e−z(A+B) .

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3 Linear Evolution Equations

2 Cauchy Problems 2.1 Construction of Solutions Let us consider the Cauchy problem for a linear evolution equation dU 0 < t ≤ T, dt + AU = F (t), U (0) = U0 ,

(3.9)

in a Banach space X, where 0 < T < ∞ is fixed time. Here, A is a sectorial operator of X with angle ωA < π2 , that is, A satisfies (2.92) and (2.93). Throughout this section, the notation C will stand for a universal constant which is determined in each occurrence by the exponents and constants in (2.92)–(2.93) and by A−1 in a specific way. The function F is a given external force function defined in (0, T ] with values in X. We assume that F belongs to F ∈ Fβ,σ ((0, T ]; X),

0 0. So, it suffices only to show that Aβ U (t) is continuous at t = 0. We use the expression A [U (t) − U0 ] = [e−tA − 1]Aβ U0 +

β

t

Aβ e−(t−τ )A [F (τ ) − F (t)] dτ

0

+ [1 − e

−tA

]A

β−1

F (t)

= J1 (t) + J2 (t) + J3 (t).

2 Cauchy Problems

127

Clearly, limt→0 J1 (t) = 0 in X. In addition, it is seen that

t

J2 (t)X ≤ C

(t − τ )σ −β τ β−σ −1 dτ sup

0

0 0 as an L(X)-valued function. Hence, the proof of proposition will be completed if we verify the following lemma.

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3 Linear Evolution Equations

Lemma 3.2 For 0 ≤ θ ≤ 2, A(t)θ e−τ A(t) − A(s)θ e−τ A(s) ≤ Cτ ν−θ−1 |t − s|μ ,

τ > 0, 0 ≤ s, t ≤ T .

Proof Notice the formula (λ − A(t))−1 − (λ − A(s))−1

= −A(t)(λ − A(t))−1 A(t)−1 − A(s)−1 A(s)(λ − A(s))−1 . Then, by the same techniques as for (3.32), we observe that (λ − A(t))−1 − (λ − A(s))−1 ≤ C|λ|−ν |t − s|μ , λ ∈ / Σω , 0 ≤ s, t ≤ T . (3.33) We use (2.127) and write A(t)θ e−τ A(t) − A(s)θ e−τ A(s) =

1 2πi

λθ e−τ λ (λ − A(t))−1 − (λ − A(s))−1 dλ, Γω

where Γω : λ = ρe±iω , 0 ≤ ρ < ∞, is an integral contour. Then, using (3.33), we obtain that A(t)θ e−τ A(t) − A(s)θ e−τ A(s) ≤ C |λ|θ−ν e−τ Re λ |dλ||t − s|μ Γω

≤ Cτ ν−θ−1 |t − s|μ . When θ = 0 and ν = 1, this integral may not be convergent in a neighborhood λ = 0. But in this case, we are able to shift the integral contour Γω in the same way as for proving (2.128). Then, the assertion of lemma is proved without any change. When θ = 0, the continuity of the semigroup is extended at τ = 0. In fact, we verify the following proposition. Proposition 3.2 e−τ A(t) is strongly continuous on X for τ ≥ 0 and 0 ≤ t ≤ T . Proof Obviously, it suffices to prove the result at τ = 0 only. We already know that −τ A(t) [e − 1]A(t)−ν ≤ Cτ ν , τ ≥ 0, 0 ≤ t ≤ T . Let U ∈ D(A(0)). Then, [e−τ A(t) − 1]U ≤ [e−τ A(t) − 1]A(t)−ν A(t)ν A(0)−1 A(0)U ≤ CA(0)U τ ν ,

τ ≥ 0, 0 ≤ t ≤ T .

Furthermore, the denseness of D(A(0)) in X, together with the uniform boundedness of the norms e−τ A(t) − 1, yields that e−τ A(t) F → F in X as τ → 0 for every F ∈ X, the convergence being uniform in 0 ≤ t ≤ T .

4 Nonautonomous Abstract Evolution Equations

137

4.3 Structural Assumptions and Yosida Approximation Let a family of sectorial operators A(t) satisfy (3.27), (3.28), (3.29), (3.30), and (3.31). For each n = 1, 2, 3, . . . , let An (t), 0 ≤ t ≤ T , be a family of Yosida approximations of A(t). The purpose of this subsection is to show that An (t), 0 ≤ t ≤ T , also satisfy these conditions uniformly in n; especially ω, μ, and ν can be the same as for A(t). As already verified by (2.84) and (2.86), σ (An (t)) ⊂ Σω , and the estimate (λ − An (t))−1 ≤ M/|λ|, ˜ (3.34) λ∈ / Σω , 0 ≤ t ≤ T , holds with some constant M˜ ≥ 1 determined by M and ω only. In the meantime, it is possible to verify from (3.30) that

An (t)ν An (t)−1 − An (s)−1 ≤ N˜ |t − s|μ , 0 ≤ s, t ≤ T .

(3.35)

In fact, since An (t)−1 = n−1 + A(t)−1 due to (2.81), we have An (t)−1 − An (s)−1 = A(t)−1 − A(s)−1 . Then, (3.35) follows as a consequence of the two propositions presented below. Proposition 3.3 Let 0 < θ ≤ 1. For n = 1, 2, 3, . . . , it holds that An (t)θ A(t)−θ = (1 + n−1 A(t))−θ ,

0 ≤ t ≤ T.

(3.36)

Proof When θ = 1, the assertion is clear. So, we consider the case where 0 < θ < 1 only. Let us prove that A(t)−θ = An (t)−θ (1 + n−1 A(t))−θ . From (λ − (1 + n−1 A(t)))−1 = n(n(λ − 1) − A(t))−1 it is seen that σ (1 + −1 n A(t)) ⊂ Σω and the estimate (λ − (1 + n−1 A(t)))−1 ≤

M M ≤ , |λ − 1| |λ| sin ω

λ∈ / Σω ,

(3.37)

holds. This means that 1 + n−1 A(t) are also sectorial operators with angles < ω. Their fractional powers are then given by 1 (1 + n−1 A(t))−θ = λ−θ (λ − (1 + n−1 A(t)))−1 dλ 2πi Γλ 1 λ−θ n(n(λ − 1) − A(t))−1 dλ. = 2πi Γλ Since 0 < θ < 1, we can take an integral contour which passes the origin. Here, we choose Γλ : λ = re±iω , 0 ≤ r < ∞. Then, change of variable η = n(λ − 1) yields that 1 (1 + n−1 η)−θ (η − A(t))−1 dη, (1 + n−1 A(t))−θ = 2πi Γη where Γη = Γλ − n, and the branch of the function (1 + n−1 η)−θ is C − (−∞, −n].

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3 Linear Evolution Equations

In the meantime, we observe by (2.81) that −1 λ−θ 1 nλ dλ, An (t)−θ = (n + A(t)) − A(t) 2πi Γλ n − λ n−λ where Γλ is the same contour as above. We here use the integral formula −1 −1 nλ 1 nλ = (η − A(t))−1 dη , λ ∈ Γλ , − A(t) −η n−λ 2πi Γη n − λ

choosing a contour Γη = Γω ∪ Γδ such that Γω : η = re±iω , δ ≤ r < ∞, and Γδ : η = δeiϕ , −ω ≤ ϕ ≤ ω , where ω is an angle such that ωA(t) < ω < ω, and δ is a radius such that 0 < δ < A(t)−1 −1 . Then, on D(A(t)), we obtain that −1 λ−θ 1 nλ (η − A(t))−1 dλ dη (n + A(t)) − η An (t)−θ = (2πi)2 Γλ Γη n − λ n − λ 1 1 nη −θ = (η − A(t))−1 dη (n + A(t)). 2πi Γη n + η n + η Noting that

nη n+η

1 2πi

∈ Σω for η ∈ Γη , we used the formula Γλ

λ−θ dλ = λ − nη (n + η )−1

nη n + η

−θ ,

η ∈ Γη .

nη −θ is given by C − [−n, 0]. We easily notice that the branch of the function n+η We now consider the product 1 nη −θ 1 (η − A(t))−1 dη 2πi Γη n + η n + η 1 × (1 + n−1 η)−θ (η − A(t))−1 dη. 2πi Γη Then, it is deduced by the same techniques as for (2.98) and (2.111) (cf. also (16.4)) that this product is equal to (1 + n−1 λ)−θ nλ −θ 1 (λ − A(t))−1 dλ. 2πi Γλ n+λ n+λ nλ −θ n −θ −θ nλ Furthermore, we observe that n+λ = = n+λ λ for λ ∈ Γλ , since arg n+λ arg(nλ) − arg(n + λ) and −π < arg(nλ) − arg(n + λ) < π . We therefore arrive at (1 + n−1 λ)−θ nλ −θ 1 (λ − A(t))−1 dλ = A(t)−θ (n + A(t))−1 . 2πi Γλ n+λ n+λ Hence, we proved that A(t)−θ = An (t)−θ (1 + n−1 A(t))−θ .

5 Evolution Operators

139

Proposition 3.4 Let 0 < θ ≤ 1. For n = 1, 2, 3, . . . , it holds that An (t)θ A(t)−θ ≤ 2 M + 1 M θ , 0 ≤ t ≤ T . sin ω

(3.38)

In addition, for each t , An (t)θ A(t)−θ converges to 1 strongly on X as n → ∞. Proof Applying the moment inequality (2.118) to 1 + n−1 A(t), we have (1 + n−1 A(t))−θ ≤ 2 M + 1 (1 + n−1 A(t))−1 θ ≤ 2 M + 1 M θ sin ω sin ω due to (3.37). Hence, (3.38) is an immediate consequence of (3.36). The strong convergence of An (t)θ A(t)−θ to 1 is first verified on the domain D(A(t)1−θ ). Indeed, it is seen that An (t)θ A(t)−θ U = An (t)θ−1 (1 + n−1 A(t))−1 A(t)1−θ U → U, U ∈ D A(t)1−θ . Then, the convergence on X is verified by the uniform boundedness of An (t)θ A(t)−θ and the denseness of D(A(t)1−θ ) in X.

5 Evolution Operators In this section, we shall construct under (3.27), (3.28), (3.29), (3.30), and (3.31) the evolution operator U (t, s) for the family of sectorial operators A(t). For each n = 1, 2, 3, . . . , let Un (t, s), 0 ≤ s ≤ t ≤ T , be the evolution operator for the family of Yosida approximations An (t), 0 ≤ t ≤ T , of A(t). In fact, Un (t, s) is obtained as a unique solution to the Cauchy problem of the abstract differential equation dU n dt + An (t)Un = 0, s ≤ t ≤ T , Un (s) = 1, in the space L(X); note that An (·) ∈ C([0, T ]; L(X)) (due to (3.32) and (3.33)). Obviously, the problem is equivalent to the integral equation t An (τ )Un (τ ) dτ, s ≤ t ≤ T . Un (t) = 1 + s

Then, the solution is constructed by the recurrence method as in the proof of Theorems 1.29 and 1.30. The evolution operator Un (t, s) is a continuous function defined for 0 ≤ s ≤ t ≤ T with values in L(X). By the definition, Un (t, s) enjoys the generalized semigroup property Un (t, r)Un (r, s) = Un (t, s), 0 ≤ s ≤ r ≤ t ≤ T , (3.39) 0 ≤ s ≤ T. Un (s, s) = 1,

140

3 Linear Evolution Equations

In addition, Un (t, s) is differentiable both in t and s with the derivatives ∂ Un (t, s) = −An (t)Un (t, s), ∂t ∂ Un (t, s) = Un (t, s)An (s). ∂s

(3.40) (3.41)

Verification of (3.41) is not direct. We have to consider another family of operators U˜ n (t, s), 0 ≤ s ≤ t ≤ T , where U˜ n (t, s) is a solution to the backward Cauchy problem ˜ d Un ˜ ds − Un An (s) = 0, 0 ≤ s ≤ t, U˜ n (t) = 1, for 0 < t ≤ T . Clearly, U˜ n (t, s) satisfies (3.41). Moreover, Un (t, s) = U˜ n (t, s) for every 0 ≤ s ≤ t ≤ T . Indeed, since ∂ ˜ Un (t, τ )Un (τ, s) = 0 for all τ in (s, t), ∂τ it is deduced that U˜ n (t, τ )Un (τ, s) is constant in τ ; hence, taking τ = s and t, we conclude that Un (t, s) = U˜ n (t, s) for every 0 ≤ s ≤ t ≤ T .

5.1 Integral Equations We first derive two integral equations for Un (t, s). By using (3.41) we have t ∂ [Un (t, τ )e−(τ −s)An (s) ] dτ Un (t, s) − e−(t−s)An (s) = − ∂τ s t

Un (t, τ )An (τ ) An (τ )−1 − An (s)−1 = s

× An (s)e−(τ −s)An (s) dτ. Operating An (s)1−ν to this equality from the right-hand side and putting Vn (t, s) = Un (t, s)An (s)1−ν , 0 ≤ s ≤ t ≤ T ,

Dn (t, s) = An (t)ν An (t)−1 − An (s)−1 , 0 ≤ s ≤ t ≤ T , we obtain the first integral equation Vn (t, s) = An (s)1−ν e−(t−s)An (s) t + Vn (t.τ )Dn (τ, s)An (s)2−ν e−(τ −s)An (s) dτ. s

(3.42)

5 Evolution Operators

141

To obtain the other one, we use (3.40). Then, Un (t, s) − e−(t−s)An (t) t

=− An (t)e−(t−τ )An (t) An (t)−1 − An (τ )−1 An (τ )Un (τ, s) dτ. s

Operating An (t) to this equality from the left-hand side and putting Wn (t, s) = An (t)Un (t, s) − An (t)e−(t−s)An (t) ,

0 ≤ s ≤ t ≤ T,

we obtain the integral equation t Wn (t, s) = Rn (t, s) − An (t)2−ν e−(t−τ )An (t) Dn (t, τ )Wn (τ, s) dτ,

(3.43)

s

where

Rn (t, s) = −

t

An (t)2 e−(t−τ )An (t)

s

× [An (t)−1 − An (τ )−1 ]An (τ )e−(τ −s)An (τ ) dτ.

(3.44)

Secondly, we shall derive some norm estimates for Un (t, s) which hold uniformly in n. From (2.133) and (3.35) it is readily seen that the integral kernels of (3.42) satisfy An (s)1−ν e−(t−s)An (s) ≤ C(t − s)ν−1 , Dn (t, s)An (s)2−ν e−(t−s)An (s) ≤ C(t − s)μ+ν−2 , respectively. Then, we can apply Theorem 1.30 to (3.42). As a direct consequence, we have the following proposition. Proposition 3.5 The operator Un (t, s)An (s)1−ν satisfies Un (t, s)An (s)1−ν ≤ C(t − s)ν−1 , 0 ≤ s < t ≤ T ,

(3.45)

C > 0 being independent of n. Concerning the integral kernels of (3.43), we readily verify from (2.133) and (3.35) that An (t)2−ν e−(t−s)An (t) Dn (t, s) ≤ C(t − s)μ+ν−2 . (3.46) As for Rn (t, s), however, we need more careful calculations. Lemma 3.3 Rn (t, s) satisfies the estimates Rn (t, s) ≤ C(t − s)μ+ν−2 , C > 0 being independent of n.

0 ≤ s < t ≤ T,

(3.47)

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3 Linear Evolution Equations

Proof Because of

An (t)2 e−(t−τ )An (t) An (t)−1 − An (τ )−1 An (τ )e−(τ −s)An (τ ) ≤ C(t − τ )μ+ν−2 (τ − s)−1 , we cannot obtain the desired estimate by direct estimation of the integral (3.44). We have to decompose the integral suitably into several terms which admit uniform estimates. As a matter of fact, we write Rn (t, s) in the form t

An (t)2 e−(t−τ )An (t) An (τ )−1 − An (t)−1 An (τ )e−(τ −s)An (τ ) dτ Rn (t, s) = r

r

+

An (t)2 e−(t−τ )An (t) [e−(τ −s)An (τ ) − e−(τ −s)An (s) ] dτ

s r

+

An (t)e−(t−τ )An (t) [An (s)e−(τ −s)An (s) − An (τ )e−(τ −s)An (τ ) ] dτ

s

+ An (t)e−(t−r)An (t) [e−(r−s)An (s) − e−(r−s)An (t) ], where r =

t+s 2 .

(3.48)

Here, we used the formula r An (t)e−(t−τ )An (t) [An (t) − An (s)]e−(τ −s)An (s) dτ s

= An (t)e−(t−r)An (t) [e−(r−s)An (s) − e−(r−s)An (t) ]. As the family of Yosida approximations An (t) also satisfies the same structural assumptions, we can apply Lemma 3.2 to An (t). In other words, for 0 ≤ θ ≤ 2, it holds that An (t)θ e−τ An (t) − An (s)θ e−τ An (s) ≤ Cτ ν−θ−1 |t − s|μ , τ > 0, 0 ≤ s, t ≤ T . (3.49) This estimate is then available to estimate the decomposed terms in the right-hand side of (3.48). Indeed, by (2.133), (3.35), and (3.49) (with θ = 0, 1), we can obtain estimate (3.47). The details of the proof are however left to the reader. It is now ready to apply Theorem 1.29 to (3.43). In fact, we obtain the following estimate. Proposition 3.6 The operator An (t)Un (t, s) satisfies An (t)Un (t, s) ≤ C(t − s)−1 ,

0 ≤ s < t ≤ T,

(3.50)

C > 0 being independent of n. Proof Applying Theorem 1.29 to (3.43), we immediately obtain that Wn (t, s) ≤ C(t − s)μ+ν−2 .

(3.51)

5 Evolution Operators

143

Then, since An (t)e−(t−s)An (t) ≤ C(t − s)−1 , we conclude the estimate of proposition.

5.2 Convergence of Un (t, s) In this subsection, we intend to see convergence of Un (t, s) as n → ∞. To this end, we will utilize the dominated convergence theorem, Theorem 1.31, to the integral equation (3.43). From (2.102) and Proposition 3.4, we observe that, as n → ∞, An (t)2−ν e−(t−s)An (t) Dn (t, s) → A(t)2−ν e−(t−s)A(t) D(t, s) strongly on X, where

D(t, s) = A(t)ν A(t)−1 − A(s)−1 , As

0 ≤ s < t ≤ T.

A(t)2−ν e−(t−s)A(t) D(t, s) = A(t)2 e−(t−s)A(t) A(t)−1 − A(s)−1 ,

the limit function A(t)2−ν e−(t−s)A(t) D(t, s) is continuous for 0 ≤ s < t ≤ T in L(X). For verifying convergence of Rn (t, s), we return to the decomposition (3.48). Then, by (2.102), Rn (t, s) is seen to converge strongly to the bounded linear operator t

R(t, s) = A(t)2 e−(t−τ )A(t) A(τ )−1 − A(t)−1 A(τ )e−(τ −s)A(τ ) dτ r

r

+

A(t)2 e−(t−τ )A(t) [e−(τ −s)A(τ ) − e−(τ −s)A(s) ] dτ

s r

+

A(t)e−(t−τ )A(t) [A(s)e−(τ −s)A(s) − A(τ )e−(τ −s)A(τ ) ] dτ

s

+ A(t)e−(t−r)A(t) [e−(r−s)A(s) − e−(r−s)A(t) ], where r = t+s 2 , for all 0 ≤ s < t ≤ T . Furthermore, by the same techniques as in the proof of Lemma 1.3, R(t, s) is verified to be continuous in L(X) for 0 ≤ s < t ≤ T . It is now ready to apply Theorem 1.31 to (3.43). Indeed, in view of (3.46) and (3.47), Theorem 1.31 immediately provides the following convergence result. Proposition 3.7 As n → ∞, Wn (t, s) converges to a bounded linear operator W (t, s)

(3.52)

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3 Linear Evolution Equations

strongly on X for all 0 ≤ s < t ≤ T . The limit W (t, s) is characterized as a solution to the integral equation t W (t, s) = R(t, s) − A(t)2−ν e−(t−τ )A(t) D(t, τ )W (τ, s) dτ. (3.53) s

Since Un (t, s) = e−(t−s)An (t) + An (t)−1 Wn (t, s), as a consequence of the proposition it follows that Un (t, s) is also strongly convergent. We then define the operator U (t, s) as the limit of Un (t, s), namely, as n → ∞, Un (t, s) → U (t, s) = e−(t−s)A(t) + A(t)−1 W (t, s)

(3.54)

strongly on X for all 0 ≤ s < t ≤ T . We see from relation (3.54) that R(U (t, s)) ⊂ D(A(t)) and have A(t)U (t, s) = A(t)e−(t−s)A(t) + W (t, s). As a consequence, An (t)Un (t, s) → A(t)U (t, s) = A(t)e−(t−s)A(t) + W (t, s)

(3.55)

strongly on X for all 0 ≤ s < t ≤ T . If U ∈ D(A(s)1−ν ), then Un (t, s)An (s)1−ν U → U (t, s)A(s)1−ν U . But, since the norms Un (t, s)An (s)1−ν are uniformly bounded in n (due to (3.45)), we verify that U (t, s)A(s)1−ν has a bounded extension on X. Moreover, as n → ∞, Un (t, s)An (s)1−ν → U (t, s)A(s)1−ν

(3.56)

strongly on X for all 0 ≤ s < t ≤ T .

5.3 Basic Properties of U (t, s) For 0 ≤ s ≤ T , we define U (s, s) = 1. Then, by (3.39), U (t, s) also enjoys the generalized semigroup property U (t, s) = U (t, r)U (r, s), 0 ≤ s ≤ r ≤ t ≤ T , (3.57) U (s, s) = 1, 0 ≤ s ≤ T . The operator U (t, s) has the following properties. Proposition 3.8 A(t)U (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T satisfying the estimate A(t)U (t, s) ≤ C(t − s)−1 , with some constant C > 0.

0 ≤ s < t ≤ T,

(3.58)

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145

Proof As W (t, s) has been obtained as a solution to the integral equation (3.53), W (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T . This, together with Proposition 3.1 (θ = 1), then yields the continuity of A(t)U (t, s) in the uniform topology. Estimate (3.58) is an immediate consequence of (3.50) and (3.55). Proposition 3.9 U (t, s) is an L(X)-valued strongly continuous function for 0 ≤ s ≤ t ≤ T with the estimate U (t, s) ≤ C,

0 ≤ s ≤ t ≤ T,

(3.59)

with some constant C > 0. Proof From (3.42) we have Un (t, s) = e

−(t−s)An (s)

+

t

Vn (t, τ )Dn (τ, s)An (s)e−(τ −s)An (s) dτ.

s

Therefore, by (2.102), (3.35), and (3.45), Un (t, s) is estimated by t

Un (t, s) ≤ C + C (t − τ )ν−1 (τ − s)μ−1 dτ ≤ C 1 + (t − s)μ+ν−1 ≤ C. s

This, together with (3.54), then yields estimate (3.59). As U (t, s) = A(t)−1 A(t)U (t, s), Proposition 3.8, together with (3.32), yields the continuity of U (t, s) in L(X) for 0 ≤ s < t ≤ T . So, it suffices to verify that U (t, s) → 1 strongly as (t, s) → (r, r) with s ≤ t , where 0 ≤ r ≤ T . By the same reason as above, we have Un (t, s) − e−(t−s)An (s) ≤ C(t − s)μ+ν−1 . So, letting n → ∞, U (t, s) − e−(t−s)A(s) ≤ C(t − s)μ+ν−1 . Therefore, as (t, s) → (r, r), U (t, s) − e−(t−s)A(s) → 0. Then, the desired strong convergence of U (t, s) is deduced from Proposition 3.2. We finally show the differentiability of U (t, s) in t . Let 0 ≤ s < t ≤ T . In view of (3.50), let n → ∞ in the formula t+h −1 −1 h [Un (t + h, s) − Un (t, s)] = −h An (τ )Un (τ, s) dτ. t

By (3.55) we obtain that −1

h

−1

t+h

[U (t + h, s) − U (t, s)] = −h

A(τ )U (τ, s) dτ, t

146

3 Linear Evolution Equations

which shows that U (t, s) is differentiable in t in the uniform topology with the derivative ∂ U (t, s) = −A(t)U (t, s), 0 ≤ s < t ≤ T . (3.60) ∂t We here sum up the obtained results concerning the evolution operator. Theorem 3.8 Let A(t) be a family of sectorial operators of X satisfying the structural assumptions (3.27), (3.28), (3.29), (3.30), and (3.31). Then, there exists a unique evolution operator U (t, s) satisfying (3.57) and having the following properties. A(t)U (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T with estimate (3.58), and U (t, s) is an L(X)-valued strongly continuous function for 0 ≤ s ≤ t ≤ T with estimate (3.59). Furthermore, for t > s, U (t, s) is differentiable in t with the derivative (3.60). Proof It remains only to show the uniqueness of the evolution operator. Let U˜ (t, s) be any other evolution operator for the family A(t). For s < τ ≤ t , we have ∂ Un (t, τ )U˜ (τ, s) = Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ, s). ∂τ Integrating this equality in τ on the interval [s + ε, t], where ε > 0 is a small positive number, we obtain that t U˜ (t, s) − Un (t, s + ε)U˜ (s + ε, s) = Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ, s) dτ. s+ε

Letting n → ∞, U˜ (t, s) − U (t, s + ε)U˜ (s + ε, s) = 0 (due to (2.89)). Finally, letting ε → 0, we conclude that U˜ (t, s) = U (t, s). Hence, the evolution operator is uniquely determined from A(t).

6 Cauchy Problems We consider the Cauchy problem dU dt + A(t)U = F (t),

0 < t ≤ T,

U (0) = U0 .

(3.61)

Here, A(t), 0 ≤ t ≤ T , is a family of sectorial operators of X which satisfies all the structural assumptions (3.27), (3.28), (3.29), (3.30), and (3.31). The function F is a given function on (0, T ] which belongs to the function space: F ∈ Fβ,σ ((0, T ]; X), The initial value U0 is taken in X.

0 < σ < β ≤ 1.

(3.62)

6 Cauchy Problems

147

Let U (t, s) be the evolution operator for the family of the operators A(t) constructed in the preceding section. We show the following existence and uniqueness result. Theorem 3.9 Under (3.27), (3.28), (3.29), (3.30), and (3.31), for any F belonging to (3.62) and any U0 ∈ X, there exists a unique solution U to (3.61) in the function space: U ∈ C([0, T ]; X) ∩ C1 ((0, T ]; X),

AU ∈ C((0, T ]; X),

with the estimate dU U (t) + t (t) + tA(t)U (t) ≤ C(U0 + F Fβ,σ ), dt Moreover, U is necessarily given by the formula t U (t, τ )F (τ ) dτ, U (t) = U (t, 0)U0 +

(3.63)

0 < t ≤ T . (3.64)

0 ≤ t ≤ T.

(3.65)

0

Proof In order to prove this theorem, we will go back to the stage of constructing the evolution operator U (t, s). Let An (t) be the Yosida approximations for A(t) for each n = 1, 2, 3, . . . , and let Un (t, s) be the evolution operator for An (t). Using the operator Un (t, s), we define an X-valued function of the form t Un (t) = Un (t, 0)U0 + Un (t, τ )F (τ ) dτ, 0 ≤ t ≤ T . 0

In view of (3.54), as n → ∞, this function converges to the function U given by (3.65) in X pointwise. It is easy to see that U is an X-valued continuous function on [0, T ]. By a direct calculation, we verify from (3.40) that Un is differentiable for t > 0 and the derivative is given by dUn (t) = −An (t)Un (t) + F (t), dt

0 < t ≤ T.

Therefore, for t > 0, −1

h

−1

[Un (t + h) − Un (t)] = h

t+h

[F (τ ) − An (τ )Un (τ )] dτ.

(3.66)

t

In the meantime, we consider the function An (t)Un (t). This function can be decomposed in the form t An (t)Un (t) =An (t)Un (t, 0)U0 + An (t)Un (t, τ )[F (τ ) − F (t)] dτ + 0

0 t

Wn (t, τ ) dτ F (t) + [1 − e−tAn (t) ]F (t).

148

3 Linear Evolution Equations

Here, we used the formula

t

An (t)e−(t−τ )An (t) dτ = 1 − e−tAn (t)

0

(which follows from (2.130)). By (1.8), (3.51), and (3.50), we then obtain the estimate An (t)Un (t) ≤ CF,U0 t −1 ,

0 < t ≤ T.

(3.67)

The constant CF,U0 > 0 depends on F and U0 . Furthermore, by (3.52) and (3.55), the function is seen to converge to the function W (t) = A(t)U (t, 0)U0 +

t

A(t)U (t, τ )[F (τ ) − F (t)] dτ

0

t

+

W (t, τ ) dτ F (t) + [1 − e−tA(t) ]F (t)

0

in X pointwise. The limit function is also verified to be an X-valued continuous function for 0 < t ≤ T by arguing in a similar way as in the proof of Lemma 1.3. Since Un (t) = An (t)−1 An (t)Un (t), it follows that U (t) = A(t)−1 W (t); therefore, U (t) ∈ D(A(t)) and A(t)U (t) = W (t) for each 0 < t ≤ T . In view of (3.67), let n → ∞ in (3.66). Then, h−1 [U (t + h) − U (t)] = h−1

t+h

[F (τ ) − A(τ )U (τ )] dτ,

0 < t ≤ T.

t

Hence, dU (t) = −A(t)U (t) + F (t), dt

0 < t ≤ T,

which shows that U is a C1 function on (0, T ] and satisfies the equation of (3.61). It is easy to verify that U satisfies (3.64). It now remains only to show the uniqueness of solution. Let U˜ be any other solution in the space (3.63). Let 0 < ε < t ≤ T . By (3.41), ∂ Un (t, τ )U˜ (τ ) = Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ ) + Un (t, τ )F (τ ). ∂τ So, U˜ (t) = Un (t, ε)U˜ (ε) +

t

Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ ) dτ

ε

t

+

Un (t, τ )F (τ ) dτ. ε

7 Conditions (3.29) and (3.30)

149

Letting n → ∞, we obtain that U˜ (t) = U (t, ε)U˜ (ε) +

t

U (t, τ )F (τ ) dτ. ε

Finally, letting ε → 0, we conclude that U˜ (t) = U (t) for every 0 ≤ t ≤ T .

Remark 3.3 We needed the derivative of Un (t, s) in s in establishing the uniqueness of solution to problem (3.61). On the other hand, we used the derivative of Un (t, s) in t in constructing the solution.

7 Conditions (3.29) and (3.30) The objective of this section is to see how one can verify conditions (3.29) and (3.30) in concrete examples. The case where ν = 1 is the most favorable one. In this case, since

(3.68) A(t) A(t)−1 − A(s)−1 = −[A(t) − A(s)]A(s)−1 , we can calculate directly the difference A(t) − A(s) in order to verify (3.30). On the other hand, when ν < 1, there is no convenient representation for the term A(t)ν [A(t)−1 − A(s)−1 ], for we have to consider the fractional power A(t)ν .

7.1 Case of Hilbert Spaces Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. We consider a family of sesquilinear forms a(t; U, V ), 0 ≤ t ≤ T , which are all defined on Z and satisfy (2.6) and (2.7) with some uniform constants M > 0 and δ > 0, respectively. In addition, we assume the Hölder condition |a(t; U, V ) − a(s; U, V )| ≤ N |t − s|μ U V ,

0 ≤ s, t ≤ T ; U, V ∈ Z, (3.69)

with some fixed exponent 0 < μ ≤ 1 and some constant N > 0. We already know (see Chap. 1) that, from the sesquilinear forms, sectorial operators A(t), 0 ≤ t ≤ T , of Z ∗ and equally sectorial operators A(t), 0 ≤ t ≤ T , of X are defined. Consider first the family of A(t)’s. By Theorem 1.24, D(A(t)) coincides with Z, namely, (3.29) is fulfilled with ν = 1. In addition, using the formula (3.68) for A(t), we observe that

A(t) A(t)−1 − A(s)−1 Φ, U

= − a t; A(s)−1 Φ, U − a s; A(s)−1 Φ, U , Φ ∈ Z ∗ , U ∈ Z. (3.70)

150

3 Linear Evolution Equations

So, (3.69) yields directly that

A(t) A(t)−1 − A(s)−1 Φ ≤ N δ −1 |t − s|μ Φ∗ , ∗

Φ ∈ Z∗.

Thus, under (3.69), A(t)’s satisfy (3.29) and (3.30) with ν = 1 and the μ.

(3.71)

Secondly, consider the family of A(t)’s. According to Theorem 2.32, we have D(A(t)∗θ ) ⊂ Z for any 12 < θ ≤ 1. Therefore, for any ν such that 0 < ν < 12 , we observe from (3.70) that

A(t)ν A(t)−1 − A(s)−1 F, G = − a t; A(s)−1 F, A(t)∗(ν−1) G

(3.72) − a s; A(s)−1 F, A(t)∗(ν−1) G , F, G ∈ X. Hence, (3.69) implies that

A(t)ν A(t)−1 − A(s)−1 F, G ≤ C|t − s|μ |F ||G|, namely,

A(t)ν A(t)−1 − A(s)−1 F ≤ N |t − s|μ |F |,

F ∈ X, G ∈ X,

F ∈ X.

Thus, under (3.69), A(t) satisfy (3.29) and (3.30) with any 0 < ν

0

7 Conditions (3.29) and (3.30)

151

uniformly with respect to the variable t, that c(x, t) satisfies condition (2.20), and that aij ∈ C([0, T ]; C1 (Ω)) ∩ Cμ ([0, T ]; Lp (Ω)), (3.74) c ∈ C([0, T ]; L∞ (Ω)) ∩ Cμ ([0, T ]; Lp (Ω)), with some exponents 0 < μ ≤ 1 and 1 < p < ∞. We already know that one can define, from the forms a(t; ·, ·), realizations

Ap (t) of the differential operators − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in the space Lp (Ω), 1 < p < ∞, under the Neumann-type boundary conditions ∂ν∂u = 0 on A(t) ∂Ω. We are concerned with the family of sectorial operators Ap (t), 0 ≤ t ≤ T , in the Banach space Lp (Ω). It is known by (2.77) and (2.78) that D(Ap (t)) = {u ∈ Hp2 (Ω); ∂ν∂u = 0 on ∂Ω} and A(t) uHp2 ≤ CAp (t)uLp ,

u ∈ D(Ap (t)).

(3.75)

Here, the constant C > 0 can be uniform with respect to t , because aij (·, t) vary only in some relatively compact subset of C1 (Ω). We consider also the adjoint operators Bp (t) = Ap (t)∗ (see Theorem 2.14) of Ap (t) with respect to the adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1, Bp (t) being determined by the adjoint forms a ∗ (t; ·, ·). As will be shown by Theorems 16.7 and 16.10 in Chap. 16, we have, in favorable cases, that D(Bp (t)θ ) ⊂ Hp2θ (Ω) for any 0 < θ ≤ 1, and uH 2θ ≤ C Bp (t)θ uL , p

p

u ∈ D Bp (t)θ ,

(3.76)

C > 0 being uniform for t . In the general case, according to (16.36), it is at least true that D(Bp (t)θ ) ⊂ Hp2θ (Ω) for any 0 < θ < θ ≤ 1 with the estimate uH 2θ ≤ Cθ ,θ Bp (t)θ uL , p

p

u ∈ D Bp (t)θ ,

(3.77)

Cθ ,θ > 0 being uniform for t . In what follows, let us consider the optimal case (3.76). For the general case, we shall have some weaker result, see Remark 3.4. ν Let 12 ≤ ν < p+1 2p . Then, by (16.36) again, we see that D(A(s)) ⊂ D(A(t) ) for all 0 ≤ s, t ≤ T ; hence, (3.29) is fulfilled. When p = 2, the formula (3.72) reads as

A2 (t)ν A2 (t)−1 − A2 (s)−1 f, g n [ai,j (t, x) − aij (s, x)]Di A2 (s)−1 f · Dj A2 (t)∗(ν−1) g dx =− i,j =1 Ω

−

[c(x, t) − c(s, x)]A2 (s)−1 f · A2 (t)∗(ν−1) g dx,

Ω

f ∈ L2 (Ω), g ∈ D(A2 (t)∗ ).

152

3 Linear Evolution Equations

For the general 1 < p < ∞, this formula yields that

Ap (t)ν Ap (t)−1 − Ap (s)−1 f, g L

p ×Lp

n

=−

[aij (t, x) − aij (s, x)]Di Ap (s)−1 f · Dj Bp (t)ν−1 g dx

i,j =1 Ω

[c(x, t) − c(s, x)]Ap (s)−1 f · Bp (t)ν−1 g dx,

− Ω

f ∈ Lp (Ω), g ∈ D(Bp (t)). Since 0 ≤ 2ν − 1 < p1 , we observe by (1.80) and (1.82) that [aij (t, x) − aij (s, x)]Di Ap (s)−1 f · Dj Bp (t)ν−1 g dx Ω ≤ [aij (t) − aij (s)]Di Ap (s)−1 f H˚ 2ν−1 Dj Bp (t)ν−1 g H 1−2ν p

p

≤ aij (t) − aij (s)L(H 1 ,H 2ν−1 ) p

p

Di Ap (s)−1 f

ν−1 g 1 Dj Bp (t)

Hp

Hp1−2ν

.

(3.78) In addition, by (3.75), Di Ap (s)−1 f Hp1 ≤ Cf Lp . By the boundedness of dif2(1−ν)

ferentiation Dj ∈ L(Hp (Ω), Hp1−2ν (Ω)) (due to Theorem 1.43) and estimate ν−1 (3.76), we have Dj Bp (t) gH 1−2ν ≤ Cν gLp . Finally, Lemma 3.4 below, top

gether with assumption (3.74), provides that ⎧ 2pμ(1−ν)/n ⎪ ⎨C|t − s| aij (t) − aij (s)L(H 1 ,Hp2ν−1 ) ≤ Cμ |t − s|2μ (1−ν) p ⎪ ⎩ C|t − s|2μ(1−ν)

if 1 < p < n, if p = n, if n < p < ∞,

where μ can be taken arbitrarily in such a way that 0 < μ < μ. In this way, under (3.74) and (3.76), we have shown that, for any condition (3.30) is fulfilled as ⎧ 2pμ(1−ν)/n ⎪ ⎨Cν |t − s|

ν −1 −1 Ap (t) Ap (t) − Ap (s) ≤ Cν,μ |t − s|2μ (1−ν) L(Lp ) ⎪ ⎩ Cν |t − s|2μ(1−ν) μ being an arbitrary exponent such that 0 < μ < μ.

1 2

≤ν

0. Let 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1. We write A(t)θ [e−τ A(t) − e−τ A(s) ]A(s)−ϕ 1 = e−τ λ A(t)θ−ν+1 (λ − A(t))−1 D(t, s)A(s)1−ϕ (λ − A(s))−1 dλ. 2πi Γ By assumption, it holds that −2 ≤ θ − ϕ − ν ≤ 0. In the case where −1 < θ − ϕ − ν, A(t)θ [e−τ A(t) − e−τ A(s) ]A(s)−ϕ ≤C |λ|θ−ϕ−ν e−τ Re λ |dλ| |t − s|μ ≤ Cτ ϕ−θ+ν−1 |t − s|μ . Γω

In the case where −2 ≤ θ − ϕ − ν ≤ −1, we can always use the shifting techniques to obtain (3.91). For 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1, we write

A(t)θ e−τ A(t) A(t)−ϕ − e−τ A(s) A(s)−ϕ 1 λ−ϕ e−τ λ A(t)θ−ν+1 (λ − A(t))−1 D(t, s)A(s)(λ − A(s))−1 dλ. = 2πi Γ When −1 < θ − ϕ − ν, (3.92) is verified by the same arguments as before. In the critical case where θ − ϕ − ν = −1, the shifting of integral contour is not available except the special case θ = ν and ϕ = 1. When θ − ϕ − ν < −1, we have |λ|θ−ϕ−ν e−τ Re λ |dλ| ≤ C(ϕ + ν − θ − 1)−1 . Γδ,ω Let 0 < ϕ ≤ 1. To see (3.83), we notice that A(t)U (t, s)A(s)−ϕ is a solution to the integral equation A(t)U (t, s)A(s)−ϕ = A(t)e−(t−s)A(t) A(s)−ϕ t − A(t)2−ν e−(t−τ )A(t) D(t, τ )A(τ )U (τ, s)A(s)−ϕ dτ. s

(3.93) Moreover, by (3.89), A(t)e−(t−s)A(t) A(s)−ϕ L(X) −(t−s)A(t) ≤ [A(t)e − A(s)e−(t−s)A(s) ]A(s)−ϕ L(X) + A(s)1−ϕ e−(t−s)A(s) L(X) ≤ C(t − s)ϕ−1 .

158

3 Linear Evolution Equations

Hence, (3.83) is verified for 0 < ϕ ≤ 1 and θ = 1. When ϕ = 0, (3.83) is already known. Let 0 ≤ θ < μ + ν and 0 < ϕ ≤ 1. From (3.93) it also follows that A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ t =− A(t)θ−ν+1 e−(t−τ )A(t) D(t, τ )A(τ )U (τ, s)A(s)−ϕ dτ. s

Therefore, A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ

L(X)

t

≤C

(t − τ )−θ+μ+ν−1 (τ − s)ϕ−1 dτ ≤ C(t − s)ϕ−θ+μ+ν−1 ,

s

and hence (3.84) is verified for 0 < ϕ ≤ 1, too. Let 0 ≤ ϕ ≤ 1 and 0 ≤ ϕ ≤ θ < μ + ν. The estimate just established, together with Lemma 4.1, yields that A(t)θ U (t, s)A(s)−ϕ − A(s)θ−ϕ e−(t−s)A(s) L(X) ≤ A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ L(X)

+ A(t)θ e−(t−s)A(t) − A(s)θ e−(t−s)A(s) A(s)−ϕ ≤ Cθ log (t − s)−1 + 1 (t − s)ϕ−θ+μ+ν−1 .

L(X)

(3.94)

Therefore, (3.83) holds. Let θ = ϕ = 0 in (3.84). Then, U (t, s) − e−(t−s)A(t) L(X) ≤ C(t − s)μ+ν−1 . This, together with (3.91) (θ = ϕ = 0), therefore yields (3.85) in the case k = 0. Similarly, let θ = ϕ = 1 in (3.84); then (3.85) of the case k = 1 is verified by (3.89) (θ = ϕ = 1). Equation (3.86) is also verified in the same manner. Indeed, for 0 < θ = ϕ < 1, A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−θ ≤ C(t − s)μ+ν−1 . L(X) This, together with (3.90) (θ = ϕ), then yields (3.86). For 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1, (3.84), together with (3.91), yields (3.87). We have thus established all the properties of U (t, s) to be shown. Remark 3.5 In some problems we may also need the property (3.82) for θ ≥ μ. In order to extend (3.82) to such an exponent θ , say θ = 1, we have to assume a dual condition of (3.30). That is, for some exponent 0 < ν ≤ 1, the operators

8 Maximal Regularity

159

[A(t)−1 − A(s)−1 ]A(s)ν have bounded extensions on X and satisfy the Hölder condition

A(t)−1 − A(s)−1 A(s)ν ≤ N |t − s|μ , 0 ≤ s, t ≤ T , (3.95) L(X) with another exponent 0 < μ ≤ 1 with the relation 1 < μ + ν .

(3.96)

Furthermore, we assume that X is a reflexive Banach space. In the adjoint space X ∗ , let us consider the family of sectorial operators A∗ (t) ≡ A(T − t)∗ , 0 ≤ t ≤ T . By Theorem 2.32, the family of A∗ (t)’s is easily seen to satisfy (3.27) and (3.28). In addition, by (2.126), (3.95), together with (3.96), implies that A∗ (t) satisfies (3.29), (3.30), and (3.31) with μ = μ and ν = ν . Consequently, there exists an evolution operator U ∗ (t, s), 0 ≤ s ≤ t ≤ T , for A∗ (t). Then, it is easily verified that U ∗ (t, s) = U (T − s, T − t)∗ for any 0 ≤ s ≤ t ≤ T . Indeed, for τ such that s ≤ τ ≤ t, we have ∂ [U (T − τ, T − t)∗ U ∗ (τ, s)] = [−A(T − τ )U (T − τ, T − t)]∗ U ∗ (τ, s) ∂τ + U (T − τ, T − t)∗ A∗ (τ )U ∗ (τ, s) = 0. Since A∗ (t)U ∗ (t, s) = A(T − t)∗ U (T − s, T − t)∗ = [U (T − s, T − t)A(T − t)]∗ , estimate (3.58) for A∗ (t) shows that U (T − s, T − t)A(T − t)θ L(X) ≤ C(t − s)−θ for 0 < θ < μ + ν . We have thus seen that, when X is reflexive, (3.95), together with (3.96), yields property (3.82) for μ ≤ θ < μ + ν , too.

8.2 Maximal Regularity We return to problem (3.61). Utilizing the refined properties of U (t, s), we shall show the maximal regularity of solutions. Theorem 3.10 Under (3.27), (3.28), (3.29), (3.30), and (3.31), let F ∈ Fβ,σ ((0, T ]; X), and let

0 < σ < min{β, μ + ν − 1},

U0 ∈ D A(0)β ,

0 < β ≤ 1.

(3.97)

(3.98)

Then, the solution U to (3.61) given by (3.65) possesses the following regularity:

160

3 Linear Evolution Equations

A(t)β U ∈ C([0, T ]; X), dU , dt

(3.99)

A(t)U ∈ Fβ,σ ((0, T ]; X),

with the estimates

A(t)β U ≤ C A(0)β U0 + F Fβ,σ , C dU

β dt β,σ + A(t)U Fβ,σ ≤ C A(0) U0 + F Fβ,σ . F

(3.100)

(3.101) (3.102)

Proof Let us first verify (3.99). For 0 < t ≤ T , we notice that A(t)β U (t) = A(t)β−1 A(t)U (t). Since A(·)β−1 ∈ C([0, T ]; L(X)), (3.63) yields that A(t)β U (t) is continuous for t > 0. So, it suffices to show that A(t)β U (t) is continuous at t = 0. We use the expression

A(t)β U (t) = A(0)β U0 + A(t)β U (t, 0)A(0)−β − 1 A(0)β U0 t + A(t)β U (t, τ )[F (τ ) − F (t)] dτ 0

+

t

A(t)β [U (t, τ ) − e−(t−τ )A(t) ] dτ F (t)

0

+t

β−1

[1 − e−tA(t) ]A(t)β−1 t 1−β F (t).

Then, by the same arguments as in the proof of Theorem 3.5, we directly see that A(t)β U (t) → A(0)β U0 as t → 0. Hence, (3.99) is proved. Estimate (3.101) is also verified in the same way. Let us now prove (3.100). We use the expression A(t)U (t) = A(0)e−tA(0) U0

+ A(t)U (t, 0)A(0)−β − A(0)1−β e−tA(0) A(0)β U0 t t + A(t)U (t, τ )[F (τ ) − F (t)] dτ + W (t, τ ) dτ F (t) 0

0

+ [1 − e−tA(t) ]F (t) = W1 (t) + W2 (t) + W3 (t) + W4 (t) + W5 (t), where W (t, s) = A(t)[U (t, s) − e−(t−s)A(t) ]. By (2.132), it is already known that W1 ∈ Fβ,σ ((0, T ]; X). So, it remains to verify Wi ∈ Fβ,σ ((0, T ]; X) for 2 ≤ i ≤ 5. Proof for W2 We use (3.94) to observe that t 1−β W2 (t)X ≤ Ct μ+ν−1 log(t −1 + 1) → 0 as t → 0. In the meantime, W2 (t) − W2 (s) is seen to be decomposed as

8 Maximal Regularity

161

W2 (t) − W2 (s) = A(t)U (t, s)A(s)−1 − e−(t−s)A(s) A(s)U (s, 0)A(0)−β A(0)β U0 + [e−(t−s)A(s) − 1]A(s)−σ A(s)1+σ U (s, 0)A(0)−β A(0)β U0 − [e−(t−s)A(0) − 1]A(0)−σ A(0)1−β+σ e−sA(0) A(0)β U0 . Then, by (2.129), (3.83), and (3.85),

W2 (t) − W2 (s)X ≤ C s β−1 (t − s)μ+ν−1 + s β−σ −1 (t − s)σ A(0)β U0 X . Hence, since σ < μ + ν − 1, W2 satisfies (1.6). It is also readily verified by (2.130) and (3.94) that W2 satisfies (1.7). Proof for W3 We shall argue as in the proof of Theorem 3.5. It is easy to see that limt→0 t 1−β W3 (t) = 0. In addition, W3 (t) − W3 (s) is decomposed as W3 (t) − W3 (s) = s

t

A(t)U (t, τ )[F (τ ) − F (t)] dτ

s

+

W (t, τ ) dτ + [e−(t−s)A(t) − e−tA(t) ] [F (s) − F (t)]

0

+ A(t)U (t, s)A(s)−1 − e−(t−s)A(s) s × A(s)U (s, τ )[F (τ ) − F (s)] dτ 0

+ [e−(t−s)A(s) − 1]A(s)−σ + [e−(t−s)A(s) − 1]

s

A(s)σ W (s, τ )[F (τ ) − F (s)] dτ

0 s

A(s)e−(s−τ )A(s) [F (τ ) − F (s)] dτ.

0

Then, using (2.129), (3.81), (3.84), and (3.85) and especially using (3.21) for the last term, we can verify that

W3 (t) − W3 (s)X ≤ C s 1−β (t − s)μ+ν−1 + s 1−β+σ (t − s)σ F Fβ,σ .

Hence, W3 satisfies (1.6). Similarly, (1.7) is also verified. Proof for W4 It is clear that limt→0 t 1−β W4 (t) = 0. We use the decomposition

t

W4 (t) − W4 (s) =

s

W (t, τ ) dτ F (t) +

W (t, τ ) dτ [F (t) − F (s)]

0

s

+ A(t)U (t, s)A(s)−1 − e−(t−s)A(s) + [e

−(t−s)A(s)

−σ

− 1]A(s)

0

t

W (s, τ ) dτ F (s) s

s

A(s)σ W (s, τ ) dτ F (s)

162

3 Linear Evolution Equations

+

s

[A(s)e−(t−τ )A(s) − A(t)e−(t−τ )A(t) ] dτ F (s)

0

+ A(t)U (t, s)A(s)−1 − e−(t−s)A(s) [1 − e−sA(s) ]F (s). Then, by (2.129), (3.49), (3.84), and (3.85),

W4 (t) − W4 (s)X ≤ C s β−1 (t − s)μ+ν−1 + t μ+ν−1 s β−σ −1 (t − s)σ F Fβ,σ .

Hence, W4 satisfies (1.6) and (1.7). Proof for W5 We see that W5 is decomposed as W5 (t) − W5 (s) = [1 − e−tA(t) ][F (t) − F (s)] − [e−tA(t) − e−tA(s) ]F (s) − [e−(t−s)A(s) − 1]A(s)−σ A(s)σ e−sA(s) F (s). Then, by (2.129) and (3.49), W5 (t) − W5 (s)X ≤ Cs β−σ −1 (t − s)σ F Fβ,σ .

It is also verified that W5 satisfies (1.7). We have thus verified that AU belongs to Fβ,σ ((0, T ]; X) and satisfies estimate (3.102). Since dU dt = −A(t)U + F (t), we have accomplished the proof of (3.100) and (3.102). Remark 3.6 When 0 < β < 1, t 1−β A(t)U (t) tends to 0 as t → 0. In the meantime, when β = 1, A(t)U (t) tends to A(0)U0 as t → 0.

8.3 Spatial Regularity We remark that, if the external force function F ∈ Fβ,σ ((0, T ]; X) has the spatial regularity β−1−ρ

A(t)ρ F ∈ B{0}

((0, T ]; X),

0 < ρ < σ,

(3.103)

the solution U to (3.61) has the same regularity. Theorem 3.11 Let U0 ∈ D(A(0)β ), 0 < β ≤ 1, and let F ∈ Fβ,σ ((0, T ]; X), 0 < 1+ρ U also belong to σ < min{β, μ + ν − 1}, satisfy (3.103). Then, A(t)ρ dU dt , A(t) β−1−ρ

B{0}

((0, T ]; X) with the estimate A(t)ρ dU dt

β−1−ρ

B{0}

+ A(t)1+ρ U Bβ−1−ρ {0}

≤ C A(0)β U0 + F Fβ,σ + A(t)ρ F Bβ−1−ρ . {0}

9 Variational Methods

163

Proof The result is easily verified from the representation A(t)1+ρ U (t) = A(t)1+ρ U (t, 0)A(0)−β A(0)β U0 t + A(t)1+ρ U (t, τ )[F (τ ) − F (t)] dτ 0

+

t

A(t)1+ρ [U (t, τ ) − e−(t−τ )A(t) ] dτ F (t)

0

+ [1 − e−tA(t) ]A(t)ρ F (t). Note that 1 + ρ < 1 + σ < μ + ν.

9 Variational Methods Let Z and X be two separable Hilbert spaces such that Z ⊂ X densely and continuously with norms · and | · |, respectively. Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces, the norm of Z ∗ being denoted by · ∗ . Consider a family of sesquilinear forms a(t; U, V ), 0 ≤ t ≤ T , defined on Z and assume that the following conditions are satisfied: a(t; ·, ·) is a measurable function with respect to t,

(3.104)

|a(t; U, V )| ≤ MU V ,

(3.105)

Re a(t; U, U ) ≥ δU , 2

U, V ∈ Z,

U ∈ Z,

(3.106)

with some constant M > 0 and positive constant δ > 0 independent of t . Let A(t) be a family of sectorial operators of Z ∗ which are associated with the forms a(t; U, V ). We then consider the Cauchy problem for an abstract evolution equation dU dt + A(t)U = F (t), 0 < t ≤ T , (3.107) U (0) = U0 , in the space Z ∗ . The following theorem is known as the fundamental existence and uniqueness result. For the proof, see Theorems 1, 2, and 3 of Dautray–Lions [DL88, Chap. XVIII]. Theorem 3.12 Let A(t) be a family of sectorial operators of Z ∗ associated with sesquilinear forms satisfying (3.104), (3.105), and (3.106). For any F ∈ L2 (0, T ; Z ∗ ) and any U0 ∈ X, there exists a unique solution U to (3.107) in the function space: U ∈ L2 (0, T ; Z) ∩ C([0, T ]; X) ∩ H 1 (0, T ; Z ∗ ). In addition, U satisfies the estimates

164

3 Linear Evolution Equations

U L2 (0,T ;Z) + U C([0,T ];X) + U H 1 (0,T ;Z ∗ ) ≤ C(|U0 | + F L2 (0,T ;Z ∗ ) ), where C > 0 is a constant determined by M, δ, and T only.

10 Nonautonomous Parabolic Equations Let us consider a nonautonomous parabolic equation of the form n ∂u Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ) = ∂t i,j =1

in a domain Ω ⊂ Rn , where 0 < T < ∞ is fixed time. It is possible to treat a number of different initial-boundary-value problems concerning this equation. In this section, however, we will describe only a few examples. As for the concrete characterization of the domains of fractional powers of strongly elliptic operators, see Chap. 16.

10.1 Problems in Hilbert Spaces Case where Ω = Rn . Consider the initial-value problem ∂u n n i,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in R × (0, T ), ∂t = u(x, 0) = u0 (x) Here, aij and c are such that aij ∈ Cμ ([0, T ]; L∞ (Rn ; R)), c ∈ Cμ ([0, T ]; L∞ (Rn ; R)),

in Rn .

1 ≤ i, j ≤ n,

(3.108)

(3.109)

with some exponent 0 < μ ≤ 1. We assume further that, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.15) with some uniform constant δ > 0, and c(x, t) satisfies (2.16) with some uniform constant c0 > 0. Let us handle (3.108) in the space H −1 (Rn ) by formulating (3.108) as an abstract problem of the form (3.61) in which the coefficient linear operator is a realization A(t) of the differential operator − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in H −1 (Rn ). By Theorem 2.2, A(t) is a sectorial operator of H −1 (Rn ) with uniform domain D(A(t)) ≡ H 1 (Rn ) and with angle ωA(t) ≤ ωA < π2 , ωA being independent of t . Therefore, the family of operators A(t) fulfills conditions (3.27), (3.28), and (3.29) with ν = 1. In addition, it is clear that (3.109) implies (3.69); therefore, by virtue of (3.71), condition (3.30) is also fulfilled. Since ν = 1 in the present case, (3.31) is trivially fulfilled.

10 Nonautonomous Parabolic Equations

165

It is now ready to apply Theorem 3.10 to (3.108) to conclude the following result. For any f ∈ Fβ,σ ((0, T ]; H −1 (Rn )), where 0 < σ < β ≤ 1, and any u0 ∈ D(A(0)β ), (3.108) possesses a unique solution in the function space: u ∈ C((0, T ]; H 1 (Rn )) ∩ C1 ((0, T ]; H −1 (Rn )), β,σ ((0, T ]; H −1 (Rn )), A(t)β u ∈ C([0, T ]; H −1 (Rn )); du dt , A(t)u ∈ F with the estimates

β A(t)β u + du dt β,σ + A(t)uFβ,σ ≤ C A(0) u0 H −1 + f Fβ,σ . C F

Dirichlet conditions. Consider the initial-boundary-value problem ⎧ ∂u n ⎪ ⎨ ∂t = i,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ), u=0 on ∂Ω × (0, T ), (3.110) ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a domain Ω ⊂ Rn . Here, aij and c are such that aij ∈ Cμ ([0, T ]; L∞ (Ω; R)), 1 ≤ i, j ≤ n, c ∈ Cμ ([0, T ]; L∞ (Ω; R)),

(3.111)

with some exponent 0 < μ ≤ 1. In addition, it is assumed that, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.19) with some uniform constant δ > 0, and c(x, t) satisfies (2.20) with some uniform constant c0 > 0. Let us handle (3.110) again in H −1 (Ω) = [H˚ 1 (Ω)]∗ . Indeed, (3.110) is formulated as an abstract problem of the form (3.61) in which the coefficient linear operator is a realization A(t) of the differential operator − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in H −1 (Ω) under the Dirichlet boundary conditions. By Theorem 2.3, A(t) is a sectorial operator of H −1 (Ω) with uniform domain D(A(t)) ≡ H˚ 1 (Ω) and with angles ωA(t) ≤ ωA < π2 , ωA being independent of t . Therefore, the family of operators A(t) fulfills conditions (3.27), (3.28), and (3.29) with ν = 1. In addition, (3.111) implies (3.69); therefore, by (3.71), condition (3.30) is also fulfilled. Clearly, (3.31) is also fulfilled. Theorem 3.5 readily provides that, for any f ∈ Fβ,σ ((0, T ]; H −1 (Ω)), where 0 < σ < β ≤ 1, and any u0 ∈ D(A(0)β ), (3.110) possesses a unique solution in the function space: u ∈ C((0, T ]; H˚ 1 (Ω)) ∩ C1 ((0, T ]; H −1 (Ω)), β,σ ((0, T ]; H −1 (Ω)), A(t)β u ∈ C([0, T ]; H −1 (Ω)); du dt , A(t)u ∈ F with the estimates

β A(t)β u + du dt β,σ + A(t)uFβ,σ ≤ C A(0) u0 H −1 + f Fβ,σ . C F

166

3 Linear Evolution Equations

Neumann-type conditions. Consider the initial-boundary-value problem ⎧

n ∂u ⎪ ⎪ i,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ), ∂t = ⎨ n on ∂Ω × (0, T ), (3.112) i,j =1 νj (x)aij (x, t)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a domain Ω ⊂ Rn . Here, aij and c are as in (3.111). In addition, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.19) with some uniform constant δ > 0, and c(x, t) satisfies (2.20) with some uniform constant c0 > 0. We handle this problem in H 1 (Ω)∗ . Indeed, (3.112) is written as (3.61) in which

the coefficient linear operator is a realization A(t) of the differential operator − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in H 1 (Ω)∗ under the Neumann-type boundary conditions. By Theorem 2.4, A(t) is a sectorial operator of H 1 (Ω)∗ with uniform domain D(A(t)) ≡ H 1 (Ω) and with angle ωA(t) ≤ ωA < π2 , ωA being independent of t . Therefore, the family of operators A(t) fulfills conditions (3.27), (3.28), and (3.29) with ν = 1. In addition, (3.111) implies (3.69); therefore, by (3.71), condition (3.30) is also fulfilled. Clearly, (3.31) is also fulfilled. Theorem 3.5 readily provides that, for any f ∈ Fβ,σ ((0, T ]; H 1 (Ω)∗ ), where 0 < σ < β ≤ 1, and any u0 ∈ D(A(0)β ), (3.112) possesses a unique solution in the function space: u ∈ C((0, T ]; H 1 (Ω)) ∩ C1 ((0, T ]; H 1 (Ω)∗ ), β,σ ((0, T ]; H 1 (Ω)∗ ), A(t)β u ∈ C([0, T ]; H 1 (Ω)∗ ); du dt , A(t)u ∈ F with the estimate

β A(t)β u + du + A(t)u β,σ ≤ C A(0) u0 1∗ + f F β,σ . F C H dt Fβ,σ

10.2 Problems in Lp Spaces Consider the initial-boundary-value problem ⎧

∂u ⎪ = ni,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ), ⎪ ∂t ⎨ n on ∂Ω × (0, T ), (3.113) i,j =1 νj (x)aij (x, t)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω. Here, aij and c are as in (3.74). In addition, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.19) with some uniform constant δ > 0, and c(x, t) satisfies (2.20) with some uniform constant c0 > 0. We now want to handle (3.113) in the space Lp (Ω), where n < p < ∞. Let Ap (t) be a realization of the differential operator − i,j Dj [aij (x, t)Di ] + c(x, t)

10 Nonautonomous Parabolic Equations

167

in Lp (Ω) under the Neumann-type boundary conditions ∂ν∂u = 0 on ∂Ω. Using A(t) Ap (t), we formulate (3.113) as an abstract problem of the form (3.61). Fix an exponent ν in such a way such that 12 ≤ ν < p+1 2p . As discussed in Sect. 7.2, a family of the operators Ap (t) can fulfill condition (3.29) with the ν. Similarly, by Remark 3.4, we know that the family can fulfill condition (3.30) with the Hölder exponent 2μ(1 − ν ), where ν is an arbitrarily fixed exponent such that ν < ν < p+1 2p . Therefore, if μ in (3.74) is such that μ > 12 (and if ν is taken sufficiently close to ν), then relation (3.31) can be fulfilled. In this way, we have verified that all the structural assumptions are fulfilled in Lp (Ω), too. Theorem 3.10 readily provides the following result. For any f ∈ Fβ,σ ((0, T ]; Lp (Ω)), where 0 < σ < β ≤ 1, and any u0 ∈ D(Ap (0)β ), (3.113) possesses a unique solution in the function space:

u ∈ C((0, T ]; Hp2 (Ω)) ∩ C1 ((0, T ]; Lp (Ω)), β,σ ((0, T ]; L (Ω)), Ap (t)β u ∈ C([0, T ]; Lp (Ω)); du p dt , Ap (t)u ∈ F

with the estimate β Ap (t)β u + du dt β,σ + Ap (t)uFβ,σ ≤ C Ap (0) u0 Lp + f Fβ,σ . C F

10.3 Some Example of the Critical Case ν = 0 Let us consider a very particular example such that A(t) is a multiplicative operator of the form A(t)f (x) = (x − t)−2 f (x) in Lp (0, 1) (1 < p < ∞) for 0 ≤ t ≤ 1 = T . It is easy to see that A(t) is a sectorial operator of Lp (Ω). The domain D(A(t)) of A(t) consists of the functions (x − t)2 f (x), where f ∈ Lp (0, 1). It is also verified that, for 0 < θ < 1, A(t)θ f (x) = |x − t|−2θ f (x) with D(A(t)θ ) = {|x − t|2θ f ; f ∈ Lp (0, 1)}. Therefore, condition (3.29) is not the case for any ν > 0. Nevertheless, there exists an evolution operator for A(t) which is given by

U (t, s) = e−

t s

(x−τ )−2 dτ

⎧ [(t−x)−1 +(x−s)−1 ] , ⎪ ⎨e = 0, ⎪ ⎩ [(t−x)−1 +(x−s)−1 ] e ,

t < x ≤ 1, s ≤ x ≤ t, 0 ≤ x < s,

for 0 ≤ s < t ≤ 1 and by U (s, s) = 1 for 0 ≤ s ≤ 1. We can in fact extend our abstract theory built in this chapter to the critical case where ν = 0. Let X be a Banach space. Consider a family of sectorial operators A(t), 0 ≤ t ≤ T , of X satisfying (3.27) and (3.28). We assume that A(·)−1 is strongly continuously differentiable on X for 0 ≤ t ≤ T

(3.114)

168

3 Linear Evolution Equations

d and that there exists some fixed exponent 0 < ν ≤ 1 such that R( dt A(t)−1 ) ⊂ ν D(A(t) ) for all t with the estimates A(t)ν d A(t)−1 ≤ N , 0 ≤ t ≤ T , (3.115) dt

N > 0 being a constant. Under (3.27), (3.28), (3.114), and (3.115), we can construct a unique evolution operator U (t, s) for A(t) and can furthermore show similar results as Theorems 3.8, 3.9, and 3.10, see Tanabe [Tan64] (cf. also [Yag76] and [Wat77]).

11 Perturbed Problems We are concerned with the perturbed problem dU dt + A(t)U + B(t)U = F (t),

0 < t ≤ T,

U (0) = U0 ,

(3.116)

in a Banach space X. Here, A(t), 0 ≤ t ≤ T , is a family of sectorial operators of X satisfying (3.27), (3.28), (3.29), (3.30), and (3.31). Meanwhile, B(t), 0 < t ≤ T , is a family of linear operators of X with singularity at t = 0. We assume that D(A(t)1−˜ν ) ⊂ D(B(t)) for all 0 < t ≤ T with the estimates ˜ B(t)A(t)ν˜ −1 ≤ Dt ˜ μ−1 , 0 < t ≤ T, (3.117) D˜ > 0 being a constant. Here, μ˜ > 0 and ν˜ > 0 are some fixed exponents such that 0 < μ˜ ≤ 1,

0 < ν˜ ≤ 1 and 1 < μ˜ + ν˜ .

(3.118)

In addition, B(t)A(t)ν˜ −1 is assumed to satisfy a Hölder condition of the form ˜ σ˜ −1 B(t)A(t)ν˜ −1 − B(s)A(s)ν˜ −1 ≤ Ls ˜ μ− (t − s)σ˜ , 0 < s ≤ t ≤ T , (3.119) with some fixed exponent 0 < σ˜ < min{μ + ν − 1, μ˜ + ν˜ − 1},

(3.120)

L˜ > 0 being a constant. The function F is a given external force function, and U0 is an initial value in X.

11.1 Evolution Operator The purpose of this subsection is to construct an evolution operator U˜ (t, s), 0 ≤ s ≤ t ≤ T , for the family of operators A(t) + B(t).

11 Perturbed Problems

169

We begin with introducing the approximation A˜ n (t) = An (t) + Bn (t),

n = 1, 2, 3, . . . , 0 < t ≤ T ,

where An (t) is the Yosida approximation of A(t), and Bn (t) = B(t)Jn (t)1−˜ν = B(t)A(t)ν˜ −1 An (t)1−˜ν ,

(3.121)

see Proposition 3.3. For any 0 < ε < T , A˜ n (t) is an L(X)-valued continuous function on [ε, T ]. So, we have an evolution operator U˜ n (t, s), ε ≤ s ≤ t ≤ T , for A˜ n (t) such that ∂ ˜ Un (t, s) = −A˜ n (t)U˜ n (t, s), ε ≤ s ≤ t ≤ T , ∂t ∂ ˜ Un (t, s) = U˜ n (t, s)A˜ n (s), ε ≤ s ≤ t ≤ T . ∂s Of course, U˜ n (t, s) does not depend on the choice of ε > 0. Therefore, the evolution operator is defined for any (t, s) such that 0 < s ≤ t ≤ T . The evolution operator U˜ n (t, s) satisfies integral equations. Due to (3.40), t ∂ ˜ U˜ n (t, s) − Un (t, s) = − Un (t, τ )Un (τ, s) dτ s ∂τ t =− U˜ n (t, τ )Bn (τ )Un (τ, s) dτ, s

where Un (t, s), 0 ≤ s ≤ t ≤ T , is the evolution operator for An (t). Therefore, U˜ n (t, s) is connected with Un (t, s) by t ˜ Un (t, s) = Un (t, s) − U˜ n (t, τ )Bn (τ )Un (τ, s) dτ, 0 < s ≤ t ≤ T . (3.122) s

Operating An (t) to this equation, we obtain also an integral equation for W˜ n (t, s) = An (t)U˜ n (t, s) − An (t)Un (t, s) such that t W˜ n (t, τ )Bn (τ )Un (τ, s) dτ, 0 < s ≤ t ≤ T , (3.123) W˜ n (t, s) = R˜ n (t, s) + s

where R˜ n (t, s) = −

t

An (t)Un (t, τ )Bn (τ )Un (τ, s) dτ. s

Let us now show the convergence of U˜ n (t, s) as n → ∞. From (3.121) we have Bn (τ )Un (τ, s) = B(τ )A(τ )ν˜ −1 An (τ )1−˜ν Un (τ, s). Therefore, by (3.81) and (3.117), ˜ ˜ ν −2 (τ − s)ν˜ −1 ≤ C(τ − s)μ+˜ , Bn (τ )Un (τ, s) ≤ Cτ μ−1

0 < s < τ ≤ T,

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3 Linear Evolution Equations

with some constant C > 0 independent of n. In addition, as n → ∞, Bn (τ )Un (τ, s) → B(τ )A(τ )ν˜ −1 A(τ )1−˜ν U (τ, s)

strongly on X.

We can then apply the dominated convergence theorems, Theorems 1.30 and 1.32, to (3.122) to conclude that U˜ n (t, s) satisfies the uniform estimate U˜ n (t, s) ≤ C0 and is strongly convergent on X to a bounded operator, which will be denoted by U˜ (t, s) for all 0 < s ≤ t ≤ T . Furthermore, U˜ (t, s) is characterized as a solution to the integral equation t ˜ U (t, s) = U (t, s) − U˜ (t, τ )B(τ )U (τ, s) dτ, 0 < s ≤ t ≤ T . s

In the meantime, we notice for 0 < s < s < T that U˜ (t, s ) − U˜ (t, s) = U (t, s ) − U (t, s) + +

s

U˜ (t, τ )B(τ )U (τ, s) dτ

s t s

U˜ (t, τ )B(τ )U (τ, s ) dτ [1 − U (s , s)].

This implies that U˜ (t, s) is uniformly continuous in s ∈ (0, t] in the strong topology of L(X) (t > 0 being fixed). Therefore, as s → 0, U˜ (t, s) is strongly convergent on X, and the strong convergence is uniform in t . In this way, we conclude that U˜ (t, s) is defined even at s = 0, namely, for all 0 ≤ s ≤ t ≤ T , and is strongly continuous on X for 0 ≤ s ≤ t ≤ T . Let us next consider (3.123). We have the following proposition. Proposition 3.10 For any n = 1, 2, 3, . . . , R˜ n (t, s) satisfies ˜ ν −2 , R˜ n (t, s) ≤ C(t − s)μ+˜

0 ≤ s < t ≤ T,

with some uniform constant C > 0. As n → ∞, Rn (t, s) converges strongly to a ˜ s) on X. R(t, ˜ s) satisfies the same estimate and is bounded linear operator R(t, strongly continuous on X for 0 ≤ s < t ≤ T . Proof We can decompose R˜ n (t, s) in the form t ˜ Rn (t, s) = − [An (t)Un (t, τ ) − An (t)e−(t−τ )An (t) ]Bn (τ )Un (τ, s) dτ s

−

t

An (t)e−(t−τ )An (t) B(τ )A(τ )ν˜ −1 − B(t)A(t)ν˜ −1 An (τ )1−˜ν Un (τ, s) dτ

t

An (t)e−(t−τ )An (t) B(t)A(t)ν˜ −1 1 − An (t)1−˜ν Un (t, τ )An (τ )ν˜ −1

s

− s

× An (τ )1−˜ν Un (τ, s) dτ + [e−(t−s)An (t) − 1]B(t)A(t)ν˜ −1 An (t)1−˜ν Un (t, s).

11 Perturbed Problems

171

We used the formula −

t

An (t)e−(t−τ )An (t) dτ = e−(t−s)An (t) − 1.

s

We notice by (3.81), (3.85), and (3.120) that

1 − An (t)1−˜ν Un (t, τ )An (τ )ν˜ −1 An (τ )1−˜ν Un (τ, s)

≤ 1 − An (t)1−˜ν Un (t, τ )An (τ )ν˜ −1 An (τ )−σ˜ An (τ )1−˜ν +σ˜ Un (τ, s) ≤ C(t − τ )σ˜ (τ − s)ν˜ −σ˜ −1 ,

0 ≤ s < τ < t ≤ T.

Using this together with (3.81), (3.84), (3.117), and (3.119), we deduce that

˜ ν −3 ˜ ν −2 . + (t − s)μ+˜ R˜ n (t, s) ≤ C (t − s)μ+ν+μ+˜ Hence, the desired estimate is proved. The strong convergence of R˜ n (t, s) is also verified by using the decomposition presented above, since each term is easily seen to be strongly convergent on X as n → ∞ by (3.55) and Proposition 3.4. It is now ready to apply Theorem 1.32 to (3.123) to deduce that W˜ n (t, s) is strongly convergent to a bounded linear operator W˜ (t, s) for each 0 ≤ s < t ≤ T , which is characterized as a solution to the integral equation ˜ s) + W˜ (t, s) = R(t,

t

W˜ (t, τ )B(τ )U (τ, s) dτ,

0 < s ≤ t ≤ T.

s

Consequently, An (t)U˜ n (t, s) = An (t)Un (t, s) + W˜ n (t, s) is strongly convergent to A(t)U (t, s) + W˜ (t, s). Since U˜ (t, s) = U (t, s) + A(t)−1 W˜ (t, s), it follows that R(U˜ (t, s)) ⊂ D(A(t)) and A(t)U˜ (t, s) = A(t)U (t, s) + W˜ (t, s). We can repeat the same arguments as in Sect. 5.3 for proving the analogous results to (3.57), (3.58), (3.59), and (3.60) for A(t) + B(t) and U˜ (t, s). In particular, since U˜ (t, s) ≤ C0 and A(t)U˜ (t, s) ≤ C1 (t − s)−1 hold, the moment inequality yields that for any 0 ≤ θ ≤ 1, A(t)θ U˜ (t, s) ≤ Cθ (t − s)−θ , 0 ≤ s < t ≤ T . (3.124) It is also seen that for 0 ≤ θ ≤ μ, ˜ A(t)U˜ (t, s)A(s)−θ ≤ Cθ (t − s)θ−1 ,

0 ≤ s < t ≤ T.

(3.125)

In fact, in the proof of Proposition 3.10, it is possible to utilize (3.83) with θ = 1 − ν˜ to obtain that for 0 ≤ θ ≤ 1 − ν˜ , ˜ ν +θ−2 R˜ n (t, s)An (s)−θ ≤ C(t − s)μ+˜ , 0 ≤ s < t ≤ T.

172

3 Linear Evolution Equations

˜ s)A(s)−θ . Then, for 0 ≤ θ ≤ 1 − ν˜ , It is the same for R(t, ˜ ν +θ−2 W˜ (t, s)A(s)−θ ≤ C(t − s)μ+˜ , 0 ≤ s < t ≤ T. This means that (3.125) is valid for 0 ≤ θ ≤ 1 − ν˜ . For 1 − ν˜ < θ ≤ μ, ˜ it is observed that ˜ W˜ (t, s)A(s)−θ ≤ C W˜ (t, s)A(s)ν˜ −1 ≤ C(t − s)μ−1 , 0 ≤ s < t ≤ T. This then means that (3.125) is extensively valid for 1 − ν˜ < θ ≤ μ, ˜ too. We sum up the results obtained for the evolution operator U˜ (t, s). Theorem 3.13 Let (3.27), (3.28), (3.29), (3.30), and (3.31) be satisfied for A(t), and let (3.117), (3.118), (3.119), and (3.120) be satisfied for B(t). Then, there exists a unique evolution operator U˜ (t, s) for A(t) + B(t) satisfying (3.57), having the regularity that A(t)U˜ (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T , and U˜ (t, s) is strongly continuous on X for 0 ≤ s ≤ t ≤ T , satisfies estimates (3.124) and (3.125), and is differentiable in t for t > s with ∂t∂ U˜ (t, s) = −[A(t) + B(t)]U˜ (t, s).

11.2 Cauchy Problem Let us consider problem (3.116) under conditions (3.27), (3.28), (3.29), (3.30), and (3.31) and (3.117), (3.118), (3.119), and (3.120). Let U˜ (t, s) be an evolution operator for A(t) + B(t) constructed above. As a matter of fact, we can repeat the same arguments as in Sect. 2 to prove the following theorem. So, the proof is left to the reader. Theorem 3.14 Let (3.27), (3.28), (3.29), (3.30), and (3.31) be satisfied, and let (3.117), (3.118), (3.119), and (3.120) be also satisfied. Then, for any function F satisfying (3.62) and any initial value U0 ∈ X, (3.116) possesses a unique solution U in the function space: U ∈ C([0, T ]; X) ∩ C1 ((0, T ]; X),

A(t)U ∈ C((0, T ]; X).

Moreover, U is given by the formula t U˜ (t, s)F (s) ds, U (t) = U˜ (t, 0)U0 +

0 ≤ t ≤ T.

(3.126)

(3.127)

0

Concerning the regularity of B(t)U (t), we have the following result. Corollary 3.1 Let U be the solution obtained in Theorem 3.14. Then, B(t)U (t) satisfies the following estimate: ˜ ν −2 B(t)U (t) ≤ C U0 + F Bβ−1 t μ+˜ , 0 < t ≤ T, (3.128) {0}

11 Perturbed Problems

173

˜ ν −σ˜ −2 B(t)U (t) − B(s)U (s) ≤ C U0 + F Bβ−1 (t − s)σ˜ s μ+˜ , {0}

0 < s < t ≤ T.

(3.129)

Proof The first estimate is verified directly from (3.127). Indeed, t ν˜ −1 1−˜ν ˜ ˜ U (t, s)F (s) ds B(t)U (t) ≤ B(t)A(t) U (t, 0)U0 + A(t) 0

˜ ν −2 ≤ C U0 + F Bβ−1 t μ+˜ , {0}

due to (3.124). To prove the second one, we shall use rather a formula of the form t U (t, s)[F (s) − B(s)U (s)] ds, 0 ≤ t ≤ T , U (t) = U (t, 0)U0 + 0

where U (t, s) denotes the evolution operator for A(t). Then, from

B(t)U (t, 0) − B(s)U (s, 0) = B(t)A(t)ν˜ −1 − B(s)A(s)ν˜ −1 A(s)1−˜ν U (s, 0)

+ B(t)A(t)ν˜ −1 A(t)1−˜ν U (t, s)A(s)ν˜ −1 − 1 A(s)−σ˜ × A(s)1−˜ν +σ˜ U (s, 0) we observe that ˜ ν −σ˜ −2 [B(t)U (t, 0) − B(s)U (s, 0)]U0 ≤ CU0 (t − s)σ˜ s μ+˜ .

Similarly, from t s B(t) U (t, τ )F (τ ) dτ − B(s) U (s, τ )F (τ ) dτ 0

= B(t)A(t)ν˜ −1

0 t

A(t)1−˜ν U (t, τ )F (τ ) dτ

s

+ B(t)A(t)ν˜ −1 A(t)1−˜ν U (t, s)A(s)ν˜ −1 − 1 A(s)−σ˜ s

× A(s)1−˜ν +σ˜ U (s, τ )F (τ ) dτ + B(t)A(t)ν˜ −1 − B(s)A(s)ν˜ −1 0

×

s

A(s)1−˜ν U (s, τ )F (τ ) dτ

0

we obtain that t s B(t) U (t, τ )F (τ ) dτ − B(s) U (s, τ )F (τ ) dτ 0

0

˜ ν −σ˜ −2 ≤ CF Bβ−1 (t − s)σ˜ s μ+˜ . {0}

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3 Linear Evolution Equations

It is the same for B(t) μ+˜ ˜ ν −2 ((0, T ]; X) B{0}

t 0

U (t, τ )B(τ )U (τ ) dτ , since (3.128) means that B(·)U (·) ∈

with B(t)U Bμ+˜ ˜ ν −2 ≤ C U0 + F β−1 . B {0}

{0}

˜

We notice from (3.128) and (3.129) that B(·)U (·) belongs to Fβ,σ˜ ((0, T ]; X) ˜ with any β˜ such that σ˜ < β˜ < μ+ ˜ ν˜ − 1. Therefore, if F ∈ Fβ,σ˜ ((0, T ]; X) and U0 ∈ ˜ D(A(0)β ), then Theorem 3.10 provides the maximal regularity that dU dt , A(t)U ∈ ˜

Fβ,σ˜ ((0, T ]; X) with the estimate dU β˜ dt β,˜ σ˜ + A(t)U Fβ,˜ σ˜ ≤ C A(0) U0 + F Fβ,˜ σ˜ . F

(3.130)

Remark 3.7 Theorems 3.13 and 3.14 are applicable even for the autonomous case. Namely, A(t) ≡ A is a sectorial operator satisfying (2.92) and (2.93), and B(t) ≡ B is a linear operator satisfying (3.8). Then, it is easily verified that the evolution operator for A + B obtained by Theorem 3.13 is nothing more than the analytic semigroup e−t (A+B) given by Theorem 3.3. Therefore, in this case, we can read all the results in Theorem 3.14 by substituting e−(t−s)(A+B) for U˜ (t, s).

Notes and Further Researches There are three main approaches for linear abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. The semigroup methods have originated in the birth of the notion of analytic semigroups (see Notes of Chap. 2). The analytic semigroup was first used to study autonomous abstract parabolic evolution equations by Solomyak [Sol58], and those abstract results were then applied for solving concrete autonomous parabolic differential equations, see Hille [HP57], Yosida [Yos80], Krein [Kre67], and so on. Afterward, Sobolevskii [Sob61a] and Tanabe [Tan60] independently began to study nonautonomous abstract parabolic evolution equations. On the basis of analytic semigroups, they could construct an evolution operator for the Cauchy problem of the form (3.33). Sobolevskii and Tanabe considered first the case where the domains D(A(t)) of coefficient linear operators are independent of the variable t . Sobolevskii [Sob61b] and Kato [Kat61c] then considered the case where the domains D(A(t)) are variable with t but, for some exponent 0 < θ < 1, the domains D(A(t)θ ) of their fractional powers A(t)θ are independent of t . Tanabe [Tan64] then considered the case where the domains D(A(t)) vary completely with t in the sense that, for any exponent, D(A(t)θ ) are possibly variant (see also Kato–Tanabe [KT62]). The semigroup methods are the most constructive ones and provide us the solution formulas (3.13) and (3.51) which furthermore give us much information on the solutions of equations. For the details, see the books by Friedman

Notes and Further Researches

175

[Fri69] and Tanabe [Tan75, Tan97]. In particular, we can get the maximal regularity with respect to the initial values and the external force functions, i.e., Theorems 3.5 and 3.10. The variational methods have originated with Lions [Lio61]. Abstract equations are considered in Hilbert spaces. For L2 external force functions with values in a Hilbert space, unique H 1 solutions are immediately constructed together with the L2 maximal regularity, Theorem 3.12. For details, see Lions–Magenes [LM72] and Dautray–Lions [DL88]. Da Prato–Grisvard [DaPG75] (see also [DaPG84]) devised a method for solving operational equations of the form AU + BU = F given by the sum of two linear operators A and B. Acquistapace–Terreni [AcT86, AcT87] used this method for constructing C1 solutions for abstract parabolic evolution equations in Banach spaces. Lunardi [Lun95] studied the maximal regularity of C 1 solutions. In the meantime, Dore–Venni [DV87] used the method for constructing Wp1 (1 < p < ∞) solutions for Lp external force functions with values in a Banach space enjoying HT property (namely, a Banach space in which the Hilbert transform is bounded) under the condition that the coefficient linear operators have bounded purely imaginary powers. This result gives the Lp (1 < p < ∞) maximal regularity (see also Yamamoto [Yamm93]). Afterward, many authors have studied extensions of this result in the context of H∞ functional calculus in various directions. We refer the reader to the monograph due to Prüss [Pru02] and references therein. There is of course a great deal of important literature which deals with the parabolic differential equations directly by the functional analytical methods. However, we will here quote only some monographs by Ladyzhenskaya–Solonnikov– Uraltseva [LSU68], Amann [Ama95], and so forth. The main Theorems 3.8 and 3.9 were proved when D(A(t)) ≡ D(A(0)) (i.e., ν = 1) in [Sob61a] and [Tan60]. When the domain condition (3.29) fails for any ν > 0 (i.e., ν = 0, see Sect. 10.3), as shown by [Tan64], we have to assume condition (3.115). So, (3.29), (3.30), and (3.31) can be regarded as those of interpolating these conditions of the two limit cases that ν = 1 (with μ > 0) and ν = 0 (with μ > 1). In [Sob61b] and [Kat61c], it was assumed that D(A(t)θ ) are independent of t for some suitable exponent 0 < θ < 1, but we need not to assume such an independence condition. Theorems 3.8 and 3.9 were essentially presented by the paper [Yag88]. So far, maximal regularity was mainly investigated for the correspondence F → dU dt (from the external force function to the derivative of solution), and the initial value U0 was usually assumed to be such that U0 ∈ D(A(0)). Theorem 3.5 or 3.10 newly provides the maximal regularity for both initial value U0 ∈ D(A(0)β ) and external force function F ∈ Fβ,σ ((0, T ]; X). The results obtained in this chapter can be extended in various directions. For example, we are able to get rid of the density of domains D(A(t)) (cf. [Yag88]). Conditions (3.27) and (3.28) can be relaxed to σ (A(t)) ⊂ Σα = λ ∈ C; c|Im λ|α < Re λ , (3.131) (λ − A(t))−1 ≤ M , |λ|β

λ∈ / Σα ,

(3.132)

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3 Linear Evolution Equations

respectively, with some exponents 0 < β ≤ α ≤ 1, see [Yag89, Yag90]. We are able to generalize the results to the abstract parabolic evolution equations of the form dU + A(t)U F (t), dt

0 < t ≤ T,

where A(t) is a family of sectorial multivalued linear operators (Chap. 1, Sect. 3.4), see Favini–Yagi [FY99]. Muramatsu–Tojima [MT01] have relaxed (3.30) to some integrability condition.

Chapter 4

Semilinear Evolution Equations

This chapter is devoted to the Cauchy problem for a semilinear evolution equation of the form (4.1) in a Banach space X. We want to present existence and uniqueness results in a sophisticated way enough to apply them to various models considered in the subsequent chapters (Chaps. 8, 9, 10, 11, 12, and 13). The equation in (4.1) contains two operators, a linear operator A and a nonlinear operator F . The linear operator A is a sectorial operator of X with angle ωA < π2 , and hence −A generates an analytic semigroup on X. Meanwhile, the nonlinear operator F is dominated by the fractional power Aη of A with some exponent 0 ≤ η < 1, and F satisfies a Lipschitz condition described by two fractional powers Aη and Aβ , where β is another exponent such that 0 ≤ β ≤ η < 1. In applications, we intend to take the β as small as possible in view of the fact that η is allowed to be arbitrarily close to 1 by the following two reasons. First, the initial values of (4.1) can be taken in D(Aβ ). Secondly, as Corollaries 4.1 and 4.3 show, the a priori estimates for local solutions of (4.1) with respect to the Aβ graph norm ensure extension of local solutions without limit so as to construct the global solutions. So, the case β = 0 is the optimal one.

1 Semilinear Abstract Evolution Equations 1.1 Cauchy Problems Let X be a Banach space with norm · . We consider the Cauchy problem for a semilinear abstract evolution equation dU 0 < t ≤ T, dt + AU = F (U ) + G(t), (4.1) U (0) = U0 , in X. Here, A is a sectorial operator of X satisfying (2.92) and (2.93). Meanwhile, F is a nonlinear operator from D(Aη ) into X, where 0 < η < 1. F is assumed to A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_4, © Springer-Verlag Berlin Heidelberg 2010

177

178

4 Semilinear Evolution Equations

satisfy a Lipschitz condition of the form F (U ) − F (V ) ≤ ϕ(Aβ U + Aβ V )[Aη (U − V ) + (Aη U + Aη V )Aβ (U − V )],

U, V ∈ D(Aη ), (4.2)

where β is a second exponent such that 0 < β ≤ η < 1,

(4.3)

ϕ(·) being some increasing continuous function. In particular, (4.2) implies the estimate F (U ) ≤ ψ(Aβ U )(Aη U + 1),

U ∈ D(Aη ),

(4.4)

where ψ(ξ ) = F (0) + ϕ(ξ )(ξ + 1). The function G(t) is an external force function given in the space Fβ,σ ((0, T ]; X), 0 < σ < β. The initial value U0 is taken in D(Aβ ). We shall show the following result. Theorem 4.1 Let (2.92), (2.93), (4.2), and (4.3) be satisfied. Then, for any G ∈ Fβ,σ ((0, T ]; X), where 0 < σ < 1 − η, and any U0 ∈ D(Aβ ), (4.1) possesses a unique local solution U in the function space: U ∈ C((0, TG,U0 ]; D(A)) ∩ C([0, TG,U0 ]; D(Aβ )) ∩ C1 ((0, TG,U0 ]; X), (4.5) dU AU ∈ Fβ,σ ((0, TG,U0 ]; X), dt , TG,U0 > 0 depending only on the norms GFβ,σ and Aβ U0 . In addition, U satisfies the estimates dU (4.6) Aβ U C + dt β,σ + AU Fβ,σ ≤ CG,U0 F with some constant CG,U0 > 0 depending on the norms GFβ,σ and Aβ U0 .

1.2 Proof of Theorem 4.1 For each S ∈ (0, T ], we set the following Banach space which consists of D(Aη ) valued continuous functions on (0, S]: X(S) = U ∈ C((0, S]; D(Aη )) ∩ C([0, S]; D(Aβ )); sup t η−β Aη U (t) < ∞ , 0 A0 Aβ U0 + Aβ B(1 − β, β)Gβ,σ , and if S is sufficiently small, then Φ(U ) satisfies (4.7) and (4.8). For 0 < s < t ≤ S, we see by the semigroup property that t −(t−s)A −sA {ΦU }(t) = e e U0 + e−(t−τ )A [F (U (τ )) + G(τ )] dτ + e−(t−s)A

s s

e−(s−τ )A [F (U (τ ) + G(τ )] dτ.

0

From this it is verified that {ΦU }(t) − {ΦU }(s) = [e−(t−s)A − 1]{ΦU }(s) t + e−(t−τ )A [F (U (τ )) + G(τ )] dτ. s

Then, by (4.10) and (4.11) (θ = η + σ < 1), we obtain that Aη [{ΦU }(t) − {ΦU }(s)] ≤ [e−(t−s)A − 1]A−σ Aη+σ {ΦU }(s) t + Aη e−(t−τ )A [F (U (τ )) + G(τ )] dτ s

≤ CG,U0 A1−σ (t − s)σ s β−σ −η t + CG,U0 Aη (t − τ )−η (τ β−1 + 1) dτ. s

Here, we also used property (2.129). Writing β − 1 as β − 1 = (η + σ − 1) + (β − σ − η), we have t t (t − τ )−η τ β−1 dτ ≤ (t − τ )−η (τ − s)η+σ −1 s β−σ −η dτ s

s

= B(1 − η, η + σ )(t − s)σ s β−σ −η . Hence, we have obtained that Aη [{ΦU }(t) − {ΦU }(s)] ≤ CG,U0 (t − s)σ s β−σ −η ,

0 < s < t ≤ S.

(4.12)

This in particular shows that ΦU ∈ C((0, S]; D(Aη )). It is also verified, by a similar calculation, that σ −σ

A [{ΦU }(t) − {ΦU }(s)] ≤ CG,U0 (t − s) s β

+ CG,U0

s

t

(t − τ )−β τ β−1 dτ.

1 Semilinear Abstract Evolution Equations

181

Using now the decomposition β − 1 = (β + σ − 1) − σ , we have t t (t − τ )−β τ β−1 dτ ≤ (t − τ )−β (τ − s)β+σ −1 s −σ dτ s

s

= B(1 − β, β + σ )(t − s)σ s −σ . Therefore, Aβ [{ΦU }(t) − {ΦU }(s)] ≤ CG,U0 (t − s)σ s −σ ,

0 < s < t ≤ S.

(4.13)

t We already know by Theorem 3.5 that e−tA U0 + 0 e−(t−s)A G(s)ds is continuous at t = 0 in the graph norm of D(Aβ ). In addition, it follows from (4.10) that, as t → 0, t t β −(t−s)A A ≤ Aβ e F (U (s)) ds (t − s)−β (s β−η + 1) ds 0

0

≤ CAβ (t 1−η + t 1−β ) → 0. Therefore, ΦU ∈ C([0, S]; D(Aβ )). The mapping Φ has been shown to map K(S) into itself, provided that S > 0 is sufficiently small. Step 2. We next verify that Φ is a contraction mapping of X(S). Let U, V ∈ K(S) be any two functions. For θ such that β ≤ θ < 1, we can write t θ A [{ΦU }(t) − {ΦV }(t)] = Aθ e−(t−s)A [F (U (s)) − F (V (s))] ds. 0

From (4.2), (4.7), and (4.8) it is seen that t

θ−β

A [{ΦU }(t) − {ΦV }(t)] ≤ Aθ ϕ(2C2 )t θ

θ−β

t

(t − s)−θ

0

× {A [U (s) − V (s)] + (2C1 s + 1)A [U (s) − V (s)]} ds t ≤ CAθ (C1 + 1)ϕ(2C2 )t θ−β (t − s)−θ (s β−η + 1) dsU − V X(S) . η

β−η

β

0

Applying these estimates with θ = η and θ = β, we easily conclude that ΦU − ΦV X(S) ≤ C(C1 + 1)ϕ(2C2 )S 1−η U − V X(S) , which shows that Φ is a contraction in X(S), provided that S > 0 is sufficiently small. Step 3. Let S > 0 be sufficiently small in such a way that Φ maps K(S) into itself and is a contraction with respect to the norm X(S). By the arguments of Steps 1 and 2, S = TG,U0 > 0 can be determined by the two norms Gβ,σ and Aβ U0

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4 Semilinear Evolution Equations

only. Thanks to the fixed point theorem for contraction mappings, Theorem 1.11, there exists a unique function U ∈ K(TG,U0 ) such that U = ΦU . From now on, U stands for this function. Clearly, t e−(t−s)A [F (U (s)) + G(s)] ds, 0 ≤ t ≤ TG,U0 . (4.14) U (t) = e−tA U0 + 0

We next verify that F (U ) ∈ Fβ,σ ((0, TG,U0 ]; X).

(4.15)

In fact, by(4.10), as t → 0, t 1−β F (U (t)) ≤ CG,U0 (t 1−η + t 1−β ) → 0. Furthermore, from (4.2), (4.12), and (4.13) it follows that F (U (t)) − F (U (s)) = F ({ΦU }(t)) − F ({ΦU }(s)) ≤ CG,U0 (t − s)σ s β−σ −η + (s β−η + 1)(t − s)σ s −σ ≤ CG,U0 (t − s)σ s β−σ −η . Consequently, s 1−β+σ F (U (t)) − F (U (s)) ≤ CG,U0 s 1−η , (t − s)σ

0 < s < t ≤ TG,U0 .

This shows that (1.6) and (1.7) are valid for F (U (t)). Hence, (4.15) is verified. We now know that F (U ) + G ∈ Fβ,σ ((0, T ]; X). In view of (4.14), we can apply Theorem 3.5 to conclude that U belongs to the space (4.5) with estimates (4.6) and is a solution to problem (4.1). Step 4. Let us finally verify the uniqueness of solutions. Let U˜ be any other solution to (4.1) on the interval [0, TG,U0 ] which belongs to the space (4.5). By the definition, t 1−β AU˜ (t) + Aβ U˜ (t) ≤ CU˜ ,

0 < t ≤ TG,U0 .

Moreover, we can verify the intermediate estimate t η−β Aη U˜ (t) ≤ CU˜ ,

0 < t ≤ TG,U0 .

Indeed, by the moment inequality (2.117) for the fractional power A1−β applied with θ = η−β 1−β , we have η−β

1−η

Aη−β V ≤ CA1−β V 1−β V 1−β ,

V ∈ D(A1−β ).

Then, taking V = Aβ U˜ (t), we obtain the desired estimate.

1 Semilinear Abstract Evolution Equations

183

The formula U˜ (t) = e−tA U0 +

t

e−(t−s)A [F (U˜ (s)) + G(s)] ds,

0

0 ≤ t ≤ TG,U0 ,

together with (4.14), yields that U (t) − U˜ (t) =

t

e−(t−s)A [F (U (s)) − F (U˜ (s))] ds,

0

0 ≤ t ≤ TG,U0 .

We can then repeat the same arguments as in Step 2 to deduce that U − U˜ X(S) ≤ CU,U˜ S 1−η U − U˜ X(S) ,

0 < S ≤ TG,U0 .

This means that, if S > 0 is sufficiently small, then U (t) = U˜ (t) for every 0 ≤ t ≤ S. Let S˜ = sup{S; U (t) = U˜ (t) for all 0 ≤ t ≤ S}, and suppose that S˜ < TG,U0 . Then, ˜ = U˜ (S). ˜ We repeat the same procedure with the initial time S˜ and we have U (S) ˜ ˜ to derive that U (S˜ + t) = U˜ (S˜ + t) for every sufficiently initial value U (S) = U˜ (S) small t > 0, which is a contradiction. Hence, U (t) = U˜ (t) for every 0 < t ≤ TG,U0 .

1.3 Regularity for More Regular Initial Data For an external force function G ∈ Fγ ,σ ((0, T ]; X), β < γ ≤ 1, and an initial value U0 ∈ D(Aγ ), β < γ ≤ 1, we can verify a stronger regularity than (4.5) for the local solution of (4.1). Theorem 4.2 Let (2.92), (2.93), (4.2), and (4.3) be satisfied. Then, for any G ∈ Fγ ,σ ((0, T ]; X), 0 < σ < 1 − η, β < γ ≤ 1, and any U0 ∈ D(Aγ ), β < γ ≤ 1, the local solution U of (4.1) constructed in Theorem 4.1 has the following regularity: U ∈ C((0, TG,U0 ]; D(A)) ∩ C([0, TG,U0 ]; D(Aγ )) ∩ C1 ((0, TG,U0 ]; X), (4.16) dU AU ∈ Fγ ,σ ((0, TG,U0 ]; X). dt , In addition, U satisfies the estimates dU Aγ U C + dt γ ,σ + AU Fγ ,σ ≤ CG,U0 F

(4.17)

with some constant CG,U0 > 0 depending on the norms GFγ ,σ and Aγ U0 . Proof From (4.14) we have A U (t) = A η

η−γ −tA

e

A U0 + γ

0

t

Aη e−(t−s)A [F (U (s)) + G(s)] ds.

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4 Semilinear Evolution Equations

Therefore, in view of (4.4), Aη U (t) ≤ Aη−γ e−tA Aγ U0 t +C (t − s)−η [Aη U (s) + GFγ ,σ s γ −1 + 1] ds. 0

t When η ≤ γ ≤ 1, we see that Aη U (t) ≤ CG,U0 + C 0 (t − s)−η Aη U (s) ds. Theorem 1.27 then provides that Aη U (t) ≤ CG,U0 . When β < γ < η, however, we see that t η γ −η +C (t − s)−η Aη U (s) ds. A U (t) ≤ CG,U0 t 0

In this case, the function

p(t) = t η−γ Aη U (t)

p(t) ≤ CG,U0 + C

t

satisfies

t η−γ (t − s)−η s γ −η p(s) ds.

0

Let us solve the integral equation as follows. Let ε > 0 denote a small parameter. For 0 ≤ t ≤ ε, t p(t) ≤ CG,U0 + C t η−γ (t − s)−η s γ −η ds sup p(s) 0

0≤s≤ε

≤ CG,U0 + Ct 1−η sup p(s) ≤ CG,U0 + Cε 1−η sup p(s). 0≤s≤ε

0≤s≤ε

Hence, (1 − Cε 1−η ) sup0≤s≤ε p(s) ≤ CG,U0 . If ε is taken sufficiently small so that Cε1−η ≤ 12 , then we conclude that p(t) ≤ CG,U0 , 0 ≤ t ≤ ε. Meanwhile, for ε < t ≤ TG,U0 , we have t ε p(t) ≤ CG,U0 + C t η−γ (t − s)−η s γ −η ds + CT η−γ ε γ −η (t − s)−η p(s) ds 0

≤ CG,U0 + C

ε

t

(t − s)−η p(s) ds.

ε

Then, Theorem 1.27 again provides that p(t) ≤ CU0 ,G , ε ≤ t ≤ TG,U0 . We have thus shown that Aη U (t) ≤ CG,U0 (t γ −η + 1),

0 < t ≤ TG,U0 .

(4.18)

Using (4.18) (instead of (4.7)) and (4.8), we will repeat the same argument as in Step 1 of the proof of Theorem 4.1. Then, it is concluded that Aη [U (t) − U (s)] ≤ CG,U0 (t − s)σ (s γ −η−σ + 1),

0 < s ≤ t ≤ TG,U0 ,

Aβ [U (t) − U (s)] ≤ CG,U0 (t − s)σ (s γ −β−σ + 1),

0 < s ≤ t ≤ TG,U0 .

and

1 Semilinear Abstract Evolution Equations

185

Furthermore, these estimates immediately imply that F (U ) ∈ Fγ ,σ ((0, TG,U0 ]; X) with the estimate F (U )Fγ ,σ ≤ CG,U0 . The desired regularity (4.16) is then an immediate consequence of Theorem 3.5.

1.4 Global Existence We will return to the case where G ∈ Fβ,σ ((0, T ]; X) and U0 ∈ D(Aβ ). We recall that, according to Theorem 4.1, the interval [0, TG,U0 ] on which the local solution was constructed is determined by the norms GFβ,σ and Aβ U0 only. This fact immediately provides the following global existence of solutions which is utilized very often in applications. Corollary 4.1 Under the assumptions of Theorem 4.1, let G ∈ Fβ,σ ((0, T ]; X), 0 < σ < 1 − η, and U0 ∈ D(Aβ ). Assume that any local solution U of (4.1) in the function space C((0, TU ]; D(A)) ∩ C([0, TU ]; D(Aβ )) ∩ C1 ((0, TU ]; X) satisfies the estimate Aβ U (t) ≤ CG,U0 ,

0 ≤ t ≤ TU ,

(4.19)

with some constant CG,U0 > 0 independent of TU . Then, (4.1) possesses a unique global solution on the interval [0, T ]. Proof We extend the function G to a function G defined on the whole interval [0, ∞) by putting G(t) ≡ G(T ) for T < t < ∞. Clearly, we have GFβ,σ ((a,b];X) ≤ GFβ,σ ((0,T ];X) for any interval (a, b]. Let U1 be any vector in D(Aβ ) satisfying Aβ U1 ≤ CG,U0 . We here consider the Cauchy problem dV dt + AV = F (V ) + G(t), t1 < t < ∞, (4.20) V (t1 ) = U1 , with initial time t1 such that 0 ≤ t1 < ∞. Theorem 4.1 then provides that there exists a length τ > 0 determined by GFβ,σ and CG,U0 only such that (4.20) always possesses a local solution on an interval [t1 , t1 + τ ]. On account of this fact, we can prove the assertion of the corollary. Let us start with the local solution to (4.1) on [0, TG,U0 ] obtained in Theorem 4.1. Clearly, TG,U0 ≥ τ . Setting t1 = TG,U0 − τ2 and U1 = U (t1 ), we consider problem (4.20) to obtain a local solution V on an interval [t1 , t1 + τ ]. By the uniqueness of solution, we have U (t) = V (t) for t1 ≤ t ≤ TG,U0 ; this means that we have constructed

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4 Semilinear Evolution Equations

a local solution to (4.1) on the interval [0, TG,U0 + τ2 ]. The a priori estimate (4.19) allows us to continue this procedure unlimitedly. Each time, the local solution is extended over the fixed length τ2 of interval. So, by finite times, the extended interval can cover the given interval [0, T ].

1.5 Lipschitz Continuity of Solutions in Initial Data Let G be a bounded ball G = {G ∈ Fβ,σ ((0, T ]; X); Gβ,σ ≤ R1 },

0 < R1 < ∞,

of the space Fβ,σ ((0, T ]; X), and let B be a bounded ball B = {U ∈ D(Aβ ); Aβ U ≤ R2 },

0 < R2 < ∞,

of D(Aβ ). Then, for all G ∈ G and U0 ∈ B, there exists a local solution of (4.1) on some interval [0, TG,U0 ]. There is then time TG,B > 0 such that [0, TG,B ] ⊂ [0, TG,U0 ] for all pairs of (G, U0 ) ∈ G × B. We shall show the continuous dependence of solutions on (G, U0 ). Theorem 4.3 Under (2.92), (2.93), (4.2), and (4.3), let U and V be the solutions of (4.1) for the data (G, U0 ) and (H, V0 ) in G × B, respectively. Then, t η Aη [U (t) − V (t)] + t β Aβ [U (t) − V (t)] + U (t) − V (t) ≤ CG,B [U0 − V0 + t β G − H β,σ ],

0 < t ≤ TG,B .

Proof This theorem is proved by analogous arguments as in the proof of Theorem 4.1. Indeed, for 0 ≤ θ < 1, we see that t Aθ [U (t) − V (t)] = Aθ e−tA (U0 − V0 ) + Aθ e−(t−s)A 0

× {[F (U (s)) − F (V (s))] + [G(s) − H (s)]} ds, 0 < t ≤ TB,G . As a consequence, t θ Aθ [U (t) − V (t)] ≤ Aθ U0 − V0 + Aθ t θ + CG,B Aθ t θ

t

(t − s)−θ G(s) − H (s) ds

0 t

(t − s)−θ {Aη [U (s) − V (s)]

0

+ (s β−η + 1)Aβ [U (s) − V (s)]} ds

1 Semilinear Abstract Evolution Equations

187

≤ Aθ U0 − V0 + Aθ G − H β,σ t θ + CG,B Aθ t

θ

t

(t − s)−θ s β−1 ds

0 t

(t − s)−θ s −η p(s) ds,

0

where p(t) = t η Aη [U (t) − V (t)] + t β Aβ [U (t) − V (t)],

0 < t ≤ TG,B .

Using these estimates with θ = β and η, we obtain the integral inequality p(t) ≤ C[U0 − V0 + t β G − H β,σ ] t η + CG,B t (t − s)−η + t β (t − s)−β s −η p(s) ds,

0 < t ≤ TG,B .

0

To solve this inequality, we use the same techniques as in the proof of Theorem 4.2. Arguing first in a small interval [0, ε] and then in the other interval [ε, TG,B ], we in fact conclude that p(t) ≤ CG,B [U0 − V0 + t β G − H β,σ ],

0 < t ≤ TG,B .

Finally, taking θ = 0, we conclude also that U (t) − V (t) ≤ CG,B [U0 − V0 + t β G − H β,σ ],

0 ≤ t ≤ TG,B .

Corollary 4.2 Under the conditions of Theorem 4.3, t η−β Aη [U (t) − V (t)] + Aβ [U (t) − V (t)] ≤ CG,B [Aβ (U0 − V0 ) + G − H β,σ ],

0 < t ≤ TG,B .

Proof The techniques used in the proof of Theorem 4.3 are available for verifying the assertion. It indeed suffices to substitute the estimate Aθ e−tA (U0 − V0 ) ≤ Aθ−β t β−θ Aβ (U0 − V0 ) with θ = β and η for Aθ e−tA (U0 − V0 ) ≤ Aθ t −θ U0 − V0 .

1.6 Case β = 0 It is possible to generalize some results of Theorem 4.1 to the critical case where β = 0. Let us assume that (4.2) is valid with β = 0, i.e., F (U ) − F (V ) ≤ ϕ(U + V )[Aη (U − V ) + (Aη U + Aη V )U − V ],

U, V ∈ D(Aη ),

(4.21)

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4 Semilinear Evolution Equations

with 0 ≤ η < 1.

(4.22)

We can then verify the following result. Theorem 4.4 Let (2.92), (2.93), (4.21), and (4.22) be satisfied. Then, for any G ∈ Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1 − η, and any U0 ∈ X, (4.1) possesses a unique local solution U in the function space: U ∈ C([0, TG,U0 ]; X) ∩ C1 ((0, TG,U0 ]; X), AU ∈ C((0, TG,U0 ]; X),

(4.23)

TG,U0 > 0 depending only on the norms GFβ,σ and U0 . In addition, U satisfies the estimates dU (4.24) (t) U (t) + t + tAU (t) ≤ CG,U0 , 0 < t ≤ TG,U0 , dt with some constant CG,U0 > 0 depending on the norms GFβ,σ and U0 . Proof The method of proving this theorem is quite analogous to that of Theorem 4.1. We set for 0 < S ≤ T , the basic Banach space X(S) = U ∈ C((0, S]; D(Aη )) ∩ C([0, S]; X); sup t η Aη U (t) < ∞ , 0 0 (i = 1, 2) are chosen appropriately and S > 0 is sufficiently small, then Φ maps the set K(S) into itself and is a contraction with respect to the norm X(S). Consequently, Φ possesses a unique fixed point U ∈ K(S). Since F (U ) = F (ΦU ) ∈ F β,σ ((0, S]; X),

(4.27)

we conclude that U is a unique solution of (4.1) in the function space (4.23). Estimate (4.24) is verified by Theorem 3.4.

2 Variational Methods

189

By the same argument as in Corollary 4.1, we can similarly construct a global solution to (4.1). Corollary 4.3 Under the assumptions of Theorem 4.4, let G ∈ Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1 − η, and U0 ∈ X. Assume that any local solution U of (4.1) in the function space C((0, TU ]; D(A)) ∩ C([0, TU ]; X) ∩ C1 ((0, TU ]; X)

(4.28)

satisfies the estimate U (t) ≤ CG,U0 ,

0 ≤ t ≤ TU ,

(4.29)

with some constant CG,U0 > 0 independent of TU . Then, (4.1) possesses a unique global solution in the function space: U ∈ C((0, T ]; D(A)) ∩ C([0, T ]; X) ∩ C1 ((0, T ]; X). Let us finally verify the Lipschitz continuity of solutions in initial data. Let G be a bounded ball G = {G ∈ Fβ,σ ((0, T ]; X); Gβ,σ ≤ R1 },

0 < R1 < ∞,

of the space Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1 − η, and let B be a bounded ball B = {U ∈ X; U ≤ R2 },

0 < R2 < ∞,

of X. Then, there is an interval [0, TG,B ] on which (4.1) has a unique local solution for any pair (G, U0 ) ∈ G × B. Then, we have the following result. Theorem 4.5 Under (2.92), (2.93), (4.21), and (4.22), let U and V be the solutions of (4.1) for the data (G, U0 ) and (H, V0 ) in G × B, respectively. Then, t η Aη [U (t) − V (t)] + U (t) − V (t) ≤ CG,B [U0 − V0 + t β G − H β,σ ],

0 < t ≤ TG,B .

As the proof of this theorem is quite analogous to that of Theorem 4.3, we may omit it.

2 Variational Methods We here present existence and uniqueness results for problem (4.1) by the variational methods. Let Z and X be two separable Hilbert spaces with dense and compact embedding Z ⊂ X. Let · and | · | be the norms of Z and X, respectively. Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Let · ∗ be the norm of Z ∗ . Let a(U, V ) be a sesquilinear

190

4 Semilinear Evolution Equations

form defined on Z satisfying (2.6) and (2.7), and let A be a sectorial operator of Z ∗ associated with the form. We are concerned with the Cauchy problem for a semilinear evolution equation dU dt + AU = F (U ) + G(t), 0 < t ≤ T , (4.30) U (0) = U0 , in Z ∗ . Here, F is a nonlinear operator from Z to Z ∗ and is assumed to satisfy the following conditions. For any positive number ζ > 0, there exist continuous increasing functions φζ (·) ≥ 0 and ψζ (·) ≥ 0 such that the following estimates hold: F (U )∗ ≤ ζ U + φζ (|U |),

U ∈ Z;

(4.31)

F (U ) − F (V )∗ ≤ ζ U − V + (U + V + 1)ψζ (|U | + |V |)|U − V |, U, V ∈ Z.

(4.32)

The external force function G is a Z ∗ -valued function belonging to L2 (0, T ; Z ∗ ), and the initial value U0 is taken in X. We then obtain the following theorem. For the proof, see Lions [Lio69] (see also [RY01]). Theorem 4.6 Let (4.31) and (4.32) be satisfied. Then, for any G ∈ L2 (0, T ; Z ∗ ) and any U0 ∈ X, there exists a unique local solution U to (4.30) in the function space: U ∈ L2 (0, TU0 ,G ; Z) ∩ C([0, TU0 ,G ]; X) ∩ H 1 (0, TU0 ,G ; Z ∗ ), where TU0 ,G > 0 is determined by the norms |U0 | and GL2 (0,T ;Z ∗ ) . In addition, U satisfies the estimates U L2 (0,TG,U0 ;Z) + U C([0,TG,U0 ];X) + U H 1 (0,TG,U

0

;Z ∗ )

≤ CG,U0 ,

where CG,U0 > 0 is a constant determined by the norms GL2 (0,T ;Z ∗ ) and |U0 |.

3 Semilinear Parabolic Equations We consider the initial-boundary-value problem for a semilinear parabolic equation ⎧ n ∂u ⎪ ⎪ i,j =1 Dj [aij (x)Di u] + f (x, u, ∇u) + g(x, t) in Ω × (0, T ), ∂t = ⎨ n on ∂Ω × (0, T ), (4.33) i,j =1 νj (x)aij (x)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω, where 0 < T < ∞ is fixed time. We assume that aij (x) satisfy (2.19) and (2.28). The function f (x, u, ζ ) is

4 Competition System with Diffusions

191

a complex-valued function for x ∈ Ω, u ∈ R + iR, and ζ ∈ (R + iR)n , which is a smooth function with respect to the real variables x ∈ Ω, Re u, Im u, Re ζ , and Im ζ . The function g(x, t) is an external force function, and u0 (x) is an initial function. We want to handle (4.33) in a Banach space X = Lp (Ω), where n < p < ∞. Let Ap be a realization of the differential operator − i,j Dj [aij (x)Di ] + 1 in Lp (Ω) ∂u under the Neumann-type boundary conditions ∂ν = 0 on ∂Ω. It is known that the A domain D(Ap ) is characterized by (2.77). Using this operator, we obtain an abstract formulation of (4.33) of the form (4.1). Here, the nonlinear operator F (u) is given by F (u) = u + f (x, u, ∇u),

u ∈ Hps (Ω),

where the exponent s is taken as 1 + pn < s < 2. We notice by (1.76) that Hps (Ω) ⊂ C1 (Ω) with continuous embedding. We choose exponents β and η as 2s < β = η < 1. Then, according to (16.36) in η Chap. 16, it is observed that D(Ap ) ⊂ Hps (Ω). In addition, by a direct calculation, F (u) is shown to fulfill condition (4.2). So, the abstract results obtained above are available to (4.33). By Theorem 4.1, for any g ∈ Fβ,σ ((0, T ]; Lp (Ω)), 0 < σ < β 1 − η, and any u0 ∈ D(Ap ), there exists a unique local solution u to (4.33) in the function space: β u ∈ C((0, Tg,u0 ]; D(Ap )) ∩ C([0, Tg,u0 ]; D(Ap )) ∩ C1 ((0, Tg,u0 ]; Lp (Ω)), du Ap u ∈ Fβ,σ ((0, Tg,u0 ]; Lp (Ω)), dt , β

Tg,u0 > 0 depending only on the norms gFβ,σ and Ap u0 . Furthermore, u satisfies the estimates β A U + du p dt β,σ + Ap uFβ,σ ≤ Cg,u0 . C F

4 Competition System with Diffusions Consider the initial-boundary-value problem for the competition system ⎧ ∂u ⎪ = aΔu + cu − γ11 u2 − γ12 uv in Ω × (0, ∞), ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂v = bΔv + dv − γ21 uv − γ22 v 2 in Ω × (0, ∞), ∂t ∂u ∂v ⎪ on ∂Ω × (0, ∞), ⎪ ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x) in Ω, 0

(4.34)

0

in a C2 or convex, bounded domain Ω ⊂ R3 . We consider two competing species of biological individuals in Ω. Their densities are denoted by u(x, t) and v(x, t), respectively. We consider diffusions of individuals, and the diffusion rates of species are denoted by a and b, respectively.

192

4 Semilinear Evolution Equations

We assume that a, b, c, d, and γij (i, j = 1, 2) are positive constants (> 0). Initial functions u0 , v0 are assumed to satisfy 0 ≤ u0 , v0 ∈ L2 (Ω).

4.1 Construction of Local Solutions We want to handle (4.34) in the product space X = L2 (Ω) (see (1.84)). The problem is then formulated as the Cauchy problem for an abstract equation dU 0 < t < ∞, dt + AU = F (U ), (4.35) U (0) = U0 , in X. Here, A is a diagonal matrix operator of X given by A = diag {A1 , A2 } with domain D(A) = H2N (Ω), where A1 and A2 are realizations of operators −aΔ + 1 and −bΔ + 1, respectively, in L2 (Ω) under the Neumann boundary conditions on ∂Ω. It is known by (2.34) that A1 and A2 are positive definite self-adjoint operators of L2 (Ω) with domains D(A1 ) = D(A2 ) = HN2 (Ω). Furthermore, according to Theorems 16.7 and 16.9 in Chap. 16, D(Aθ1 ) = D(Aθ2 ) = H 2θ (Ω) if 0 ≤ θ < 34 and D(Aθ1 ) = D(Aθ2 ) = HN2θ (Ω) if 34 < θ ≤ 1. As a consequence, A is a positive definite self-adjoint operator of X, and the domains of fractional powers are given 3 by D(Aθ ) = H2θ (Ω) if 0 ≤ θ < 34 and D(Aθ ) = H2θ N (Ω) if 4 < θ ≤ 1. The nonlinear operator F : L4 (Ω) → L2 (Ω) is defined by u (c + 1)u − γ11 u2 − γ12 uv , U= ∈ L4 (Ω). F (U ) = (d + 1)v − γ21 uv − γ22 v 2 v The space of initial functions is given by u K = U0 = 0 ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ v0 ∈ L2 (Ω) . v0 It is immediate to verify that the nonlinear operator F fulfills the condition (4.21) 2η with 34 < η < 1. Indeed, since D(Aη ) = HN (Ω) ⊂ L∞ (Ω), we have F (U ) − F (V )L2 ≤ C(U L∞ + V L∞ + 1)U − V L2 ≤ C(Aη U L2 + Aη V L2 + 1)U − V L2 ,

U, V ∈ D(Aη ).

Theorem 4.4 then provides the existence of local solutions. Indeed, for any U0 ∈ L2 (Ω), (4.35) possesses a unique local solution U in the function space: U ∈ C (0, TU0 ]; H2N (Ω) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)). (4.36) Here, TU0 > 0 is determined by the norm U0 L2 only. Furthermore, the solution satisfies the estimates U (t)L2 + tAU (t)L2 ≤ CU0 .

(4.37)

4 Competition System with Diffusions

193

4.2 Nonnegativity of Local Solutions We shall show that the local solution constructed above is nonnegative. For U0 ∈ K, let U = t(u, v) be the local solution in the function space (4.36) with estimate (4.37), and let us prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . We first notice that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (4.35) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) must be real valued. We will employ the truncation method. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞ 0 such that, for any local solution U in (4.38) with U0 ∈ K, it holds that U (t)L2 ≤ C(e−t U0 L2 + 1),

0 ≤ t ≤ TU .

(4.39)

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4 Semilinear Evolution Equations

Proof Multiply the first equation of (4.34) by u and integrate the product in Ω. Then, 1 d 2 dt

u2 dx + a Ω

|∇u|2 dx =

Ω

(cu2 − γ11 u3 − γ u2 v) dx

Ω

≤

(cu2 − γ11 u3 ) dx. Ω

Since 1 (2c + 1)3 cu2 − γ11 u3 ≤ − u2 + 2 2 27γ11

for all u ≥ 0,

it follows that 1 d 2 dt

1 u dx ≤ − 2 Ω

u2 dx +

2

Ω

|Ω|(2c + 1)3 . 2 27γ11

Solving this differential inequality (cf. (1.58)), we obtain that 2|Ω|(2c + 1)3 , 2 27γ11

u(t)2L2 ≤ e−t u0 2L2 +

0 ≤ t ≤ TU .

We verify by the similar arguments that v(t)2L2 ≤ e−t v0 2L2 +

2|Ω|(2d + 1)3 , 2 27γ22

0 ≤ t ≤ TU .

This, together with the above, yields the desired estimate (4.39).

The global existence is now obtained immediately by Corollary 4.3. Indeed, (4.39) shows that estimate (4.29) holds for any nonnegative local solution with initial value U0 ∈ K. We can then repeat the same arguments as in Corollary 4.3 to conclude the global existence of a solution. We have thus deduced that, for any U0 ∈ K, there exists a unique global solution of (4.34) in the function space: 0 ≤ u, v ∈ C (0, ∞); HN2 (Ω) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)).

5 Some Model in Immunology We are concerned with the initial-boundary-value problem for some mathematical model describing the immune system. The problem is given by

5 Some Model in Immunology

195

⎧ ∂u ⎪ ⎪ ∂t = c − du − αuρ ⎪ ⎪ ⎪ ∂v ⎪ ⎪ ⎪ ∂t = αuρ − ev − βvw ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ⎨ ∂t = aΔw − f w + γ (x) ∂ρ ∂t = bΔρ − gρ ⎪ ⎪ ⎪ ∂ρ ∂w ⎪ ⎪ = ∂n =0 ⎪ ⎪ ∂n

+ hv

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞),

(4.40)

on ∂Ω × (0, ∞), ⎪ ⎪ ⎪ ⎪ v(x, 0) = v0 (x), u(x, 0) = u0 (x), ⎪ ⎪ ⎩ ρ(x, 0) = ρ0 (x) in Ω, w(x, 0) = w0 (x), in a C2 or convex, bounded domain Ω ⊂ R3 . Here, Ω denotes an organism or a body. The function u(x, t) denotes the density of uninfected target cells in Ω at time t, and v(x, t) denotes the density of infected cells by viruses. By w(x, t) the density of killer cells is denoted. The killer cells are produced at a rate γ (x) in Ω by the effect of the immune system and kill the infected cells. By ρ(x, t) the density of viruses is denoted. Viruses are produced by the infected cells and diffuses in Ω with diffusion rate b. We assume that 0 ≤ γ ∈ L2 (Ω)

(4.41)

and that a, b, c, d, e, f , and g are positive constants (>0) and α, β are also positive constants (>0). We will handle this problem in the product space X = L2 (Ω) = t(u, v, w, ρ); u, v, w and ρ ∈ L2 (Ω) . The space of initial functions is given by K = t(u0 , v0 , w0 , ρ0 ); 0 ≤ u0 , v0 , w0 and ρ0 ∈ L2 (Ω) .

(4.42)

5.1 Construction of Local Solutions Problem (4.40) is formulated in X as the Cauchy problem of the form (4.35). Here, the linear operator A is a diagonal matrix operator of X given by A = diag{d, e, A1 , A2 }, where A1 and A2 are realizations of differential operators −aΔ + f and −bΔ + g, respectively, in L2 (Ω) under the Neumann boundary conditions on ∂Ω. So, D(A) is given by D(A) = t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ HN2 (Ω) (see (2.34)). Clearly, A is a positive definite self-adjoint operator of X. For 0 < θ < 1, its fractional powers Aθ are also diagonal operators diag{d θ , eθ , Aθ1 , Aθ2 } and, thanks to Theorems 16.7 and 16.9, their domains are characterized by D(Aθ ) = {t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ H 2θ (Ω)}, 0 ≤ θ < 34 , D(Aθ ) = {t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ HN2θ (Ω)}, 34 < θ ≤ 1.

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4 Semilinear Evolution Equations

The nonlinear operator F is given by F (U ) = t (c − αuρ, αuρ − βvw, γ (x), hv) with domain D(F ) = t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ L∞ (Ω) . Fix an exponent η such that 34 < η < 1. Then, by virtue of (1.76), we have D(Aη ) ⊂ D(F ). Furthermore, it is immediate to see that F satisfies condition (4.21). Thanks to Theorem 4.4, we can state that, for any U0 = t(u0 , v0 , w0 , ρ0 ) ∈ K, there exists a unique local solution U = t(u, v, w, ρ) to (4.40) in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; X) ∩ C1 ((0, TU0 ]; X),

(4.43)

where TU0 > 0 is determined by the norm U0 X only. In addition, the local solution satisfies the estimates dU U (t)L2 + t (4.44) dt + tAU (t)L2 ≤ CU0 , 0 < t ≤ TU0 , L2 CU0 > 0 being determined by U0 L2 .

5.2 Nonnegativity of Local Solutions For U0 ∈ K, let U be the local solution of (4.40) constructed above. We want to show that u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, and ρ(t) ≥ 0 for every 0 < t ≤ TU0 . For this purpose, however, we have to introduce the modified nonlinear operator F˜ (U ) = t (c − αuρ, αuρ − βvw, γ (x), hχ(Re v)),

U ∈ D(F˜ ) = D(F ),

where χ(v) denotes a cutoff function such that χ(v) ≡ 0 for −∞ < v < 0 and χ(v) = v for 0 ≤ v < ∞. We have to consider the auxiliary problem

d U˜ dt

+ AU˜ = F˜ (U ),

U˜ (0) = U0 ,

0 < t < ∞,

(4.45)

in X. It is clear that the new nonlinear operator F˜ also satisfies (4.21) with the same ˜ L2 for v, v˜ ∈ L2 (Ω). So, exponent η because of χ(Re v) − χ(Re v) ˜ L2 ≤ v − v ˜ v, ˜ w, ˜ ρ) ˜ on an interval [0, T˜U0 ] in (4.45) possesses a unique local solution U˜ = t(u, the same function space (4.43). Then, it is seen that u(t) ˜ ≥ 0, v(t) ˜ ≥ 0, w(t) ˜ ≥ 0, ρ(t) ˜ ≥ 0 for every 0 < t ≤ T˜0 . In fact, let us first verify that U˜ (t) is real valued. It is clear that the complex conjugate U˜ (t) of U˜ (t) is also a local solution to (4.45) with the same initial value; so, by the uniqueness of solution, U˜ (t) = U˜ (t); hence, U˜ (t) is real valued.

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197

Let us next use a cutoff function H (ρ) ˜ such that H (ρ) ˜ = 12 ρ˜ 2 for −∞ < ρ˜ < 0 and H (ρ) ˜ ≡ 0 for 0 ≤ ρ˜ < ∞. Since ρ˜ ∈ C([0, T˜U0 ];L2 (Ω)) ∩ C1 ((0, T˜U0 ]; L2 (Ω)) due to (4.43), we observe by (1.100) that ψ(t) = Ω H (ρ(t)) ˜ dx is continuously differentiable with the derivative

ψ (t) = b H (ρ)Δ ˜ ρ˜ dx − g H (ρ) ˜ ρ˜ dx + h H (ρ)χ( ˜ v) ˜ dx. Ω

Ω

Ω

By property (1.96), we have H (ρ)Δ ˜ ρ˜ dx = − ∇H (ρ) ˜ · ∇ ρ˜ dx = − |∇H (ρ)| ˜ 2 dx ≤ 0. Ω

Ω

Ω

˜ ≤ 0 and H (ρ) ˜ ρ˜ ≥ 0, it follows that ψ (t) ≤ 0; conseIn addition, since H (ρ) quently, ψ(t) ≤ ψ(0) for 0 < t ≤ T˜U0 . Thus, ψ(0) = 0 implies ψ(t) ≡ 0, namely, ρ(t) ˜ ≥ 0 for 0 ≤ t ≤ T˜U0 . We can repeat the same argument for u, ˜ v, ˜ w˜ also. In fact, knowing that ρ(t) ˜ ≥ 0, we are able to prove that u(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . Since ˜ ρ(t) ˜ ≥ 0 and w(t) ˜ ≥ 0, we γ (x) ≥ 0, w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . Finally, since u(t) conclude that v(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . We now notice that v(t) ˜ ≥ 0 implies that χ(v(t)) ˜ = v(t); ˜ in other words, U˜ is a local solution of the original problem (4.40), too. The uniqueness of solution then implies that U (t) ≡ U˜ (t). Hence, u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, ρ(t) ≥ 0 for 0 < t ≤ T˜U0 . If T˜U0 ≥ TU0 , we finish the proof. If not, we put T0 = sup{0 < T ≤ TU0 ; u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, ρ(t) ≥ 0 for every 0 < t ≤ T }. Since Ω H (u(T0 )) dx = limtT0 Ω H (u(t)) dx = 0, we see that u(T0 ) ≥ 0. By similar reasons, we have v(T0 ) ≥ 0, w(T0 ) ≥ 0, ρ(T0 ) ≥ 0. So, if T0 = TU0 , we finish the proof. If T0 < TU0 , we will consider again problem (4.45) but with the initial time T0 and the initial value U (T0 ). Repeating the same arguments as above, we conclude that there is τ > 0 such that u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, ρ(t) ≥ 0 for every t such that TU0 ≤ t ≤ TU0 + τ . This is a contradiction; hence, T0 = TU0 .

5.3 Global Solutions In order to construct a global solution, let us obtain a priori estimates for local solutions. For U0 ∈ K, let U = t(u, v, w, ρ) be a local solution of (4.40) on [0, TU ] in the function space: 0 ≤ U ∈ C((0, TU ]; D(A)) ∩ C([0, TU ]; X) ∩ C1 ((0, TU ]; X). Integrate the first equation of (4.40) in Ω. Then, d u dx = (c − du − αuρ) dx. dt Ω Ω

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Similarly, by the second equation of (4.40), d v dx = (αuρ − ev − βvw) dx. dt Ω Ω d Therefore, dt Ω (u + v) dx ≤ c|Ω| − δ Ω (u + v) dx, where δ = min{d, e}. Consequently (cf. (1.58)), it follows that u(t) + v(t)L1 ≤ e−δt (u0 + v0 L1 ) + cδ −1 |Ω|,

0 ≤ t ≤ TU .

(4.46)

Multiply the first equation by u and integrate the product in Ω. Then, 1 d u2 dx = (cu − du2 − αu2 ρ) dx. 2 dt Ω Ω Meanwhile, multiplying the first equation by v and integrating the product in Ω, we have du v dx = (cv − duv − αuvρ) dx. Ω dt Ω In a similar way, we can obtain from the second equation on v the equalities dv u dx = (αu2 ρ − euv − βuvw) dx Ω dt Ω and 1 d 2 dt

v 2 dx = Ω

(αuvρ − ev 2 − βv 2 w) dx. Ω

Therefore, summing up these four equalities, we have 1 d c(u + v) − δ(u + v)2 dx (u + v)2 dx ≤ 2 dt Ω Ω ≤ C(u0 + v0 L1 + 1) − δ (u + v)2 dx Ω

(due to (4.46)). Hence, solving this differential inequality, we conclude that u(t) + v(t)2L2 ≤ e−2δt u0 + v0 2L2 + C(u0 + v0 L1 + 1),

0 ≤ t ≤ TU .

Since u(t) ≥ 0 and v(t) ≥ 0, it then follows that u(t)2L2 + v(t)2L2 ≤ e−2δt u0 + v0 2L2 + C(u0 + v0 L1 + 1), 0 ≤ t ≤ TU .

(4.47)

Multiply the fourth equation on ρ by ρ and integrate the product in Ω. Then, after some calculations, 1 d g h2 2 2 2 ρ dx + b |∇ρ| dx + g ρ dx ≤ ρ 2 dx + v(t)2L2 . 2 dt Ω 2 Ω 2g Ω Ω

6 Nonautonomous Semilinear Abstract Evolution Equations

199

Using (4.47), we obtain that g 1 d ρ 2 dx + ρ 2 dx ≤ C u0 + v0 2L2 + 1 . 2 dt Ω 2 Ω Hence, ρ(t)2L2 ≤ e−gt ρ0 2L2 + C u0 + v0 2L2 + 1 ,

0 ≤ t ≤ TU .

(4.48)

In view of (4.41), we can use the similar techniques to conclude that w(t)2L2 ≤ e−f w0 2L2 + C, 0 ≤ t ≤ TU . Hence, this, together with (4.47) and (4.48), yields that U (t)L2 ≤ C(U0 L2 + 1),

0 ≤ t ≤ TU .

(4.49)

Now, when we have the a priori estimates (4.29), the global existence is concluded by applying Corollary 4.3. Indeed, for any U0 ∈ K, (4.40) possesses a unique global solution in the function space: 0 ≤ U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); X) ∩ C1 ((0, ∞); X).

6 Nonautonomous Semilinear Abstract Evolution Equations In this section, we will present some device for treating nonautonomous semilinear abstract evolution equations. Consider the Cauchy problem for a nonautonomous evolution equation dU dt

+ AU = F (t, U ) + G(t),

0 < t ≤ T,

U (0) = U0 ,

(4.50)

in a Banach space X. Here, A is a sectorial operator of X satisfying (2.92) and (2.93). For each 0 ≤ t ≤ T , F (t, ·) is a nonlinear operator from D(Aη ) into X, where η is some fixed exponent. We assume the following Lipschitz condition: F (t, U ) − F (s, V ) ≤ ϕ(Aβ U + Aβ V ) × Aη (U − V ) + (Aη U + Aη V + 1)[|t − s| + Aβ (U − V )] , (t, U ), (s, V ) ∈ [0, T ] × D(Aη ),

(4.51)

where β is a second fixed exponent satisfying the relation 0 ≤ β ≤ η < 1. The function G is a given external force function in Fβ,σ ((0, T ]; X). The initial value U0 is taken in D(Aβ ).

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We consider the product Banach space X = C × X and rewrite problem (4.50) in the form d τ 1 0 τ 0 τ +1 dt U + 0 A U = F (χ(Re τ ),U ) + G(t) , 0 < t ≤ T , (4.52) τ 0 U (0) = U0 , where χ(t) is a C0,1 cutoff function for −∞ < t < ∞ such that χ(t) ≡ 0 for −∞ < t < 0, χ(t) = t for 0 ≤ t ≤ T , and χ(t) ≡ T for T < t < ∞. We can consider (4.52) as an autonomous abstract evolution equation in X for the unknown function t(τ, U ). Since τ (t) = t, problem (4.52) is certainly equivalent to (4.50). It is easy to see that the operator A = diag{1, A} is a sectorial operator of X satisfying (2.92) and (2.93). Furthermore, for 0 < θ < 1, Aθ = diag{1, Aθ } with domain D(Aθ ) = C × D(Aθ ). Therefore, condition (4.51) implies (4.2) if β > 0 and (4.21) if β = 0 for the nonlinear F(τ, U ) = t(τ + 1, F (χ(Re τ ), U )), which is an operator from D(Aη ) into X. In addition, U0 ∈ D(Aβ ) implies that t(0, U0 ) ∈ D(Aβ ). Hence, Theorems 4.1 if β > 0 and 4.4 if β = 0 are available to (4.52).

Notes and Further Researches The theory of semilinear abstract parabolic evolution equations has been developed in the study of the Navier–Stokes equations by the functional analytical methods. As pioneering works, we quote Fujita–Kato [FK64], Giga [Gig81, Gig85, Gig86], and Giga–Miyakawa [GM85]. For the variational methods, we quote Lions [Lio69] and Temam [Tem84]. When the Lipschitz condition (4.2) (or (4.21)) is valid with α = η, Theorem 4.1 (or 4.4) is quite standard (e.g., [Zei90, Theorem 19.I]). The generalization for the case where α < η was first done by von Wahl [vWa85] and Hoshino–Yamada [HY91]. The proof of Theorem 4.1 was presented by Osaki–Yagi [OY02]. Theorem 4.4 can be obtained by a slight modification of Theorem 4.1. In this chapter, we have described applications of the abstract theory only for the Lotka–Volterra competition model and some model in immunology. It is actually possible to apply our abstract results to enormous reaction–diffusion models presented from the real world. (We will indeed treat some of these in Chaps. 8, 9, 10, 11, 12, and 13). A historical study on the Lotka–Volterra model is seen in Israel– Gasca [IG02]. Stationary problem of (4.34) were studied by Dancer–Du [DD95a, DD95b], Kan-on [Kan02, Kan03, Kan06, Kan07], and others. Nakashima–Wakasa [NW07] and Dancer–Zhang [DZ02] studied the asymptotic behavior of solutions of the evolutional problem of (4.34). In addition, there is a great deal of literature which treats related problems to (4.34) including the homogeneous Dirichlet boundary conditions, variable coefficients, nonautonomous equations, and so forth. For mathematical models describing the immune system against the threat of invasion by bacteria and viruses, we refer the reader to Murray [Mur02, Chap. 10]. The reaction–diffusion model in (4.40) was obtained by incorporating spreading of killer cells and viruses to the ordinary differential equation model presented by Britton [Bri03, Sect. 6.5].

Chapter 5

Quasilinear Evolution Equations

This chapter is devoted to the Cauchy problem for quasilinear evolution equations of the forms (5.1) and (5.43) in Banach spaces. The linear operators in the equations are sectorial operators with angles < π2 but are allowed to depend on the unknown functions. Besides an underlying space X in which the problem is given, we shall introduce two more Banach spaces Y and Z such that Z ⊂ Y ⊂ X. The sectorial operators A(U ) are defined for U ∈ Z. Like (3.29), we make the assumption that D(A(V )) ⊂ D(A(U )ν ) for any U, V ∈ Z with some fixed exponent 0 < ν ≤ 1. In addition, instead of (3.30), A(U )ν [A(U )−1 − A(V )−1 ] is estimated by the Y norm of U − V , i.e., U − V Y . We shall use two more exponents α and β as graded scales to measure the spaces Y and Z, respectively. That means that we assume the inclusions D(A(U )α ) ⊂ Y and D(A(U )β ⊂ Z for all U ∈ Z with 0 ≤ α < β < 1. The exponents are also assumed to satisfy the relation 1 + α < β + ν. The optimal one is of course the case where ν = 1, which means that the domains D(A(U )) are independent of U . When ν ≤ β, we have to add the assumption of reflexibility of Z. The initial value U0 is taken in Z; more precisely, U0 must satisfy a compatibility condition of the form U0 ∈ D(A(U0 )β ) (⊂ Z). In applying the abstract results of this chapter, we have to set the three spaces Z ⊂ Y ⊂ X in a suitable way. The space X is a function space in which we want to handle a quasilinear equation to be considered. The space Z is chosen as large as possible with the condition that, for all U ∈ Z, A(U ) are well defined as sectorial operators of X. Recall that the initial values can be taken only in Z. The space Y is also chosen as close as possible to X in order that the exponent condition 1 + α < β + ν is valid. The optimal one is the case where Y = X, i.e., α = 0. The norm which we have to estimate a priori for obtaining global solutions is the Z norm of the local solutions, more precisely, the norm A(U (t))β U (t)X . The abstract results are actually applicable to various quasilinear problems. In Chaps. 14 and 15, we shall apply them to physical and biological models. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_5, © Springer-Verlag Berlin Heidelberg 2010

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1 Quasilinear Abstract Evolution Equations 1.1 Structural Assumptions We consider the Cauchy problem of an abstract quasilinear evolution equation dU dt

+ A(U )U = F (t),

0 < t ≤ T,

(5.1)

U (0) = U0 ,

in a Banach space X. Let Z be a second Banach space continuously embedded in X, and let K be the open ball of Z, K = B Z (0; R) = {U ∈ Z; U Z < R},

0 < R < ∞.

For U ∈ K, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 and with domain D(A(U )) which is allowed to depend on U . The function F is a given external force function on (0, T ]. The initial value U0 is taken in K(⊂ Z). We make the following assumptions. The spectrum σ (A(U )) is contained in a fixed open sectorial domain, i.e., σ (A(U )) ⊂ Σω = {λ ∈ C; |arg λ| < ω},

U ∈ K,

(5.2)

λ∈ / Σω , U ∈ K,

(5.3)

with some angle 0 < ω < π2 , and the resolvent satisfies (λ − A(U ))−1 ≤ M/|λ|, L(X)

with a constant M ≥ 1. The domain D(A(U )) may vary with U ∈ K, but there is an exponent 0 < ν ≤ 1 such that D(A(V )) ⊂ D A(U )ν for any pair of U, V ∈ K. (5.4) In addition, A(U ) satisfies a Lipschitz condition of the form A(U )ν A(U )−1 − A(V )−1 ≤ N U − V Y , U, V ∈ K, L(X)

(5.5)

with a constant N > 0. Here, Y is a third Banach space such that Z ⊂ Y ⊂ X with continuous embeddings. There are two exponents 0 ≤ α < β < 1 such that, for any U ∈ K, D(A(U )α ) ⊂ Y and D(A(U )β ) ⊂ Z with the estimates U˜ Y ≤ D1 A(U )α U˜ X , U˜ ∈ D(A(U )α ), U ∈ K, (5.6) U˜ ∈ D(A(U )β ), U ∈ K, U˜ Z ≤ D2 A(U )β U˜ X , Di > 0 (i = 1, 2) being some constants. The exponents satisfy the relations 0≤α 0 is determined by F and U0 . In addition, U satisfies the estimates A(U )β U + dU (5.11) dt β,σ + A(U )U Fβ,σ ≤ CF,U0 , C F where CF,U0 > 0 is a constant depending on F and U0 . Remark 5.1 It is very important to investigate the dependence of TF,U0 on the initial datum. We must notice by the proof of the theorem described below (see Steps 3 and 4) that TF,U0 depends not only on the norms F Fβ,σ and A(U0 )β U0 X but also on the following moduli of continuity: ωF (t) = sup 0≤s 0 be small enough to make the mapping Φ : K(S) → K(S) a contraction. The fixed point theorem, Theorem 1.11, ensures the existence of a unique fixed point U ∈ K(S). Since U satisfies the formula

t UU (t, s)F (s) ds, 0 ≤ t ≤ TF,U0 , (5.28) U (t) = UU (t, 0)U0 + 0

U is shown to be a local solution to (5.1) on the interval [0, TF,U0 ]. Since 0 < σ < μ + ν − 1, U must satisfy all the conditions of (5.10), except that U ∈ Cβ−α ([0, TF,U0 ]; Y ) and U ∈ C([0, TF,U0 ]; Z); see Theorem 3.10. The Hölder condition was however verified by (5.22). Similarly, by (5.19) it is already seen that U (t) is continuous at t = 0 in the Z norm. Meanwhile, for 0 < t ≤ TU0 ,F , we have AU (t + h)β [U (t + h) − U (t)] = AU (t + h)β U (t + h) − AU (t)β U (t) + AU (t + h)β AU (t + h)−1 − AU (t)−1 AU (t)U (t) − AU (t + h)β−1 − AU (t)β−1 AU (t)U (t). Because of AU (·)β−1 ∈ C([0, TF,U0 ]; L(X)), we verify that U (t + h) − U (t)Z → 0

as h → 0.

Therefore, U (t) is also continuous in the topology of Z for t > 0. We already know that the family of operators AU (t), 0 ≤ t ≤ TF,U0 , satisfies (3.27)–(3.31) and that U (t) is represented by (5.28). Therefore, (5.11) is an immediate consequence of Theorem 3.10. Step 6. Let us finally verify the uniqueness of solutions of (5.1) in the space (5.10). Under (5.8) and (5.9), let U˜ be any other local solution belonging to the function space (5.10) on an interval [0, TU˜ ]. We assume that TU˜ ≤ TU0 ,F and prove that U˜ (t) = U (t) for every 0 ≤ t ≤ TU˜ . Since there exists an evolution operator UU˜ (t, s) for the family of operators AU˜ (t) = A(U˜ (t)), 0 ≤ t ≤ TU˜ , U˜ is written by the formula

t ˜ UU˜ (t, s)F (s) ds, 0 ≤ t ≤ TU˜ . U (t) = UU˜ (t, 0)U0 + 0

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211

Then, we can repeat a quite similar argument as in Step 4 to obtain the estimate U˜ − U Z(S) ≤ CU˜ S μ+ν−1 U˜ − U Z(S) ,

0 < S ≤ TU˜ .

This then shows that U˜ (t) = U (t) for every t ∈ [0, S] if S > 0 is sufficiently small. That is, the uniqueness is true in a neighborhood of the initial time. We here consider the interval I = {S ∈ (0, TU˜ ]; U˜ (t) = U (t) for every t ∈ [0, S]}. By the discussion above, I is nonempty. In addition, I is open in (0, TU˜ ]. Indeed, if S ∈ I and S < TU˜ , then VS = U˜ (S) = U (S) ∈ K satisfies the condition VS ∈ D(A(VS )). It is clear that F ∈ F1,σ ([S, TU˜ ]). Then, we can construct a local solution to the problem dV dt + A(V )V = F (t), S < t ≤ T , V (S) = VS . As the solution is unique in a neighborhood of the initial time S, we conclude that V (t) = U˜ (t) = U (t) for every t > S which is sufficiently close to S. Therefore, I is open. In the meantime, it is clear that I is closed in (0, TU˜ ]. Hence, I = (0, TU˜ ]. We have thus accomplished the proof of Theorem 5.1.

1.3 Stronger Results for More Regular Data When F and U0 in problem (5.1) satisfy stronger conditions than (5.8) and (5.9), respectively, we can strengthen the results of Theorem 5.1. Let β < γ ≤ 1. Let F belong to F ∈ Fγ ,σ ((0, T ]; X),

0 < σ < β − α + ν − 1.

Let U0 satisfy the stronger compatibility condition U0 ∈ D A(U0 )γ , β < γ ≤ 1.

(5.29)

(5.30)

Theorem 5.2 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied. Let F and U0 ∈ K satisfy (5.29) and (5.30), respectively. Then, there exists a unique local solution to (5.1) in the function space: U ∈ C([0, TF,U0 ]; Z) ∩ Cγ −α ([0, TF,U0 ]; Y ) ∩ C1 ((0, TF,U0 ]; X), (5.31) γ ,σ ((0, T A(U )γ U ∈ C([0, TF,U0 ]; X); dU F,U0 ]; X), dt , A(U )U ∈ F where TF,U0 > 0 is determined by the norms F Fγ ,σ ((0,T ];X)) and A(U0 )γ U0 X only. Furthermore, U satisfies the estimates A(U )γ U + dU (5.32) dt γ ,σ + A(U )U Fγ ,σ ≤ CF,U0 , C F

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5 Quasilinear Evolution Equations

where CF,U0 > 0 is a constant determined by the norms F Fγ ,σ and A(U0 )γ U0 X . Proof By the definition (5.13), we have ωU0 (t) ≤ sup [e−sA(U0 ) − 1]A(U0 )β−γ A(U0 )γ U0 X ≤ Ct γ −β A(U0 )γ U0 X . 0≤s≤t

Similarly, by the definitions (5.12) and (5.14), ωF (t) + ωU0 ,F (t) ≤ Ct γ −β F Fγ ,σ . These estimates mean that (5.20) is valid if S γ −β (F Fγ ,σ + A(U0 )γ U0 X ) is sufficiently small. Hence, TF,U0 depends only on these quantities. By the same arguments as in Step 3 of the proof of Theorem 5.1, (5.29) and (5.30) yield that, for γ ≤ θ ≤ 1, A(U (t))θ U (t) ≤ CU ,F t γ −θ , 0 < t ≤ TF,U . 0 0 X Then, the other properties of U can also be verified by the same techniques as in Steps 3 and 5 of the proof of Theorem 5.1. As an immediate consequence, we are able to state the global existence of solutions. Corollary 5.1 Under the assumptions of Theorem 5.2, let F and U0 ∈ K satisfy (5.29) and (5.30), respectively. Assume that any local solution U of (5.1) in the function space (5.31), with TF,U0 replaced by TU , satisfies the estimates U (t)Z ≤ R1 < R, 0 ≤ t ≤ TU , (5.33) A(U (t))γ U (t)X ≤ CF,U0 , 0 ≤ t ≤ TU , with some constants R1 and CF,U0 independent of TU . Then, (5.1) possesses a unique global solution on the interval [0, T ]. Proof We extend the function F to a function F defined on the whole interval [0, ∞) by putting F (t) ≡ F (T ) for T < t < ∞. Taking an initial value U1 which satisfies U1 Z ≤ R1 and A(U1 )γ U1 X ≤ CF,U0 and setting the initial time 0 ≤ T1 < ∞ arbitrarily, we consider the following problem dV dt + A(V )V = F (t), T1 < t < ∞, (5.34) V (T1 ) = U1 . Theorem 5.2 then ensures the existence of a unique local solution on an interval [T1 , T1 + τ ], where τ > 0 is uniform with respect to U1 and T1 . Note that F Fγ ,σ ((a,b];X) ≤ F Fγ ,σ ((0,T ];X) for any finite interval (a, b].

1 Quasilinear Abstract Evolution Equations

213

This fact proves the assertion of the corollary. Consider first the local solution U constructed in Theorem 5.2 on the interval [0, TF,U0 ], where TF,U0 ≥ τ . Set T1 = TF,U0 − τ2 and U1 = U (T1 ). Then, since U1 satisfies condition (5.33), there exists a local solution V to (5.34) on the interval [T1 , T1 +τ ]. By the uniqueness of solutions, U (t) = V (t) for every T1 ≤ t ≤ TF,U0 . This then means that U has been extended to the interval [0, TF,U0 + τ2 ] as a local solution. Due to (5.33), we can repeat this procedure unlimitedly. Each time, the local solution is extended over the length τ2 of interval. Hence, we can arrive at the full interval [0, T ] by finite times.

1.4 Case ν ≤ β We assumed the relation β < ν in (5.7) as one of the structural conditions of our abstract theory. But, this is not an indispensable one. We shall show that the lack of this relation can be recovered by the reflexivity of Z. Instead of (5.7), let the relations 0 ≤ α < ν ≤ β < 1 and 1 + α < β + ν

(5.35)

be satisfied. We assume the following condition: Z is a reflexive Banach space.

(5.36)

Theorem 5.3 Let (5.3), (5.4), (5.5), (5.6), (5.35), and (5.36) be satisfied. Let F be as in (5.8), and let U0 satisfy condition (5.9) with A(U0 )β U0 < R/D2 . X Then, there exists a unique local solution to (5.1) on an interval [0, TF,U0 ] in the function space: U ∈ B([0, TF,U0 ]; Z) ∩ Cβ−α ([0, TF,U0 ]; Y ) ∩ C1 ((0, TF,U0 ]; X), (5.37) β,σ ((0, T A(U )β U ∈ C([0, TF,U0 ]; X); dU F,U0 ]; X). dt , A(U )U ∈ F Here, TF,U0 > 0 is determined by F and U0 . Proof In the present case, Proposition 5.1 is not applicable with θ = β in showing (5.26). This fact makes us change the underlying Banach spaces from Z(S) to the space μ

Y(S) = C{0} ([0, S]; Y )

with 1 − ν + σ < μ < β − α.

By the new condition (5.36), the closed ball {U ∈ Z; U Z ≤ R1 }, where R1 is a fixed number such that D2 A(U0 )β U0 X < R1 < R,

214

5 Quasilinear Evolution Equations

is weakly compact. As a consequence, this ball is a closed subset of Y . (In fact, let Un Z ≤ R1 and Un → U in Y ; then, we can assume that Un → V weakly in Z with V Z ≤ R1 . Since Z ⊂ Y with continuous embedding, Φ ∈ Y implies Φ|Z ∈ Z ; therefore, Φ(Un ) = Φ|Z (Un ) → Φ(V ), i.e., Un → V weakly in Y , too. Hence, U = V . Therefore, the set K(S) introduced in the proof of Theorem 5.1 is a closed subset of Y(S). In showing that Φ maps K(S) into itself, we will utilize the following formula. For U = Φ(V ), by (5.18) it holds that AV (t)β U (t) = A(U0 )β U0 + [e−tA(U0 ) − 1]A(U0 )β U0 + AV (t)β UV (t, 0)AV (0)−β − e−tAV (0) A(U0 )β U0

t + AV (t)β UV (t, s)[F (s) − F (t)] ds 0

+

t

AV (t)β [UV (t, s) − e−(t−s)AV (t) ] ds F (t)

0

− [e−tAV (t) − 1]AV (t)β−1 F (t). Then, it is verified by (2.129), (3.80), (3.84), and (3.86) that AV (t)β U (t) ≤ A(U0 )β U0 + C[ωU (t) + ωF (t)] 0 X X μ+ν−1 β + Ct A(U0 ) U0 X + F Fβ,σ + ω˜ F,U0 (t), where

ω˜ F,U0 (t) = t β−1 [e−tAV (t) − 1]AV (t)β−1 t 1−β F (t)X .

Therefore, by (5.6), U (t)Z ≤ D2 A(U0 )β U0 X + C[ωU0 (t) + ωF (t)] + Ct μ+ν−1 A(U0 )β U0 X + F Fβ,σ + ω˜ F,U0 (t).

(5.38)

As will be seen by the lemma below, we have limt→0 ω˜ F,U0 (t) = 0. Hence, if S > 0 is sufficiently small, then sup U (t)Z ≤ R1 . 0≤t≤S

Lemma 5.1 As t → 0, ω˜ F,U0 (t) → 0. Proof We have ω˜ F,U0 (t) ≤ t β−1 [e−tAV (t) − 1]AV (t)β−1 [t 1−β F (t) − F0 ]X + t β−1 [e−tAV (t) − 1]AV (t)β−1 F0 X ,

2 Lipschitz Continuity of Solutions in Initial Data

215

where F0 = limt→0 t 1−β F (t). The first term in the right-hand side is seen to converge to 0. For the second term, we use the similar argument as for (2.131). When F0 ∈ D(A(0)), it follows that t β−1 [e−tAV (t) − 1]AV (t)β−1 F0 ≤ t β−1 [e−tAV (t) − 1]AV (t)β−ν−1 X

× AV (t)ν A(0)−1

L(X)

L(X)

A(0)F0 X → 0.

For general F0 ∈ X, the desired convergence is deduced by the uniform boundedness of t β−1 [e−tAV (t) − 1]AV (t)β−1 L(X) and the density of D(A(0)) in X. The same Lipschitz condition as (5.22) is verified. Hence, we conclude that Φ is a mapping from K(S) into itself if S > 0 is sufficiently small. As estimate (5.27) was proved without using the relation β < ν, Φ is a contraction with respect to the Y(S) norm if S > 0 sufficiently small. We can apply again the fixed point theorem to Φ : K(S) → K(S) to obtain the desired solution U . By analogous arguments as before, U is shown to belong to the space (5.37). In the present case, U may not be a continuous function with values in Z. We know only that A(U )β U is an X-valued continuous function on [0, TF,U0 ] by Theorem 3.10.

2 Lipschitz Continuity of Solutions in Initial Data Under (5.2)–(5.7), we shall show the Lipschitz continuity of the local solutions with respect to initial data (F, U0 ). We introduce the set of functions F by E(T ) = {F ∈ Fγ ,σ (T ); F Fγ ,σ ((0,T ];X) ≤ L1 },

0 < σ < ν + β − α − 1,

with some constant L1 > 0, and also introduce the set of initial values

B = U0 ∈ Z; U0 Z ≤ R1 and A(U0 )γ U0 X ≤ R2 , β < γ ≤ 1, with some constants 0 < R1 < R and 0 < R2 < ∞. By Theorem 5.1, there exists a unique local solution of (5.1) for each pair (F, U0 ) ∈ E(T ) × B on some interval [0, TF,U0 ]. Let TE,B = inf(F,U0 )∈E(T )×B TF,U0 ; then TE,B is positive. Our goal is to prove the following theorem. Theorem 5.4 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied. Let U (resp. V ) be the local solution of (5.1) with initial data (F, U0 ) ∈ E(T ) × B (resp. (G, V0 ) ∈ E(T ) × B). Then, there exist some constants CE,B > 0 depending only on E(T ) and B such that t β U (t) − V (t)Z + t α U (t) − V (t)Y + U (t) − V (t)X ≤ CB,E (U0 − V0 X + t β F − GFβ,σ ),

0 < t ≤ TE,B .

216

5 Quasilinear Evolution Equations

Proof Let UU (t, s) (resp. UV (t, s)) denote the evolution operator for a family of AU (t) = A(U (t)) (resp. AV (t) = A(V (t))). It follows from (5.28) that U (t) − V (t) = UU (t, 0)(U0 − V0 ) + [UU (t, 0) − UV (t, 0)]V0

t + [UU (t, s) − UV (t, s)]G(s) ds 0

t

+

UU (t, s)[F (s) − G(s)] ds.

0

Then, by (3.80) and (5.6) we have UU (t, 0)(U0 − V0 )Y ≤ D1 AU (t)α UU (t, 0)(U0 − V0 )X ≤ Ct −α U0 − V0 X . Similarly, t t α UU (t, s)[F (s) − G(s)] ds ≤ D1 AU (t) UU (t, s)[F (s) − G(s)] ds 0 0 Y X

t ≤ CF − GFβ,σ (t − s)−α s β−1 ds 0

≤ Ct

β−α

F − GFβ,σ .

Repeating the same arguments as in the proof of Proposition 5.1, we can verify for 0 ≤ θ < ν that AU (t)θ [UU (t, 0) − UV (t, 0)]V0 X

t ≤ CV0 β (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds

(5.39)

0

and

t AU (t)θ [UU (t, s) − UV (t, s)]G(s) ds 0 X

t ≤ CGFβ,σ (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds. 0

Therefore, applying these estimates with θ = α, we observe that t [UU (t, 0) − UV (t, 0)]V0 Y + [UU (t, s) − UV (t, s)]G(s) ds 0 Y

t ≤ CE,B (t − s)ν−α−1 s β−1 U (s) − V (s)Y ds. 0

(5.40)

2 Lipschitz Continuity of Solutions in Initial Data

217

In this way, we have obtained the integral inequality

t ϕ(t) ≤ CE(t) + CE,B t α (t − s)ν−α−1 s β−α−1 ϕ(s) ds

(5.41)

0

for the function ϕ(t) which is defined by ϕ(t) = t α U (t) − V (t)Y ,

0 ≤ t ≤ TE,B ,

where E(t) = U0 − V0 X + t β F − GFβ,σ ,

0 ≤ t ≤ TE,B .

To solve this integral inequality, we first consider small variable t . For any s such that 0 ≤ s ≤ t , we have

s ϕ(s) ≤ CE(s) + CE,B s α (s − σ )ν−α−1 σ β−α−1 dσ sup ϕ(σ ) 0≤σ ≤s

0

≤ CE(t) + CE,B t ν+β−α−1 sup ϕ(s). 0≤s≤t

Therefore, it follows that 1 − CE,B t ν+β−α−1 sup ϕ(s) ≤ CE(t). 0≤s≤t

Fix an ε > 0 small enough to satisfy CE,B ε ν+β−α−1 < 1. Then, for 0 ≤ t ≤ ε, ϕ(t) ≤ sup ϕ(s) ≤ CE(t),

0 ≤ t ≤ ε.

0≤s≤t

Secondly, let us consider variable t such that t ≥ ε. We observe that

ε (t − s)ν−α−1 s β−α−1 ds ϕ(t) ≤ CE(t) + CE,B t α E(ε) 0

+ CE,B t α ε β−α−1

≤ CE,B E(t) +

t

t

(t − s)ν−α−1 ϕ(s) ds

ε

(t − s)

ν−α−1

ϕ(s) ds .

ε

Applying Theorem 1.27, we conclude that ϕ(t) ≤ CE,B E(t),

ε ≤ t ≤ TE,B .

Thus, for all 0 ≤ t ≤ TE,B , ϕ(t) ≤ CE,B E(t),

i.e.,

t α U (t) − V (t)Y ≤ CE,B E(t),

0 ≤ t ≤ TE,B . (5.42)

218

5 Quasilinear Evolution Equations

Estimation for Z and X norms of U (t) − V (t) is now immediate. By (3.80) and (5.6), UU (t, 0)(U0 − V0 )Z ≤ Ct −β U0 − V0 X . Similarly, t UU (t, s)[F (s) − G(s)] ds 0 Z t β ≤ D2 AU (t) UU (t, s)[F (s) − G(s)] ds 0

≤ CF − GFβ,σ

0

t

X

(t − s)−β s β−1 ds ≤ CF − GFβ,σ .

In addition, applying (5.39) and (5.40) with θ = β, we have t [UU (t, 0) − UV (t, 0)]V0 Z + [UU (t, s) − UV (t, s)]G(s) ds

≤ CE,B

0

t

Z

(t − s)ν−β−1 s β−1 U (s) − V (s)Y ds

0

and, therefore, by (5.42),

t ≤ CE,B U0 − V0 X (t − s)ν−β−1 s β−α−1 ds ≤ CE,B t ν−α−1 U0 − V0 X . 0

Summing up these estimates, we conclude that t β U (t) − V (t)Z ≤ CE,B E(t),

0 ≤ t ≤ TE,B .

It is similar for the estimation of U (t) − V (t)X . It suffices to apply (5.39) and (5.40) with θ = 0.

3 Equations Including Semilinear Terms In this section, we consider the Cauchy problem for a quasilinear equation of the form dU dt + A(U )U = F (U ) + G(t), 0 < t ≤ T , (5.43) U (0) = U0 , in a Banach space X. For U ∈ K, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 , where K is an open ball K = {U ∈ Z; U Z < R}, 0 < R < ∞, of a second Banach space Z continuously embedded in X. The operator F is a nonlinear operator from another Banach space W into X, W being continuously embedded in Z. The function G is a given external force function. The initial value U0 is taken in K.

3 Equations Including Semilinear Terms

219

3.1 Existence Theorem For the linear operators A(U ), we make the same structural assumptions as in Sect. 1.1. That is, we use the three Banach spaces Z ⊂ Y ⊂ X. The operator A(U ), U ∈ K, is a sectorial operator of X satisfying conditions (5.2) and (5.3) uniformly. The domain D(A(U )) is allowed to vary with U ∈ K, but A(U ) is assumed to satisfy (5.4) and the Lipschitz condition (5.5). Furthermore, the domains of fractional powers D(A(U )α ) and D(A(U )β ) are contained in Y and Z with (5.6), respectively, with the exponents α and β satisfying relations (5.7). For the nonlinear operator F , we assume the Lipschitz condition F (U ) − F (V )X ≤ ϕ(U Z + V Z )[U − V W + (U W + V W ) × U − V Z ],

U, V ∈ W,

(5.44)

where ϕ(·) is some continuous increasing function. This condition obviously implies that F (U )X ≤ F (0)X + ϕ(U Z )(1 + U Z )U W ,

U ∈ W.

(5.45)

For the space W , we assume an analogous condition to that for Y and Z. There is an exponent 0 < η < 1 such that D(A(U )η ) ⊂ W with the estimate U˜ W ≤ D3 A(U )η U˜ X , U˜ ∈ D A(U )η for U ∈ K, (5.46) D3 > 0 being some constant. Finally, the exponents satisfy the relation β < η < ν.

(5.47)

As in the preceding section, the notation C will stand for a universal constant which is determined by the angle, the exponents, the norm A(0)−1 , and the constants appearing in (5.2)–(5.7). In this section, C may depend also on the function ϕ(·), the norm F (0)X , the exponent η, and the constant D3 . We shall prove the following theorem. Theorem 5.5 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let G belong to (5.8) with 0 < σ < min{β − α + ν − 1, 1 − η}. Let U0 ∈ K satisfy (5.9). Then, there exists a unique local solution U to (5.43) on an interval [0, TG,U0 ] in the function space: ⎧ β−η ⎪ ⎨U ∈ B{0} ((0, TG,U0 ]; W ) ∩ C([0, TG,U0 ]; Z) ∩Cβ−α ([0, TG,U0 ]; Y ) ∩ C1 ((0, TG,U0 ]; X), ⎪ ⎩ β,σ ((0, T A(U )β U ∈ C([0, TG,U0 ]; X); dU G,U0 ]; X); dt , A(U )U ∈ F (5.48) here TG,U0 > 0 is determined by G and U0 . In addition, U satisfies the estimates

220

5 Quasilinear Evolution Equations

A(U )β U + dU dt C

+ A(U )U Fβ,σ ≤ CG,U0 ,

F β,σ

(5.49)

where CG,U0 > 0 is a constant depending on G and U0 . Remark 5.2 The dependence of TG,U0 on G and U0 is quite similar to that in Theorem 5.1. As will be seen in the proof below (see Steps 3 and 4), TG,U0 is determined by GFβ,σ and A(U0 )β U0 X and by the moduli of continuity ωG (t), ωU0 (t), and ωG,U0 (t) as t → 0, which are already defined by (5.12), (5.13), and (5.14), respectively.

3.2 Proof of Theorem 5.5 The proof consists of several steps. Step 1. For each 0 < S ≤ T , we set the underlying Banach space −ρ

μ

W(S) = B{0} ((0, S]; W ) ∩ B([0, S]; Z) ∩ C{0} ([0, S]; Y ) with some fixed exponents μ and ρ such that 1 − ν + σ < μ < β − α (cf. the proof of Theorem 5.1) and 1 − σ − β < ρ < 1 − β. Therefore, we have the relations η − β < 1 − σ − β < ρ < 1 − β < 1 − μ − α. −ρ

(5.50)

μ

For the function spaces B{0} ((0, S]; W ) and C{0} ([0, S]; Y ), see Chap 1, Sect. 2. We also set a subset F(S) of W(S) as follows: F(S) = U ∈ W(S); U (0) = U0 , sup U (t)Z ≤ R1 , sup t ρ U (t)W ≤ 1, 0≤t≤S

sup 0≤s 0 is sufficiently small, then sup

sup {Ψ (V )}(t)Z ≤ R1 .

V ∈F(S) 0≤t≤S

We next apply the moment inequality (2.119) for estimating AU (t)θ U (t)X . Indeed, due to (3.101) and (3.102), for β ≤ θ ≤ 1, AU (t)θ U (t) ≤ Ct β−θ (FV + GFβ,σ + U0 β ), 0 ≤ t ≤ S. (5.57) X Therefore, in view of (5.46), t ρ U (t)W ≤ D3 t ρ AU (t)η U (t)X ≤ Ct ρ+β−η (U0 β + FV + GFβ,σ ) ≤ 1,

0 ≤ t ≤ S,

if S > 0 is sufficiently small. For proving the Hölder condition of U , we write

t

U (t) − U (s) =[UU (t, s) − 1]U (s) +

UU (t, τ )[FV (τ ) + G(τ )] dτ

s

={[UU (t, s) − e−(t−s)AU (s) ] + [e−(t−s)AU (s) − 1]}AU (s)−(η+σ )

t η+σ ×AU (s) U (s) + UU (t, τ )[FV (τ ) + G(τ )] dτ, s

0 ≤ s < t ≤ S. Then, by (2.129) and (3.87) (note that η < ν), AU (t)η [UU (t, s) − e−(t−s)AU (s) ]AU (s)−(η+σ ) ≤ C(t − s)σ +μ+ν−1 , L(X) AU (s)η [e−(t−s)AU (s) − 1]AU (s)−(η+σ ) ≤ C(t − s)σ . L(X) By (3.81),

3 Equations Including Semilinear Terms

AU (t)η UU (t, τ )

t

s

≤ C(t − s)σ

t

223

τ L(X)

β−1

dτ ≤ C

t

(t − τ )−η τ β−1 dτ

s

(t − τ )−η−σ τ β−1 dτ ≤ C(t − s)σ

t

(t − τ )−η−σ τ β−1 dτ

0

s

≤ C(t − s) t

σ β−η−σ

.

In view of (5.46) and (5.57) (θ = η + σ ), we therefore obtain that U (t) − U (s)W ≤ C(FV + GFβ,σ + U0 β )(t − s)σ s β−η−σ .

(5.58)

Hence, if S > 0 is sufficiently small, then sup 0<s 0 is sufficiently small, then sup 0<s 0 is sufficiently small. Step 4. We shall prove that Ψ : F(S) → W(S) is a contraction with respect to the W(S) norm if S > 0 is sufficiently small. In fact, let Vi ∈ F(S) and Ui = Ψ (Vi ), i = 1, 2. From (5.56) it is seen that U1 (t) − U2 (t) = [UU1 (t, 0) − UU2 (t, 0)]U0

t + [UU1 (t, s) − UU2 (t, s)][FV1 (s) + G(s)] ds 0

t

+ 0

UU2 (t, s)[FV1 (s) − FV2 (s)] ds.

For 0 ≤ θ < ν, using (5.23) of Proposition 5.1, we have AU (t)θ [UU (t, 0) − UU (t, 0)]U0 1

1

≤ Cθ t

2

β−θ+μ+ν−1

X

U0 β U1 − U2 Cμ

{0} ([0,S];Y )

Similarly, using (5.24),

t AU (t)θ [UU1 (t, s) − UU2 (t, s)][FV1 (s) + G(s)] ds 1 0

≤ Cθ (GFβ,σ + 1)t

β−θ+μ+ν−1

U1 − U2 Cμ

(5.59)

.

X

{0} ([0,S];Y )

.

(5.60)

224

5 Quasilinear Evolution Equations

By (5.44), we have

t AU (t)θ UU2 (t, s)[FV1 (s) − FV2 (s)] ds 2

≤C

0

t

X

(t − s)−θ s −ρ [s ρ V1 (s) − V2 (s)W + V1 (s) − V2 (s)Z ] ds

0

≤ Cθ t −θ−ρ+1 (V1 − V2 B−ρ ((0,S];W ) + V1 − V2 B([0,S];Z) ).

(5.61)

{0}

Applying these estimates with θ = α, we obtain that t −μ U1 (t) − U2 (t)Y ≤ C(GFβ,σ + U0 β + 1)S β−α+ν−1 U1 − U2 Cμ

{0} ([0,S];Y )

+ CS 1−μ−α−ρ (V1 − V2 B−ρ ((0,S];W ) + V1 − V2 B([0,S];Z) ), {0}

0 < t ≤ S.

(5.62)

Similarly, applying the estimates with θ = β, U1 (t) − U2 (t)Z ≤ C(GFβ,σ + U0 β + 1)S μ+ν−1 U1 − U2 Cμ

{0} ([0,S];Y )

+ CS 1−β−ρ (V1 − V2 B−ρ ((0,S];W ) {0}

+ V1 − V2 B([0,S];Z) ),

0 < t ≤ S.

(5.63)

Finally, applying the estimates with θ = η, we have t ρ U1 (t) − U2 (t)W ≤ C(GFβ,σ + U0 β + 1)S ρ+β−η+μ+ν−1 × U1 − U2 Cμ

{0} ([0,S];Y )

+ V1 − V2 B([0,S];Z) ),

+ CS 1−η (V1 − V2 B−ρ ((0,S];W ) {0}

0 < t ≤ S.

(5.64)

Therefore, summing up (5.62), (5.63), and (5.64), we have U1 − U2 W(S) ≤ C(GFβ,σ + U0 β + 1)S μ+ν−1 U1 − U2 Cμ

{0} ([0,S];Y )

+ CS 1−β−ρ (V1 − V2 B−ρ ((0,S];W ) + V1 − V2 B([0,S];Z) ) {0}

(due to (5.50)). In this way, we conclude that Ψ is a contraction with respect to the W(S) norm, provided that S > 0 is sufficiently small. Step 5. Let TG,U0 = S > 0 be small enough to make Ψ : F(S) → F(S) to be a contraction. Then, there exists a unique fixed point U ∈ F(S) of Ψ which satisfies the formula

t U (t) = UU (t, 0)U0 + UU (t, s)[FU (s) + G(s)] ds, 0 ≤ t ≤ TG,U0 . (5.65) 0

3 Equations Including Semilinear Terms

225

Then, U is a local solution to (5.43) on [0, TG,U0 ] belonging to the space (5.10). In addition, applying (5.57) with θ = η, we verify by (5.46) that U (t)W ≤ Ct β−η (FU + GFβ,σ + U0 β ). Hence, U belongs to the space (5.48). In view of (5.55), the family of operators AU (t), 0 ≤ t ≤ TG,U0 , satisfies (3.27)–(3.31). Therefore, (5.49) is readily verified by Theorem 3.10 and (5.54). The uniqueness of a local solution is also verified in an analogous way as in the proof of Theorem 5.1. We have thus accomplished the proof of Theorem 5.5.

3.3 Stronger Results for More Regular Data Let G belong to the function space (5.29). Let U0 satisfy the compatibility condition (5.30), which is stronger than (5.8). Then, Theorem 5.5 can be strengthened as follows. Theorem 5.6 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let G and U0 ∈ K satisfy (5.29) and (5.30), respectively, with 0 < σ < min{β − α + ν − 1, 1 − η}. Then, there exists a unique local solution U to (5.43) on an interval [0, TG,U0 ] in the function space (5.31). Here, TG,U0 > 0 is determined by the norms GFγ ,σ and A(U0 )γ U0 X alone. Furthermore, U satisfies the estimates A(U )γ U + dU (5.66) dt γ ,σ + A(U )U Fγ ,σ ≤ CG,U0 C F with a constant CG,U0 > 0 determined by the norms GFγ ,σ and A(U0 )γ U0 X . As noticed in Remark 5.2, the time TG,U0 > 0 is determined by ωG (t), ωU0 , and ωG,U0 (t) in addition to GFβ,σ and A(U0 )β U0 X . Then, the proof of this theorem is quite analogous to that of Theorem 5.2, and we may leave it to the reader. By the same argument as in Corollary 5.1, we verify the following global existence. The proof may now be obvious. Corollary 5.2 Under the assumptions of Theorem 5.6, let G and U0 ∈ K satisfy (5.29) and (5.30), respectively. Assume that any local solution U of (5.43) in the function space (5.31), with TG,U0 replaced by TU , satisfies the estimates 0 ≤ t ≤ TU , U (t)Z ≤ R1 < R, (5.67) A(U (t))γ U (t)X ≤ CG,U0 , 0 ≤ t ≤ TU , with some constants R1 and CG,U0 independent of TU . Then, (5.43) possesses a unique global solution on the interval [0, T ].

3.4 Case ν ≤ β The condition β < ν assumed above is not an indispensable one for the local existence. As in Sect. 1.4, the lack of this condition can be recovered by reflexivity of Z.

226

5 Quasilinear Evolution Equations

Instead of (5.44), we assume that F satisfies F (U ) − F (V )X ≤ ϕ(U Z + V Z )U − V W ,

U, V ∈ Z,

(5.68)

with some continuous increasing function ϕ(·), and that the space W satisfies the relation Z ⊂ W (⊂ Y ⊂ X) with continuous embedding. We also assume relations (5.35) and, instead of (5.47), (5.69)

α < η < ν.

Theorem 5.7 Let (5.2), (5.3), (5.4), (5.5), (5.6), (5.35), and (5.36) be satisfied, and let (5.46), (5.68), and (5.69) be satisfied. Let G belong to (5.8) with 0 < σ < min{β − α + ν − 1, 1 − η}. Let U0 satisfy condition (5.9) with A(U0 )β U0 < R D2 . X Then, there exists a unique local solution to (5.43) on an interval [0, TG,U0 ] in the function space: ⎧ β−η ([0, T ⎪ G,U0 ]; W ) ⎨U ∈ B([0, TG,U0 ]; Z) ∩ C ∩ Cβ−α ([0, TG,U0 ]; Y ) ∩ C1 ((0, TG,U0 ]; X), ⎪ ⎩ β,σ ((0, T A(U )β U ∈ C([0, TG,U0 ]; X); dU G,U0 ]; X). dt , A(U )U ∈ F (5.70) Here, TG,U0 > 0 is determined by G and U0 . Proof In the present case, (5.63) is no longer available. So, we have to change the underlying spaces from W(S) to ζ μ ˜ W(S) = C{0} ((0, S]; W ) ∩ C{0} ([0, S]; Y ),

0 < S ≤ T,

with some fixed exponents μ and ζ such that 1 − ν + σ < μ < β − α and 0 < ζ < ˜ β − η, respectively. Accordingly, we set the subset of W(S) by ˜ ˜ F(S) = U ∈ W(S); U (0) = U0 , sup U (t)Z ≤ R1 0≤t≤S

and

sup 0≤s 0. ˜

3 Equations Including Semilinear Terms

227

˜ Hence, by formula (5.56), we define a mapping U = Ψ V for V ∈ F(S) (0 < S ≤ ˜ TG,U0 ). ˜ If S > 0 is sufficiently small, then Ψ maps F(S) into itself. In fact, by (5.38), {Ψ V }(t)Z ≤ D2 A(U0 )β U0 X + C[ωU0 (t) + ωFV +G (t)] + Ct μ+ν−1 FV + GFβ,σ + A(U0 )β U0 X + ω˜ FV +G,U0 (t). This shows that, if S > 0 is sufficiently small, then sup

sup (Ψ V )(t)Z ≤ R1 .

0≤t≤S ˜ V ∈F(S)

For verifying the Hölder condition, we can use the same argument as for (5.58). In view of the relations η < ν and σ < 1 − η, estimate (5.58) is verified without any change. Therefore, if S > 0 is sufficiently small, then sup

sup

0≤s 0 is sufficiently small. Let us verify that, if S > 0 is sufficiently small, then Ψ is a contraction in the ˜ W(S) norm. This is also verified in the same way as before. In fact, let Ui = Ψ Vi ˜ with Vi ∈ F(S) (i = 1, 2). For 0 ≤ θ < ν, estimates (5.59) and (5.60) are valid for these Ui and Vi . Instead of (5.61), we verify that

t AU (t)θ U (t, s)[F (s) − F (s)] ds U2 V1 V2 2 0 X

t ≤C (t − s)−θ s ζ s −ζ V1 (s) − V2 (s)W ds 0

≤ Cθ t ζ −θ+1 V1 − V2 Cζ

{0} ((0,S];W )

.

Then, using these estimates with θ = η and α, we easily obtain that t −ζ U1 (t) − U2 (t)W + t −α U1 (t) − U2 (t)Y ≤ CG,U0 S μ+ν−1 U1 − U2 W(S) + C(S 1−η + S ζ +1−α−μ )V1 − V2 W(S) , ˜ ˜ 0 ≤ t ≤ S. ˜ This shows that Ψ is a contraction with respect to the W(S) norm, provided that S > 0 is sufficiently small (note that 1 − α − μ > 1 − β > 0). ˜ We thus conclude that Ψ has a unique fixed point U ∈ F(S) which satisfies formula (5.65). Then, U is the solution to (5.43) in the space (5.70) which enjoys all the desired properties.

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5 Quasilinear Evolution Equations

4 Lipschitz Continuity of Solutions in Initial Data We are concerned with the continuous dependence of the local solutions of (5.43) with respect to the initial data (G, U0 ). Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied, namely, we return to the case where β < ν. We consider the set of functions G given by E(T ) = {G ∈ Fγ ,σ ((0, T ]; X); GFγ ,σ ≤ L1 }, 0 < σ < min{β − α + ν − 1, 1 − η} with some constant L1 > 0. We consider also the set of initial values

B = U0 ∈ Z; U0 Z ≤ R1 and A(U0 )γ U0 X ≤ R2 , β < γ ≤ 1, with some constants 0 < R1 < R and 0 < R2 < ∞. By Theorem 5.6, there exists a unique local solution to (5.43) for each pair (G, U0 ) ∈ E(T ) × B on some interval [0, TG,U0 ]. Let TE,B = inf(G,U0 )∈E(T )×B TG,U0 > 0. Before proving the Lipschitz continuity, we shall show estimates of solutions by the initial data. Theorem 5.8 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let U be the local solution of (5.43) with (G, U0 ) ∈ E(T ) × B. Then, there exist some constants CE,B > 0 depending on E(T ) and B only such that t η U (t)W + t β U (t)Z + t α U (t)Y + U (t)X ≤ CE,B (U0 X + tF (0)X + t β GFβ,σ ),

0 ≤ t ≤ TE,B .

Proof Let FU (t) = F (U (t)), and let UU (t, s) be the evolution operator for the family of AU (t) = A(U (t)). Operating A(U (t))θ (0 ≤ θ < 1) to the solution formula (5.65), we have AU (t)θ U (t) = AU (t)θ UU (t, 0)U0

t + AU (t)θ UU (t, s)[FU (s) + G(s)] ds,

0 < t ≤ TB,E .

0

Here, by (5.45), FU (t)X ≤ F (0)X + D3 ϕ(R)(1 + R)AU (t)η U (t)X . Therefore,

t AU (t)θ U (t) ≤Cθ t −θ U0 X + (t − s)−θ F (0)X + AU (s)η U (s)X X 0

+ s β−1 GFβ,σ ds .

4 Lipschitz Continuity of Solutions in Initial Data

229

Consequently, t θ AU (t)θ U (t)X ≤ Cθ U0 + tF (0)X + t β GFβ,σ

t

+ 0

t θ (t − s)−θ s −η s η AU (s)η U (s)X ds .

If we put θ = η, then an integral inequality with respect to AU (s)η U (s)X is obtained, and the inequality can be easily solved by the same techniques used for (5.41). As a result, we conclude that t η AU (t)η U (t)X ≤ CE,B [U0 X + tF (0)X + t β GFβ,σ ], 0 < t ≤ TE,B . Hence, for 0 ≤ θ < 1, we have

t t θ AU (t)θ U (t)X ≤ Cθ U0 + tF (0)X + t β GFβ,σ + t θ (t − s)−θ s −η 0

× [U0 X + sF (0)X + s β GFβ,σ ] ds . Putting θ = β, α, and 0, we can obtain the desired estimates.

Corollary 5.3 Under the conditions of Theorem 5.8, we have t η−β U (t)W + U (t)Z ≤ CE,B A(U0 )β U0 X + t 1−β F (0)X + GFβ,σ ,

0 ≤ t ≤ TE,B .

Proof For β ≤ θ < 1, we use the estimate AU (t)θ UU (t, 0)U0 ≤ CE,B t β−θ U0 β . X Then, it follows that

t AU (t)θ U (t) ≤ Cθ CB,E t β−θ U0 β + (t − s)−θ F (0)X X 0

η β−1 + AU (s) U (s) X + s Gβ,σ ds . Putting θ = η, we have t

AU (t) U (t) X ≤ CB,E U0 β + t 1−β F (0)X + GFβ,σ

η−β

η

t

+ 0

t η−β (t − s)−η s β−η s η−β AU (s)η U (s)X ds .

230

5 Quasilinear Evolution Equations

So, solving this integral inequality, we conclude that t η−β AU (t)η U (t)X ≤ CB,E [U0 β + t 1−β F (0)X + Gβ,σ ], By putting θ = β, we also verify that AU (t)β U (t) ≤ CE,B [U0 β + t 1−β F (0)X + Gβ,σ ], X

0 < t ≤ TE,B .

0 < t ≤ TE,B .

Theorem 5.9 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let U (resp. V ) be the local solution of (5.43) with (G1 , U0 ) ∈ E(T ) × B (resp. (G2 , V0 ) ∈ E(T ) × B). Then, there exist some constants CE,B > 0 depending on E(T ) and B only such that t η U (t) − V (t)W + t β U (t) − V (t)Z + t α U (t) − V (t)Y + U (t) − V (t)X ≤ CE,B (U0 − V0 X + t β G1 − G2 Fβ,σ ),

0 ≤ t ≤ TE,B .

Proof Let FU (t) = F (U (t)) (resp. FV (t) = F (V (t))), and let UU (t, s) (resp. UV (t, s)) be the evolution operator for the family of AU (t) = A(U (t)) (resp. AV (t) = A(V (t))). By (5.65) we can write U − V as U (t) − V (t) = UU (t, 0)(U0 − V0 ) + [UU (t, 0) − UV (t, 0)]V0

t + [UU (t, s) − UV (t, s)][FV (s) + G2 (s)] ds

0 t

+

UU (t, s)[FU (s) − FV (s)] ds

0 t

+

UU (t, s)[G1 (s) − G2 (s)] ds.

(5.71)

0

For 0 ≤ θ < ν, operating AU (t)θ , we estimate the X norm of the right-hand side. By (5.44),

t AU (t)θ UU (t, s)[FU (s) − FV (s)] ds

≤C

0

t

X

(t − s)−θ [U (s) − V (s)W + s −ρ U (s) − V (s)Z ] ds.

0

The norms of other terms are estimated in the same way as in the proof of Theorem 5.4. Indeed, by (3.80), AU (t)θ UU (t, 0)(U0 − V0 ) ≤ Ct −θ U0 − V0 X (5.72) X and

t AU (t)θ UU (t, s)[G1 (s) − G2 (s)] ds ≤ Ct β−θ G1 − G2 Fβ,σ . 0

X

4 Lipschitz Continuity of Solutions in Initial Data

231

By the same arguments as for (5.39) and (5.40), we obtain that AU (t)θ [UU (t, 0) − UV (t, 0)]V0 X

t ≤ C A(V0 )β V0 X (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds 0

and

t AU (t)θ [U (t, s) − U (t, s)][F (s) + G (s)] ds U V V 2 0 X

t ≤ CFV + G2 Fβ,σ (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds. 0

Applying these estimates with θ = η, β, and α and using (5.6) and (5.46), we immediately observe that the function ϕ(t) = t η U (t) − V (t)W + t β U (t) − V (t)Z + t α U (t) − V (t)Y , 0 < t ≤ TE,B , satisfies the integral inequality ϕ(t) ≤ CE,B (U0 − V0 X + t β G1 − G2 Fβ,σ )

t θ −θ −η −ρ−β ν−1 β−α−1 ϕ(s) ds. + CE,B t (t − s) s + s + (t − s) s 0

θ=η,β,α

We can then solve this integral inequality by the same technique as for (5.41) (see (5.50)) and can in fact conclude that ϕ(t) ≤ CE,B (U0 − V0 X + t β G1 − G2 Fβ,σ ),

0 ≤ t ≤ TE,B .

As in the proof of Theorem 5.4, we can estimate the norm U (t) − V (t)X , too, directly from (5.71). Hence, we have accomplished the proof. Corollary 5.4 Under the conditions of Theorem 5.9, it holds that t η−α U (t) − V (t)W + t β−α U (t) − V (t)Z + U (t) − V (t)Y ≤ CB,E A(U0 )α (U0 − V0 )X + t β−α G1 − G2 Fβ,σ , 0 ≤ t ≤ TB,E . Proof For α ≤ θ ≤ 1, it holds that AU (t)θ UU (t, 0)(U0 − V0 ) X θ −α ≤ AU (t) UU (t, 0)AU (0) L(X) AU (0)α (U0 − V0 )X ≤ CE,B t α−θ AU (0)α (U0 − V0 )X , 0 < t ≤ TE,B .

232

5 Quasilinear Evolution Equations

We will use this instead of (5.72). Then, by a similar argument as in the proof Theorem 5.9, it is deduced that ϕ(t) ≤ CE,B t α A(U0 )α (U0 − V0 )X + t β G1 − G2 Fβ,σ , 0 ≤ t ≤ TE,B .

Hence, the desired estimate is obtained.

5 Variational Methods In this section, we will remark on the variational methods for studying problem (5.1). Let X and Z be two Hilbert spaces such that Z ⊂ X with dense and compact embedding. Let | · | (resp. · ) be the norm of X (resp. Z). Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Let · ∗ be the norm of Z ∗ . We consider the Cauchy problem for a quasilinear parabolic evolution equation dU dt + A(U )U = F (U ) + G(t), 0 < t ≤ T , (5.73) U (0) = U0 , in Z ∗ . Here, the sectorial operators A(U ) are given as follows. Let K be an open ball of X, K = {E ∈ X; |E| < R},

0 < R < ∞.

For each E ∈ K, let a(E; ·, ·) be a sectorial operator on Z and assume the following conditions: |a(E; U, V )| ≤ MU V , Re a(E; U, U ) ≥ δU , 2

E ∈ K, U, V ∈ Z,

E ∈ K, U ∈ Z,

(5.74) (5.75)

|a(E1 ; U, V ) − a(E2 ; U, V )| ≤ N |E1 − E2 |U V , Ei ∈ K (i = 1, 2), U, V ∈ Z,

(5.76)

with constants δ > 0 and M, N independent of E ∈ K. By Theorem 2.1, we know that the associated operator A(E) is defined and is a sectorial operator of Z ∗ with angle ωA(E) < π2 . The operator F : K ∩ Z → Z ∗ is a nonlinear operator satisfying the following conditions. For any ζ > 0, there exists a constant Lζ such that F (U )∗ ≤ ζ U + Lζ ,

U ∈ K ∩ Z,

(5.77)

F (U ) − F (V )∗ ≤ ζ U − V + Lζ (U + V + 1)|U − V |, U, V ∈ K ∩ Z.

(5.78)

The function G(t) is a given external force function belonging to L2 (0, T ; Z ∗ ). The initial value U0 is taken in K.

6 Quasilinear Parabolic Equations

233

We can then prove the following results in a quite analogous way to the proof of Theorem 4.6. Theorem 5.10 Let (5.74), (5.75), (5.76), (5.77), and (5.78) be satisfied. Then, for any G ∈ L2 (0, T ; Z ∗ ) and any U0 ∈ K, there exists a unique local solution U to (5.73) in the function space: U ∈ L2 ((0, TG,U0 ); Z) ∩ C([0, TG,U0 ]; X) ∩ H 1 ((0, TG,U0 ); Z ∗ ), where TG,U0 > 0 is determined by the norms GL2 (0,T ;Z ∗ ) and |U0 | only. In addition, U satisfies the estimates U L2 ((0,TG,U0 );Z) + U C([0,TG,U0 ];X) + U H 1 ((0,TG,U

0

);Z ∗ )

≤ CG,U0 ,

where CG,U0 > 0 depends on the norms |U0 | and GL2 ((0,T );Z ∗ ) .

6 Quasilinear Parabolic Equations We consider the initial-boundary-value problem for a quasilinear parabolic equation ⎧ ∂u n ⎪ ⎨ ∂t = i,j =1 Dj [aij (x, u)Di u] + f (x, u, ∇u) + g(x, t) n νj (x)aij (x, u)Di u = 0 ⎪ ⎩ i,j =1 u(x, 0) = u0 (x)

in Ω × (0, T ),

on ∂Ω × (0, T ), in Ω, (5.79) in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω, where 0 < T < ∞ is fixed time. Here, aij (x, u) are real-valued functions for (x, u) ∈ Ω × (R + iR) which are smooth functions with respect to the real variables x ∈ Ω, Re u and Im u. We assume that aij (x, u) satisfy conditions (2.19) and (2.28) for every u ∈ R + iR with some uniform constant δ > 0. The function f (x, u, ζ ) is a complex-valued function for (x, u, ζ ) ∈ Ω × (R + iR) × (R + iR)n which is a smooth function with respect to the real variables x ∈ Ω, Re u, Im u, Re ζ , and Im ζ . The function g(x, t) is an external force function, and u0 (x) is an initial function.

6.1 Abstract Formulation Let us formulate problem (5.79) as the Cauchy problem of the form (5.43) in X = Lp (Ω), where n < p < ∞. We set the Sobolev space Hps (Ω) as the space Z where s s 1 s is taken as p+n 2p < 2 < 1, Hp (Ω) being embedded in C (Ω) continuously due to (1.76). For each u ∈ K = {u ∈ Hps (Ω); uHps < R}, 0 < R < ∞, we consider a real ization Ap (u) of the differential operator − i,j Dj [aij (x, u(x))Di ·] + 1 in Lp (Ω)

234

5 Quasilinear Evolution Equations

u˜ under the Neumann-type boundary conditions ∂ν∂A(u) = 0 on ∂Ω. The domain of Ap (u) is given by (2.77) and depends on u. The nonlinear operator F (u) is given by

F (u) = u + f (x, u, ∇u),

u ∈ Hps (Ω).

We set Hp1 (Ω) as the space W . In this way, we rewrite (5.79) in the form (5.43).

6.2 Construction of Local Solutions We already know by Theorem 2.12 that Ap (u) satisfy (5.2) and (5.3). We fix an exponent ν as 12 < ν < p+1 2p . Then, according to the fact (16.36) presented in Chap. 16,

condition (5.4) is fulfilled. As for the third space Y , we set Hp2ν −1 (Ω) = Y , where

the exponent ν is fixed as ν < ν

0 being independent of u. Lemma 5.3 Let a(x, u) be a function for (x, u) ∈ Ω × (R + iR) which is a smooth function with respect to the real variables Re u and Im u. Then, for 0 ≤ s ≤ 1, a(x, u) − a(x, v)Hps ≤ ϕ(uC1 + vC1 )u − vHps ,

u, v ∈ C1 (Ω),

where ϕ(·) is some continuous increasing function determined by a(x, u). Proof Indeed,

1

a(x, u) − a(x, v) =

+ 0

0 1

∂a (x, θ Re u + (1 − θ ) Re v + i Im v) dθ · Re(u − v) ∂ Re u

∂a (x, Re v + iθ Im u + i(1 − θ ) Im v) dθ · i Im(u − v). ∂ Im u

It then suffices to apply (1.87) to the function in the right-hand side with m = 1. Let us now fix the exponents α and β so that (5.6) is verified. By (16.36), if we 1 take these exponents in such a way that 2ν 2−1 < α < 2p and 2s < β < 1, then the two space conditions in (5.6) are verified. In addition, it holds that 1+α 0).

7.1 Abstract Formulation We want to treat (5.80) in the product space ⎧⎛ ⎞ ⎫ u ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎟ ⎬ v 1 1 1 ⎟ ; u ∈ H (Ω), v ∈ H (Ω), w ∈ H (Ω), ρ ∈ L2 (Ω) , X= ⎜ p p p ⎝w ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ρ p being a fixed number such that 2 < p ≤ 4. Let 0 < ε < 1 be a fixed exponent. The space K of initial values is defined as follows. A vector U0 = t (u0 , v0 , w0 , ρ0 ) ∈ X belongs to K if and only if u0 , v0 , w0 ∈ Hp1 (Ω) and ρ0 ∈ H ε (Ω) with the conditions 0 ≤ u0 ≤ 1,

0 ≤ v0 ≤ 1,

0 ≤ w0 ≤ 1,

and 0 ≤ ρ0 , ρ0 L1 = 1.

(5.81)

7 Some Model for Honeybee Colonies

237

We then formulate (5.80) as the Cauchy problem for an abstract equation dU dt

+ A(U )U = F (U ),

0 < t < ∞,

(5.82)

U (0) = U0 ,

in X. Here, for U = t (u, v, w, ρ) ∈ X, A(U ) is a linear operator given by A(U ) = diag{1, 1, 1, A(u, v)}, where A(u, v) is a realization of the differential operator # $ k2 −d∇ · ∇· + 1 k2 + χ2 (Re u) + χ2 (Re v) in the space L2 (Ω) under the Neumann boundary conditions on ∂Ω, χ2 (u) being a smooth cutoff function for −∞ < u < ∞ such that χ2 (u) ≡ − k22 for −∞ < u ≤ − k22 , χ2 (u) = u for 0 ≤ u ≤ 1, and χ2 (u) ≡ 1 + k22 for 1 + k22 ≤ u < ∞. Actually, A(u, v) is a positive definite self-adjoint operator of L2 (Ω). Note that Theorem 2.8 is available to verify that the domain D(A(u, v)) is independent of (u, v) and is given by D(A(u, v)) = HN2 (Ω) (see (2.34)). Consequently, we observe that D(A(U )) ≡ D = Hp1 (Ω) × Hp1 (Ω) × Hp1 (Ω) × HN2 (Ω),

U ∈ X.

(5.83)

In particular, D(A(U )) is independent of U ∈ X. In addition, for 0 ≤ θ ≤ 1, we have A(U )θ = diag{1, 1, 1, A(u, v)θ }. Since A(u, v) is a self-adjoint operator, it follows that D(A(u, v)θ ) coincides with the interpolation space [L2 (Ω), HN2 (Ω)]θ characterized by Theorems 16.7 and 16.9. Therefore, for every U ∈ X,

D A(U )

θ

≡ Dθ =

[Hp1 (Ω)]3 × H 2θ (Ω) for 0 ≤ θ < 34 , [Hp1 (Ω)]3 × HN2θ (Ω) for

3 4

< θ ≤ 1.

(5.84)

The nonlinear operator F is defined by ⎛ ⎞ 1 (Re u) [1 + c(Re ˜ w)] u + a1 s˜ (Re u)θ˜1 (Re v)θ˜2 (Re w) − a2 k1χ+χ 1 (Re u) ⎜ ⎟ ⎜v + a3 s˜ (Re v)θ˜1 (Re u)θ˜2 (Re w) − a4 χ1 (Re v) [1 + c(Re ⎟ ˜ w)] ⎜ ⎟. k +χ (Re v) 1 1 F (U ) = ⎜ ⎟ ⎝ ⎠ ˜ ˜ w + a5 θ1 (Re u)θ1 (Re v)(1 − w)ρ ρ Here, s˜ (u), θ˜i (u) (i = 1, 2) denote C1,1 extensions of s(u), θi (u), respectively, to the whole real line (−∞, ∞) in such a way that s˜ (u) = s(u) for 0 ≤ u ≤ 1, s˜ (u) ≡ 1 for −∞ < u ≤ 0, and s˜ (u) ≡ 0 for 1 ≤ u < ∞. It is the same for θ˜i (u) (i = 1, 2). Similarly, c(w) ˜ is a C1,1 extension of c(w) for −∞ < w < ∞ such that c(w) ˜ = c(w) for 0 ≤ w ≤ 1 and − 12 ≤ c(w) ˜ ≤ c(1) + 12 for −∞ < w < ∞. While, χ1 (u) is a smooth cutoff function for −∞ < u < ∞ such that χ1 (u) ≡ − k21 for −∞ < u ≤ − k21 , χ1 (u) = u for 0 ≤ u ≤ 1, and χ1 (u) ≡ 1 + k21 for 1 + k21 ≤ u < ∞. Let us take 1 2θ 1 the exponent θ = p−1 p (> 2 ). Then, H (Ω) ⊂ Hp (Ω). Note also property (1.95)

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5 Quasilinear Evolution Equations

and the fact that Hp1 (Ω) is a Banach algebra (see (1.88)). As a result, we conclude that F : W ≡ Hp1 (Ω) × Hp1 (Ω) × Hp1 (Ω) × H 2θ (Ω) → X.

(5.85)

The initial values U0 are taken in K. In this way, problem (5.80) has been formulated in the form (5.82).

7.2 Construction of Local Solutions It is immediate to apply Theorem 5.6 to (5.82) for constructing local solutions. By the definition, it is clear that A(U ) is a sectorial operator of X with angle ωA(U ) ≤ ω < π2 , namely, (5.2) and (5.3) are fulfilled with some angle ω and constant M ≥ 1. In view of (5.83), (5.4) is also fulfilled with ν = 1. For verifying (5.5), we will use (3.68). Then, it is sufficient to verify that [A(U ) − A(V )]U˜ X ≤ CU − V Y U˜ D ,

U˜ ∈ D.

But, since ˜ H2, ∇ · [χ∇ u] ˜ L2 ≤ CχHp1 u

χ ∈ Hp1 (Ω), u˜ ∈ HN2 (Ω),

the estimate is observed directly by setting Y = X as the space Y . In addition, since X = Y = Z, (5.6) is also fulfilled with α = 0 and an arbitrarily fixed β such that 0 < β < 2ε , as well as (5.7). As verified by (5.85), F is an operator from W into X. Furthermore, F is seen to satisfy (5.44). Indeed, ˜s (u1 ) − s˜ (u2 )Hp1 ≤ C[∇[˜s (u1 ) − s˜ (u2 )]Lp + ˜s (u1 ) − s˜ (u2 )Lp ] ≤ C[˜s (u1 )∇u1 − s˜ (u2 )∇u2 Lp + u1 − u2 Lp ] ≤ C(u1 Hp1 + u2 Hp1 + 1)u1 − u2 Hp1 ,

u1 , u2 ∈ Hp1 (Ω).

It is the same for other functions contained in the components of F . On account of (5.84), (5.46) is fulfilled with η = θ = p−1 p , as well as (5.47) because of 0 < β < < 12 < p−1 p = η < 1. We finally verify that U0 satisfies (5.30) if γ is fixed as γ = 2ε . Thus, for any U0 ∈ K, (5.82) possesses a unique local solution in the function space: ε 2

U ∈ C((0, TU0 ]; D) ∩ C([0, TU0 ]; Dγ ) ∩ C1 ((0, TU0 ]; X), where TU0 > 0 is determined by the norm U0 D ε . By (5.66), 2 belong to dU , dt

with any 0 < σ < min 2ε , p1 .

ε

A(U )U ∈ F 2 ,σ ((0, TU0 ]; X)

dU dt

(5.86) and A(U )U

(5.87)

7 Some Model for Honeybee Colonies

239

7.3 Nonnegativity of Local Solutions For U0 ∈ K, let U (t) be the local solution of (5.82) constructed above in (5.86) and (5.87). We shall prove that the solution inherits all the properties described in (5.81). Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (5.82) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function given by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative $ #

χ1 (u) ˜ ˜ H (u) a1 s˜ (u)θ1 (v)θ2 (w) − a2 [1 + c(w)] ˜ dx. ψ (t) = k1 + χ1 (u) Ω ˜ and χ1 (u), we observe that By the definitions of s˜ (u), θ˜i (v) (i = 1, 2), c(w), # $

χ1 (u) H (u) [1 + c(w)] ˜ dx ψ (t) ≤ −a2 k1 + χ1 (u) Ω

≤C |H (u)χ1 (u)| dx ≤ C H (u) dx. Ω

Ω

Since ψ (t) ≤ Cψ(t), it follows that ψ(t) ≤ eCt ψ(0) = 0, i.e., ψ(t) ≡ 0. This shows that u(t) ≥ 0 for every 0 ≤ t ≤ TU0 . We next consider the function ψ1 (t) = Ω H (1 − u(t)) dx. By similar arguments, we observe that ψ1 (t) ≤ 0. Hence, ψ1 (t) ≤ ψ1 (0) = 0, i.e., u(t) ≤ 1 for every 0 ≤ t ≤ T U0 . We can repeat quite similar arguments as those above to prove that 0 ≤ v(t) ≤ 1 for 0 ≤ t ≤ TU0 . In the meantime, the nonnegativity of ρ(t) is easily verified by the same techniques as in Chap. 4, Sect. 4.2. From the differential equation of ρ, we d ρ(t) dx ≡ 0. Therefore, ρ(t)L1 = Ω ρ(t) dx ≡ Ω ρ0 dx = 1. Now, have dt Ω since ρ(t) ≥ 0, we can carry out the proof of 0 ≤ w(t) ≤ 1. We will in fact repeat the same arguments as for u and v in view of the known estimate 1

1−ε

1

ρ(t)L∞ ≤ Cρ(t)H 1+ε ≤ Cρ(t)H2−ε2 ρ(t)H2−εε ≤ CU t − 2 ,

0 < t ≤ TU 0 .

7.4 A Priori Estimates In this subsection, let U0 ∈ K satisfy a supplement condition that ρ0 ∈ L∞ (Ω) ∩ H 1 (Ω). Let U denote a local solution of (5.82) on [0, TU ] with U (t) ∈ K, 0 ≤ t ≤ TU , which belongs to the function space: U ∈ C((0, TU ]; D) ∩ C([0, TU ]; Dγ ) ∩ C1 ((0, TU ]; X). We shall establish a priori estimates for local solutions.

(5.88)

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5 Quasilinear Evolution Equations

Proposition 5.2 There exists a continuous increasing function p(·) such that the estimate U (t)Dγ ≤ p(U0 X + ρ0 L∞ + ρ0 H 1 + t),

0 ≤ t ≤ TU ,

(5.89)

holds for any local solution of (5.82) in the function space (5.88). Proof The proof will consist of a several steps. Step 1. Let 2 < q < ∞ be any number. Multiply the equation of ρ by qρ q−1 and integrate the product in Ω. Then,

d k2 ρ q−2 q |∇ρ|2 dx ≤ 0. ρ dx = −dq(q − 1) dt Ω Ω k2 + u + v Therefore, ρ(t)Lq ≤ ρ0 Lq , 0 ≤ t ≤ TU . Letting q → ∞, it follows that ρ(t)L∞ ≤ ρ0 L∞ ,

0 ≤ t ≤ TU .

(5.90)

k2 Step 2. Multiply now the equation of ρ by ∇ · [ k2 +u+v ∇ρ] and integrate the product in Ω. Then, 2

k2 k2 ∂ρ ∇· ∇ρ dx = ∇ · ∇ρ dx. k2 + u + v k2 + u + v Ω ∂t Ω

By a direct calculation, the left-hand side is equal to

k2 d ∂ρ k2 ∇· ∇ρ dx = − |∇ρ|2 dx k2 + u + v dt Ω 2(k2 + u + v) Ω ∂t

k2 (ut + vt ) + |∇ρ|2 dx. 2 2(k + u + v) 2 Ω Here, from the equations of u and v we have ut ≤ a1 and vt ≤ a3 . Therefore, 2

k2 k2 d 2 |∇ρ| dx + ∇ · ∇ρ dx dt Ω 2(k2 + u + v) k + u + v 2 Ω

k2 a1 + a3 |∇ρ|2 dx. ≤ k2 Ω 2(k2 + u + v) Consequently,

k2 2 Ω 2(k2 +u+v) |∇ρ| dx

∇ρ(t)2L2 ≤

≤ 12 e

a1 +a3 k2 t

ρ0 2H 1 , i.e.,

k2 + 2 a1k+a3 t e 2 ρ0 2H 1 , k2

0 ≤ t ≤ TU .

As well, we obtain that 2

t a1 +a3 k2 dx ≤ k2 + 2 e k2 t − 1 ρ0 2 1 , ∇ · ∇ρ H k2 + u + v 2k2 0 Ω

(5.91)

0 ≤ t ≤ TU . (5.92)

7 Some Model for Honeybee Colonies

241

Step 3. We shall use the notation p(ρ0 ) = p(ρ0 L∞ + ρ0 H 1 ), where p(·) in the right-hand side denotes a suitable continuous increasing function. Differentiating the equation of u with respect to the variable xi (i = 1, 2), we observe that ∂ Di u = f1 (u, v, w)Di u + f2 (u, v, w)Di v + f3 (u, v, w)Di w, ∂t where fj (u, v, w) (j = 1, 2, 3) are real smooth functions defined for 0 ≤ u ≤ 1, 0 ≤ v ≤ 1, 0 ≤ w ≤ 1. Multiply this equation by p|Di u|p−2 Di u and integrate the product in Ω. After some calculations, we have

d p p p |Di u|p dx ≤ C Di uLp + Di vLp + Di wLp . dt Ω d Note that du |u|p = p|u|p−2 u for −∞ < u < ∞. By using the equations of v and w (in view of (5.90)), we similarly obtain that

d p p p |Di v|p dx ≤ C Di uLp + Di vLp + Di wLp , dt Ω

d p p p p |Di w|p dx ≤ Cp(ρ0 ) Di uLp + Di vLp + Di wLp + Di ρLp . dt Ω

Therefore,

d |Di u|p + |Di v|p + |Di w|p dx dt Ω p p p p ≤ Cp(ρ0 ) Di u(t)Lp + Di v(t)Lp + Di w(t)Lp + Di ρ(t)Lp . Here, by (5.90) and (5.91), p

p−2

p−2

Di ρLp ≤ CDi ρH 1 Di ρ2L2 ≤ Cp(ρ0 )eμt ρ(t)H 2 , where μ =

a1 +a3 k2 .

In addition, from

$ k2 k2 k2 ∇ρ = Δρ − (∇u + ∇v) · ∇ρ ∇· k2 + u + v k2 + u + v (k2 + u + v)2 #

we see that ΔρL2

# $ k2 k2 + 2 1 ∇· ∇ρ ≤ + (∇uLp + ∇vLp )∇ρL 2p k2 k2 + u + v k2 p−2 L2 # $ k2 k2 + 2 ∇ · ∇ρ ≤ k2 k2 + u + v L2

242

5 Quasilinear Evolution Equations 2

p−2

+ C(∇uLp + ∇vLp )ρHp 2 ρHp1 # $ k2 k2 + 2 ∇· ≤ ∇ρ + ζ ρH 2 k2 k2 + u + v L2 p p & % μ p−2 + Cζ p(ρ0 )e 2 t ∇uLp−2 + ∇v Lp p with any number ζ > 0 and some constant Cζ > 0 depending on ζ . Since ρH 2 ≤ C(ΔρL2 + ρL2 ) for all ρ ∈ HN2 (Ω) (see (2.33)), it follows that # $ p p & ' μ % ( k2 p−2 p−2 2 t ∇u + Cp(ρ ) e + ∇v ∇ρ + 1 ρH 2 ≤ C ∇ · 0 Lp Lp k2 + u + v L2 and that p−2 ρH 2

# $p−2 k2 ≤ C ∇ · ∇ρ k2 + u + v L2 μ(p−2) t p p ∇uLp + ∇vLp + 1 . + Cp(ρ0 ) e 2

In this way, we arrive at the differential inequality

d |∇u|p + |∇v|p + |∇w|p dx dt Ω pμ p p p t 2 ∇uLp + ∇vLp + ∇wLp ≤ Cp(ρ0 )e # $p−2 k2 + ∇ · +1 . ∇ρ k2 + u + v L2 By virtue of (5.92) (recall that p was taken as 2 < p ≤ 4), we can solve this differential inequality to conclude that ∇u(t)Lp + ∇v(t)Lp + ∇w(t)Lp ≤ p(U0 X + ρ0 L∞ + ρ0 H 1 + t),

0 ≤ t ≤ TU .

This, together with (5.91), then yields the desired estimate (5.89).

7.5 Global Solutions Utilizing the a priori estimates (5.89), we shall construct a global solution to (5.82). For U0 ∈ K, we know that there exists a local solution having the properties (5.81) at least on an interval [0, TU0 ]. Let 0 < t1 < TU0 . Then, U1 = U (t1 ) ∈ K

8 Some Chemotaxis Model

243

and ρ1 = ρ(t1 ) ∈ HN2 (Ω). We next consider problem (5.82) with the initial value U1 on an interval [t1 , T ], where the end time T > 0 is any finite time. Proposition 5.2 ensures estimate (5.89) for any local solution V , i.e., V (t)Dγ ≤ p(U1 X + ρ1 L∞ + ρ1 H 1 + T ), t1 ≤ t ≤ TV . Then, the local solution V can always be extended over an interval [t1 , TV + τ ] as local solution, τ > 0 being dependent only on p(U1 X + ρ1 L∞ + ρ1 H 1 + T ) and hence being independent of the extreme time TV (cf. Corollary 5.2). This means that our Cauchy problem possesses a global solution on the interval [t1 , T ]. This argument is meaningful for any finite time T > 0. So, we conclude the global existence of solution. For any initial value U0 ∈ K, there exists a unique global solution to (5.82) with U (t) ∈ K, 0 ≤ t < ∞, in the function space: U ∈ C((0, ∞); D) ∩ C([0, ∞); Dγ ) ∩ C1 ((0, ∞); X).

8 Some Chemotaxis Model Let us consider the initial-boundary-value problem for a chemotaxis model in biology which is given by ⎧ ∂u 2 in Ω × (0, ∞), ⎪ ∂t = Δ(au + αu ) − μ∇ · (u∇ρ) ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu in Ω × (0, ∞), ∂t (5.93) ∂ρ ⎪ ∂u ⎪ = = 0 on ∂Ω × (0, ∞), ⎪ ∂n ∂n ⎪ ⎩ ρ(x, 0) = ρ0 (x) in Ω, u(x, 0) = u0 (x), in a C2 or convex, bounded domain Ω ⊂ R2 . In the domain Ω, biological individuals (bacteria) exist. The density of bacteria at time t is denoted by u(x, t). The function ρ(x, t) denotes the concentration of chemical substance at time t . Bacteria diffuse not only by random walking with diffusion rate a but also by self-diffusion with rate α. They also have a directed movement which is called the chemotaxis in the sense that they have a tendency to move toward higher concentration of the chemical substance. The flow by this effect is given by μu∇ρ with some mobility constant μ. Bacteria produce the chemical substance by themselves at a rate ν. The chemical substance diffuses with diffusion rate b. For the details, see Chap. 12. We assume that a, b, d, α, μ, and ν are positive constants (>0). Let 0 < ε < 12 be arbitrarily fixed. The initial functions are taken as 0 ≤ u0 ∈ H 1+ε (Ω) and 0 ≤ ρ0 ∈ H 1+ε (Ω).

(5.94)

8.1 Abstract Formulation Let us set X = L2 (Ω) (see (1.84)) as the underlying space X. We want to formulate (5.93) as the Cauchy problem for an abstract equation

244

5 Quasilinear Evolution Equations

+ A(U )U = F (U ), U (0) = U0 , dU dt

0 < t < ∞,

(5.95)

in X. Here, A(U ) is a linear operator of X defined for U ∈ Z = H1+ε1 (Ω), where 0 < ε1 < ε, which is given by # $ $ # u A1 (u) B(u) ∈ Z. with D(A(U )) = H2N (Ω), U = A(U ) = ρ 0 A2 The operator A1 (u) is a realization of the differential operator −{∇ · [a + 2αχ(Re u)]∇·} + 1 in L2 (Ω) under the Neumann boundary conditions on ∂Ω, where χ(u) is a smooth cutoff function such that χ(u) = u for u ≥ 0 and χ(u) ≡ −δ for u ≤ −δ, δ > 0 being some small positive constant such that 4αδ ≤ a. By Theorem 2.4, A1 (u) is a positive definite self-adjoint operator of L2 (Ω), and by Theorem 2.8, their domain is given by D(A1 (u)) = HN2 (Ω) (see (2.34)). The operator A2 is a realization of the operator −bΔ + d in L2 (Ω) under the same boundary conditions on ∂Ω and is a positive definite self-adjoint operator of L2 (Ω) with domain D(A2 ) = HN2 (Ω); and the operator B(u) is given by μ∇ · (u∇·). Since ∇ · (u∇v) = ∇u · ∇v + uΔu, we observe from (1.74), (1.75), and (1.76) that ∇ · (u∇v)L2 ≤ CuH 1+ε1 vH 2 , This means that D(B(u)) ⊃ H 2 (Ω). The operator F : X → X is given by # $ u F (U ) = , νu

u ∈ H 1+ε1 (Ω), v ∈ H 2 (Ω).

U=

# $ u ∈ X. ρ

The initial value U0 is given by U0 = t (u0 , ρ0 ), u0 and ρ0 satisfying (5.94).

8.2 Basic Properties of Operators A(U ) Let 0 < R < ∞, and let KR = {U ∈ Z; U Z < R} be an open ball of Z. Proposition 5.3 For any U ∈ Z, we have D(A(U )) = H2N (Ω). For each 0 < R < ∞, there is a constant CR > 0 such that CR−1 U˜ H2 ≤ A(U )U˜ X ≤ CR U˜ H2 , U˜ ∈ D(A(U )), holds for every U ∈ KR . Proof This result is easily verified by (2.33) which is ensured by Theorem 2.8. Proposition 5.4 For U ∈ KR , A(U ) is a sectorial operator of X with angle ωA(U ) ≤ ωR < π2 and satisfies (5.3) with MR ≥ 1 uniformly. Proof As A1 (u) and A2 are positive definite self-adjoint operators of L2 (Ω), this result is obtained by applying Theorem 1.8 to each operator A(U ).

8 Some Chemotaxis Model

245

For s > 32 , we put HsN (Ω) =

∂U U ∈ H (Ω); = 0 on ∂Ω . ∂n s

Proposition 5.5 For all U ∈ Z, we have D A(U )θ = H2θ (Ω), D A(U )θ = H2θ N (Ω),

3 0≤θ < , 4 3 < θ ≤ 1. 4

Let 0 ≤ θ ≤ 1, θ = 34 . For 0 < R < ∞, there is a constant CR,θ > 0 such that −1 ˜ U H2θ ≤ A(U )θ U˜ X ≤ CR,θ U˜ H2θ , CR,θ

U˜ ∈ D A(U )θ ; U ∈ KR .

Proof In order to prove the assertion of theorem, we will appeal to Theorem 16.4 presented in Chap. 16. For applying Theorem 16.4, it is in fact sufficient to verify that for each u, the adjoint operator B(u)∗ : D(B(u)∗ ) ⊂ L2 (Ω) → L2 (Ω) of B(u) has a domain such that D(B(u)∗ ) ⊃ HN2 (Ω) and is a bounded operator from HN2 (Ω) into L2 (Ω). However, this is easily verified. It is clear that HN2 (Ω) ⊂ D(B(u)∗ ) and B(u)∗ v = μ∇ · (u∇v) for v ∈ HN2 (Ω). Hence, B(u)∗ is bounded from HN2 (Ω) into L2 (Ω). We thus know that D(A(U )θ ) = [X, H2N (Ω)]θ for 0 ≤ θ ≤ 1. Then, the desired characterization of D(A(U )θ ) is verified by Theorem 16.7.

8.3 Construction of Local Solutions We are ready to apply Theorem 5.6 to construct local solutions to (5.95). Let U0 be any initial value satisfying (5.94). We can then choose a radius R > 0 in such a way that U0 ∈ KR . It is already known that, for U ∈ KR , A(U ) fulfills (5.2) and (5.3). Since D(A(U )) = H2N (Ω) for every U ∈ KR , (5.4) is also fulfilled with ν = 1. In view of (3.68), (5.5) is equivalent to showing that [A(U ) − A(V )]U˜ X ≤ CR U − V Y U˜ H2 ,

U˜ ∈ H2N (Ω),

with a suitable space Y . We here notice that ˜ ∇ · [χ(Re u)∇ u] ˜ = χ(Re u)Δu˜ + χ (Re u)∇(Re u) · ∇ u. From this we observe that ˜ H2 ∇ · [χ(Re u)∇ u] ˜ L2 ≤ (χ(Re u)L∞ + χ (Re u)L∞ uH θ+1 )u ≤ (χ(Re u)H θ+1 + χ (Re u)H θ+1 uH θ+1 )u ˜ H2, (5.96)

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5 Quasilinear Evolution Equations

whatever 0 < θ < 1 is. In view of this fact, we set Y = H1+ε0 (Ω), where ε0 is a third exponent such that 0 < ε0 < ε1 < ε. Then, (5.5) is verified immediately with the aid of (1.89) and (1.94). By virtue of Proposition 5.5, (5.6) is then fulfilled with 1+ε1 0 α = 1+ε 2 and β = 2 . Relation (5.7) is now obvious. The nonlinear operator F is defined on W = H1+ε2 (Ω), where ε < ε2 < 12 . It is trivial that (5.44) is fulfilled. Conditions (5.46) and (5.47) are also verified, since 2 Proposition 5.5 provides that D(A(U )η ) = W with η = 1+ε 2 for every U ∈ KR . The 1+ε initial value U0 satisfies (5.30) if γ is taken as γ = 2 . Theorem 5.6 then provides that, for any U0 ∈ KR satisfying (5.94), (5.95) possesses a unique local solution in the function space: U ∈ C (0, TU0 ]; H2N (Ω) ∩ C([0, TU0 ]; H1+ε (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), (5.97) where TU0 > 0 is determined by the norm U0 H1+ε . By (5.66), A(U )U belongs to dU , dt

A(U )U ∈ F

1+ε 2 ,σ

((0, TU0 ]; L2 (Ω))

(5.98)

with any 0 < σ < min{ε1 − ε0 , 1−ε 2 }.

8.4 Nonnegativity of Local Solutions For U0 satisfying (5.94), let U (t) be the local solution of (5.95) constructed above in the function space (5.97) and (5.98). We want to prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . Let us notice first that U (t) is real valued. The complex conjugate U (t) of U (t) is also a local solution of (5.94) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. We then utilize the truncation method as in Sect. 7.3. Let H (u) be a C1,1 cutoff function given by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative

H (u)∇ · {[a + 2αχ(u)]∇u} dx − μ H (u)∇ · (u∇ρ) dx. ψ (t) = Ω

Ω

Using property (1.96), we have

H (u)∇ · {[a + 2αχ(u)]∇u} dx = − [a + 2αχ(u)]∇H (u) · ∇u dx Ω

=−

Ω

[a + 2αχ(u)]|∇H (u)|2 dx ≤ −a

Ω

By a similar reason,

|∇H (u)|2 dx. Ω

(5.99)

9 Nonautonomous Quasilinear Abstract Evolution Equations

−μ

H (u)∇ · (u∇ρ) dx = μ

Ω

247

u∇H (u) · ∇ρ dx

Ω

=μ

H (u)∇H (u) · ∇ρ dx

Ω

≤ μH (u)Lp ∇H (u)L2 ∇ρLq 2(1− p1 )

≤ CH (u)H 1 ≤

2

H (u)Lp 2 ∇ρLq

a p H (u)2H 1 + CH (u)2L2 ∇ρLq , 2

where 2 < p, q < ∞ are any numbers satisfying

1 p

+

1 q

= 12 . Therefore, it follows C

t

p

[∇ρ(s)

+1]ds

Lq that ψ (t) ≤ C[∇v(t)Lq +1]ψ(t). Consequently, ψ(t) ≤ ψ(0)e 0 for 0 < t < TU0 . p p From the regularity (5.98), we notice that ∇ρ(s)Lq ≤ Cq ρ(s)H 2 ≤ Cq CU0 ×

p

s p(ε−1)/2 . It is possible to choose p > 2 arbitrarily close to 2, which shows that p ∇ρ(·)Lq is integrable in (0, TU0 ). Then, ψ(0) = 0 implies ψ(t) ≡ 0. Thus, u(t) ≥ 0 for every 0 < t ≤ TU0 . It is similar (actually easier) for the proof of ρ(t) ≥ 0 for 0 < t ≤ TU0 . Hence, we have accomplished the proof of nonnegativity of U (t). Now, since u(t) ≥ 0, it holds that χ(Re u(t)) = u(t); this then means that the local solution of (5.95) is regarded as a local solution to the original problem (5.93).

8.5 Global Solutions We are able to build up a priori estimates for local solutions of (5.95) with initial values satisfying (5.94) and an additional condition U0 ∈ H2 (Ω). In fact, there exists a continuous increasing function p(·) such that, for any local solution of (5.95) in (5.97) (in which TU0 must be replaced by TU ) with initial value satisfying (5.94) and U0 ∈ H2 (Ω), it holds that U (t))H2 ≤ p(U0 H2 ),

0 < t ≤ TU .

(5.100)

For showing these estimates step-by-step, we need more complicated calculations for the norms of U (t) in each step. We will not describe the proof here, cf. Chap. 12, Sect. 3.2, Chap. 15, Sects. 3.1 and 3.2. It is of course immediate to conclude global existence of solutions for (5.95) utilizing (5.100) (cf. Corollary 5.2). Note also that, for any initial value satisfying (5.94), its local solution always satisfies U (t) ∈ H2 (Ω) for any t > 0.

9 Nonautonomous Quasilinear Abstract Evolution Equations In this section, we will show how to reduce nonautonomous problems to autonomous ones as in Chap. 4, Sect. 5. Consider the Cauchy problem for a quasilinear

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5 Quasilinear Evolution Equations

abstract evolution equation dU dt + A(t, U )U = F (t, U ) + G(t),

0 < t ≤ T,

U (0) = U0 ,

(5.101)

in a Banach X. Let Z be a second Banach space continuously embedded in X, and let K = {U ∈ Z; U Z < R}, 0 < R < ∞, be an open ball of Z. For (t, U ) ∈ [0, T ] × K, A(t, U ) is a sectorial operator of X with angle ωA(t,U ) < π2 . For each 0 ≤ t ≤ T , F (t, ·) is a nonlinear operator from W into X, where W is another Banach space continuously embedded in X. The function G is a given X-valued function. The initial value U0 is taken in K. For the sectorial operators A(t, U ), we make the same structural assumptions as in Sect. 1.1. We use three Banach spaces Z ⊂ Y ⊂ X. The operator A(t, U ) is a sectorial operator of X satisfying (5.2) and (5.3). The domain D(A(t, U )) is allowed to vary with (t, U ) ∈ [0, T ] × K, but it is assumed that there exists a fixed exponent 0 < ν ≤ 1 such that D(A(s, V )) ⊂ D A(t, U )ν for any pair (t, U ), (s, V ) ∈ [0, T ] × K (5.102) (analogously to (5.4)). The inverse A(t, U )−1 satisfies the Lipschitz condition A(t, U )ν A(t, U )−1 − A(s, V )−1 L(X) ≤ N (|t − s| + U − V Y ),

(t, U ), (s, V ) ∈ [0, T ] × K,

(5.103)

with a constant N > 0 (instead of (5.5)). Furthermore, the domains of fractional powers D(A(t, U )α ) and D(A(t, U )β ) are contained in Y and Z with estimate (5.6), respectively, and with exponents 0 ≤ α < β < 1 satisfying relations (5.7). Similarly, D(A(t, U )η ) is contained in W with estimate (5.46). When (5.47) is the case, i.e., β < ν, we assume the Lipschitz condition F (t, U ) − F (s, V )X ≤ ϕ(U Z + V Z )[U − V W + (U W + V W + 1)(|t − s| + U − V Z )], (t, U ), (s, V ) ∈ [0, T ] × W,

(5.104)

where ϕ(·) is some continuous increasing function. On the other hand, when (5.69) is the case, i.e., ν ≤ β, we have to assume not only the Lipschitz condition F (t, U ) − F (s, V )X ≤ ϕ(U Z + V Z )(|t − s| + U − V W ), (t, U ), (s, V ) ∈ [0, T ] × Z, but also the reflexivity of Z (see (5.36)). We now rewrite problem (5.101) as the following one: d τ 1 τ 0 0 τ +1 dt U + 0 A(χ(Re τ ),U ) U = F (χ(Re τ ),U ) + G(t) , 0 τ U (0) = U0 ,

(5.105)

(5.106)

Notes and Further Researches

249

in the product space X = C × X for the unknown function t (τ, U ), where χ(t) is a C0,1 cutoff function for −∞ < t < ∞ such that χ(t) ≡ 0 for −∞ < t < 0, χ(t) = t for 0 ≤ t ≤ T , and χ(t) ≡ T for T < t < ∞. Since τ (t) = t , (5.106) is certainly equivalent to (5.101). It is not difficult to verify that the conditions assumed above, including (5.102), (5.103), (5.104), and (5.105), imply all the structural conditions for the linear operators A(τ, U ) = diag{1, A(χ(Re τ ), U )},

(τ, U ) ∈ C × K,

of X and for the nonlinear operator F(τ, U ) = t (τ + 1, F (χ(Re τ ), U )) from C × W into X. Hence, Theorems 5.5, 5.6, 5.7, and others are available to (5.106).

Notes and Further Researches The abstract problems of the form (5.1) or (5.43) were first studied by Sobolevskii [Sob61b], where the independence of the domains D(A(t, U )ν ) is assumed for some 0 < ν ≤ 1. The analyticity of solutions was studied by Furuya [Fur81, Fur83], and the global existence of solutions was studied by Lunardi [Lun84, Lun85]. Under a similar assumption as (5.5), especially in the case where β ≤ ν (together with (5.36)), the problems were handled by Yagi [Yag91, Yag02]. For the nonautonomous problem (5.101), see [Yag04]. In Favini–Yagi [FY04], the Cauchy problem for the multivalued quasilinear equation dU + A(U )U F (U ), dt

0 < t ≤ T,

was handled. For the study of (5.1) or (5.43) by the variational methods, we quote Lions [Lio69, DL88]. Theorem 5.10 can be proved by similar arguments as in Ryu–Yagi [RY01]. By the methods of operational equations (see [DaPG75]), the abstract problems can also be studied; we quote Prüss [Pru02]. There is of course a great deal of important literature which deals with the quasilinear parabolic differential equations directly by the functional analytic methods. But we will here quote only Ladyzhenskaya–Solonnikov–Uraltseva [LSU68], Amann [Ama86, Ama93, Ama95], Quittner–Souplet [QS07], and so forth. The abstract results obtained in this chapter are available for various model equations presented from the real world. In Sects. 7 and 8, some of them were represented. The honeybee colony model (5.80) was introduced in Camazine–Sneyd– Jenkins–Murray [CSJM90] for describing pattern formation created spontaneously by the egg-laying queen. For more detailed observations and arguments for modeling, see Camazine et al. [CDFSTB, Chap. 16]. The chemotaxis system with selfdiffusion (5.93) is one of specific forms of the chemotaxis model presented by Keller–Segel [KS70]. Moreover, these abstract results will be utilized in Chaps. 14 and 15.

Chapter 6

Dynamical Systems

In this chapter, we study dynamical systems determined from the Cauchy problems for abstract semilinear and quasilinear evolution equations in Banach spaces. The theory of infinite-dimensional dynamical systems was developed by Ladyzhenskaya, Vishik, Temam, and many other mathematicians. The first half of the chapter is mainly devoted to studying two basic matters of the dynamical systems, namely, stable and unstable manifolds of the equilibria and attractors. We shall use the notation (S(t), X, X) to denote a dynamical system. Here, X is a Banach space and denotes the universal space in which a dynamical system is considered. The notation X denotes a subset of X and is a metric space. This is the space consisting of all possible states the system can take and is called the phase space. Meanwhile, S(t) (0 ≤ t < ∞) is a family of nonlinear operators acting on X with the semigroup property S(t)S(s) = S(t + s) and with S(0) = 1 (the identity operator on X). It is assumed that S(t) is continuous in both space and time variables in the sense that the correspondence (t, U0 ) → S(t)U0 is continuous from [0, ∞) × X into X. The operators S(t) give a propagation law of states of the system. For U0 ∈ X, the continuous function S(·)U0 for 0 ≤ t < ∞ with values in X denotes a transition of states starting from U0 at t = 0. The trace of points S(·)U0 which vary in the phase space X as time t goes on from 0 to ∞ is called the trajectory starting from U0 . The semigroup property implies S(t + s)U0 = S(t)S(s)U0 for any 0 ≤ s, t < ∞; this means that, s being fixed, a trajectory passing by S(s)U0 at time s coincides with that starting from S(s)U0 at time 0. The totality of such trajectories is called a dynamical system and is denoted by the triplet (S(t), X, X). A point U ∈ X which satisfies S(t)U = U is called a fixed point of S(t). A point U which is a fixed point for every S(t), that is, S(t)U = U for every 0 ≤ t < ∞, is called an equilibrium of dynamical system. For each equilibrium U , one can define a stable manifold M− (U ) and an unstable manifold M+ (U ) by (6.11) and (6.12), respectively. We then want to construct stable and unstable manifolds which are smooth in a neighborhood of U . If M− (U ) is a neighborhood of U , then U is stable. In the meantime, if M+ (U ) = {U }, then U is unstable. We shall indeed show that, if S(t) is Fréchet differentiable in a neighborhood of U with Hölder continuous derivative (6.19) and if the differential S(t) U satisfies a spectrum separation condition (6.20), then one can construct the desired smooth stable and unstable manifolds in a neighborhood of U . A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_6, © Springer-Verlag Berlin Heidelberg 2010

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For (S(t), X, X), a subset A of X is called an attractor if A is invariant (see (6.4)) and if A attracts, as t → ∞, all the trajectories starting in a neighborhood of A. Among others, an attractor A is called a global attractor of dynamical system if A is a compact set of X, if A is strictly invariant (see (6.5)), and if A attracts all trajectories in X. We know however that the global attractor is not necessarily robusts. Consider a dynamical system (Sξ (t), X, X) containing a parameter ξ and assume that (Sξ (t), X, X) possesses a global attractor Aξ , even if the semigroup Sξ (t) depends on ξ smoothly, Aξ may be not continuous in ξ . It is only upper semicontinuous in general. In addition, the global attractor may not be rich enough to describe the nature of the system, for, since A is strictly invariant, A can easily miss some important trajectories which correspond to a specific transition of states like pattern formations. As a more robust attractor, Foias–Sell–Temam introduced the notion of inertial manifolds which are hyperbolic (and thus robust), compact, invariant, finite-dimensional C1 manifolds that attract all the trajectories at exponential rates. But one can ensure the existence of inertial manifolds only under a very restrictive assumption called the spectral gap condition. Consequently, we do not know whether the dynamical systems constructed from many important models have inertial manifolds or not. In 1994, Eden–Foias–Nicolaenko–Temam have founded the new concept of exponential attractors (sometimes called inertial sets) to overcome this difficulty. An attractor M containing the global attractor is called an exponential attractor if M is a compact set of X with finite fractal dimension and if M attracts all the trajectories at exponential rates. (So, M need not to be a smooth manifold.) We notice that the exponential attractor is not uniquely determined and usually exists as a family. But the exponential attractor Mξ can depend continuously on a parameter ξ in a certain specific sense if the semigroup Sξ (t) depends on ξ smoothly, see Theorem 6.17. Two kinds of a sufficient condition on S(t) are known that ensure the existence of exponential attractors for (S(t), X, X). One is a squeezing property presented by the founders of the notion themselves. The squeezing property is meaningful only when X is a Hilbert space. When X is a Hilbert space, the operator S(t) is said to enjoy a squeezing property if, for any pair (U, V ) of points U, V ∈ X, it holds that either S(t)U − S(t)V ≤ δ U − V with some fixed exponent 0 ≤ δ < 14 or (1 − P )[S(t)U − S(t)V ] ≤ P [S(t)U − S(t)V ] with some fixed finite-rank orthogonal projection P (see (6.45)–(6.46)). The other is a compact perturbation of a contraction operator presented first by Efendiev–Miranville–Zelik. This property is meaningful in general Banach spaces, and there are two types. The operator S(t) is said to be a compact perturbation of a contraction operator of type I if the difference S(t)U − S(t)V is divided into two parts S0 (U, V ) and K(U, V ) such that S0 (U, V ) X ≤ δ U − V X with some fixed exponent 0 ≤ δ < 12 and that K(U, V ) Z ≤ L U − V X with some constant L > 0 and some fixed subspace Z which is compactly embedded in X (see (6.28)–(6.29)). Similarly, S(t) is said to be a compact perturbation of a contraction operator of type II if the norm S(t)U − S(t)V X is estimated by the sum of norms δ U − V X and KU −KV X with some fixed exponent 0 ≤ δ < 12 and with some Lipschitz continuous nonlinear operator K from X into a subspace Z which is compactly embedded in X (see (6.38)–(6.39)). As will be seen, the squeezing property provides sharper

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dimension estimates for exponential attractors than those of the compact perturbation condition. On the other hand, we have to notice that the property can easily be verified only when (S(t), X, X) are constructed from semilinear problems in Hilbert spaces of the form (6.53). Unfortunately, no convenient method is known so far for dynamical systems from quasilinear problems. In contrast, the method of using the compact perturbation condition on S(t) can easily be available to many concrete models. In the second half of the chapter, we shall consider dynamical systems which are determined by the Cauchy problems of semilinear and quasilinear abstract evolution equations. We are concerned with describing general methods for constructing stable and unstable manifolds for equilibria, i.e., stationary solutions. Similarly, we are concerned with general methods for constructing exponential attractors. For semilinear problems, stable–unstable manifolds are constructed under conditions (6.68), (6.69), and (6.76) and exponential attractors under (6.59) and (6.60), respectively. For quasilinear problems, stable–unstable manifolds are constructed under (6.115), (6.116) and (6.127) and exponential attractors under (6.103) and (6.104), respectively. We remark also that all these conditions are described mainly in terms of linear and nonlinear operators in the equations.

1 Basics of Dynamical Systems In this section, we shall prepare the basic matters of dynamical systems in Banach spaces. For details, see [Lad91, BV92, Tem97, Rob01] and others.

1.1 Equilibria and Attractors Let X be a Banach space with norm · . Let X be a subset of X, X being a metric space with the distance d(U, V ) = U − V induced by · . A family of nonlinear operators S(t), 0 ≤ t < ∞, from X into itself is called a (nonlinear) semigroup on X if S(t) satisfies (1) S(0) = 1 (identity mapping on X). (2) S(t)S(s) = S(t + s), 0 ≤ t, s < ∞ (semigroup property). When a semigroup S(t) on X has the property that the mapping G(t, U ) = S(t)U is continuous from [0, ∞) × X to X,

(6.1)

S(t) is called a continuous (nonlinear) semigroup on X. Let {Un }n=0,1,2,... be any sequence of X which converges to U in X. Then, for any fixed 0 < T < ∞, the set [0, T ] × {U0 , U1 , U2 , . . . , U } is a compact set of [0, ∞) × X. Therefore, if S(t) is a continuous semigroup, G is uniformly continuous on [0, T ] × {U0 , U1 , U2 , . . . , U }.

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6 Dynamical Systems

Thus, for any fixed 0 < T < ∞, Un → U

sup S(t)Un − S(t)U → 0.

implies

(6.2)

0≤t≤T

Let S(t) be a continuous semigroup on X. For each U0 ∈ X, S(·)U0 defines an X-valued continuous function for 0 ≤ t < ∞. The trace of the function S(·)U0 in X is called a trajectory starting from U0 . The totality of trajectories starting from the points in X are denoted by the triplet (S(t), X, X) and is called a dynamical system. The set X is called a phase space, and X a universal space. Let (S(t), X, X) be a dynamical system. A point U ∈ X is called a fixed point of S(t) if S(t)U = U . If U is a fixed point for all operators S(t), i.e., S(t)U = U

for every t ≥ 0,

(6.3)

then U is called an equilibrium of (S(t), X, X). More generally, a set A ⊂ X is said to be invariant under S(t) if S(t)A ⊂ A. If A is invariant under all operators S(t), i.e., S(t)A ⊂ A

for every t ≥ 0,

(6.4)

then A is called an invariant set of (S(t), X, X). Similarly, a set A ⊂ X is said to be strictly invariant under S(t) if S(t)A = A. If S(t)A = A

for every t ≥ 0,

(6.5)

then A is called a strictly invariant set of (S(t), X, X). Two important notions of absorbing and attraction are defined as follows. A set A is said to absorb a set B if there is a certain time t0 ≥ 0 such that S(t)B is contained in A for every t ≥ t0 , i.e., S(t)B ⊂ A

for every t ≥ t0 .

We call a subset A ⊂ X an absorbing set of (S(t), X, X) if A ⊂ X absorbs every bounded set of X, namely, if, for any bounded set B ⊂ X, there is a time tB ≥ 0 (depending on B) such that S(t)B ⊂ A

for every t ≥ tB .

In the meantime, a set A is said to attract a set B if for any ε > 0, the εneighborhood Wε (A) of A absorbs B. Using the Hausdorff pseudo-distance h(·, ·) (see (1.1)), this definition can be described as lim h(S(t)B, A) = 0.

t→∞

(6.6)

Indeed, if (6.6) holds true, then for any ε > 0, there exists some tε > 0 such that h(S(t)B, A) ≤ 2ε for t ≥ tε ; therefore, for any U ∈ S(t)B, d(U, A) ≤ 2ε ; hence, U ∈ Wε (A), i.e., S(t)B ⊂ Wε (A)

for every t ≥ tε .

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255

Conversely, this condition implies h(S(t)B, A) ≤ ε for every t ≥ tε . So, if A attracts B, then (6.6) is valid. An equilibrium U of (S(t), X, X) is said to be asymptotically stable if some neighborhood W of U is attracted by U , i.e., lim h(S(t)W, U ) = 0.

t→∞

(6.7)

We simply say that U is stable if for any positive number ε > 0, there exists a neighborhood W of U which is attracted by an ε-neighborhood of U in such a way that lim sup h(S(t)W, U ) ≤ ε. t→∞

Clearly, if U is asymptotically stable, it is stable. Asymptotic stability is therefore stronger than mere stability. When U is not stable, U is said to be unstable. In other words, U is unstable if and only if there exists a positive number ε0 > 0 such that, for any neighborhood W of U , it holds that lim sup h(S(t)W, U ) > ε0 .

(6.8)

t→∞

That is, for any neighborhood W , there are a time sequence tn → ∞ and a sequence of points Un ∈ W such that d(S(tn )Un , U ) ≥ ε0 for all n. An invariant set A of (S(t), X, X) is called an attractor if some open neighborhood W of A is attracted by A, namely, A is an attractor if S(t)A ⊂ A for every t and lim h(S(t)W, A) = 0

t→∞

for some neighborhood W of A. An attractor A is called a global attractor of (S(t), X, X) if (1) A is a compact subset of X. (2) S(t)A = A for every t > 0. (3) A attracts every bounded subset of X, i.e., for every bounded B ⊂ X, lim h(S(t)B, A) = 0.

t→∞

(6.9)

Let B be a bounded subset of X which contains the global attractor A. Then, A = S(t)A ⊂ S(t)B, i.e., h(A, S(t)B) = 0 for any t > 0; hence, limt→∞ d(S(t)B, A) = 0, where d(·, ·) is the distance of two sets. By definition, if it exists, the global attractor A is always unique.

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1.2 ω-limit Sets and Global Attractor Consider a dynamical system (S(t), X, X). For each U0 ∈ X, we put {S(τ )U0 ; t ≤ τ < ∞} (closure in X). ω(U0 ) = 0≤t 0. There exists a sequence tn → ∞ such that S(tn )U0 → U ; for tn ≥ t, S(tn )U0 = S(t)S(tn − t)U0 , and S(tn − t)U0 contains a convergent subsequence with a limit U ∈ ω(U0 ); hence, U = S(t)U . In this way, S(t)ω(U0 ) = ω(U0 ) for every t ≥ 0. Consider two open sets Oi (i = 1, 2) of X such that ω(U0 ) ⊂ O1 ∪ O2 and O1 ∩ O2 = ∅. Suppose that there exist points Ui ∈ ω(U0 ) ∩ Oi (i = 1, 2); then, there exist sequences sm → ∞ and tn → ∞ such that S(sm )U0 → U1 and S(tn )U0 → U2 , respectively; here, it is possible to assume that S(sm )U0 ∈ O1 and S(tn )U0 ∈ O2 . If sm < tn , then there exists a time τ ∈ (sm , tn ) such that S(τ )U0 ∈ / O1 ∪ O2 because of the connectedness of the arc {S(t)U0 ; sm ≤ t ≤ tn } of the trajectory S(t)U0 ; similarly, if tn < sm , then there is a time τ ∈ (tn , sm ) such that S(τ )U0 ∈ / O1 ∪ O2 ; therefore, we can take a sequence of times τk → ∞ for which S(τk )U0 ∈ / O1 ∪ O2 holds. Furthermore, this implies that there exists a point U ∈ ω(U0 ) such that U∈ / O1 ∪ O2 , which is a contradiction. Assume that (S(t), X, X) possesses an global attractor A. Then, for any U0 ∈ X, ω(U0 ) = ∅ and ω(U0 ) ⊂ A. In fact, since {U0 } is a bounded set, the trajectory S(t)U0 is attracted by A; therefore, for each integer n ≥ 1, there exists a time tn ≥ n such that d(S(tn )U0 , A) ≤ n1 and that d(S(tn )U0 , Vn ) ≤ n2 with some Vn ∈ A. Since the sequence Vn contains a subsequence which converges to a limit V ∈ A, it follows that S(tn )U0 → V and hence V ∈ ω(U0 ). Moreover, it is easy to see that ω(U0 ) ⊂ A. The following theorem is very convenient for constructing global attractors. Theorem 6.2 Let (S(t), X, X) be a dynamical system. Assume that there exists a compact absorbing set B.

(6.10)

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257

Then, (S(t), X, X) has a global attractor A. In addition, if B is a connected set, then it is the same for A. Proof Since B itself is absorbed by B, there exists a time tB ≥ 0 such that S(t)B ⊂ B for every t ≥ tB . We then consider the set ˜ = X

S(t)B ⊂ B (closure in X).

tB ≤t 0. There exist ˜ such that S(tn )Un → U . a time sequence tn → ∞ and a sequence Un of points of X We write S(tn )Un = S(t)S(tn − t)Un for tn ≥ t; then, there exists a subsequence ˜ i.e., U = S(t)V ; by the of S(tn − t)Un which is convergent to some point V ∈ X, verified fact, V must be in A; therefore, U ∈ S(t)A and hence A ⊂ S(t)A. Thus, we verify that S(t)A = A for every t > 0.

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˜ A) = 0. Indeed, suppose the contrary; Let us verify now that limt→∞ h(S(t)X, then, there would exist some α > 0 such that, for any T > 0, we can find some t ≥ T ˜ A) > α. This then means that there exist a time sequence tn → ∞ such that h(S(t)X, ˜ such that d(S(tn )Un , A) ≥ α. By the compactness of X, ˜ we and a sequence Un ∈ X ˜ can assume that S(tn )Un is convergent to some point V ∈ X. Clearly, d(V , A) ≥ α. Meanwhile, as noticed above, V must be a point of A; hence, d(V , A) = 0, a contradiction. Let us finally show that A is connected under the condition that B is. Suppose the contrary. Then, there would exist two open sets Oi (i = 1, 2) of X such that A ⊂ O1 ∪ O2 , O1 ∩ O2 = ∅, and A ∩ Oi = ∅ for i = 1, 2. Since A is compact, there exists an ε-neighborhood of A such that Wε (A) ⊂ O1 ∪ O2 . Applying (6.9) to B, we have that S(t)B ⊂ Wε (A) ⊂ O1 ∪ O2 for sufficiently large t. Since A ⊂ S(t)B, S(t)B ∩ Oi = ∅ for i = 1, 2. Therefore, for such t, S(t)B is neither connected. But this contradicts the fact that any image of a connected set by a continuous mapping is connected. In the proof of this theorem, we have already obtained the following result. Corollary 6.1 Let (S(t), X, X) possess a compact absorbing set B. Then, there ˜ ⊂ B. The asymptotic behavior of exists a compact, absorbing, and invariant set X the original dynamical system (S(t), X, X) is therefore reduced to that of a smaller ˜ X) having a compact phase space. one (S(t), X, Remark 6.1 In the proof of Theorem 6.2, it is possible to construct the global at˜ In fact, A can be set directly by tractor A directly without using the set X. A = ω(B) =

S(s)B

(closure in X).

0≤t 0, let t = nt ∗ + r with

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integer n ≥ 0 and 0 ≤ r < t ∗ . Then, S(t)U0 − U = S(r)S n U0 − U ≤ sup0≤r≤t ∗ S(r)S n U0 − S(r)U . As t → ∞, n also tends to ∞; therefore, (6.4) yields limt→∞ S(t)U0 − U = 0, which means that U0 ∈ M− (U ). Hence, W− (U ) ⊂ M− (U ). The coincidence of unstable manifolds is also verified in a similar way. By definition, it is obvious that M+ (U ) ⊂ W+ (U ). Let next U0 ∈ W+ (U ). From a sequence {U−n }n=1,2,3,... such that SU−n = U−n+1 , we define the function U : (−∞, 0] → X as U (−t) = S((n + 1)t ∗ − t)U−(n+1) for t ∈ (nt ∗ , (n + 1)t ∗ ], where n = 0, 1, 2, . . . , and as U (0) = U0 . Then, it is seen that the relation S(t)U (−τ ) = U (t − τ ) holds for any 0 ≤ t ≤ τ . Indeed, if τ = 0, then t = 0; so, the relation is clear. Let nt ∗ < τ ≤ (n + 1)t ∗ with some integer n ≥ 0. If τ − t = mt ∗ with some integer m ≥ 0, then S(t)U (−τ ) = S(t)S((n + 1)t ∗ − τ )U−(n+1) = S((n + 1 − m)t ∗ )U−(n+1) = U−m = U (t − τ ). Otherwise, mt ∗ < τ − t < (m + 1)t ∗ with some integer m ≥ 0 (it is clear that n ≥ m); then, S(t)U (−τ ) = S(t + (n + 1)t ∗ − τ )U−(n+1) = S((m + 1)t ∗ + t − τ )S n−m U−(n+1) = S((m + 1)t ∗ + t − τ )U−(m+1) = U (t − τ ). Therefore, in both cases, the function U (−t) satisfies the relation S(t)U (−τ ) = U (t − τ ). We now verify the convergence limt→∞ U (−t) = U . Let nt ∗ < t ≤ (n + 1)t ∗ . Then, U (−t) − U = S((n + 1)t ∗ − t)U−(n+1) − S((n + 1)t ∗ − t)U ≤ sup0≤r≤t ∗ S(r)U−(n+1) − S(r)U . Since n → ∞ as t → ∞, it is observed by (6.4) that limt→∞ U (−t) − U = 0. Therefore, U0 ∈ M+ (U ). Hence, W+ (U ) ⊂ M+ (U ). Let O be any neighborhood of U in X. Localizing the stable and unstable manifolds W∓ (U ) in this neighborhood, we will consider the sets W− (U ; O) = U0 ∈ O; S n U0 ∈ O for n ≥ 1 and lim S n U0 = U (6.13) n→∞

and

W+ (U ; O) = U0 ∈ O; ∃{U−n }n=1,2,3,... ⊂ O, SU−n = U−n+1 for n ≥ 1 and lim U−n = U , n→∞ (6.14) respectively. The set W− (U ; O) (resp. W+ (U ; O)) is called a local stable manifold (resp. a local unstable manifold) of U in a neighborhood O. By definition, it is clear that S(W− (U , O)) ⊂ W− (U , O))

and S(W+ (U ; O)) ⊃ W+ (U , O))

and that W− (U ; O) ⊂ W− (U )

and W+ (U ; O) ⊂ W+ (U ).

Moreover, it is verified without difficulty that W− (U ) =

∞ n=0

S −n (W− (U ; O)) and W+ (U ) =

∞ n=0

S n (W+ (U ; O)). (6.15)

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These facts, together with Theorem 6.3, then yield the following results. Theorem 6.4 We have M− (U ) =

∞

S −n (W− (U ; O)) and M+ (U ) =

n=0

∞

S n (W+ (U ; O)).

n=0

The global attractor, if it exists, always contains the unstable manifold. Theorem 6.5 Assume that (S(t), X, X) has a global attractor A. Then, M+ (U ) ⊂ A. Proof Let U0 ∈ M+ (U ). Then, there exists a negative trajectory U (−t) in X ending at U0 . The set B = {U (−t); 0 ≤ t < ∞} is a bounded set. Since A attracts B, we have limt→∞ h(S(t)B, A) = 0. In view of S(t)U (−t) = U (0) = U0 , we have d(U0 , A) = d(S(t)U (−t), A) ≤ h(S(t)B, A) → 0 as t → ∞. Hence, d(U0 , A) = 0, i.e., U0 ∈ A.

1.4 Exponential Attraction An equilibrium U of (S(t), X, X) is said to be exponentially stable if there exists some neighborhood W of U which U attracts exponentially in the sense that h(S(t)W, U ) ≤ C0 e−kt ,

0 ≤ t < ∞,

(6.16)

with some constant C0 > 0 and some exponent k > 0. In a neighborhood of an exponentially stable equilibrium, one can find an invariant set to define a small dynamical system. Theorem 6.6 Let U be an exponentially stable equilibrium of (S(t), X, X). Then, there exists some neighborhood W of U such that W is an invariant set (therefore, (S(t), W, X) defines a new dynamical system) and W is attracted exponentially by U in the sense (6.16). Proof There is some open ball W = B(U ; r), r > 0, for which (6.16) holds. Then r −kT take a time T > 0 such that C0 e = 2 . Then, for every t ≥ T , S(t)W ⊂ W . So, if we set W = 0≤t 0. (3) There exists some exponent k > 0 such that, for any bounded set B ⊂ X, it holds that h(S(t)B, M) ≤ CB e−kt

for 0 ≤ t < ∞

(6.17)

with a suitable constant CB > 0 depending on B. We must notice that, even if it exists, the exponential attractor is not determined uniquely in general. Usually, they exist as a family. To see this, assume that a dynamical system (S(t), X, X) possesses an exponential attractor M and that, for each t > 0, S(t) satisfies the Lipschitz condition S(t)U − S(t)V ≤ Lt U − V ,

U, V ∈ X.

(6.18)

For 0 < T < ∞, set MT = S(T )M. Then, MT is seen to be an exponential attractor of (S(t), X, X). In fact, since M is a compact set, its image MT by the continuous operator S(T ) is also a compact set of X. Since MT ⊂ M, MT has also a finite fractal dimension with dF (MT ) ≤ dF (M). By the semigroup property, MT is also an invariant set. Let us finally verify (6.17) for MT . For t ≥ T , it follows from (6.18) that d(S(t)U, S(T )V ) = d(S(T )S(t − T )U, S(T )V ) ≤ LT d(S(t − T )U, V ),

U ∈ X, V ∈ M.

Taking the infimum in the right-hand side with respect to V ∈ M, we have d(S(t)U, MT ) = d(S(t)U, S(T )M) ≤ LT d(S(t − T )U, M),

U ∈ X.

Let B be any bounded subset of X. Taking now the supremum in the left-hand side with respect to U ∈ B, we have h(S(t)B, MT ) ≤ LT h(S(t − T )B, M). Therefore, (6.17) implies that h(S(t)B, MT ) ≤ LT CB e−k(t−T ) = CB LT ekT e−kt ,

T ≤ t < ∞.

Meanwhile, since MT ⊂ M, h(S(t)B, MT ) ≤ h(S(t)B, M) + δ(M) ≤ CB + δ(M),

0 ≤ t ≤ T,

where δ(M) is the diameter of M. Therefore, MT also attracts B at an exponential rate. Hence, MT becomes a smaller exponential attractor than M.

2 Representation Theorem of Stable and Unstable Manifolds

263

Theorem 6.7 If (S(t), X, X) possesses an exponential attractor M and if S(t) satisfies (6.18), then in any ε neighborhood Wε (A) of the global attractor A, there exists another exponential attractor Mε , i.e., A ⊂ Mε ⊂ Wε (A). Proof Since M is a bounded set, limT →∞ h(S(T )M, A) = 0. So, for any ε > 0, there is a time T > 0 such that h(S(T )M, A) < ε, namely, MT = S(T )M ⊂ Wε (A). In addition, A = S(T )A ⊂ S(T )M = MT .

2 Representation Theorem of Stable and Unstable Manifolds Throughout this section, we handle a dynamical system whose phase space is a whole space X, namely, a dynamical system of the form (S(t), X, X).

2.1 Discrete Case Let (S n , X, X) be a discrete dynamical system in a Banach space X, where S is a continuous nonlinear operator acting on X. We assume that S has a fixed point U ∈ X, so U is an equilibrium of (S n , X, X). We assume also that S is Fréchet ˜ of U and the derivative satisfies the Hölder differentiable in an open neighborhood O condition S U − S V ≤ D U − V α ,

˜ U, V ∈ O,

(6.19)

with some exponent 0 < α ≤ 1 and constant D > 0. When the spectrum σ (S U ) of S U does not intersect with the unit circle, i.e., σ (S U ) ∩ {λ ∈ C; |λ| = 1} = ∅,

(6.20)

U is called a hyperbolic equilibrium. Let U be hyperbolic. According to Theorem 1.10, the space X and the operator S U are decomposed in a unique way into a direct sum X = Xi ⊕ Xe of two closed subspaces and into a direct sum S U = Si ⊕ Se in such a way that (1) Si and Se are bounded operators of Xi and Xe , respectively. (2) The spectra of Si and Se are given by σ (Si ) = σ (S U ) ∩ {λ ∈ C; |λ| < 1} and σ (Se ) = σ (S U ) ∩ {λ ∈ C; |λ| > 1}, respectively. The spectral radius of Si then satisfies r(Si ) = 1 − 2δi with some small δi > 0, and Se is invertible on Xe with r(Se −1 ) = 1 − 2δe with some small δe > 0. Using these numbers, we introduce the new norm on X by n Ui 0 = sup Si Ui /(1 − δi )n , Ui ∈ Xi , (6.21) n≥0

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6 Dynamical Systems

−n Ue 0 = sup Se Ue /(1 − δe )n ,

Ue ∈ X e ,

(6.22)

n≥0

Ui + Ue 0 = max{ Ui 0 , Ue 0 },

U = Ui + Ue ∈ X.

(6.23)

It is immediate to verify that · 0 defines a norm on X which is equivalent to the original one · . Furthermore, as seen in (1.14), it is true that S Ui ≤ (1 − δi ) Ui 0 , Ui ∈ Xi , i 0 −1 S Ue ≤ (1 − δe ) Ue 0 , Ue ∈ Xe . e 0 Under such a situation, we consider the local stable and unstable manifolds ˜ W∓ (U , O) defined by (6.13) and (6.14), respectively, in a neighborhood of O ⊂ O. When O is sufficiently small, we can establish the following representation theorem. Theorem 6.8 Let U be an equilibrium of (S n , X, X). Let S be Fréchet differentiable ˜ of U with derivative satisfying (6.19). Assume that U is in an open neighborhood O hyperbolic (see (6.20)) and let X be given by the continuous direct sum of two closed subspaces Xi and Xe with the properties described above. Then, for a sufficiently ˜ of U , there exists a one-to-one correspondence small open neighborhood O ⊂ O between the points of W− (U ; O) and the points of Oi ⊂ Xi such that W− (U ; O) U0 = U + Ui + Ge (Ui ), Xi ⊃ Oi Ui → Ge (Ui ) ∈ Xe , Ge ∈ C1+α

Ge (0) = 0, Ge (0) = 0,

when 0 < α < 1 and

Ge ∈ C1,1

when α = 1.

Similarly, there exists a one-to-one correspondence between the points of W+ (U ; O) and the points of Oe ⊂ Xe such that W+ (U ; O) U0 = U + Gi (Ue ) + Ue , Xe ⊃ Oe Ue → Gi (Ue ) ∈ Xi , Gi ∈ C1+α

Gi (0) = 0, Gi (0) = 0,

when 0 < α < 1 and

Gi ∈ C1,1

when α = 1.

Here, Oi and Oe are some suitable open neighborhoods of the origin of Xi and Xe , respectively. Proof Let us first prove the representation result for W+ (U ; O). For simplicity, we assume that U = 0, so that S0 = 0. Pi and Pe denote the projections from X onto Xi and Xe , respectively. By (6.23), we have Pi 0 ≤ 1 and Pe 0 ≤ 1. Let us consider the Banach space X = U = {Un }n=0,1,2,... ⊂ X; lim Un = 0 n→∞

2 Representation Theorem of Stable and Unstable Manifolds

265

with norm U X = sup Un 0 . n

The zero vector of X is denoted by 0. We define the linear operator L : X → X by (LU )0 = Si Pi U1 + 0e , (LU )n = Si Pi Un+1 + Se

−1

Pe Un−1 ,

n ≥ 1.

In addition, the nonlinear operator R : X → X is defined by (RU )0 = Pi (S − S 0)U1 + 0e , (RU )n = Pi (S − S 0)Un+1 − Se

−1

Pe (S − S 0)Un ,

n ≥ 1.

Obviously, R is an operator of class C1+α (resp. C1,1 ) when 0 < α < 1 (resp. α = 1) with R0 = 0. We finally put Φ = L + R. Contractibility of Φ. Let O denote any neighborhood of 0 in X. From O we can always define a neighborhood O of 0 = (0, 0, 0, . . .) in X in such a way that O = {U = {Un }n ∈ X; Un ∈ O}. Then, O is said to be a corresponding neighborhood of O. Let us show that, if O is sufficiently small, then Φ is a contraction mapping from O into X. By (6.21) and (6.22), L satisfies LU X ≤ max{1 − δi , 1 − δe } U X ,

U ∈ X.

In the meantime, R can satisfy RU − RV X ≤ C( U X + V X )α U − V X ,

U , V ∈ O ,

in a suitable neighborhood O , C > 0 being some constant. Indeed, this is readily verified by the following estimate: 1 (S − S 0)U − (S − S 0)V = {S (θ U + (1 − θ )V ) − S 0}(U − V ) dθ 0

≤ C sup θ U + (1 − θ )V α U − V ,

U, V ∈ O ;

0≤θ≤1

here O is a sufficiently small neighborhood of 0 in which any segment connecting ˜ appearing in (6.19). Theretwo points of O is contained in the neighborhood of O fore, we conclude that, if O is sufficiently small, then R satisfies RU − RV X ≤

1 min{δi , δe } U − V X , 2

in the corresponding neighborhood O.

U , V ∈ O,

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6 Dynamical Systems

In this way, Φ is shown to be a contraction on such a neighborhood. One-to-one correspondence. Let O be a sufficiently small open neighborhood of 0, and let O be the corresponding neighborhood on which Φ is a contraction. We can then apply the inverse mapping theorem, Theorem 1.14, to I − Φ, where I is the identity operator of X, to conclude that I − Φ is a homeomorphism from O onto some open neighborhood O of 0, both I − Φ and (I − Φ)−1 being of class C1+α (resp. C1,1 ) when 0 < α < 1 (resp. α = 1). Let U0 ∈ W+ (0; O), and let {Un }n=0,1,2,... be a sequence in O such that SUn = Un−1 for n ≥ 1 and limn→∞ Un = 0. Then, U = {Un }n is an element of O. So, let (I − Φ)U = U = {Un }n . Then, by definition, Pi (U0 − SU1 ) + Pe U0 = U0 Pi (Un − SUn+1 ) − Se

−1

for n = 0,

(6.24)

Pe (Un−1 − SUn ) = Un

for n ≥ 1.

(6.25)

Since SUn = Un−1 , it follows that U0 = Pe U0 ∈ Xe

and Un = 0 for n ≥ 1.

Conversely, let Ue ∈ Xe and U = (Ue , 0, 0, . . .) ∈ O . Then, there exists an element U = {Un }n ∈ O such that (I − Φ)U = U . As relations (6.24) (U0 = Ue ) and (6.25) (Un = 0) are still valid, it follows that Pi (U0 − SU1 ) = 0 and Pe U0 = Ue , Pi (Un − SUn+1 ) = 0

and Pe (Un−1 − SUn ) = 0 for n ≥ 1.

This indeed shows that U0 ∈ W+ (0; O) and Pe U0 = Ue . Set Oe = {Ue ∈ Xe ; (Ue , 0, 0, . . .) ∈ O }. Obviously, Oe is an open neighborhood of the origin 0e of Xe . As shown above, there is one-to-one correspondence between the points in W+ (0; O) and the points in Oe . Mapping Gi . We observe more precisely that the point Ue ∈ Oe corresponds to the point U0 = Ue + Gi (Ue ) ∈ W+ (0; O) in such a way that Gi (Ue ) = Pi Π0 (I − Φ)−1 (Ue , 0, 0, . . .),

Ue ∈ O e ,

where Π0 is the projection from X to X such that Π0 ({Un }n ) = U0 . Clearly, Gi is a C1+α (resp. C1,1 ) mapping when 0 < α < 1 (resp. α = 1) from Oe into Xi with Gi (0) = 0. Since (I − Φ) 0 = I − L, we have {(I − Φ)−1 } 0 = (I − L)−1 . Therefore, (Gi 0)Ue = Pi Π0 (I − L)−1 (Ue , 0, 0, . . .),

Ue ∈ X e .

2 Representation Theorem of Stable and Unstable Manifolds

267

We easily verify that (I − L)−1 (Ue , 0, 0, . . .) = {Un }n implies Pi Un = Si Pi Un+1 for n ≥ 0, and hence Pi Un = 0 for every n. This then shows that Gi 0 = 0. The representation result for W− (0; O) is also deduced in a quite analogous way. We shall use a contraction mapping Ψ instead of Φ which is given by Ψ = M + Q. Here, M : X → X is the linear operator given by (MU )0 = 0i + Se

−1

Pe U1 ,

(MU )n = Si Pi Un−1 + Se

−1

Pe Un+1 ,

n ≥ 1,

and Q : X → X is the nonlinear operator given by (QU )0 = 0i − Se

−1

Pe (S − S 0)U0 ,

(QU )n = Pi (S − S 0)Un−1 − Se

−1

Pe (S − S 0)Un ,

n ≥ 1.

The equivalent condition to (I − Ψ )U = U is written as Pi U0 + Se

−1

Pe (SU0 − U1 ) = U0

−Pi (SUn−1 − Un ) + Se

−1

The rest of proof is left to the reader.

for n = 0,

Pe (SUn − Un+1 ) = Un

for n ≥ 1.

Remark 6.2 Since X is the continuous direct sum of Xi and Xe , we can identify X as the product space Xi × Xe by the natural isomorphism U = Ui + Ue → (Ui , Ue ). If we use this identification, then the points Ui + Ge (Ui ) correspond to (Ui , Ge (Ui )). The theorem obtained above then shows that the local stable manifold W− (U ; O) can be represented as the sum of U and the graph of the mapping Ge : Oi → Xe . Therefore, W− (U ; O) is tangent to U + Xi × {0} at the equilibrium U . It is the same for the local unstable manifold W+ (U ; O). That is, W+ (U ; O) ⊂ Xi × Xe can be represented as the sum of U and the graph of the mapping Gi : Oe → Xi . Therefore, W+ (U ; O) is tangent to U + {0} × Xe at the U . ˜ This theorem shows that, for sufficiently small open neighborhoods O ⊂ O, W− (U ; O) and W+ (U ; O) are C1+α (resp. C1,1 ) manifolds, when 0 < α < 1 (resp. α = 1), with dimensions dim Xi and dim Xe , respectively. We also remark that the whole stable and unstable manifolds W− (U ) and W+ (U ) are given by the images of these localized manifolds by the operators S −n and S n , respectively, due to (6.15).

2.2 Continuous Case We shall now consider a continuous dynamical system (S(t), X, X), where S(t) is a continuous semigroup acting on the whole space X. We assume that, for some

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6 Dynamical Systems

fixed 0 < t ∗ < ∞, S = S(t ∗ ) is Fréchet differentiable in an open neighborhood ˜ of U and its derivative S U satisfies the same condition as in the discrete case O (6.19). In addition, we assume that U is a hyperbolic equilibrium of the discrete dynamical system (S n , X, X) (S = S(t ∗ )) in the sense described above. Let Xi = Xi (U ) and Xe = Xe (U ) be invariant subspaces of S U such that X = Xi + Xe and the parts (S U )|Xi and (S U )|Xe of S U have their spectra in {λ ∈ C; |λ| < 1} and in {λ ∈ C; |λ| > 1}, respectively. The following result is then a direct consequence of Theorem 6.8. Theorem 6.9 Let U be an equilibrium of (S(t), X, X). Let, for some fixed ˜ of U with 0 < t ∗ < ∞, S(t ∗ ) be Fréchet differentiable in an open neighborhood O derivative satisfying (6.19), and let U be hyperbolic with invariants subspaces Xi and Xe of the derivative [S(t ∗ )] U . Then, for sufficiently small open neighborhoods ˜ the local stable and unstable manifolds in O for (S(t ∗ )n , X, X) can be O ⊂ O, represented in such a way that W− (U ; O) U0 = U + Ui + Ge (Ui ), Xi ⊃ Oi Ui → Ge (Ui ) ∈ Xe , Ge ∈ C1+α

Ge (0) = 0, Ge (0) = 0,

when 0 < α < 1 and Ge ∈ C1,1

when α = 1

and W+ (U ; O) U0 = U + Gi (Ue ) + Ue , Xe ⊃ Oe Ue → Gi (Ue ) ∈ Xi , Gi ∈ C1+α

Gi (0) = 0, Gi (0) = 0,

when 0 < α < 1 and Gi ∈ C1,1

when α = 1,

respectively. Here, Oi and Oe are some suitable open neighborhoods of the origin of Xi and Xe , respectively. ˜ Theorem 6.9 shows that, for sufficiently small open neighborhoods O ⊂ O, 1+α 1,1 (resp. C ) manifolds when 0 < α < 1 (resp. W− (U ; O) and W+ (U ; O) are C α = 1), with dimensions dim Xi and dim Xe , respectively. In addition, as shown by Theorem 6.4, the whole stable and unstable manifolds M− (U ) and M+ (U ) are given by the images of these localized manifolds by the operators S(t ∗ )−n and S(t ∗ )n .

3 Exponentially Stable Equilibria Consider a dynamical system (S(t), X, X) in a Banach space X. Let U ∈ X be an equilibrium. We assume that S(t) is a contraction at the equilibrium in the following sense. There exist a time t ∗ > 0 and a constant 0 < δ < 1 such that S(t ∗ )U − U = S(t ∗ )U − S(t ∗ )U ≤ δ U − U , for some R > 0.

U ∈ B(U ; R) ∩ X, (6.26)

4 Exponential Attractors

269

Theorem 6.10 For an equilibrium U , let (6.26) be satisfied with some t ∗ > 0 and 0 < δ < 1. Then, there exists a radius r > 0 such that 1

S(t)U − U ≤ Rδ −1 e−( t ∗ log δ

−1 )t

,

U ∈ B(U ; r) ∩ X.

Proof We first notice that there exists an r > 0 such that {S(τ )U ; 0 ≤ τ ≤ t ∗ , U − U ≤ r} ⊂ B(U ; R). Indeed, suppose the contrary. Then, for any integer n ≥ 1, there exist some τn , 0 ≤ τn ≤ t ∗ , and some Un ∈ X such that Un − U ≤ n1 and S(τn )Un − U > R. We can then assume that τn → τ and Un → U as n → ∞. From (6.1) it follows that S(τ )U − U ≥ R, which clearly contradicts S(τ )U = U . Let next U ∈ B(U ; r) ∩ X. For any t > 0, there exists an integer n ≥ 0 such that t = nt ∗ + τ with some 0 ≤ τ < t ∗ . Then, S(τ )U − U ≤ R. Furthermore, using (6.26) repeatedly, we verify that S(t ∗ )n S(τ )U − U ≤ δ n R. Since S(t)U = S(t ∗ )n S(τ )U , we conclude that 1

S(t)U − U ≤ Rδ − t ∗ δ t ∗ ≤ Rδ −1 e−( t ∗ log δ τ

t

−1 )t

.

4 Exponential Attractors 4.1 Contraction Semigroups Consider a dynamical system (S(t), X, X) in a Banach space X, where the phase X is a closed bounded subset of X. We assume for some t ∗ > 0 that S(t ∗ ) is a contraction operator from X into itself, i.e., S(t ∗ )U − S(t ∗ )V ≤ δ U − V ,

U, V ∈ X,

(6.27)

with some exponent 0 < δ < 1. Let us consider under (6.27) a discrete dynamical system (S n , X, X), where S = S(t ∗ ). By virtue of the fixed point theorem for contraction mappings, Theorem 1.11, there exists a unique equilibrium U of (S n , X, X). Furthermore, for any integer n ≥ 0, S n U − U = S n U − S n U ≤ δ n U − U ,

U ∈ X.

We then verify that U is an equilibrium of (S(t), X, X), too. Indeed, for any t > 0, SS(t)U = S(t ∗ + t)U = S(t)SU = S(t)U ; that is, S(t)U is also a fixed point of S; by the uniqueness of fixed point, S(t)U = U . Furthermore, let t = nt ∗ + τ with integer n ≥ 0 and 0 ≤ τ < t ∗ . Then, S(t)U − U = S n S(τ )U − S n U ≤ Rδ

t−τ t∗

1

= Rδ −1 e−( t ∗ log δ

where R = δ(X) = supU,V ∈X U − V is the diameter of X. We have thus obtained the following result.

−1 )t

,

U ∈ X,

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6 Dynamical Systems

Theorem 6.11 Let X be a bounded closed subset of X, and let (6.27) be satisfied. Then, (S(t), X, X) has a unique equilibrium U which attracts every trajectory at the 1 −1 exponential rate d(S(t)X, U ) ≤ Rδ −1 e−( t ∗ log δ )t , where R is the diameter of X.

4.2 Compact Perturbation of Contraction Operator, I Let (S(t), X, X) be a dynamical system in a Banach space X, where X is a closed bounded subset of X. In the subsequent two subsections, we shall consider compact perturbations of condition (6.27) by two ways. In the first way, the semigroup S(t ∗ ) is divided into two parts in which the first part is a contraction like (6.27), and the second one is a Lipschitz continuous compact operator (see (6.29)). In the second way, we divide the norm S(t ∗ )U − S(t ∗ )V X into two parts in which the first part is again a contraction and the second one is estimated by the norm of a Lipschitz continuous compact operator (see (6.38)). In this subsection, we assume for some fixed t ∗ > 0 that S(t ∗ ) is divided into two parts as follows. There are two operators S0 : X × X → X and K : X × X → Z, where Z is a second Banach space compactly embedded in X and such that for each pair (U, V ) ∈ X2 , it holds that S(t ∗ )U − S(t ∗ )V = S0 (U, V ) + K(U, V ). In addition, S0 and K satisfy S0 (U, V ) X ≤ δ U − V X ,

U, V ∈ X,

(6.28)

K(U, V ) Z ≤ L U − V X ,

U, V ∈ X,

(6.29)

with some constants 0 ≤ δ < 12 and L > 0, respectively. When conditions (6.28)– (6.29) are satisfied, S(t ∗ ) is said to be a compact perturbation of contraction operator of type I. Putting S = S(t ∗ ), let us consider the discrete dynamical system (S n , X, X). We can prove the following theorem. Theorem 6.12 Let (6.28) and (6.29) be satisfied with 0 ≤ δ < 12 . Let θ be any ∗ exponent such that 0 < θ < 1−2δ 2L . Then, there exists an exponential attractor Mθ n for (S , X, X) with the following properties: (1) M∗θ contains a global attractor A∗ of (S n , X, X). (2) M∗θ is a compact set of X with finite fractal dimension dF (M∗θ ) ≤ log Nθ / log

1 . aθ

(6.30)

(3) M∗θ is invariant under S, i.e., SM∗θ ⊂ M∗θ . (4) h(S n X, M∗θ ) ≤ Raθn for all integers n ≥ 0. Here, R = δ(X) is the diameter of X, 0 < aθ < 1 is the exponent given by aθ = 2(δ + θ L), and Nθ is the minimal number of balls of X with radius θ which cover Z the closed unit ball B (0; 1) of Z.

4 Exponential Attractors

271

Proof The proof is divided into three steps. Step 1. For n = 0, 1, 2, . . . , let us construct inductively a finite covering of S n X such that n

SnX ⊂

Nθ

B(Wn,i ; Ra n ),

where a = aθ ,

(6.31)

i=1

with centers Wn,i ∈ S n X, 1 ≤ i ≤ Nθn . For n = 0, it is clear that S 0 X = X ⊂ B(W0 ; R) with an arbitrarily fixed W0 ∈ X. So, (6.31) is trivial. Assume that the covering (6.31) is defined for n − 1. Then, we have Nθn−1

S X = S(S n

n−1

X) ⊂

S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X).

i=1

So, it suffices to cover each set S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) by Nθ closed balls with centers in S n X and radius Ra n . We consider the image K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ). From (6.29) we see that Z

K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ) ⊂ B (0; LRa n−1 ). Then, by the compactness of closed bounded balls of Z in X, the last ball can be covered by Nθ closed balls B(V˜n−1,j ; θ LRa n−1 ), 1 ≤ j ≤ Nθ , of X with centers V˜n−1,j ∈ X and radius θ LRa n−1 . Therefore, we obtain that K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ) ⊂

Nθ

B(V˜n−1,j ; θ LRa n−1 ).

j =1

Furthermore, K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ) + SWn−1,i ⊂

Nθ

B(V˜n−1,j + SWn−1,i ; θ LRa n−1 ).

(6.32)

j =1

Let U ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, and let SU − SWn−1,i = S0 (U, Wn−1,i ) + K(U, Wn−1,i ). By (6.32), there is some j such that K(U, Wn−1,i ) + SWn−1,i ∈ B(V˜n−1,j + SWn−1,i ; θ LRa n−1 ).

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6 Dynamical Systems

Since SU = S0 (U, Wn−1,i )+K(U, Wn−1,i )+SWn−1,i and since S0 (U, Wn−1,i ) X ≤ δ U − Wn−1,i X ≤ δRa n−1 due to (6.28), it follows that SU ∈ B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ). Hence, S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂

Nθ

B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ).

j =1

We are here allowed to assume that S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ∩ B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ) = ∅ for every j , since, if not for some j , we can exclude such balls from the covering. So, we can choose for each j , a point Vn−1,i,j such that Vn−1,i,j ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, SVn−1,i,j ∈ B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ).

(6.33)

Substituting for each j , the central point SVn−1,i for V˜n−1,j +SWn−1,i and doubling the radius, we obtain the covering S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂

Nθ

B(SVn−1,i,j ; aθ Ra n−1 ).

(6.34)

j =1

From (6.33) we notice that SVn−1,i,j ∈ S n X. It is of course possible to construct the same covering as (6.34) for all other balls B(Wn−1,i ; Ra n−1 ), 1 ≤ i ≤ Nθn−1 . The desired covering (6.31) for n is thus obtained just locating central points as Wn,i ; 1 ≤ i ≤ Nθn = SVn−1,i,j ; 1 ≤ i ≤ Nθn−1 , 1 ≤ j ≤ Nθ ⊂ S n X.

Step 2. Let

P = Wn,i ; 0 ≤ n < ∞, 1 ≤ i ≤ Nθn

be a collection of all central points of the covering (6.31) and set M∗θ in such a way that M∗θ =

∞

S n P.

n=0

Then, as will be shown, M∗θ is an exponential attractor for (S n , X, X). Let us verify in this step that Mθ has a finite fractal dimension.

4 Exponential Attractors

273

We will first cover M∗θ by closed balls with radius Ra n . We know by (6.31) that is covered by Nθn balls with radius Ra n . We then decompose M∗θ as

n−1 ∗ p m S Wm,i ; 0 ≤ m ≤ n − 1, 1 ≤ i ≤ Nθ Mθ =

SnX

p=0

n−1

Sp

Wm,i ; n ≤ m < ∞, 1 ≤ i

≤ Nθm

∞

SpP

.

p=n

p=0

Then,

n−1

∞ S p Wm,i ; n ≤ m < ∞, 1 ≤ i ≤ Nθm S p P ⊂ S n X. p=n

p=0

In the meantime, the number of points of Nθm } is calculated by

n−1

p=0 S

p {W

m,i ;

0 ≤ m ≤ n − 1, 1 ≤ i ≤

n(Nθn − 1) nNθn < . n 1 + Nθ + Nθ2 + · · · + Nθn−1 = Nθ − 1 Nθ − 1 N n+1 +(n−1)N n

Therefore, M∗θ is covered at least by θ Nθ −1 θ balls of radius Ra n . Let next 0 < ε < R be any number. By N (ε) we denote the minimal number of balls with radius ε which cover M∗θ . Let Ra n+1 ≤ ε ≤ Ra n . Then, we have N (ε) ≤ N (Ra n+1 ) ≤

Nθn+2 + nNθn+1 . Nθ − 1

(6.35)

In particular, this means that M∗θ is a precompact set of X. As M∗θ is closed, M∗θ is a compact set of X. Furthermore, it follows from (6.35) and the estimate log Ra1 n ≤ log 1ε that log N (ε) log 1ε

≤

log(Nθn+2 + nNθn ) − log(Nθ − 1) n log a1 − log R1

.

Consequently, we obtain that dF (M∗θ ) = lim sup

log N (ε)

ε→0

log 1ε

≤

log Nθ log a1

.

Step 3. Let us verify that M∗θ enjoys the other properties as an exponential attractor. Since S is a continuous mapping, we have S

∞ n=0

SnP ⊂ S

∞ n=0

SnP ⊂

∞ n=0

S n P.

274

6 Dynamical Systems

Hence, M∗θ is an invariant set of S. If U ∈ S n X, we see from (6.31) that d U, M∗θ ≤ d(U, P) ≤ Ra n . Consequently,

h S n X, M∗θ ≤ Ra n .

(6.36)

∗ attracts the phase space X at an exponential rate. This shows that M θ ∗ n X. Since X ⊃ M∗ and since M∗ is an invariant comWe set A = ∞ n=0 S θ ∞θ n ∗ n pact set, it follows that ∞ n=0 S X ⊃ n=0 S Mθ = ∅. On the other hand, let n n+m U U∈ ∞ n+m = n=0 S X. Then, for any integers n, m ≥ 1, we can write U = S ∗ n m m S (S Un+m ) with Un+m ∈ X; since d(S Un+m , Mθ ) → 0 as m → ∞, we can assume that S m Un+m → V ∈ M∗θ as m → used the compactness of M∗θ ); ∞∞ (we ∞ ∗ n n n therefore, U =S V ∈ S Mθ . Hence, n=0 S X ⊂ n=0 S n M∗θ . So that A∗ = ∞ ∞ n n ∗ ∗ n=0 S X = n=0 S Mθ . In particular, by Theorem 6.2, A is a strictly invari-

ant compact set. Let us finally verify that A∗ attracts the phase space X. Let · X ≤ c · Z . Then, (6.28) and (6.29) give SU − SV X ≤ (δ + cL) U − V X , By (1.2),

U, V ∈ X.

(6.37)

h(S m+n X, A∗ ) ≤ h S m+n X, S m M∗θ + h S m M∗θ , A∗ .

Here, (6.36) and (6.37) yield that h S m+n X, S m M∗θ ≤ (δ + cL)m h S n X, M∗θ ≤ (δ + cL)m Ra n . Therefore,

h(S m+n X, A∗ ) ≤ R(δ + cL)m a n + h S m M∗θ , A∗ .

Letting n → ∞, we have lim supn→∞ h(S n X, A∗ ) ≤ h(S m M∗θ , A∗ ). By Theorem 6.2, we already know that limm→∞ h(S m M∗θ , A∗ ) = 0. Hence, it follows that limn→∞ h(S n X, A∗ ) = 0. Thus, we have shown that A∗ is a global attractor of (S n , X, X).

4.3 Compact Perturbation of Contraction Operator, II Let (S(t), X, X) be a dynamical system in a Banach space X, where X is a closed bounded subset of X. In this subsection, we assume for some fixed t ∗ > 0 that S = S(t ∗ ) is a compact perturbation of a contraction operator in the sense that SU − SV X ≤ δ U − V X + KU − KV X ,

U, V ∈ X.

(6.38)

4 Exponential Attractors

275

Here, δ is a constant such that 0 ≤ δ < 12 , and K is an operator from X into a second Banach Z which is compactly embedded in X and satisfies the Lipschitz condition KU − KV Z ≤ L U − V X ,

U, V ∈ X,

(6.39)

with some L > 0. When conditions (6.38)–(6.39) are satisfied, S(t ∗ ) is said to be a compact perturbation of contraction operator of type II. We are then able to prove a similar result as Theorem 6.12 for the discrete dynamical system (S n , X, X). Theorem 6.13 Let (6.38) and (6.39) be satisfied with 0 ≤ δ < 12 . Let θ be any ∗ exponent such that 0 < θ < 1−2δ 2L . Then, there exists an exponential attractor Mθ n for (S , X, X) having the same properties as those announced in Theorem 6.12. Proof We can follow the similar procedure as in the proof of Theorem 6.12. Let us construct under (6.38)–(6.39) the same covering as before. For n = 0, 1, 2, . . . , there is a covering of S n X such that n

Nθ

S X⊂ n

B(Wn,i ; Ra n ),

where a = aθ ,

(6.40)

i=1

with centers Wn,i ∈ S n X, 1 ≤ i ≤ Nθn . Here, as before, R = δ(X) is the diameter of X, 0 < aθ < 1 is the exponent given by aθ = 2(δ + θ L), and Nθ is the minimal Z number of balls of X with radius θ which cover the closed unit ball B (0; 1) of Z. For n = 0, it is clear that S 0 X = X ⊂ B(W0 ; R) with arbitrarily fixed W0 ∈ X. Assume that the covering (6.40) is defined for n − 1. Then, we have Nθn−1

S X = S(S n

n−1

X) ⊂

S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X).

i=1

So, it suffices to cover each set S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) by Nθ closed balls with centers in S n X and the radius Ra n . From (6.39) we see that Z

K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂ B (KWn−1,i ; LRa n−1 ). Then, by the compactness of closed bounded balls of Z in X, the last ball can be covered by Nθ closed balls of X in such a way that Z

B (KWn−1,i ; LRa n−1 ) ⊂

Nθ j =1

B(V˜n−1,i,j ; θ LRa n−1 )

276

6 Dynamical Systems

with centers V˜n−1,i,j ∈ X, 1 ≤ j ≤ Nθ , and radius θ LRa n−1 . Therefore, K(B(Wn−1,i ; Ra

n−1

)∩S

n−1

X) ⊂

Nθ

B(V˜n−1,i,j ; θ LRa n−1 ).

(6.41)

j =1

We are here allowed to assume that K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ∩ B(V˜n−1,i,j ; θ LRa n−1 ) = ∅ for every j , since, if not for some j , we can exclude these balls from the covering. So, we can choose for each j , a point Vn−1,i,j such that Vn−1,i,j ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, KVn−1,i,j ∈ B(V˜n−1,i,j ; θ LRa n−1 ).

(6.42)

Therefore, from (6.41) it is deduced that K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂

Nθ

B(KVn−1,i,j ; 2θ LRa n−1 ).

j =1

Let now U ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X. Then, there is some j such that KU ∈ B(KVn−1,i,j ; 2θ LRa n−1 ). As a consequence, it follows from (6.38) that SU − SVn−1,i,j X ≤ δ U − Vn−1,i,j X + KU − KVn−1,i,j X ≤ δ U − Vn−1,i,j X + 2θ LRa n−1 . In addition, by (6.42), U − Vn−1,i,j X ≤ U − Wn−1,i X + Wn−1,i,j − Vn−1,i,j X ≤ 2Ra n−1 , so that SU − SVn−1,i,j X ≤ 2δRa n−1 + 2Lθ Ra n−1 = Ra n . Hence, it holds that S(B(Wn−1,i ; Ra

n−1

)∩S

n−1

X) ⊂

Nθ

B(SVn−1,i,j ; Ra n ).

(6.43)

j =1

We observe that SVn−1,i,j ∈ S n X due to (6.42). Covering of the form (6.43) can of course be constructed for all other balls B(Wn−1,i ; Ra n−1 ), 1 ≤ i ≤ Nθn−1 . Therefore, the desired covering (6.40) for n is obtained by locating central points as Wn,i ; 1 ≤ i ≤ Nθn = SVn−1,i,j ; 1 ≤ i ≤ Nθn−1 , 1 ≤ j ≤ Nθ ⊂ S n X. We here recall that in Steps 2 and 3, the proof of Theorem 6.12 was carried out only on the basis of the covering (6.31) and conditions (6.28)–(6.29) were never used. As (6.40) has been established, we shall repeat the same arguments as in Steps 2 and 3 of the proof of the previous theorem to complete our proof.

4 Exponential Attractors

277

4.4 Squeezing Property In this subsection, let X be a Hilbert space. We consider a dynamical system (S(t), X, X) in X, where X is a compact subset of X. For some fixed 0 < t ∗ < ∞, assume that S = S(t ∗ ) satisfies the Lipschitz condition SU − SV ≤ L U − V ,

U, V ∈ X,

(6.44)

and the following condition: either SU − SV ≤ δ U − V

(6.45)

or (1 − P )(SU − SV ) ≤ P (SU − SV )

(6.46)

holds for any pair (U, V ) in X × X with some fixed constant 0 ≤ δ < and some fixed orthogonal projection P from X onto a finite-dimensional subspace Z in X, say dim Z = N < ∞. Especially, the second condition that either (6.45) or (6.46) is valid is called a squeezing property for S. Since 1 4

SU − SV 2 = P (SU − SV ) 2 + (1 − P )(SU − SV ) 2 , it is easy to observe from the squeezing property that √ SU − SV ≤ δ U − V + 2 P (SU − SV ) for any pair (U, V ) ∈ X × X. In addition, (6.44) implies that P (SU − SV ) ≤ L U − V ,

U, V ∈ X.

As the finite-dimensional space Z is compactly embedded in X, we notice that the squeezing property (6.45)–(6.46), together with (6.44), √ implies that S fulfills the compact perturbation condition (6.38)–(6.39) with K = 2P S. As a consequence √ , there exists an exponential attractor M∗ for of Theorem 6.13, for 0 < θ < 1−2δ θ 2 2L the discrete dynamical system (S n , X, X). It is, however, known that we can acquire a more precise dimension estimate for M∗θ than (6.30) without using the covering number Nθ . Theorem 6.14 Let X be a Hilbert space, and let X be a compact phase space. Let (6.44) and (6.45)–(6.46) be satisfied with 0 ≤ δ < 14 . Then, for any 0 < θ < 1 − 2δ, there exists an exponential attractor M∗θ for (S n , X, X) with fractal dimension 3L 1 ∗ +1 log , 1 dF (Mθ ) ≤ N max log θ aθ and the same properties as (1), (3), and (4) announced in Theorem 6.12, where aθ = 2δ + θ .

278

6 Dynamical Systems

The proof of this theorem requires us more refined techniques than those for proving Theorem 6.13. We refer the reader to Eden, Foias, Nicolaenko, and Temam [EFNT94, Theorem 2.1]. We here remark only the following fact. Remark 6.3 The squeezing property implies a sharper covering than (6.31) or (6.40). Let W ∈ X and r > 0. We are concerned with the image S(B(W ; r) ∩ X) of a closed ball B(W ; r) by S. So, let U ∈ B(W ; r) ∩ X. If (6.45) takes place for the pair (U, W ), then SU ∈ B(SW ; δr). On the other hand, if (6.46) takes place, then it follows that SU − SW 2 ≤ 2 Re(SU − SW, P (SU − SW )). Therefore, SU − SW P (SU − SW ) ≤ 2 Re(SU − SW, P (SU − SW )).

(6.47)

In view of this, we consider the cone of X defined by C = U˜ ∈ X − {0};

max

V ∈Z, V Z =1

Re

1 (U˜ , V ) ≥ . U˜ V 2

Then, (6.47) shows that SU − SW ∈ C. Hence, (6.46), together with (6.44), implies that SU ∈ (SW + C) ∩ B(SW ; Lr). We thus observed that (6.44) and (6.45)–(6.46) imply that S(B(W ; r) ∩ X) ⊂ B(SW ; δr) ∪ [(SW + C) ∩ B(SW ; Lr)].

4.5 Continuous Dynamical System Let us return to the continuous dynamical system (S(t), X, X) in a Banach space X, where X is a closed bounded subset of X. We assume for some fixed t ∗ > 0 that S(t ∗ ) satisfies the compact perturbation condition either (6.28)–(6.29) or (6.38)– (6.39). In addition, we assume that G(t, U0 ) = S(t)U0 satisfies a Lipschitz condition on [0, t ∗ ] × X, i.e., G(t, U ) − G(s, V ) ≤ L1 (|t − s| + U − V ),

t, s ∈ [0, t ∗ ], U, V ∈ X. (6.48)

Then, on the basis of the exponential attractors M∗θ for (S(t ∗ )n , X, X), we can readily construct exponential attractors for (S(t), X, X). Theorem 6.15 Let (S(t), X, X) be a dynamical system in a Banach space X with closed bounded phase space X. For some fixed t ∗ > 0, let S(t ∗ ) satisfy either (6.28)– (6.29) or (6.38)–(6.39), and let (6.48) be satisfied. Then, for any 0 < θ < 1−2δ 2L , there exists an exponential attractor Mθ for (S(t), X, X) with the following properties: (1) Mθ contains a global attractor A of (S(t), X, X).

4 Exponential Attractors

279

(2) Mθ is a compact set of X with finite fractal dimension dF (Mθ ) ≤ dF M∗θ + 1. (3) Mθ is an invariant set, i.e., S(t)Mθ ⊂ Mθ for every 0 < t < ∞. −1 1 (4) h(S(t)X, Mθ ) ≤ Raθ−1 e−( t ∗ log aθ )t for every 0 < t < ∞. Here, M∗θ is the exponential attractor of (S(t ∗ )n , X, X) constructed in Theorem 6.12 or Theorem 6.13, R = δ(X) is the diameter of X, and aθ = 2(δ + θ L). Proof Let M∗θ be the exponential attractor of (S(t ∗ )n , X, X) constructed by Theorem 6.12 or 6.13. We consider the image G([0, t ∗ ] × M∗θ ) of a set [0, t ∗ ] × M∗θ by G and denote it by Mθ , i.e., S(τ )M∗θ . Mθ = G [0, t ∗ ] × M∗θ = 0≤τ ≤t ∗

Our goal is then to show that Mθ is the desired exponential attractor. Since [0, t ∗ ]×M∗θ is a compact set of C×X and since G is a continuous mapping on [0, t ∗ ] × M∗θ , its image Mθ is also a compact set of X. In addition, since G is Lipschitz continuous, we obtain by Theorem 1.1 that dF (Mθ ) ≤ dF [0, t ∗ ] × M∗θ = dF M∗θ + 1. Let 0 < t < ∞ and write t = nt ∗ + τ with integer n ≥ 0 and 0 ≤ τ < t ∗ . Then, S(t)M∗θ = S(τ )S(nt ∗ )M∗θ = S(τ )S(t ∗ )n M∗θ ⊂ S(τ )M∗θ . ∗ S(τ )M∗θ = Mθ . Conversely, it is clear that Therefore, 0≤t 0 (depending only on U (s; U0 ) Dβ and hence only on p( U0 Dβ )) and a continuous increasing function p1 (·) such that AU (t) X ≤ (t − s)β−1 p1 ( U (s; U0 ) Dβ ),

s < t ≤ s + τ.

First, applying this with s = 0, we have AU (t) X ≤ t β−1 p1 ( U0 Dβ ),

0 < t ≤ τ.

Secondly, taking s = t − τ , we have AU (t) X ≤ τ β−1 p1 ( U (t − τ ) Dβ ) ≤ τ β−1 p1 (p0 ( U0 Dβ )),

τ < t < ∞.

Combining these estimates, we obtain the desired one.

Under (6.55), (6.53) possesses a unique global solution U (t; U0 ) in the function space: U (·; U0 ) ∈ C((0, ∞); D(A)) ∩ C([0, ∞); Dβ ) ∩ C1 ((0, ∞); X),

U0 ∈ D β ,

and (6.56) and (6.57) hold true. We next set S(t)U0 = U (t; U0 ), 0 ≤ t < ∞. Then, the uniqueness of solutions readily implies that S(t) is a nonlinear semigroup acting on Dβ . Let us establish

5 Dynamical Systems for Semilinear Evolution Equations

283

its continuity on Dβ . For any 0 < R < ∞, let KR = {U ∈ Dβ ; U Dβ < R} be an open ball. By Corollary 4.2 (when β > 0) and Theorem 4.5 (when β = 0), we know that there exists tR > 0 such that S(t)U0 − S(t)V0 Dβ ≤ LR U0 − V0 Dβ , 0 ≤ t ≤ tR ; U0 , V0 ∈ KR . In view of 0≤t 0, it is often very important to construct a dynamical system in the universal space X (instead of Dβ ). To this end, we introduce for any 0 < R < ∞, the phase space {S(t)U0 ; 0 ≤ t < ∞} ⊃ KR . XR = U0 ∈KR

Clearly, S(t) maps XR into itself, namely, XR is invariant under S(t). In addition, since XR ⊂ Kp(R) due to (6.56), XR is a bounded set of Dβ . With the aid of Theorem 4.3, we can argue in a similar way as for the proof of (6.58) to deduce the following proposition. Proposition 6.3 For any 0 < R < ∞, it holds that S(t)U0 − S(t)V0 X ≤ Ln+1 p(R) U0 − V0 X , t ∈ [ntp(R) , (n + 1)tp(R) ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ XR ; with some constant Lp(R) > 0. This ultimately implies that G : [0, ∞) × XR → X is continuous, where XR is equipped with the induced distance from X. Thus, problem (6.53) determines a dynamical system (S(t), XR , X) even in the universal space X, too.

284

6 Dynamical Systems

5.2 Absorbing and Invariant Compact Set We shall now make (in addition to the above) the assumptions that X is a reflexive Banach space; D(A) is compactly embedded in X.

(6.59)

As a consequence, we observe the following fact. Proposition 6.4 Let D = D(A) be a Banach space with the graph norm · D = D A · . Then, any bounded closed ball B (0; R), 0 < R < ∞, of D is a compact subset of X. D

Proof Indeed, it suffices to prove that the ball is closed in X. Let Un ∈ B (0; R) and Un → U in X. Since D is reflexive (note that D and X are isomorphic), D B (0; R) is weakly compact; therefore, we can assume that Un → V weakly in D with V D ≤ R. Since D ⊂ X densely and continuously, we have X ⊂ D ; D hence, also Un → V weakly in X. Consequently, U = V ∈ B (0; R). We assume also a dissipative condition, that is, there exists a number C˜ > 0 for which the following assertion is valid. For any bounded subset B of Dβ , there is a time tB such that ˜ sup sup S(t)U0 Dβ ≤ C.

U0 ∈B t≥tB

(6.60)

A stronger dissipative condition is then deduced by the same argument as for (6.57). In fact, there exists a number C˜ 1 > 0 such that, for any bounded subset B of Dβ , it holds that sup sup S(t)U0 D ≤ C˜ 1

U0 ∈B t≥tB

(6.61)

with a suitable time tB depending on B. Using this constant C˜ 1 , we introduce the closed ball D B = B (0; C˜ 1 ).

Estimate (6.61) obviously means that, for any bounded subset B of Dβ , there exists a time tB such that S(t)B ⊂ B for all t ≥ tB , that is, B is an absorbing set. As noticed above, B is a compact set of X. We furthermore consider the set X=

S(t)B ⊂ B

(closure in X).

tB ≤t 0 such that the images S(t)B are also absorbed by itself for all t ≥ tB . As B is an absorbing set, it is the

5 Dynamical Systems for Semilinear Evolution Equations

285

same for X. In addition, since S(t) is continuous with respect to the X norm (recall Proposition 6.3), it holds that S(t) S(τ )B ⊂ S(t)S(τ )B ⊂ S(τ )B. tB ≤τ 0. As X is a bounded set of D, t S(t)V0 − S(s)V0 = [F (S(τ )V0 ) − AS(τ )V0 ] dτ s

≤ CX (t − s),

0 ≤ s < t ≤ t ∗ , V0 ∈ X.

These two estimates obviously yield (6.48). We have thus deduced, under (2.92), (2.93), either (4.2) or (4.21), (6.55), (6.59), and (6.60), that the dynamical system (S(t), X, X) possesses a family of exponential attractors M. Let us finally verify that each M is an exponential attractor of the dynamical system (S(t), X, Dβ ), too, where X is equipped with the induced distance from Dβ . By the moment inequality (2.117), it is readily seen that hDβ (S(t)X, M) ≤ CX h(S(t)X, M)1−β ≤ CX e−(1−β)kt ,

0 ≤ t < ∞,

where hDβ (·, ·) denotes the Hausdorff pseudo-distance in Dβ . This shows that M attracts S(t)X at an exponential rate even in Dβ . Meanwhile, by Theorem 1.3, the fractal dimension dDβ (M) of M in Dβ is estimated by (1 − β)−1 dX (M). Hence, M keeps its properties as an exponential attractor in (S(t), X, Dβ ).

286

6 Dynamical Systems

5.4 Squeezing Property of S(t) Let X be a Hilbert space. For the operator A in (6.53), we now assume that A is a positive definite self-adjoint operator of X; (6.62) D(A) is compactly embedded in X. For the nonlinear operator F , we assume (4.2) (when β > 0) or (4.21) (when β = 0). We assume also that problem (6.53) determines a dynamical system (S(t), X, X) in which X is a compact set of X and is a bounded set of D(A). Consequently, F satisfies the Lipschitz condition F (U ) − F (V ) ≤ D Aη (U − V ) ,

U, V ∈ X, 0 ≤ η < 1.

(6.63)

On account of (6.62), there exist positive eigenvalues 0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · → ∞ of A, and the corresponding eigenvectors U1 , U2 , U3 , . . . compose an orthonormal basis of X. We consider a finite-dimensional subspace Z = Span{U1 , U2 , U3 , . . . , UN }. Let P : X → Z be the orthogonal projection. We can show that the squeezing property (6.45)–(6.46) is fulfilled on X if N is sufficiently large. Let t ∗ > 0 be a time such that the estimate t η Aη [S(t)U − S(t)V ] + S(t)U − S(t)V ≤ C U − V ,

U, V ∈ X,

holds for any 0 < t ≤ t ∗ (see Theorem 4.3 or 4.5) and set S = S(t ∗ ). In view of (6.45) and (6.46), it suffices to prove that, if (1 − P )[SU − SV ] > P [SU − SV ] for U, V ∈ X, then (6.45) is valid with a suitable constant 0 ≤ δ < 14 . So, let (1 − P )[SU − SV ] > SU − SV be satisfied. Then, SU − SV 2 = P [SU − SV ] 2 + (1 − P )[SU − SV ] 2 < 2 (1 − P )[SU − SV ] 2 . In the meantime, we have (1 − P )SU = e

−t ∗ A

(1 − P )U +

t∗

e−(t

∗ −s)A

(1 − P )F (S(s)U ) ds.

0

It is the same for (1 − P )SV . We then observe that e−t

∗A

(1 − P )(U − V ) ≤ e−t

∗λ N+1

1 U − V ≤ √ U − V 2 34

if N is sufficiently large. Similarly, due to (6.63),

6 Stationary Solutions to Semilinear Equations

t∗

e−(t

0

∗ −s)A

≤D

t∗

287

(1 − P )[F (S(s)U ) − F (S(s)V )] ds

e−(t

∗ −s)λ N+1

F (S(s)U ) − F (S(s)V ) ds

0

t∗

1 s −η ds U − V ≤ √ U − V 2 34 0 t∗ ∗ if N is sufficiently large. We here used the fact that limλ→∞ 0 e−(t −s)λ s −η ds = 0. 1 Hence, it follows that SU − SV 2 ≤ 17 U − V 2 , namely, (6.45) is fulfilled. We have thus shown under (6.62) that S(t) enjoys the squeezing property. ≤D

e−(t

∗ −s)λ N+1

6 Stationary Solutions to Semilinear Equations 6.1 Equilibria of Dynamical System Consider problem (6.53) in a Banach space X. We assume that the linear operator A and the nonlinear operators F satisfy (2.92), (2.93), and either (4.2) (when β > 0) or (4.21) (when β = 0). In addition, we assume that the a priori estimate (6.55) holds for any U0 ∈ Dβ . As shown, a dynamical system (S(t), Dβ , Dβ ) is constructed, where Dβ = D(Aβ ) (and D0 = X when β = 0). Let (6.53) have a stationary solution U (t) ≡ U , i.e., AU = F (U ). Clearly, U is an equilibrium of (S(t), Dβ , Dβ ). Let R > 0 be such that U Dβ < R. In this section, we shall investigate the stability and instability of U by applying Theorems 6.9 and 6.10, respectively, in which the Fréchet differential of S(t) at U played an essential role.

6.2 Exponential Stability of U Let us assume that F : D(Aη ) → X is Fréchet differentiable at U , i.e., ΔF (h) = F (U + h) − F (U ) − F (U )h, F (U ) ∈ L(D(Aη ), X), limh→0 oF (h)/ Aη h = 0. oF (h) = ΔF (h) ,

(6.64)

Here, oF (h) is defined for all h ∈ D(Aη ). In view of (4.2) or (4.21), we observe that oF (h) ≤ C Aη h for all h such that Aη h ≤ 1. Let now h ∈ Dβ denote a small variable satisfying h Dβ < R − U Dβ . For such an h, let Uh (t) = S(t)(U + h). By Theorem 4.3 or 4.5, there exists T > 0 such that t η Aη [Uh (t) − U ] + t β Aβ [Uh (t) − U ] ≤ C h ,

0 < t ≤ T.

(6.65)

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6 Dynamical Systems

We here introduce the linear problem dV h dt + AVh = F (U )Vh ,

0 < t ≤ T,

Vh (0) = h, in X. Since F (U ) ∈ L(D(Aη ), X), we know by Theorem 3.3 that A − F (U ) generates an analytic semigroup on X (cf. Remark 3.7). By Theorem 3.14 the solution of the problem is given by Vh (t) = e−tA h, where A = A − F (U ). Put ΔU (t) = Uh (t) − U − Vh (t). Then, since F (Uh (t)) − F (U ) − F (U )Vh (t) = F (U )ΔU (t) + ΔF (Uh (t) − U ), it is observed that dΔ U dt + AΔU = F (U )ΔU + ΔF (Uh (t) − U ),

0 < t ≤ T,

ΔU (0) = 0. Using the semigroup e−tA , ΔU (t) is written by

t e−(t−s)A ΔF (Uh (s) − U ) ds, ΔU (t) =

0 < t ≤ T.

0

Note that ΔF (Uh − U ) ∈ Fβ,σ ((0, T ]; X) due to (4.15) (when β > 0) and ΔF (Uh − U ) ∈ F1−η,σ ((0, T ]; X) due to (4.27) (when β = 0). Therefore, for any 0 ≤ θ < 1,

t (t − s)−θ oF (Uh (s) − U ) ds. Aθ ΔU (t) ≤ C 0

Furthermore, by (6.65),

t θ A ΔU (t) ≤ C (t − s)−θ s −η oF (Uh (s) − U )/ Aη (Uh (s) − U ) ds h . 0

Since Aη (Uh (t) − U ) → 0 as h → 0, we conclude by the dominate convergence theorem that, as h → 0 (a fortiori as h Dβ → 0), 0 ≤ Aθ ΔU (t) / h Dβ ≤ Aθ ΔU (t) / h → 0.

(6.66)

Thus, under (6.64), the operator S(t) : Dβ → D(Aθ ), where 0 < t ≤ T and 0 ≤ θ < 1, is Fréchet differentiable at U with differential S(t) U = e−tA ,

where A = A − F (U ).

Moreover, since S(2t)U = S(t)S(t)U and S(t)U = U , we verify that S(2t) is also Fréchet differentiable at U . Repeating this argument, S(t) : Dβ → D(Aθ ) is Fréchet differentiable at U for every time 0 < t < ∞.

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289

We can now give a sufficient condition for exponential stability of U . Let us assume that the spectrum of A is contained in the right half-plane, i.e., σ (A) = σ (A − F (U )) ⊂ {λ ∈ C; Re λ > 0}.

(6.67)

This condition, together with Theorem 3.3, yields that A is a sectorial operator, namely, A satisfies (2.92) and (2.93). Proposition 2.5 then provides that e−tA ≤ Ce−dt , 0 ≤ t < ∞, with some positive exponent d > 0. As a consequence, e−tA U Dβ ≤ Aβ e−A e−(t−1)A U ≤ Ce−d(t−1) U Dβ . ∗

This implies that e−t A L(Dβ ) ≤ δ < 1 for any sufficiently large t ∗ > 0. Let K˜ r = B Dβ (0; r) be an open ball of Dβ . Then, (6.66) (θ = β) implies that S(t ∗ )(U + h) − S(t ∗ )U − S(t ∗ ) h Dβ = ΔU (t ∗ ) Dβ ≤

1−δ h Dβ , 2

if r > 0 is sufficiently small. Hence, 1−δ δ+1 S(t ∗ )(U + h) − U Dβ ≤ δ + h Dβ = h Dβ , 2 2

h ∈ K˜ r ,

h ∈ K˜ r .

This shows that (6.26) holds true, that is, S(t ∗ ) is a contraction at U . We have thus proved by Theorem 6.10 that, under (6.64) and (6.67), U is an exponentially stable equilibrium of (S(t), Dβ , Dβ ).

6.3 Unstable Manifold of U Let U ∈ D be again a stationary solution of (6.53) and be an equilibrium of (S(t), Dβ , Dβ ). In this subsection, we assume that F : D(Aη ) → X is of class C1,1 in a neighborhood of U . More precisely, there is a number r > 0 such that, for any U ∈ D(Aη ) ∩ B Dβ (U ; r), F is Fréchet differentiable and the derivative satisfies F (U )h ≤ L1 Aη h ,

U ∈ D(Aη ) ∩ B Dβ (U ; r), h ∈ D(Aη ),

(6.68)

with some constant L1 > 0. In addition, [F (U1 ) − F (U2 )]h ≤ L2 Aβ (U1 − U2 ) Aη h , U1 , U2 ∈ D(Aη ) ∩ B Dβ (U ; r), h ∈ D(Aη ),

(6.69)

with some constant L2 > 0. Put ΔF (U ; h) = F (U + h) − F (U ) − F (U )h. Then, (6.69) implies that ΔF (U ; h) ≤ L2 Aβ h Aη h ,

U + h, U ∈ D(Aη ) ∩ B Dβ (U ; r).

(6.70)

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Consider an open neighborhood B Dβ (U ; r1 ) of U . Let U0 ∈ B Dβ (U ; r1 ) be an initial value, and let U (t) = S(t)U0 . By Corollary 4.2 (when β > 0) or Theorem 4.5 (when β = 0), we have t η−β Aη (U (t) − U ) + Aβ (U (t) − U ) ≤ C Aβ (U0 − U ) , 0 ≤ t ≤ T1 , U0 ∈ B Dβ (U ; r1 ), for sufficiently small time T1 > 0. So, if r1 > 0 and T1 > 0 are sufficiently small, then U (t) ∈ B Dβ (U ; r) for 0 ≤ t ≤ T1 . Consequently, F (U (t)) is defined. Consider also a small vector variable h ∈ B Dβ (0; r2 ) and let Uh (t) = S(t)(U0 + h). By Theorem 4.3 or 4.5, we have t η Aη [Uh (t) − U (t)] + t β Aβ [Uh (t) − U (t)] ≤ C h , 0 < t ≤ T2 , U0 ∈ B Dβ (U ; r1 ), h ∈ B Dβ (0; r2 ),

(6.71)

for sufficiently small time T2 > 0. We here introduce the linear problem dV

h

dt

+ AVh = F (U (t))Vh ,

0 < t ≤ T,

Vh (0) = h,

(6.72)

in X, where T = min{T1 , T2 }. To solve this problem, we will employ Theorem 3.14. Let B(t) = −F (U (t)), 0 < t ≤ T . Then, it is obvious from (6.68) that B(t)A−η ≤ L1 ,

0 < t ≤ T.

(6.73)

Furthermore, noticing (4.13), we have [B(t) − B(s)]A−η ≤ L2 Aβ [U (t) − U (s)] ≤ Cs −σ˜ (t − s)σ˜ ,

0 < s ≤ t ≤ T,

(6.74) (6.75)

with any exponent σ˜ such that 0 < σ˜ < 1 − η. These estimates (6.73) and (6.74) show that (3.117), (3.118), and (3.119) are satisfied with μ˜ = 1 and ν˜ = 1 − η. Therefore, Theorems 3.13 and 3.14 provide the existence of the evolution operator U˜ (t, s) for the family of operators A − F (U (t)) which describes the solution of (6.72) in the form Vh (t) = U˜ (t, 0)h. Let ΔU (t) = Uh (t) − U (t) − Vh (t). Since F (Uh (t))−F (U (t))−F (U (t))Vh (t) = F (U (t))ΔU (t)+ΔF (U (t); Uh (t)−U (t)), we have dΔ

U

dt

+ AΔU = F (U (t))ΔU + ΔF (U (t); Uh (t) − U (t)),

ΔU (0) = 0.

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291

Then, ΔU is also described by U˜ (t, s) in the form

t ΔU (t) = U˜ (t, s)ΔF (U (s); Uh (s) − U (s)) ds. 0

Note that ΔF (U ; Uh − U ) ∈ Fβ,σ ((0, T ]; X) (when β > 0) due to (4.15) and ΔF (U ; Uh − U ) ∈ F1−η,σ ((0, T ]; X) (when β = 0) due to (4.27). Therefore, on account of (3.124) and (6.70), for any 0 ≤ θ < 1,

t θ A ΔU (t) ≤ C (t − s)−θ Aβ [Uh (s) − U (s)] Aη [Uh (s) − U (s)] ds. 0

Hence, by (6.71),

t

Aθ ΔU (t) ≤ C

(t − s)−θ s −η Aβ [Uh (s) − U (s)] ds h .

0

Aβ [U

Since h (s) − U (s)] → 0 as h → 0, it follows that, as h → 0 (a fortiori as h Dβ → 0), 0 ≤ Aθ ΔU (t) / h Dβ ≤ Aθ ΔU (t) / h → 0. Thus, under (6.68) and (6.69), S(t) : Dβ → D(Aθ ) is Fréchet differentiable in the neighborhood B Dβ (U ; r1 ) for 0 ≤ t ≤ T , where r1 > 0 and T > 0 are sufficiently small. Furthermore, its derivative is given by S(t) U0 = U˜ (t, 0),

0 < t ≤ T , U0 ∈ B Dβ (U ; r1 ),

where U˜ (t, s) is the evolution operator for the family of operators A − F (S(t)U0 ). Let U1 (t, 0) = S(t) U1 and U2 (t, 0) = S(t) U2 . Then,

t Aθ U2 (t, s)[F (S(s)U1 ) − F (S(s)U2 )]A−η Aθ [U1 (t, 0) − U2 (t, 0)]h = 0

× Aη U1 (s, 0)A−β Aβ h ds. By (3.124), (3.125), and (6.69), it follows that

t θ (t − s)−θ s β−η ds h Dβ A [U1 (t, 0) − U2 (t, 0)]h ≤ C 0

× sup Aβ [S(s)U1 − S(s)U2 ] . 0≤s≤T

Therefore, on account of (6.58), S(t) U1 − S(t) U2 L(Dβ ,D(Aθ )) = U1 (t, 0) − U2 (t, 0) L(Dβ ,D(Aθ )) ≤ Ct 1+β−θ−η U1 − U2 Dβ ,

0 ≤ t ≤ T ; U1 , U2 ∈ B Dβ (U ; r1 ).

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6 Dynamical Systems

Let us assume now that the spectrum of A = A − F (U ) is separated by the imaginary axis, i.e., σ (A) ∩ {λ ∈ C; Re λ = 0} = ∅.

(6.76)

Denote σ− (A) = σ (A) ∩ {λ ∈ C; Re λ < 0} and σ+ (A) = σ (A) ∩ {λ ∈ C; Re λ > 0}. Since F (U ) satisfies (6.68), A satisfies (3.2). This necessarily implies that σ− (A) is a bounded closed set. We can then decompose the space X into X = X− ⊕ X+ and the operator A into A = A− ⊕ A+ by applying Theorem 1.10. Here, X∓ are closed subspaces of X with continuous projection, and A∓ are the parts of A in X∓ , respectively. The operator A− is a bounded operator of X− with σ (A− ) = σ− (A); in particular, X− ⊂ D(A). In the meantime, A+ is a densely defined, closed linear operator of X+ with D(A+ ) = D(A) ∩ X+ and σ (A+ ) = σ+ (A). For λ ∈ ρ(A), the resolvent (λ − A)−1 is also decomposed into (λ − A)−1 = (λ − A− )−1 ⊕ (λ − A+ )−1 ; in particular, we verify that A+ is a sectorial operator of X+ . As a consequence, we obtain a decomposition of semigroup e−tA of the form e−tA = e−tA− ⊕ e−tA+ . Since A− is a bounded operator, we can use the spectral mapping theorem, Theorem 1.9, to conclude that σ (e−tA− ) = e−tσ− (A) ⊂ {λ ∈ C; |λ| > 1} for any 0 < t < ∞. In the meantime, on account of Proposition 2.5, we have e−tA+ ≤ Ce−δt with some δ > 0. Therefore, 1 the spectral radius of e−tA+ is estimated by limn→∞ e−ntA+ n ≤ e−δt ; hence, σ (e−tA+ ) ⊂ {λ ∈ C; |λ| < 1}. In this way, we have verified from (6.76) that σ (e−tA ) ∩ {λ ∈ C; |λ| = 1} = ∅,

0 < t < ∞,

and σ (e−tA ) ∩ {λ ∈ C; |λ| > 1} = e−tσ− (A) ,

0 < t < ∞.

Moreover, the subspace Xe of X corresponding to the exterior spectrum σ (e−tA ) ∩ {λ ∈ C; |λ| > 1} is the space X− . When β > 0, let us consider e−tA as a bounded operator from Dβ into itself. We observe the following fact. Lemma 6.1 Let t > 0. For λ = 0, λ ∈ σ (e−tA ) if and only if λ ∈ σ Dβ (e−tA ), where σ Dβ (·) denotes the spectrum of an operator in the space Dβ . Proof Let λ ∈ ρ(e−tA ). Then, (λ − e−tA )−1 = λ−1 e−tA (λ − e−tA )−1 + 1 . From this it is deduced that (λ − e−tA )−1 is a bounded operator from Dβ into itself. Therefore, (λ − e−tA )−1 is a bounded inverse of λ − e−tA in the space Dβ , too, i.e., λ ∈ ρ Dβ (e−tA ). Conversely, let λ ∈ ρ Dβ (e−tA ). Taking a λ0 ∈ ρ(A), we have (λ − e−tA )−1 = λ−1 A−β Aβ (λ − e−tA )−1 A−β Aβ (λ0 − A)−1 (λ0 − A)e−tA + 1 .

6 Stationary Solutions to Semilinear Equations

293

Since Aβ (λ − e−tA )−1 A−β ∈ L(X), this shows that (λ − e−tA )−1 is continuous in the norm of X, too. Since Dβ is dense in X, it follows that (λ − e−tA )−1 is extended over X as a bounded operator of X, and hence λ ∈ ρ(e−tA ). Thus, the assertion of lemma is verified. In view of this lemma, we conclude that (6.76) equally implies σ Dβ (e−tA ) ∩ {λ ∈ C; |λ| = 1} = ∅,

0 < t < ∞,

and σ Dβ (e−tA ) ∩ {λ ∈ C; |λ| > 1} = e−tσ− (A) ,

0 < t < ∞.

Moreover, the subspace Dβ,e of Dβ which corresponds to the exterior spectrum σ Dβ (e−tA ) ∩ {λ ∈ C; |λ| > 1} is nothing more than X− . Indeed, the last assertion is verified as follows. Let X = Xe ⊕ Xi and Dβ = Dβ,e ⊕ Dβ,i be the decompositions of X and Dβ , respectively, corresponding to the spectrum separation of the semigroup e−tA by the unit circle. Let P and Pβ be the projections from X to Xi and from Dβ to Dβ,i . By (1.15), if U ∈ Dβ , then obviously Pβ U = P U ; consequently, (1 − Pβ )U = (1 − P )U for U ∈ Dβ ; in particular, Dβ,e ⊂ Xe . While, as shown above, Xe = X− ⊂ D(A) ⊂ Dβ ; hence, Xe ⊂ Dβ,e , i.e., Xe = Dβ,e . In this way, under (6.68), (6.69), and (6.76), we have shown that S(t) U satisfies (6.20), that is, U is a hyperbolic equilibrium of (S(t), Dβ , Dβ ). The exterior spec trum of S(t) U is given by e−tσ− (A−F (U )) . By virtue of Theorem 6.9, there exists a smooth local unstable manifold W+ (U ; O) of U which has a dimension dim X− and is tangent to the subspace U + X− of Dβ at U .

6.4 Reaction–Diffusion Systems Let us apply our general results obtained above to the initial-boundary-value problem ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu + f (u, v) ⎪ ⎪ ⎪ ⎨ ∂v = bΔv + g(u, v) in Ω × (0, ∞), ∂t (6.77) ∂u ∂v ⎪ = ∂n = 0 on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x) in Ω, 0

0

in a C2 or convex, bounded domain Ω ⊂ Rd (d = 1, 2, 3). Here, a > 0 and b > 0 are diffusion constants. The kinetic functions f (u, v) and g(u, v) are real smooth functions defined for (u, v) ∈ R2 . Let (u, v) ≡ (u, v) be a stationary solution to (6.77), that is, (u, v) ∈ R2 satisfies f (u, v) = g(u, v) = 0.

(6.78)

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6 Dynamical Systems

We assume that f (u, v) and g(u, v) can be extended over a complex neighborhood D ⊂ C2 of (u, v) and the extended functions, still denoted by f (u, v) and g(u, v), are both differentiable as complex variable functions from D into C: f (u, v) and g(u, v) are complex differentiable functions in D.

(6.79)

We set the underlying space by f X= ; f ∈ L2 (Ω) and g ∈ L2 (Ω) , g and formulate problem (6.77) as the Cauchy problem for an abstract equation of the form (6.53). The linear operator A of X is defined by u A1 0 2 2 , D(A) = ; u ∈ HN (Ω) and v ∈ HN (Ω) , A= v 0 A2 where A1 and A2 are realizations of the differential operators −aΔ+1 and −bΔ+ 1 in L2 (Ω), respectively, under the homogeneous Neumann boundary conditions on ∂Ω. Then, A is a positive definite self-adjoint operator of X. By virtue of Theorems 16.7 and 16.9, the domains of its fractional powers are given by 3 u θ 2θ 2θ ; u ∈ H (Ω) and v ∈ H (Ω) for 0 ≤ θ < D(A ) = v 4 and D(Aθ ) =

u ; u ∈ HN2θ (Ω) and v ∈ HN2θ (Ω) v

for

3 < θ ≤ 1. 4

The nonlinear operator F from D(Aη ) into X is defined by u + f (u, v) u F (U ) = , U= ∈ D(Aη ), v + g(u, v) v where the exponent η is taken in such a way that d4 < η < 1. Since H s (Ω) ⊂ C(Ω) for s > d2 (due to (1.76)), we observe that D(Aη ) ⊂ C(Ω). We take another exponent β so that d4 < β < η < 1. Then, D(Aη ) ⊂ D(Aβ ) ⊂ C(Ω). By direct calculations, it is easily verified from (6.79) that F is Fréchet differentiable in a neighborhood of U = t (u, v) with derivative 1 + fu (u, v) fv (u, v) h, F (U )h = 1 + gv (u, v) gu (u, v) u U= ∈ B Dβ (U ; r), h ∈ D(Aη ), v where Dβ = D(Aβ ) and r > 0. Furthermore, the derivative fulfills conditions (6.68) and (6.69). Let us then investigate how the spectral condition (6.67) or the separation condition (6.76) is fulfilled by A = A − F (U ).

6 Stationary Solutions to Semilinear Equations

295

We here consider the self-adjoint operator Λ of L2 (Ω) which is a realization of −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. As is well known, Λ possesses denumerable nonnegative eigenvalues, and the corresponding real eigenfunctions can constitute an orthonormal basis of L2 (Ω), see Corollary 2.1. So, let 0 = μ 0 < μ1 ≤ μ 2 ≤ · · · → ∞ 1

be eigenvalues of Λ, and let φ0 = |Ω|− 2 , φ1 , φ2 , . . . , be the corresponding real eigenfunctions which constitute an orthonormal basis. We notice that X is decomposed into an infinite orthogonal direct sum of subspaces Xk , where each Xk is a two-dimensional subspace given by φk 0 + ηk ; ξk , ηk ∈ C , k = 0, 1, 2, . . . , Xk = U = ξ k 0 φk 0 φk X= ∞ k=0 Xk . Obviously, Φk = 0 and Ψk = φk are orthonormal bases of Xk with components of real functions. Since aΛ + 1 0 1+fu fv A= and F (U ) = , 0 bΛ + 1 gu 1 + gv where f u = fu (u, v), f v = fv (u, v), g u = gu (u, v), and g v = gv (u, v), we notice each Xk is invariant also that A maps the subspace Xk into itself. In other words, under A. In this way, A is also decomposed into the sum A = ∞ k=0 Ak , where Ak is the part of A in Xk . Let us consider the part Ak in Xk . Obviously, Ak Φk = (aμk − f u )Φk − g u Ψk ,

Ak Ψk = −f v Φk + (bμk − g v )Ψk .

The transformation matrix M k of Ak with respect to the bases Φk , Ψk is given by aμk − f u −g u Mk = . −f v bμk − g v So, the characteristic equation of M k is given by |λ − M k | ≡ λ2 − [(a + b)μk − (f u + g v )]λ + abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0. If the discriminant Δk is nonnegative, i.e., Δk = (a − b)2 μ2k + 2(a − b)(g v − f u )μk + (f u − g v )2 + 4f v g u ≥ 0, then the characteristic equation has two real solutions λk ≤ λk satisfying 2[abμ2k − (ag v + bf u )μk + (f u g v − f v g u )] ≤ λk ≤ λk . √ (a + b)μk − (f u + g v ) + Δk

(6.80)

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6 Dynamical Systems

On the other hand, if Δk < 0, then the solutions λk and λk are complex and satisfy Re λk = Re λk = [(a + b)μk − (f u + g v )]/2, √ |Im λk | = |Im λk | = |a − b|μk + O( μk ). Both cases, as k → ∞, λk and λk lie in Σω + 1 with some fixed angle ω such π that arctan |a−b| a+b < ω < 2 ; in addition, |λk | = O(μk ) and |λk | = O(μk ). Let ψk : Xk → C2 be a natural isomorphism such that ψk (ξk Φk + ηk Ψk ) = t (ξk , ηk ) ( ψk = ψk−1 = 1); then, Ak and its transformation matrix M k satisfy the relation Ak = ψk−1 M k ψk . Therefore, (λ − Ak )−1 = ψk−1 (λ − M k )−1 ψk

for λ = λk , λk .

(6.81)

Hence, σ (Ak ) = {λk , λk } for each k = 0, 1, 2, . . . . Let us now consider (λ − A)−1 . Take an angle ω such that ω < ω < π2 , and let λ∈ / Σω + 1. For all sufficiently large k such that λk , λk ∈ Σω + 1, we have 1 λ − bμk + g v −g u (λ − M k )−1 = . −f v λ − aμk + f u (λ − λk )(λ − λk ) In addition, |λ − λk |2 ≥ |λ|2 + |λk |2 − 2|λ||λk | cos(ω − ω), |λ − λk |2 ≥ |λ|2 + |λk |2 − 2|λ||λk | cos(ω − ω). Therefore, in view of (6.81), we obtain that (λ − Ak )−1 L(Xk ) ≤ Cλ for all such k, where Cλ > 0 are independent of k. This then means that λ ∈ ρ(A) if and only if λ ∈ ρ(Ak ) for each k = 0, 1, 2, . . . . By these considerations, we can now state the following. The operator A fulfills (6.67) or (6.76) if and only if the part Ak fulfills the same for each k = 0, 1, 2, . . . . Condition (6.67). A necessary and sufficient condition that A0 (μ0 = 0) satisfies (6.67) is given by f u + g v < 0 and f u g v > f v g u . Therefore, a necessary and sufficient condition that every Ak satisfies (6.67) is that f u + g v < 0, f u g v > f v g u , and that (ag v + bf u )μk < abμ2k + (f u g v − f v g u ) for every μk . As (ag v + bf u )μ < abμ2 + (f u g v − f v g u ) for all μ ≥ 0, provided that ag v + bf u < 2 ab(f u g v − f v g u ), we find that the triplet of inequalities f u + g v < 0, f u g v > f v g u , and ag v + bf u < 2 ab(f u g v − f v g u ) (6.82) is a sufficient condition for (6.67).

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297

Condition (6.76). A necessary and sufficient condition that equation (6.80) has a purely imaginary solution is given by abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0, or by (a + b)μk = f u + g v and abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) > 0. So, it is equivalent to the condition that abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0, or μk = (f u + g v )/(a + b) and −(a + b)2 f u g v > (ag v − bf u )2 . Therefore, a necessary and sufficient condition in order that (6.76) is fulfilled (in other words, (6.80) has no purely imaginary solution) is that abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0 for all μk , and μk = (f u + g v )/(a + b) for all μk ’s or − (a + b)2 f v g u ≤ (ag v − bf u )2 . (6.83)

Condition σ (A) ∩ {λ ∈ C; Re λ < 0} = ∅. Assume that, for some μk , abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) < 0.

(6.84)

For such a μk , the characteristic equation (6.80) has solutions λk < 0 < λk . There ξ fore, λk ∈ σ (A) ∩ {λ ∈ C; Re λ < 0}. Let ηk be an eigenvector of M k correspondk

ing to the negative eigenvalue λk . Since M k is a real matrix, ξk and ηk are real num ξ bers. It then follows that ψk−1 ηk ∈ Xk is an eigenfunction of Ak corresponding to k ξ ξ φ the eigenvalue λk . Since the eigenfunction is given by ψk−1 ηk = ηk φk , we observe k

k k

that the eigenfunction of Ak corresponding to the negative eigenvalue λk is a realvalued function. Thus, under (6.84), we deduce that A has a negative eigenvalue and the corresponding eigenfunction is a real-valued function componentwise.

7 Dynamical Systems for Quasilinear Evolution Equations 7.1 Structural Assumptions We consider the Cauchy problem for a quasilinear abstract parabolic evolution equation dU dt + A(U )U = F (U ), 0 < t < ∞, (6.85) U (0) = U0 , in a Banach space X. Let Z be a second Banach space continuously embedded in X. For U ∈ Z, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 . The domain D(A(U ) is uniform, i.e., D(A(U )) ≡ D,

U ∈ Z.

(6.86)

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6 Dynamical Systems

The domain D is a Banach space equipped with the graph norm A(0) · X of the operator A(0). The operator F is a nonlinear operator from a third Banach space W into X, where W ⊂ Z ⊂ X with continuous embeddings. We make the following structural assumptions. For each 0 < R < ∞, let KR = {U ∈ Z; U Z < R} denote an open ball of Z. For U ∈ KR , A(U ) satisfies σ (A(U )) ⊂ ΣR = {λ ∈ C; | arg λ| < ωR },

U ∈ KR ,

(6.87)

with some uniform angle 0 < ωR < π2 , and the resolvent satisfies (λ − A(U ))−1 L(X) ≤ MR |λ|,

λ∈ / ΣR , U ∈ KR ,

(6.88)

with some uniform constant MR ≥ 1. In addition, A(U ) satisfies the Lipschitz condition A(U ) A(U )−1 − A(V )−1 ≤ NR U − V Y , U, V ∈ KR , (6.89) L(X) with some constant NR > 0, where Y is another Banach space such that Z ⊂ Y ⊂ X with continuous embeddings. It is verified from (6.89) that there is a constant CR > 0 such that CR−1 A(0)U˜ X ≤ A(U )U˜ X ≤ CR A(0)U˜ X ,

U ∈ KR , U˜ ∈ D.

(6.90)

For the nonlinear operator F : W → X, we assume the Lipschitz condition F (U ) − F (V ) X ≤ ϕ( U Z + V Z )[ U − V W + ( U W + V W ) × U − V Z ],

U, V ∈ W,

(6.91)

with some continuous increasing function ϕ(·). We may recall that (6.91) implies (5.45). For the spaces W ⊂ Z ⊂ Y ⊂ X, we assume that there exist exponents 0 ≤ α < β < η < 1 such that, for every U ∈ Z, D(A(U )α ) ⊂ Y, D(A(U )β ) ⊂ Z, D(A(U )η ) ⊂ W with continuous embeddings such that ⎧ α α ⎪ ⎨ U˜ Y ≤ D1,R A(U ) U˜ X , U˜ ∈ D(A(U ) ), U ∈ KR , (6.92) U˜ Z ≤ D2,R A(U )β U˜ X , U˜ ∈ D(A(U )β ), U ∈ KR , ⎪ ⎩ ˜ U W ≤ D3,R A(U )η U˜ X , U˜ ∈ D(A(U )η ), U ∈ KR , with some constants Di,R > 0 (i = 1, 2, 3). We furthermore introduce another exponent γ such that β < γ < η and assume that the domain D(A(U )γ ) of the fractional power is independent of U ∈ Z with the equivalence estimate CR−1 A(0)γ U˜ X ≤ A(U )γ U˜ X ≤ CR A(0)γ U˜ X , U˜ ∈ D(A(0)γ ), U ∈ KR , (6.93)

7 Dynamical Systems for Quasilinear Evolution Equations

299

with some constant CR > 0. The domain D(A(U )γ ) is denoted by Dγ , which is a Banach space with the graph norm · Dγ = A(0)γ · X . This space is then used as a space of initial values for problem (6.85). Since D(A(0)γ ) ⊂ D(A(0)β ) ⊂ Z due to (6.92), Dγ is continuously embedded in Z; so, let · Z ≤ c · Dγ . For ˜ be an open ball of Dγ . Then, 0 < R˜ < ∞, let K˜ R˜ = {U ∈ Dγ ; U Dγ < R} K˜ R˜ ⊂ KcR˜ ,

0 < R˜ < ∞.

(6.94)

7.2 Dynamical System Applying the results obtained in Chap. 5, we can see the local existence of solution ˜ to (6.85) for each initial value U0 ∈ Dγ . In fact, let U0 ∈ Dγ with U0 Dγ < R; ˜ then, U0 ∈ Z with U0 Z < cR = R due to (6.94). The family of operators A(U ), U ∈ KR , and the nonlinear operator F satisfy all the structural assumptions (5.2), (5.3), (5.4), (5.5), (5.6), (5.7), (5.46), and (5.47) with ν = 1. In view of U0 ∈ D(A(0)γ ) = D(A(U0 )γ ), we get by Theorem 5.6 the existence of a unique local solution U (t) on some interval [0, TU0 ] in the function space: U ∈ C((0, TU0 ]; D) ∩ C([0, TU0 ]; Dγ ) ∩ C1 ((0, TU0 ]; X),

(6.95)

where TU0 > 0 is determined by the norm A(0)γ U0 X . In order to obtain the global existence, we have to establish a priori estimates for any local solution with the initial value U0 . We here assume that there exists some fixed continuous increasing function p(·) > 0 such that the estimate U (t) Dγ ≤ p( U0 Dγ ),

0 ≤ t ≤ T U , U0 ∈ D γ ,

(6.96)

holds for any local solution U on [0, TU ] to (6.85). This means that any local solution lies in a fixed open ball Kcp( U0 Dγ ) of Z. Since the family of operators A(U ), U ∈ Kcp( U0 Dγ ) , also satisfies all the structural assumptions and since A(U (t))γ U (t) ≤ Ccp( U0 Dγ ) p( U0 Dγ ) due to (6.93), we can use Corollary 5.2 to conclude the existence of a global solution U (t; U0 ) in the function space (6.95) in which TU0 is an arbitrary finite time. Clearly, (6.96) implies that U (t; U0 ) Dγ ≤ p( U0 Dγ ),

0 ≤ t < ∞, U0 ∈ Dγ .

(6.97)

Furthermore, by the same arguments as in Proposition 6.1, the following stronger estimate is verified. Proposition 6.5 For U (t; U0 ), it holds that U (t; U0 ) D ≤ (t γ −1 + 1)p1 ( U0 Dγ ), with some continuous increasing function p1 (·).

0 < t < ∞, U0 ∈ Dγ ,

(6.98)

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6 Dynamical Systems

In view of γ < η < 1, we can use the moment inequality (2.119). Then, (6.97) and (6.98) yield that U (t; U0 ) W ≤ (t γ −η + 1)pη ( U0 Dγ ),

0 ≤ t < ∞, U0 ∈ Dγ ,

(6.99)

with some continuous increasing function pη (·). For U0 ∈ Dγ , we set S(t)U0 = U (t; U0 ). The uniqueness of solutions implies that S(t) is a nonlinear semigroup acting on Dγ . Let us show that S(t) is continuous on Dγ . Let 0 < R˜ < ∞. For U0 ∈ K˜ R˜ , it holds that U0 Z < cR˜ and A(U0 )γ U0 X ≤ CcR˜ . So, Corollary 5.4 is applicable for the solutions S(t)U0 , U0 ∈ K˜ R˜ , to see that t η−α S(t)U0 − S(t)V0 W + t β−α S(t)U0 − S(t)V0 Z + S(t)U0 − S(t)V0 Y 0 < t ≤ tR˜ , U0 , V0 ∈ K˜ R˜ .

≤ CR˜ U0 − V0 Dγ ,

(6.100)

On the basis of this estimate, we can deduce a Hölder condition of S(t). Proposition 6.6 With the exponent δ = min{ γ1−η −α , 1}, S(t) satisfies S(t)U0 − S(t)V0 Dγ ≤ LR˜ U0 − V0 δDγ ,

0 ≤ t ≤ tR˜ , U0 , V0 ∈ K˜ R˜ .

Proof For simplicity, let us use the notation U (t) = S(t)U0 and V (t) = S(t)V0 , and let UU (t, s) and UV (t, s) denote the evolution operators for the families of operators AU (t) = A(U (t)) and AV (t) = A(V (t)), respectively. We repeat the similar argument as in the proof of Theorem 5.9. Indeed, as shown in (5.71), U (t) − V (t) = UU (t, 0)(U0 − V0 ) + [UU (t, 0) − UV (t, 0)]V0

t + [UU (t, s) − UV (t, s)]F (V (s)) ds 0

+

t

UU (t, s)[F (U (s)) − F (V (s))] ds.

0

Here, by (6.93), AU (t)γ UU (t, 0)(U0 − V0 ) ≤ C ˜ A(U0 )γ (U0 − V0 ) ≤ C ˜ U0 − V0 D . γ R R X X By the same arguments as for (5.39) but substituting γ for β, we verify that AU (t)γ [UU (t, 0) − UV (t, 0)]V0 X

t ≤ CR˜ V0 Dγ (t − s)−γ s γ −1 U (s) − V (s) Y ds. 0

Hence, by (3.83), AU (t)γ [UU (t, 0) − UV (t, 0)]V0 ≤ C ˜ U0 − V0 D . γ R X

7 Dynamical Systems for Quasilinear Evolution Equations

301

Similarly, by the same arguments as for (5.40), we verify that

t AU (t)γ [UU (t, s) − UV (t, s)]F (V (s)) ds 0

≤ CR˜ F (V (·)) Fγ ,σ

X

t

(t − s)−γ s γ −1 U (s) − V (s) Y ds.

0

Hence, by (6.100),

t AU (t)γ [UU (t, s) − UV (t, s)]F (V (s)) ds ≤ CR˜ U0 − V0 Dγ . 0

X

In the meantime, by (6.91),

t AU (t)γ U (t, s)[F (U (s)) − F (V (s))] ds U 0

≤ CR˜

t

X

(t − s)−γ [ U (s) − V (s) W

0

+ ( U (s) W + V (s) W ) U (s) − V (s) Z ] ds. We notice from (6.99) and (6.100) that U (s) − V (s) W ≤ ( U (s) W + V (s) W )1−δ U (s) − V (s) δW ≤ CR˜ s γ (1−δ)+αδ−η U0 − V0 δDγ with any 0 < δ ≤ 1. So, taking δ = min{ γ1−η −α , 1}, we have

t

(t − s)−γ s γ (1−δ)+αδ−η ds ≤ C.

0

Similarly, ( U (s) W + V (s) W ) U (s) − V (s) Z

≤ CR˜ ( U (s) W + V (s) W )( U (s) Z + V (s) Z )1−δ U (s) − V (s) δZ

≤ CR˜ s γ −η+(α−β)δ U0 − V0 δDγ 1−η with any 0 < δ ≤ 1. So, taking δ = min{ β−α , 1}, we have

t

(t − s)−γ s γ −η+(α−β)δ ds ≤ C.

0

Hence, we conclude that

t AU (t)γ ≤ C ˜ U0 − V0 δ U (t, s)[F (U (s)) − F (V (s))] ds U Dγ R 0

X

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6 Dynamical Systems

with the exponent δ = min{δ, δ }. Summing up these estimates, we obtain the desired one.

The a priori estimate (6.96) implies that

{S(t)U0 ; 0 ≤ t < ∞} ⊂ K˜ p(R) ˜ .

U0 ∈K˜ R˜

We now apply Proposition 6.6 on the ball K˜ p(R) ˜ . Proposition 6.7 For any 0 < R˜ < ∞, it holds that n+1

S(t)U0 − S(t)V0 Dγ ≤ Ln+1˜ U0 − V0 δDγ , p(R)

˜ t ∈ [ntp(R) ˜ , (n + 1)tp(R) ˜ ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ KR˜ .

(6.101)

Proof In fact, when n = 0, this is already verified by Proposition 6.6 because of ˜ Assume that this is true for n − 1. For nt ˜ < t ≤ (n + 1)t ˜ , we R˜ ≤ p(R). p(R) p(R) have S(t) = S(t − ntp(R) ˜ )S(ntR˜ ); then (6.101) is easily obtained since S(ntp(R) ˜ )U0 and S(nt ˜ )V0 are in K˜ ˜ . p(R)

p(R)

Estimate (6.101) means that S(t) is Hölder continuous on the ball K˜ R˜ and the Hölder exponent is uniform in t on any finite time interval. Meanwhile, for each U0 ∈ K˜ R˜ , S(·)U0 is continuous from [0, ∞) to Dγ . Then, the mapping G : [0, ∞) × K˜ R˜ → Dγ , where G(t, U0 ) = S(t)U0 , is continuous. As R˜ can be taken arbitrarily, we conclude that S(t) is a continuous semigroup on Dγ . Thus, under (6.87), (6.88), (6.89), (6.91), (6.92), (6.93), and (6.96), problem (6.85) determines a dynamical system (S(t), Dγ , Dγ ), where Dγ = D(A(0)γ ). It is often important to construct a dynamical system in the universal space X instead of Dγ . To define such a dynamical system, we introduce the phase space XR˜ =

{S(t)U0 ; 0 ≤ t < ∞},

0 < R˜ < ∞.

(6.102)

U0 ∈K˜ R˜

Clearly, XR˜ is invariant under S(t) for every 0 < t < ∞, and XR˜ ⊂ K˜ p(R) ˜ due to (6.96). With the aid of Theorem 5.9, we can then repeat the same arguments as in Proposition 6.3 to deduce the Lipschitz continuity of S(t) in the X norm. Proposition 6.8 For any 0 < R˜ < ∞, it holds that S(t)U0 − S(t)V0 X ≤ Ln+1˜ U0 − V0 X , p(R)

t ∈ [ntp(R) ˜ , (n + 1)tp(R) ˜ ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ XR˜ .

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303

Consequently, G : [0, ∞) × XR˜ → X is continuous, where XR˜ is equipped with the induced distance from X. Thus, problem (6.85) defines a dynamical system (S(t), XR˜ , X) in the space X, too. Clearly, K˜ R˜ ⊂ XR˜ for 0 < R˜ < ∞.

7.3 Absorbing and Invariant Compact Set We now add two assumptions that X is a reflexive Banach space, D is compactly embedded in X.

(6.103)

As for Proposition 6.4, these assumptions immediately imply the following result. D

Proposition 6.9 Any closed bounded ball B (0; R), 0 < R < ∞, of D is a compact set of X. We assume also the dissipative condition that there exists a number C˜ > 0 for which the following estimate is valid. For any bounded subset B of Dγ , there is a time tB such that ˜ sup sup S(t)U0 Dγ ≤ C.

U0 ∈B t≥tB

(6.104)

As for Proposition 6.5 (actually Proposition 6.1), this condition implies a stronger one. There exists a number C˜ 1 > 0 such that, for any bounded subset B of Dγ , it holds that sup sup S(t)U0 D ≤ C˜ 1

U0 ∈B t≥tB

(6.105)

with a suitable time tB depending on B. Using this constant C˜ 1 , we introduce the closed ball D B = B (0; C˜ 1 ).

Then (6.105) obviously means that B is an absorbing set. As noticed above, B is a compact set of X. We next introduce the set X=

S(t)B ⊂ B

(closure in X).

(6.106)

tB ≤t 0 that the operator S(t ∗ ) from X into Dγ satisfies the compact perturbation condition (6.38)–(6.39) (with δ = 0 and Z = Dγ ). Similarly, by Proposition 6.8, S(t)U0 − S(t)V0 X ≤ CX U0 − V0 X ,

0 ≤ t ≤ t ∗ , U0 , V0 ∈ X,

with some constant CX > 0. Meanwhile, by (5.45) and (6.90), t [F (S(τ )V ) − A(S(τ )V )S(τ )V ] dτ S(t)V0 − S(s)V0 X = 0 0 0 s

≤ CX (t − s),

X

∗

0 ≤ s < t ≤ t , V0 ∈ X.

Hence, condition (6.48) is fulfilled. Under (6.87), (6.88), (6.89), (6.91), (6.92), (6.93), (6.96), (6.103), and (6.104), we have proved that (S(t), X, X) possesses a family of exponential attractors M. Let us equally verify that M are exponential attractors of the dynamical system (S(t), X, Dγ ), too, where X is equipped with the induced distance from Dγ . Let M be an exponential attractor of (S(t), X, X) satisfying (6.17). By the moment inequality, we have hDγ (S(t)X, M) ≤ CX h(S(t)X, M)1−γ ≤ CX e−(1−γ )kt ,

0 ≤ t < ∞,

where hDγ (·, ·) denotes the Hausdorff pseudo-distance in Dγ . In addition, by Theorem 1.3, the fractal dimension dDγ (M) of M in Dγ is estimated by (1 − γ )−1 dX (M), where dX (M) denotes that of M in X. These facts show that M is an exponential attractor of (S(t), X, Dγ ), too.

8 Stationary Solutions to Quasilinear Equations 8.1 Equilibria of Dynamical System Let U be a stationary solution of (6.85), i.e., U ∈ D and A(U )U = F (U ). We assume that (6.85) determines a dynamical system (S(t), Dγ , Dγ ). More precisely, all the structural conditions (6.87), (6.88), (6.89), (6.91), (6.92), and (6.93) are satisfied and the a priori estimates (6.96) hold for all initial values in Dγ . So, U is an equi˜ Then, U Z < R, librium of (S(t), Dγ , Dγ ). Let R˜ > 0 be such that U Dγ < R. Z where R = cR˜ (see (6.94)), and let KR be the open ball B (0; R) of Z. In this section, we shall investigate the stability and instability of U by applying Theorems 6.9 and 6.10, respectively.

8 Stationary Solutions to Quasilinear Equations

305

8.2 Exponential Stability of U Let us assume the following conditions. The operator A : Z → L(D, X) is Fréchet differentiable at U with respect to the Y norm. Its differential A (U ) is a bounded operator from Y to L(D, X) and, equivalently, is a bilinear operator from Y × D into X. So, the derivative is written by A (U )[U, U˜ ] ∈ X for [U, U˜ ] ∈ Y × D. Therefore, ΔA (h) = A(U + h) − A(U ) − A (U )[h, ·], A (U ) ∈ L(Y, L(D, X)), (6.107) limh→0 oA (h)/ h Y = 0. oA (h) = ΔA (h) L(D,X) , Here, oA (h) is defined for h ∈ Z such that U + h ∈ KR , and oA (h) is uniformly bounded for h such that h ∈ KR − U . The nonlinear operator F : W → X is also Fréchet differentiable at U , i.e., ΔF (h) = F (U + h) − F (U ) − F (U )h, F (U ) ∈ L(W, X), (6.108) limh→0 oF (h)/ h W = 0. oF (h) = ΔF (h) X , Here, oF (h) is defined for all h ∈ W . In view of (6.91), oF (h) ≤ C h W for all h ∈ W such that h W < 1. From now on, let h denote a small variable in Dγ such that h Dγ < R˜ − U Dγ , and let Uh (t) = S(t)(U + h) be the solution to (6.85) with initial value U + h. According to Theorem 5.9, if T > 0 is sufficiently small, then t η Uh (t) − U W + t β Uh (t) − U Z + t α Uh (t) − U Y ≤ C h X ,

0 ≤ t ≤ T. (6.109)

We consider the linear problem dV

h

dt

+ A(U )Vh + A (U )[Vh , U ] = F (U )Vh ,

0 < t ≤ T,

Vh (0) = h,

(6.110)

in X with the initial value h. The linear operator of this problem can be regarded as a perturbation of A(U ) in a sense that condition (3.8) is fulfilled with ν˜ = 1 − η. Therefore, by Theorem 3.3, A(U ) + A (U )[·, U ] − F (U ) generates an analytic semigroup e−tA on X, where A = A(U ) + A (U )[·, U ] − F (U ). In addition, by Theorem 3.14 (cf. Remark 3.7), the solution of the problem is given by Vh (t) = e−tA h. In particular, we notice by maximal regularity (3.130) that ˜

A(U )Vh ∈ Fβ,σ˜ ((0, T ]; X) ˜ σ˜ such that 0 < σ˜ < β˜ < min{1 − η, γ } and that with some exponents β, Vh (t) D ≤ Ct β−1 A(U )β h , 0 < t ≤ T .

(6.111)

(6.112)

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6 Dynamical Systems

We set ΔU (t) = Uh (t) − U − Vh (t), 0 < t ≤ T . It is observed that A(Uh )Uh − A(U )U − A(U )Vh − A (U )[Vh , U ] = A(Uh )ΔU + A (U )[ΔU , U ] + A (U )[Uh − U , Vh ] + ΔA (Uh − U )(U + Vh ) and F (Uh ) − F (U ) − F (U )Vh = F (U )ΔU + ΔF (Uh − U ). This then means that ΔU (t) is regarded as a solution to the linear problem dΔ U dt + A(Uh (t))ΔU + A (U )[ΔU , U ] − F (U )ΔU = G(t), ΔU (0) = 0, where the external force function is given by G(t) = −A (U )[Uh (t) − U , Vh (t)] − ΔA (Uh (t) − U )(U + Vh (t)) + ΔF (Uh (t) − U ). The linear operators in the equation are regarded as perturbations of the operators A(t) = A(Uh (t)), 0 ≤ t ≤ T . More precisely, we can show that A(t) = A(Uh (t)) and B(t) ≡ B = A (U )[·, U ] − F (U ) satisfy the assumptions in Theorems 3.13 and 3.14. In fact, it is already known that A(t) satisfy assumptions (3.27), (3.28), (3.29), (3.30), and (3.31) with ν = 1 and with some exponent μ = β − α > 0 (due to (6.89) and (5.48)). In the meantime, B is a bounded operator from W into X. Therefore, it satisfies (3.117) and (3.118) with μ˜ = 1 and with some exponent 0 < ν˜ < 1 − η. Moreover, by using (3.92) with θ = η and ϕ = 1 − ν˜ , we observe that B A(t)ν˜ −1 − A(s)ν˜ −1 ≤ B L(W,X) A(t)η A(t)ν˜ −1 − A(s)ν˜ −1 ≤ C|t − s|η . Hence, (3.119) and (3.120) are satisfied. Consequently, there exists an evolution operator U˜ (t, s) for the operators A(t) + B. In addition, ΔU is written by

t ΔU (t) = U˜ (t, s)G(s) ds, 0 < t ≤ T . 0 ˜

Here we used a fact that G ∈ Fβ,σ˜ ((0, T ]; X) due to (6.111). Utilizing this expression, we want to estimate the norm ΔU (t) Dγ . By (3.124) and (6.93),

t ΔU (t) Dγ ≤ C A(t)γ ΔU (t) X ≤ C (t − s)−γ G(s) X ds. 0

Here, it follows from (6.109) and (6.112) that A (U )[Uh (s) − U , Vh (s)] X ≤ C Uh (s) − U Y Vh (s) D ≤ Cs −α h X s β−1 A(U )β h X ,

8 Stationary Solutions to Quasilinear Equations

307

ΔA (Uh (s) − U )(U + Vh (s)) X ≤ oA (Uh (s) − U ) U + Vh (s) D −α ≤ oA (Uh (s) − U ) Uh (s) − U −1 h X Y s β−1 A(U )β h + 1 , × s X

and −η ΔF (Uh (s) − U ) X ≤ oF (Uh (s) − U ) Uh (s) − U −1 W s h X .

Hence, G(s) X ≤ C(s β−α−1 + s −η ) A(U )β h X + oA (Uh (s) − U ) Uh (s) − U −1 Y

+ oF (Uh (s) − U ) Uh (s) − U −1 W h X . As a consequence,

ΔU (t) Dγ / h X ≤ C

0

t

(t − s)−γ (s β−α−1 + s −η ) A(U )β h X

+ oA (Uh (s) − U ) Uh (s) − U −1 Y

+ oF (Uh (s) − U ) Uh (s) − U −1 W ds. β/γ

1−β/γ

, h X → 0 implies Since A(U )β h X ≤ C A(U )γ h X h X A(U )β h X → 0. Similarly, h X → 0 implies Uh (s) − U W → 0 and Uh (s) − U Y → 0 due to (6.109). Therefore, by virtue of the dominate convergence theorem, we conclude that, as h X → 0 (a fortiori as h Dγ → 0), 0 ≤ ΔU (t) Dγ / h Dγ ≤ ΔU (t) Dγ / h X → 0.

(6.113)

Thus, under (6.107) and (6.108), the operator S(t) : Dγ → Dγ is Fréchet differentiable at U for any time t such that 0 < t ≤ T . Its differential is given by S(t) U = e−tA

where A = A(U ) + A (U )[·, U ] − F (U ).

As S(t)U = U , we verify that S(2t) = S(t)S(t) is also Fréchet differentiable at U . Repeating this argument, we conclude that S(t) is Fréchet differentiable at U for every time 0 < t < ∞. Let us now assume that U satisfies the spectral condition σ (A) = σ (A(U ) + A (U )[·, U ] − F (U )) ⊂ {λ ∈ C; Re λ > 0}.

(6.114)

This condition, together with Theorem 3.3, implies that A is a sectorial operator, i.e., A satisfies (2.92) and (2.93). Proposition 2.5 then provides the exponentially decaying estimate e−tA L(X) ≤ Ce−dt , 0 ≤ t < ∞. Then, we can repeat the same

308

6 Dynamical Systems

arguments as in Sect. 6.2 for verifying that S(t ∗ ) is a contraction at U . Hence, by Theorem 6.10, we conclude that U is exponentially stable. We have thus proved under (6.107), (6.108), and (6.114) that U is an exponentially stable equilibrium of (S(t), Dγ , Dγ ).

8.3 Unstable Manifold of U For simplicity, we will consider only the case where the semilinear term is given by F (U ) = kU with some complex number k. (For the general case, see Remark 6.4 below.) So, U is a stationary solution such that A(U )U = kU . We assume the following conditions. The operator A : Z → L(D, X) is Fréchet differentiable in a neighborhood B Z (U ; r) of U with respect to the Y norm, and the derivative satisfies the following estimates: A (U )[h, U˜ ] X ≤ N1 h Y U˜ D ,

U ∈ B Z (U ; r), h ∈ Z, U˜ ∈ D, (6.115)

{A (U1 ) − A (U2 )}[h, U˜ ] X ≤ N2 U1 − U2 Z h Y U˜ D ,

U1 , U2 ∈ B Z (U ; r), h ∈ Z, U˜ ∈ D,

(6.116)

with some constants Ni > 0 (i = 1, 2). Put ΔA (U ; h) = A(U + h) − A(U ) − A (U )[h, ·]. Then, (6.116) implies that ΔA (U ; h) L(D,X) ≤ C h Z h Y ,

U + h, U ∈ B Z (U ; r).

(6.117)

Consider an open neighborhood B Dγ (U ; r1 ) of U . Let U0 ∈ B Dγ (U ; r1 ) be an initial value, and let U (t) = S(t)U0 . By Proposition 6.6 and (6.92), we have U (t) − U Z ≤ C U0 − U δDγ ,

0 ≤ t ≤ T1 , U0 ∈ B Dγ (U ; r1 ),

for sufficiently small time T1 > 0. So, if r1 > 0 and T1 > 0 are sufficiently small, then U (t) ∈ B Z (U ; r) for 0 ≤ t ≤ T1 . Consequently, A (U (t)) is defined. Consider also a small vector variable h ∈ B Dγ (0; r2 ) and let Uh (t) = S(t)(U0 + h). By Theorem 5.9, we have t α Uh (t) − U (t) Y ≤ C h X ,

0 < t ≤ T2 , U0 ∈ B Dγ (U ; r1 ), h ∈ B Dγ (0; r2 ), (6.118)

for sufficiently small T2 > 0. We introduce the linear problem dV h dt + A(U (t))Vh + A (U (t))[Vh , U (t)] = kVh , Vh (0) = h,

0 < t ≤ T,

(6.119)

8 Stationary Solutions to Quasilinear Equations

309

in X with T = min{T1 , T2 }. The linear operators in the equation can then be treated as perturbations of operators A(t) = A(U (t)). In fact, we verify the following lemma. Lemma 6.2 Set B(t) = A (U (t))[·, U (t)] − k. Then, B(t)A(t)−β ≤ Ct γ −1 , 0 < t ≤ T , L(X) B(t)A(t)−β − B(s)A(s)−β ≤ Cs γ −1−σ (t − s)σ , L(X)

(6.120) 0 < s < t ≤ T, (6.121)

with any σ such that 0 < σ < β − α. Proof From (5.66), (6.92), and (6.115) it follows that B(t)A(t)−β ≤ B(t)A(t)−α ≤ C U (t) D ≤ Ct γ −1 . Hence, (6.120) is verified. In the meantime, B(t)A(t)−β − B(s)A(s)−β ≤ [B(t) − B(s)]A(t)−β + B(s) A(t)−β − A(s)−β . By (6.116), [B(t) − B(s)]A(t)−β ≤ C[ U (t) − U (s) Z U (t) D + U (t) − U (s) D ]. Here, by (6.92), U (t) − U (s) Z ≤ C A(t)β [U (t) − U (s)] X β

1−β

≤ C U (t) − U (s) D U (t) − U (s) X . From (5.66), in particular from A(t)U (t) ∈ Fγ ,σ ((0, T ]; X), it follows that U (t) − U (s) D ≤ C A(t)U (t) − A(s)U (s) X + A(t) A(t)−1 − A(s)−1 A(s)U (s) X

≤ Cs

γ −1−σ

with any 0 < σ < β − α. In addition,

(t − s)

dU dt

σ

∈ Fγ ,σ ((0, T ]; X) implies that

U (t) − U (s) ≤ Cs γ −1 (t − s). Therefore,

[B(t) − B(s)]A(t)−β ≤ Cs γ −1−σ (t − s)σ .

(6.122)

310

6 Dynamical Systems

We obtain from Theorem 2.25, (5.66), and (6.115) that B(s) A(t)−β − A(s)−β ≤ C B(s)A(0)−α A(0)α A(t)−β − A(s)−β ≤ C U (s) D U (t) − U (s) Y ≤ Cs γ −1 (t − s)σ .

This, together with (6.122), then yields (6.121).

We have just verified that A(t) = A(U (t)) and B(t) = A (U (t))[·, U (t)] − k satisfy (3.117), (3.118), and (3.119) with μ˜ = γ and ν˜ = 1 − β. Consequently, there exists an evolution operator U˜ (t, s) for the family of operators A(t) + B(t) by which the solution of (6.119) is written as Vh (t) = U˜ (t, 0)h. By the maximal regularity (3.130), ˜

A(U )Vh ∈ Fβ,σ˜ ((0, T ]; X)

(6.123)

˜ σ˜ such that 0 < σ˜ < β˜ < γ − β, and with some exponents β, Vh (t) D ≤ Ct β−1 A(U0 )β h , 0 < t ≤ T .

(6.124)

We now put ΔU (t) = Uh (t) − U (t) − Vh (t),

0 < t ≤ T.

It is observed that A(Uh )Uh − A(U )U − A(U )Vh − A (U )[Vh , U ] = A(Uh )ΔU + A (U )[ΔU , U ] + A (U )[Uh − U, Vh ] + ΔA (U ; Uh − U )(U + Vh ). This then means that ΔU (t) is a solution to the linear problem dΔU dt + A(Uh (t))ΔU + A (U )[ΔU , U ] − kΔU = G(t), ΔU (0) = 0,

0 < t ≤ T,

where G(t) = −A (U (t))[Uh (t) − U (t), Vh (t)] − ΔA (U (t); Uh (t) − U (t))[U (t) + Vh (t)]. By the same arguments as before, the linear operators are seen to be perturbations of the operators A(Uh (t)) with μ˜ = γ and ν˜ = 1 − β. Therefore, by using an evolution operator U˜ h (t, s) for the linear operators A(Uh (t)) + A (U (t))[·, U (t)] − k, we obtain the expression

t

ΔU (t) = 0

U˜ h (t, s)G(s) ds,

0 < t ≤ T.

8 Stationary Solutions to Quasilinear Equations

311

Then, in view of (6.117), (6.118), (6.123), and (6.124), we have

t ΔU (t) Dγ / h X ≤ C (t − s)−γ s β−α−1 A(U0 )β h X 0

+ ΔA (U (s); Uh (s) − U (s)) L(D,X) Uh (t) − U (t) −1 Y ds. We can now repeat the same arguments as in the preceding subsection to verify that, as h X → 0 (a fortiori as h Dγ → 0), 0 ≤ ΔU (t) Dγ / h Dγ ≤ ΔU (t) Dγ / h X → 0. Thus, under (6.115) and (6.116), S(t) : Dγ → Dγ is Fréchet differentiable in a neighborhood B Dγ (U ; r1 ) of U for any time t such that 0 < t ≤ T , T > 0 being sufficiently small, with the derivative S(t) U0 = U˜ (t, 0), where U˜ (t, s) is an evolution operator for a family of the linear operators A(S(t)U0 ) + A (S(t)U0 )[·, S(t)U0 ] − k,

0 ≤ t ≤ T , U0 ∈ B Dγ (U ; r1 ).

In particular, S(t) U = e−tA , where A = A(U ) + A (U )[·, U ] − k. Let us investigate next the regularity of the derivative S(t) U0 with respect to U0 . For this purpose, let Ui ∈ B Dγ (U ; r1 ), Ui (t) = S(t)Ui , and Ai (t) ≡ A(Ui (t)) for i = 1, 2. We denote also by Ui (t, s) the evolution operator for the operators Ai (t). Similarly, let A˜ i (t) ≡ A(Ui (t)) + A (Ui (t))[·, Ui (t)] − k and denote by U˜ i (t, s) the evolution operator for the operators A˜ i (t). Then, it is derived that U˜ 2 (t, 0) − U˜ 1 (t, 0) L(Dγ ) ≤ C U2 − U1 δDγ ,

0 ≤ t ≤ T,

(6.125)

where δ = min{ γ1−η −α , 1} (cf. Proposition 6.6). That is, S(t) U2 − S(t) U1 L(Dγ ) ≤ C U2 − U1 δDγ ,

0 ≤ t ≤ T ; U1 , U2 ∈ B Dγ (U ; r1 ).

In order to prove (6.125), we write A1 (t)γ [U˜ 2 (t, 0) − U˜ 1 (t, 0)]h

t = A1 (t)γ U˜ 1 (t, τ )[A(U1 (τ )) − A(U2 (τ ))] 0

× A(U2 (τ ))−1 A(U2 (τ ))U˜ 2 (τ, s)h dτ

t + A1 (t)γ U˜ 1 (t, τ ){A (U1 (τ )) − A (U2 (τ ))}[U˜ 2 (τ, 0)h, U1 (τ )] dτ 0

+

t

A1 (t)γ U˜ 1 (t, τ )A (U2 (τ ))[U˜ 2 (τ, 0)h, U1 (τ ) − U2 (τ )] dτ.

0

Then, by (6.89), (6.116), and Proposition 6.6, we immediately verify that the first and second integrals in the right-hand side are estimated by C U2 − U1 δDγ h Dγ .

312

6 Dynamical Systems

Meanwhile, the estimate for the third one is more profound (it is necessary to extend the known estimate (5.39) or (5.23) to the critical case θ = ν = 1). We can in fact show the following proposition. The proof of this proposition will be described in the next subsection. Proposition 6.10 We have U2 (t) − U1 (t) D ≤ Ct γ −1 U2 − U1 δDγ ,

0 < t ≤ T.

(6.126)

Clearly, (6.126) implies that the third integral is also estimated by C U2 − U1 δDγ h Dγ ; hence, (6.125) has been proved. Let us now make the assumption that the operator A = A(U ) + A (U )[·, U ] − k as an operator of X satisfies the spectrum separation condition σ (A) ∩ {λ ∈ C; Re λ = 0} = ∅.

(6.127)

Let σ− (A) = σ (A) ∩ {λ ∈ C; Re λ < 0} and σ+ (A) = σ (A) ∩ {λ ∈ C; Re λ > 0}, and let X = X− ⊕ X+ be the decomposition of X corresponding to the spectrum separation. By the same arguments as in Sect. 6.3, we can verify that the spectrum σ Dγ (S(t) U ) is separated by the unit circle as (6.20) and the exterior spectrum is given by e−tσ− (A) . We have thus shown that, under (6.115) and (6.116), if U satisfies (6.127), then U is a hyperbolic equilibrium of (S(t), Dγ , Dγ ). The exterior spectrum σ Dγ (S(t) U ) ∩ {λ ∈ C; |λ| > 1} is given by e−tσ− (A) . Therefore, by virtue of Theorem 6.9, there exists a smooth local unstable manifold W+ (U ; O) of U with dimension dim X− which is tangent to the subspace U + X− of Dγ at U . Remark 6.4 It is possible to derive analogous results for the general case where the term F (U ) is not necessarily of the form F (U ) = kU under the conditions F (U )h X ≤ L1 h W ,

U ∈ W ∩ B Z (U ; r), h ∈ W ;

[F (U1 ) − F (U2 )]h X ≤ L2 U1 − U2 Z h W , U1 , U2 ∈ W ∩ B Z (U ; r), h ∈ W, where Li > 0 (i = 1, 2) are some constants. Instead of (6.119), however, we have to handle the more complicated problem dV h dt + A(U (t))Vh + A (U (t))[Vh , U (t)] = F (U (t))Vh , 0 < t ≤ T , Vh (0) = h. For constructing an evolution operator for this problem, we must go back to the study of perturbed problems in Chap. 3, Sect. 10 and in fact must generalize the main theorems for the operators B(t) ≡ B1 (t) + B2 (t), each family Bi (t) satisfying conditions (3.117), (3.118), and (3.119) with some exponents 0 < μ˜ i , ν˜ i ≤ 1 depending on i = 1, 2.

8 Stationary Solutions to Quasilinear Equations

313

8.4 Proof of Proposition 6.10 We easily observe that Ui (t) = ekt Ui (t, 0)Ui for i = 1, 2. So, A1 (t)[U2 (t) − U1 (t)] = ekt A1 (t){U2 (t, 0)[U2 − U1 ] + [U2 (t, 0) − U1 (t, 0)]U1 }. Clearly, the first term in the right-hand side is estimated by A1 (t)U2 (t, 0)A2 (0)−γ A2 (0)γ [U2 − U1 ] ≤ Ct γ −1 U2 − U1 D . γ X In the same way as in the proof of Proposition 5.1, the second term is written by

t A1 (t)[U2 (t, 0) − U1 (t, 0)]U1 = A1 (t)U1 (t, τ )[A1 (t) − A2 (t)]U2 (τ, 0)U1 dτ 0

+

t

A1 (t)U1 (t, τ )[A1 (τ ) − A2 (τ )

0

− A1 (t) + A2 (t)]U2 (τ, 0)U1 dτ = J1 (t) + J2 (t). Furthermore, J1 (t) is decomposed into the form

t J1 (t) = [A1 (t)U1 (t, τ ) − A1 (t)e−(t−τ )A1 (t) ][A2 (t) − A1 (t)]U2 (τ, 0)U1 dτ 0

t

+

A1 (t)e−(t−τ )A1 (t) [A2 (t) − A1 (t)][1 − e−(t−τ )A2 (τ ) ]U2 (τ, 0)U1 dτ

0 t

+

A1 (t)e−(t−τ )A1 (t) [A2 (t) − A1 (t)][e−(t−τ )A2 (τ ) − U2 (t, τ )]

0

× U2 (τ, 0)U1 dτ + [1 − e−tA1 (t) ][A2 (t) − A1 (t)]U2 (t, 0)U1 . t Here, we used the formula 0 A1 (t)e−(t−τ )A1 (t) dτ = 1 − e−tA1 (t) . Then, by applying the estimates for the evolution operator and (6.100), we deduce the estimate J1 (t) X ≤ Ct γ −1 U2 − U1 Dγ . For estimating J2 (t), we write [Ai (t) − Ai (τ )]U˜ =

1

A (θ Ui (t) + (1 − θ )Ui (τ ))[Ui (t) − Ui (τ ), U˜ ] dθ

0

for i = 1, 2. Then, {[A2 (t) − A2 (τ )] − [A1 (t) − A1 (τ )]}U˜

1 = A (θ U1 (t) + (1 − θ )U1 (τ ))[(U2 (t) − U2 (τ )) − (U1 (t) − U1 (τ )), U˜ ] dθ 0

314

6 Dynamical Systems

+

1

{A (θ U2 (t) + (1 − θ )U2 (τ ))

0

− A (θ U1 (t) + (1 − θ )U1 (τ ))}[U2 (t) − U2 (τ ), U˜ ] dθ. Then, by (6.115) and (6.116), there exists a positive exponent ε > 0 such that {[A2 (t) − A2 (τ )] − [A1 (t) − A1 (τ )]}U˜ X ≤ C(t − τ )ε U2 − U1 δDγ U˜ D . This estimate furthermore implies that J2 (t) X ≤ Ct γ −1 U2 − U1 δDγ . We have thus accomplished the proof of Proposition 6.10.

Notes and Further Researches We refer the reader for the general matters of dynamical systems in infinitedimensional spaces to the books by Ladyzhenskaya [Lad91], Babin–Vishik [BV92], Temam [Tem97], and Robinson [Rob01]. Especially, for the attractors, we cite the literature Cholewa–Doltko [CD00], Chepyzhov–Vishik [CV02], and Miranville– Zelik [MZ08]. For the inertial manifolds, see Foias–Sell–Temam [FST88] and [Tem97, Chap. 8]. In constructing smooth stable and unstable manifolds in Theorem 6.8, we followed the methods due to Wells [Wel76], cf. also [BV92, Chap. 5] and [Tem97, Chap. VII: Part B]. The notion of exponential attractors was founded by Eden–Foias–Nicolaenko– Temam [EFNT94]. The squeezing property (6.45)–(6.46) was also introduced by the founders. Efendiev–Miranville–Zelik [EMZ00] introduced the condition of compact perturbation of a contraction operator for constructing exponential attractors. (Actually in [EMZ00], δ was taken as δ = 0. For the general case, see Takei and Yagi [TY06].) The two types of conditions (6.28)–(6.29) and (6.38)–(6.39) may seem to be very close. We see, however, that the squeezing property (6.45)–(6.46) implies (6.28)–(6.29) but not (6.38)–(6.39). In the meantime, there exists a semigroup S(t) which fulfills (6.28)–(6.29) but does not (6.38)–(6.39) (see Chap. 13). It is in fact possible to verify the existence of exponential attractors in various mathematical models. After the notion had been introduced by Eden et al., almost hundred papers were already devoted to constructing exponential attractors for dissipative nonlinear systems. We here quote only some results concerning the reaction– diffusion equations [BN95, EMZ04, GP07], the damped wave equations [FGMZ04, MPZ07], Navier–Stokes equations [EFN93, EFNS93, KB06], Ginzburg–Landau equation [Gao95, LL08], Cahn–Hilliard equations [Gal08, MZ04], the phase field systems [BHC93, MZ02, GP05], and others [Du03, EM93, FaN96, Gal98]. In Chaps. 8, 9, 10, 12, 13, 14, and 15, we will also utilize the squeezing property and the two types of compact perturbation of a contraction operator in order to construct exponential attractors for the dynamical systems determined from the model equations under consideration.

Notes and Further Researches

315

The continuous dependence of exponential attractors on a parameter was also investigated by the founders Eden–Foias–Nicolaenko–Temam in [EFNT94, Chap. 4]. Theorem 6.17 then provides us a very general result on this parameter dependence. Indeed, the theorem indicates that, if the semigroup Sξ (t) satisfies the compact perturbation condition of type I uniformly for the parameter ξ and if Sξ (t) converges to S0 (t) only on a finite time interval [0, t ∗ ], then the corresponding exponential attractor Mξ converges to M0 . This result was obtained by Efendiev–Yagi [EY05]. A similar result can be proved under the compact perturbation condition of type II, too.

Chapter 7

Numerical Analysis

This chapter is devoted to numerical analysis for nonlinear abstract parabolic evolution equations. We consider approximate problems by discretizing the spatial variable in the Cauchy problems of the forms (4.1) and (5.43) in Banach spaces. We investigate the order of convergence for the approximation problems. In the first half, we show local estimates of convergence. We introduce the approximate equations ξ dU ξ ), ξ = Fξ (U + Aξ U dt

0 < t < ∞,

and ξ dU ξ )U ξ = Fξ (U ξ ), 0 < t < ∞, + Aξ (U dt in Banach spaces Xξ , where 0 ≤ ξ ≤ 1, for the semilinear and quasilinear equations in (4.1) and (5.43), respectively. Here, Xξ denote a family of approximate spaces such that X0 = X is an underlying Banach space in which (4.1) and (5.43) are considered, and for 0 < ξ ≤ 1, Xξ are finite-dimensional subspaces of X. Under suitable structural assumptions on Xξ , Aξ , Aξ (U ), and Fξ (U ), we derive order estimates of convergence of approximate solutions in a finite interval [0, T ]. In the second half, we show some global estimates. We assume that the Cauchy problems for the approximate equations mentioned above have global solutions and determine dynamical systems (Sξ (t), Xξ , Xξ ) for 0 ≤ ξ ≤ 1. Let us assume furthermore that Sξ (t) satisfy conditions (6.38)–(6.39) uniformly for 0 ≤ ξ ≤ 1. Consequently, each dynamical system (Sξ (t), Xξ , Xξ ) has exponential attractors. We then establish that the convergence of semigroups Sξ (t) → S0 (t) as ξ → 0, together with that of phase spaces Xξ → X0 , implies the convergence of exponential attractors with respect to the distance (1.3).

1 Semilinear Evolution Equations We consider the Cauchy problem for a semilinear equation A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_7, © Springer-Verlag Berlin Heidelberg 2010

317

318

7 Numerical Analysis

dU dt

+ AU = F (U ),

0 < t < ∞,

(7.1)

U (0) = U0 ,

in a Banach space X. Here, A is a sectorial operator of X with angle ωA < π2 , that is, satisfies (2.92) and (2.93). The operator F is a nonlinear operator from D(Aβ ) into X, where 0 < β < 1. We assume that F satisfies the Lipschitz condition F (U ) − F (V ) ≤ ϕ(Aβ U + Aβ V )Aβ (U − V ),

U, V ∈ D(Aβ ), (7.2)

where ϕ(·) is some continuous increasing function. This means that (4.2) is satisfied with β = η. The initial value U0 is taken in D(Aγ ), where γ is such that β ≤ γ ≤ 1. By Theorems 4.1 and 4.2, there exists a unique solution to (7.1) on an interval [0, T ] in the function space: U ∈ C((0, T ]; D(A)) ∩ C([0, T ]; D(Aγ )) ∩ C1 ((0, T ]; X), (7.3) AU ∈ Fγ ,σ ((0, T ]; X), with 0 < σ < 1 − β.

1.1 Approximate Problems Let {Xξ }0dist(x,∂σ )

dy π = dist(x, ∂σ )−2ε , 2+2ε ε |x − y|

,

3 Two-Dimensional Finite Element Methods

we see that σ

Ω\σ

dy |x − y|2+2ε

q

333

dx ≤ σ

≤ Cε

Cε dist(x, ∂σ )−2εq dx

rσ

−rσ

rσ

η−2εq dη dξ = Cε,q rσ2−2εq .

0

uξ − ∂j uL2p (σ ) , we notice by Lemma 7.2 below and (7.49) In order to estimate ∂j that 1 1 1− p1 p p −1 ∂j uξ − ∂j uL2p (σ ) ≤ Cp uH 2 (σ u − u + r u − u 1 ξ ξ σ H (σ ) ) H 1 (σ ) 1 1 1− p1 −1 p +rp (r u ) · r u ≤ Cp uH 2 (σ 2 2 σ σ σ H (σ ) H (σ ) ) 1

= Cp rσp uH 2 (σ ) . Hence, we have σ ×(Ω\σ )

σ ∈τξ

≤

|∂j uξ (x) − ∂j u(x)|2 dx dy |x − y|2+2ε

2

2

Cε rσp u2H 2 (σ ) · rσq

−2ε

≤ Cε ξ 2−2ε u2H 2 .

σ

In view of (1.70), we have thus proved the proposition. Lemma 7.2 For 1 ≤ p < ∞, wL2p (σ ) ≤ Cp

rσ ρσ

1− 1

p

1− 1

1

1

∇wL2 (σp ) wLp 2 (σ ) + ρσp

−1

wL2 (σ ) ,

w ∈ H 1 (σ ), σ ∈ τξ ,

(7.54)

Cp > 0 being a constant independent of σ and ξ . Proof Let σ0 = {(z1 , z2 ) ∈ R2 ; z1 > 0, z2 > 0, z1 + z2 < 1} be the reference triangle, and φσ : σ0 → σ an affine mapping. Then, we have 1

1

Cq−1 |σ | q w ◦ φσ Lq (σ0 ) ≤ wLq (σ ) ≤ Cq |σ | q w ◦ φσ Lq (σ0 ) and − q1

∇(w ◦ φσ )Lq (σ0 ) ≤ Cq rσ |σ |

∇wLq (σ )

for any 1 ≤ q ≤ ∞. Now, (7.54) is readily verified by applying Theorem 1.37 (with p = n = 2) to w ◦ φσ .

334

7 Numerical Analysis

3.3 Projection Operator Pξ Consider Xξ = (Cξ (Ω), · L2 ) as a closed subspace of L2 (Ω). Let Pξ : L2 (Ω) → Xξ be the orthogonal projection. By the definition, if u ∈ C(Ω), then (1 − Pξ )uL2 = min u − vL2 ≤ u − Πξ uL2 . v∈Xξ

Therefore, by (7.53) (s = 0), if u ∈ H 2 (Ω), then (1 − Pξ )uL2 ≤ Cξ 2 uH 2 . More generally, for 0 ≤ s < 32 , we obtain by (7.50) and (7.53) that (1 − Pξ )uH s ≤ (1 − Πξ )uH s + (Πξ − Pξ )uH s ≤ Cs [ξ 2−s uH 2 + ξ −s (Πξ − Pξ )uL2 ] ≤ Cs ξ 2−s uH 2 , u ∈ H 2 (Ω).

(7.55)

As is seen, we have (1 − Pξ )uL2 ≤ Cξ 2 AuL2 for u ∈ D(A). We then use the Heinz and Kato inequality, Theorem 2.31, with X = Y = L2 (Ω) and T = 1 − Pξ , B = 1. Then, 1 − Pξ is a bounded operator from D(Aθ ) into L2 (Ω) whose operator norm is estimated by θ θ (1 − Pξ )uL2 ≤ C1 − Pξ 1−θ L(L2 ,L2 ) 1 − Pξ L(D(A),L2 ) A uL2 ,

u ∈ D(Aθ ).

Therefore, for 0 ≤ θ ≤ 1, (1 − Pξ )uL2 ≤ Cξ 2θ Aθ uL2 ,

u ∈ D(Aθ ).

(7.56)

In view of (7.48), (7.55) shows that assumptions (7.4) and (7.23) are actually fulfilled for any 0 < β < 34 with Oξ = O(ξ 2 ).

3.4 Ritz Operator Rξ Let Zξ = (Cξ (Ω), · H 1 ). Since a(u, v) is symmetric, a(u, v) can be regarded as an inner product of Zξ . Hence, for each u ∈ H 1 (Ω), there exists a unique vector Rξ u ∈ Zξ such that a(Rξ u, vξ ) = a(u, vξ ) for all v ξ ∈ Zξ . Then, Rξ uH 1 ≤ (M/ δ )uH 1 , that is, Rξ is a bounded operator from H 1 (Ω) into Zξ . Operator Rξ is called the Ritz operator. If u ∈ D(A), then a(u, vξ ) = (Au, vξ ) = (Pξ Au, vξ ) = Aξ A−1 vξ = a A−1 vξ . ξ Pξ Au, ξ Pξ Au, Therefore, Rξ u = A−1 ξ Pξ Au.

3 Two-Dimensional Finite Element Methods

335

For u ∈ H 1 (Ω), since a((1 − Rξ )u, (Πξ − Rξ )u) = 0, we have a((1 − Rξ )u, (1 − Rξ )u) = a((1 − Rξ )u, (1 − Πξ )u). By (2.6) and (2.7), − Rξ )uH 1 (1 − Πξ )uH 1 . δ (1 − Rξ )u2H 1 ≤ a((1 − Rξ )u, (1 − Rξ )u) ≤ M(1 Therefore, by (7.53) (s = 1), (1 − Rξ )uH 1 ≤ C(1 − Πξ )uH 1 ≤ Cξ uH 2 ,

u ∈ H 2 (Ω).

Furthermore, it is seen (Aubin–Nitsche’s trick) that (1 − Rξ )u2L2 = a(A−1 (1 − Rξ )u, (1 − Rξ )u) = a((1 − Rξ )A−1 (1 − Rξ )u, (1 − Rξ )u) ≤ C1 − Rξ L(H 2 ,H 1 ) A−1 L(L2 ,H 2 ) × (1 − Rξ )uL2 (1 − Rξ )uH 1 . Hence, (1 − Rξ )uL2 ≤ Cξ (1 − Rξ )uH 1 ≤ Cξ 2 uH 2 ,

u ∈ H 2 (Ω).

More generally, for 0 ≤ s < 32 , (1 − Rξ )uH s ≤ (1 − Πξ )uH s + (Πξ − Rξ )uH s ≤ Cs [ξ 2−s uH 2 + ξ −s (Πξ − Rξ )uL2 ] ≤ Cs ξ 2−s uH 2 , In addition, for 0 ≤ θ

0 being a constant independent of ξ .

u ∈ D(Aθ ),

(7.58)

336

7 Numerical Analysis

We have thus verified that assumptions (7.6) and (7.29) are fulfilled with Oξ = O(ξ 2 ).

3.5 Domains of Fractional Powers of Aξ 1

1

Since a(u, v) is symmetric, Theorem 2.34 yields that D(Aξ2 ) = Zξ and Aξ2 uξ L2 ≤ C uξ H 1 . Therefore, in view of (7.50), we have 1 1 uξ L2 ≤ C Aξ2 uξ H 1 ≤ Cξ −1 Aξ2 uξ L ≤ Cξ −1 uξ H 1 ≤ Cξ −2 uξ L2 . Aξ 2

Therefore, the moment inequality (2.117) yields that, for 0 ≤ θ ≤ 1, θ −2θ A uξ L2 , uξ ∈ Xξ . ξ uξ L ≤ Cξ 2

(7.59)

For u ∈ D(A), we have Aξ Pξ u = Aξ (Pξ − Rξ )u + Aξ Rξ u = [Aξ (Pξ − Rξ )A−1 + Pξ ]Au. Hence, by (7.56), (7.58), and (7.59), we have Aξ Pξ uL2 ≤ CAuL2 , u ∈ D(A). The Heinz and Kato inequality then provides that, for 0 ≤ θ ≤ 1, θ A Pξ u ≤ CAθ uL , u ∈ D(Aθ ). (7.60) 2 ξ L 2

Conversely, for 0 ≤ θ < 34 , we observe that

θ−1 uξ , v = a uξ , Aθ−1 v = a uξ , Rξ Aθ−1 v = Aθξ uξ , A1−θ v Aθ ξ Rξ A θ−1 θ−1 = Aθξ uξ , A1−θ v + Aθξ uξ , A1−θ v . ξ (Rξ − Pξ )A ξ Pξ A

Therefore, by (7.56), (7.58), (7.59), and (7.60), θ A uξ , v ≤ C Aθξ uξ L ξ 2(θ−1) (Rξ − Pξ )Aθ−1 vL2 + A1−θ Pξ Aθ−1 v L ξ 2 2 θ ≤ Cθ Aξ uξ L vL2 . 2

uξ L2 ≤ Cθ Aθξ uξ L2 . Thus, for 0 ≤ θ < 34 , This implies that Aθ θ A uξ

L2

≤ Cθ Aθξ uξ L , 2

uξ ∈ Xξ .

(7.61)

Estimate (7.61) then shows that assumptions (7.5) and (7.28) are fulfilled for α < β < 34 . Similarly, (7.59) shows that assumptions (7.7) and (7.31) are fulfilled with Oξ−1 = O(ξ −2 ). Finally, (7.60) means that, if u0 ∈ D(Aβ ), then β

sup0 0 being some constant. Here, d(X1 , X2 ) is the distance defined by (1.3), and Oξ stands for the rate of approximations and tends to 0 as ξ → 0. For 0 ≤ ξ ≤ 1, the operator Sξ is a nonlinear operator from Xξ into itself which satisfies uniformly the compact perturbation conditions (6.38)–(6.39) of type II, i.e., Sξ U − Sξ V X ≤ δU − V X + Kξ U − Kξ V X , Kξ U − Kξ V Z ≤ LU − V X ,

U, V ∈ Xξ , (7.63)

U, V ∈ Xξ ,

(7.64)

with some fixed 0 ≤ δ < 12 and some uniform constant L > 0. In addition, Kξ is an operator from Xξ into a second Banach space Z with norm · Z which is compactly embedded in X. Clearly, (7.63) and (7.64) imply the uniform Lipschitz condition Sξ U − Sξ V X ≤ (δ + cL)U − V X ,

U, V ∈ Xξ ,

(7.65)

where c > 0 is an embedding constant such that · X ≤ c · Z . We assume also that, as ξ → 0, Sξ converges to S0 at a rate sup Sξ U − S0 U X ≤ D2 Oξ ,

U ∈Xξ

0 < ξ ≤ 1,

(7.66)

D2 > 0 being some constant. We already know by Theorem 6.13 that, under (7.63) and (7.64), each system (Sξn , Xξ , X) has exponential attractors. We will then show that the convergence (7.62) and (7.66) provide continuous dependence of the exponential attractors as ξ → 0. Theorem 7.3 Under (7.63) and (7.64), let (7.62) and (7.66) be satisfied. Then, there exist exponential attractors M∗ξ for (Sξn , Xξ , X), 0 ≤ ξ ≤ 1, respectively, for which the estimate (7.67) d M∗ξ , M∗0 ≤ C ∗ Oξκ , 0 < ξ ≤ 1, holds with some exponent 0 < κ < 1 and some constant C ∗ > 0.

338

7 Numerical Analysis

Proof We will remember the method employed for proving Theorem 6.13 in constructing exponential attractors of the dynamical system (S0n , X0 , X). Let 0 < θ < 1−2δ 2L , and let a = 2(δ +Lθ ) < 1. Let us introduce for n = 0, 1, 2, . . . , an Ra n -radius covering of S0n X0 by a finite number of closed balls of X with centers in S0n X0 , where R is the diameter of X0 . More precisely, let n

S0n X0

⊂

Nθ

X

B (Wn,i ; Ra n ),

Wn,i ∈ S0n X0 ,

(7.68)

i=1

where Nθ is the minimal number of balls of X with radii θ which can cover the unit Z closed ball B (0; 1) of Z centered at the origin. Then, the set defined by M∗0 =

∞

S0n P0 ,

where P0 = Wn,i ; 0 ≤ n < ∞, 1 ≤ i ≤ Nθn ,

n=0

gives an exponential attractor for (S0n , X0 , X). We are then going to construct an exponential attractor for (Sξn , Xξ , X), 0 < ξ ≤ 1, on the basis of the set P0 . Since Wn,i ∈ S0n X0 , there exists Vn,i ∈ X0 such that Wn,i = S0n Vn,i . Furthermore, by (7.62), there exists Vn,i,ξ ∈ Xξ such that Vn,i − Vn,i,ξ X ≤ D1 Oξ . Setting Wn,i,ξ = Sξn Vn,i,ξ , we use Lemma 7.3 be presented below. Then, in view of (7.66), Wn,i − Wn,i,ξ X ≤ L˜ n Oξ with some constant L˜ > 1. Therefore, we obtain from (7.68) the new covering n

S0n X0 ⊂

Nθ

X B (Wn,i,ξ ; Ra n + L˜ n Oξ ),

Wn,i,ξ ∈ Sξn Xξ .

(7.69)

i=1

In the meantime, Lemma 7.3, together with (7.62) and (7.66), yields also that Sξn Xξ ⊂ W(L˜ n Oξ ) S0n X0 , (7.70) where W(L˜ n Oξ ) (S0n X0 ) denotes the L˜ n Oξ -neighborhood of S0n X0 in X. Therefore, it follows that n

Sξn Xξ ⊂

Nθ

X B (Wn,i,ξ ; Ra n + 2L˜ n Oξ ),

Wn,i,ξ ∈ Sξn Xξ .

i=1

Lemma 7.3 Let L0 = δ + cL. Then, for every n = 0, 1, 2, . . . , n S Uξ − S n U0 ≤ Ln+1 − 1 /(L0 − 1) sup Sξ V − S0 V X ξ 0 0 X

+ Uξ − U0 X ,

V ∈Xξ

Uξ ∈ Xξ , U0 ∈ X0 .

(7.71)

4 Convergence of Exponential Attractors

339

Proof We have Sξn Uξ − S0n U0 = Sξn Uξ − S0n Uξ + S0n Uξ − S0n U0 =

n

S0n−i Sξi Uξ − S0n−i+1 Sξi−1 Uξ + S0n Uξ − S0n U0 .

i=1

By the Lipschitz condition (7.65) (ξ = 0), n n S U ξ − S n U 0 ≤ Sξ S i−1 Uξ − S0 S i−1 Uξ + Ln Uξ − U0 X Ln−i ξ 0 0 ξ ξ 0 X X i=1

≤

n

Ln−i 0

i=0

sup Sξ V − S0 V X + Uξ − U0 X .

V ∈Xξ

Hence, the desired estimate is obtained.

ξ be the largest Let us accomplish the proof of theorem. For each 0 < ξ ≤ 1, let N n n n n ˜ ˜ ξ and Ra n < integer satisfying 2L Oξ ≤ Ra , i.e., 2L Oξ ≤ Ra for all 0 ≤ n ≤ N n ˜ 2L Oξ for all Nξ < n < ∞. If we use (7.71) for 0 ≤ n ≤ Nξ , then n

Sξn Xξ

⊂

Nθ

X

B (Wn,i,ξ ; 2Ra n ),

Wn,i,ξ ∈ Sξn Xξ .

i=1

ξ with central points WNξ ,i,ξ , 1 ≤ i ≤ N Nξ , and following the Starting from n = N θ same procedure as in Step 1 of the proof of Theorem 6.12, we can construct inducξ < n < ∞, a 2Ra n -radius covering of S n Xξ such that tively for every N ξ n

Sξn Xξ

⊂

Nθ

X B (W˜ n,i,ξ ; 2Ra n ),

W˜ n,i,ξ ∈ Sξn Xξ .

i=1

Put ξ , 1 ≤ i ≤ Nθn ∪ W˜ n,i,ξ ; N ξ < n < ∞, 1 ≤ i ≤ Nθn . Pξ = Wn,i,ξ ; 0 ≤ n ≤ N Then, for all n = 0, 1, 2, . . . ,

h Sξn Xξ , Pξ ≤ 2Ra n .

(7.72)

n Moreover, by the same reason as for M∗0 , the set M∗ξ = ∞ n=0 Sξ Pξ gives an expon nential attractor for (Sξ , Xξ , X). It now remains to estimate the distance d(M∗ξ , M∗0 ). By definition, M∗ξ

⊂

Nξ n=0

Sξn

Nξ +1 k ξ − n, 1 ≤ i ≤ Nθ Sξ Wk,i,ξ ; 0 ≤ k ≤ N Xξ .

340

7 Numerical Analysis

ξ + 1), together with (7.68) (n = N ξ + 1), yields that The inclusion (7.70) (n = N N +1 h Sξ ξ Xξ , M∗0 ≤ h W ≤h W

Nξ +1 ∗ X 0 , M0 S0

Nξ +1 S0 X0 , P0 ≤ L˜ Nξ +1 Oξ + Ra Nξ +1

(L˜ Nξ +1 Oξ ) (L˜ Nξ +1 Oξ )

≤ 3L˜ Nξ +1 Oξ . Meanwhile, due to Lemma 7.3, n S Wk,i,ξ − S n Wk,i ≤ Ln+1 − 1 /(L0 − 1) ξ 0 0 X × D2 Oξ + L˜ k Oξ ≤ C L˜ Nξ +1 Oξ ; therefore, it follows that d Sξn Wk,i,ξ , M∗0 ≤ C L˜ Nξ +1 Oξ ,

(7.73)

ξ − n, 1 ≤ i ≤ Nθk . 0≤k≤N

Thus, we have proved that h(M∗ξ , M∗0 ) ≤ C L˜ Nξ +1 Oξ . ξ = log(R/2)−log Oξ , we easily observe that L˜ Nξ +1 Oξ ≤ CO κ with the As N ξ ˜ exponent κ =

log L−log a − log a . ˜ log L−log a

The converse estimate h(M∗0 , M∗ξ ) ≤ COξκ is also verified in a similar way. Indeed, by definition, M∗0

⊂

Nξ

S0n

Nξ +1 k ξ − n, 1 ≤ i ≤ Nθ S0 Wk,i ; 0 ≤ k ≤ N X0 .

n=0

ξ ) to obtain that We use (7.69) (n = N N +1 N N h S0 ξ X0 , M∗ξ ≤ h S0 ξ X0 , M∗ξ ≤ h S0 ξ X0 , Pξ

≤ L˜ Nξ Oξ + Ra Nξ ≤ (1 + 2a −1 )L˜ Nξ +1 Oξ . Meanwhile, from (7.73) we conclude that ξ − n, 1 ≤ i ≤ Nθk . d S0n Wk,i , M∗ξ ≤ C L˜ Nξ +1 Oξ , 0 ≤ k ≤ N Hence, h(M∗0 , M∗ξ ) ≤ COξκ with the same exponent κ as above.

4.2 Continuous Case Let (Sξ (t), Xξ , X) be a family of continuous dynamical systems parameterized by 0 ≤ ξ ≤ 1 in a Banach space X. The phase spaces Xξ are all closed subsets of X with

4 Convergence of Exponential Attractors

341

the relation Xξ ⊂ X0 for 0 < ξ ≤ 1, and Sξ (t) are continuous nonlinear semigroups acting on Xξ , respectively. We assume that there exists a compact set B of X for which the set Xξ ∩ B is an absorbing set of (Sξ (t), Xξ , X) for every 0 ≤ ξ ≤ 1. Furthermore, we assume that Xξ ∩ B absorbs itself uniformly for ξ , that is, there exists a uniform time t ∗ > 0 such that Sξ (t)(Xξ ∩ B) ⊂ Xξ ∩ B

for every t ∗ ≤ t < ∞ and 0 ≤ ξ ≤ 1.

(7.74)

We then set Bξ = Xξ ∩ B. Clearly, Bξ is a compact set of X with the relation Bξ ⊂ B0 . Furthermore, if Sξ is set by Sξ = Sξ (t ∗ ), then (Sξn , Bξ , X) defines a discrete dynamical system for each 0 ≤ ξ ≤ 1. Let us assume that conditions (7.62), (7.63), and (7.64) are satisfied for (Sξn , Bξ , X). In addition, let Sξ (t) satisfy the Lipschitz condition Sξ (t)U − Sξ (s)V X ≤ L1 (|t − s| + U − V X ),

0 ≤ s, t ≤ t ∗ , U, V ∈ Bξ , (7.75)

with some uniform constant L1 > 0. Finally, instead of (7.66), let the convergence condition sup

sup Sξ (τ )U − S0 (τ )U X ≤ D3 Oξ ,

0≤τ ≤t ∗ U ∈Bξ

0 < ξ ≤ 1,

(7.76)

be satisfied with some constant D3 > 0. We can then prove a convergence theorem of exponential attractors for the continuous case, namely, Theorem 7.4 below. In fact, by applying Theorem 7.3 to (Sξn , Bξ , X), we deduce that there exist exponential attractors M∗ξ , 0 ≤ ξ ≤ 1, enjoying property (7.67). For each 0 ≤ ξ ≤ 1, we then set Mξ =

Sξ (τ )M∗ξ .

(7.77)

0≤τ ≤t ∗

It is seen that this Mξ gives an exponential attractor for the continuous system (Sξ (t), Xξ , X). Indeed, by the same arguments as in the proof of Theorem 6.15, Mξ is a compact set of X with finite fractal dimension less than dF (M∗ξ ) + 1 and contains the global attractor Aξ of (Sξ (t), Xξ , X), i.e., Aξ ⊂ Mξ ⊂ Xξ . Similarly, Mξ is observed to be invariant under Sξ (t) for every 0 < t < ∞. Finally, the exponential attraction is verified by the following proposition. Proposition 7.3 There exist an exponent k > 0 such that, for any bounded set B of Xξ , h(Sξ (t)B, Mξ ) ≤ CB e−kt ,

tB ≤ t < ∞,

with suitable time tB ≥ 0 and a constant CB > 0 depending on B.

(7.78)

342

7 Numerical Analysis

Proof First, let us consider the case where B = Bξ . Let 0 < t < ∞, and let t = nt ∗ + τ , where n ≥ 0 is an integer and 0 ≤ τ < t ∗ . Since Mξ ⊃ Sξ (τ )M∗ξ , we have h(Sξ (t)Bξ , Mξ ) ≤ h Sξ (t)Bξ , S(τ )M∗ξ . Using (7.75) (t = s = τ ) and noting (7.72), we see that h(Sξ (t)Bξ , Mξ ) ≤ h Sξ (τ )Sξ (t ∗ )n Bξ , S(τ )M∗ξ ≤ L1 h Sξ (t ∗ )n Bξ , M∗ξ ≤ 2L1 Ra n ≤ 2L1 Ra −1 e−(log a

−1 /t ∗ )t

,

0 ≤ t < ∞.

This shows that (7.78) is valid for Bξ with k = log a −1 /t ∗ , tBξ = 0 and CBξ = 2L1 R/a. Secondly, let us consider the general case. Let B be any bounded set of Xξ . By definition, Sξ (t)B ⊂ Bξ for any t such that t ≥ tB , tB ≥ 0 being a suitable time. Then, h(Sξ (t)B, Mξ ) = h(Sξ (t − tB )Sξ (tB )B, Mξ ) ≤ h(Sξ (t − tB )Bξ , Mξ ) ≤ CBξ e−k(t−tB ) = CBξ ektB e−kt ,

tB ≤ t < ∞.

Hence, the assertion of proposition is verified.

In this way, the set Mξ given by (7.77) has been shown to be an exponential attractor of (Sξ (t), Xξ , X) for each ξ . We can then verify that Mξ also enjoys the same property as (7.67). Theorem 7.4 Under (7.63), (7.64), (7.74), and (7.75), assume (7.62) and (7.76) for the phase spaces Bξ and the semigroups Sξ (t), respectively. Then, there exists an exponential attractor Mξ for (Sξ (t), Xξ , X) for which the estimate d(Mξ , M0 ) ≤ COξκ ,

0 < ξ ≤ 1,

(7.79)

holds with some exponent 0 < κ < 1 and constant C > 0. Proof It now suffices to verify under (7.67) that the set Mξ given by (7.77) has property (7.79). Let Uξ ∈ Mξ be any element. Then, Uξ = Sξ (τ )Uξ∗ with some 0 ≤ τ ≤ t ∗ and some Uξ∗ ∈ M∗ξ . By (7.67), there exists an element U0∗ ∈ M∗0 such that d(Uξ∗ , U0∗ ) ≤ C ∗ Oξκ . Put U0 = S0 (τ )U0∗ ∈ M0 . Then, by (7.76) and (7.75), d(Uξ , M0 ) ≤ d(Uξ , U0 ) = d Sξ (τ )Uξ∗ , S0 (τ )U0∗ ≤ d Sξ (τ )Uξ∗ , S0 (τ )Uξ∗ + d S0 (τ )Uξ∗ , S0 (τ )U0∗

Notes and Further Researches

343

≤ D3 Oξ + L1 C ∗ Oξκ . Thus, we obtain that h(Mξ , M0 ) ≤ COξκ . By the same argument, we can observe also h(M0 , Mξ ) ≤ COξκ . Hence, (7.79) has been verified.

Notes and Further Researches The local convergence theorems, Theorems 7.1 and 7.2, for semilinear and quasilinear equations, respectively, have been obtained by the semigroup methods. The methods are based on the fact that the sectorial operators with angles < π2 acting in infinite-dimensional spaces can be approximated by those with the similar angles < π2 in finite-dimensional spaces. Those were founded by Fujita–Mizutani [FM76], Baker–Bramble–Thomée [BBT77], Brenner–Thomée [BT79], and Piskarev [Pis79], cf. also Ushijima [Ush75, Ush79], and were developed by Crouzeix–Larsson– Piskarev–Thomée [CLPT93], Palencia [Pal93, Pal95], and others. In the meanwhile, the methods were applied to nonautonomous linear parabolic equations by Suzuki [Suz78, Suz82] and Sammon [Sam82, Sam83] and to semilinear parabolic equations by Keeling [Kee90]. Theorems 7.1 and 7.2 were obtained by Nakaguchi–Yagi [NY99b, NY01, NY02]. The alternative way for constructing approximate solutions for parabolic equations is the variational methods, see the books by Dautray–Lions [DL91]. For the linear problems, we quote Helfrich [Hel74], Bramble–Schatz–Thomée–Wahlbin [BSTW77] and Savaré [Sav93], and for the nonlinear problems, Lubich–Ostermann [LO93, LO95] and Nakaguchi–Yagi [NY03]. In Sects. 1 and 2, we handled only semidiscretization on the spatial variables. However, it is known that the usual discretization schemes for ordinary differential equations in finite-dimensional spaces can be properly extended for ordinary evolution equations in infinite-dimensional spaces, too, cf. Le Roux [LeR79] and Ostermann–Roche [OR92]. In the papers Lubich–Ostermann [LO95] and Nakaguchi–Yagi [NY99a], the Runge–Kutta methods (cf. Butcher [But87] and Iserles [Ise96]) were applied to the Cauchy problem for quasilinear abstract evolution equations in Banach spaces, and some convergence theorems were obtained. These results show that the Runge–Kutta methods are available for the approximate problems (7.32) in order to build full-discretization schemes for the continuous problem (7.15) as done in [NY99b, NY01, NY02]. The structural assumptions made in Sects. 1 and 2 can actually be verified by concrete discretization schemes. Section 3 was devoted to verifying some of them for the finite element methods in two-dimensional domains. Propositions 7.1 and 7.2 were first verified in [NY02]. In higher-dimensional domains, however, the verification is left to be studied. We may have to use higher-order finite element methods for obtaining a similar result as Proposition 7.1. For the finite element methods, we quote the monographs Ciarlet [Cia78], Fujita–Suzuki [FS91], and Thomée [Tho97].

344

7 Numerical Analysis

Theorems 7.3 and 7.4 were proved by modifying the arguments in Efendiev–Yagi [EY05] (cf. also [AY04b]). As seen in the subsequent chapters, the positivity of solutions of the model equations is extremely important in constructing exponential attractors. Nevertheless, the approximate equations do not admit positivity in general (cf. Aida–Yagi [AY03]). Recently, Saito [Sai04, Sai07] and Efendiev–Nakaguchi– Wendland [ENW09] have presented discretization schemes which conserve positivity. Using them, we can show that the approximate problems generate dynamical systems in which exponential attractors equally exist. Theorems 7.3 and 7.4 are then applicable to the family of dynamical systems.

Chapter 8

Semiconductor Models

In 1950, Shockley presented the drift-diffusion model ⎧ ∂u ⎪ ⎨ ∂t = aΔu − ∇ · [μu∇χ] + f (1 − uv) + g(x) ∂v = bΔv + ∇ · [νv∇χ] + f (1 − uv) + g(x) ⎪ ⎩ ∂t 0 = cΔχ − u + v + h(x)

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞),

for describing the flows of electrons and holes in a semiconductor system. Here, the unknown functions u(x, t) and v(x, t) denote the densities of electrons and holes, respectively, in a semiconductor device Ω at time t. Electrons and holes diffuse with positive diffusion rates a > 0 and b > 0. The terms −∇ · [μu∇χ] and ∇ · [νv∇χ] denote the drift-diffusions of electrons and holes, where μ and ν denote their mobilities, respectively. The function χ(x, t) denotes electrostatic potential and is determined by the Poisson equation, where c > 0 is the dielectric constant. Meanwhile, the reaction term f (1 − uv) denotes the effects of generation and recombination of electrons and holes. The functions g(x) ≥ 0 and h(x) are given, and both denote external constraints to the system. In this chapter, we show the global existence of solutions and construct a dynamical system and exponential attractors.

1 Prototype Semiconductor Equations 1.1 Model Equations

We consider the initial-boundary-value problem for a semiconductor model A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_8, © Springer-Verlag Berlin Heidelberg 2010

345

346

8 Semiconductor Models

⎧ ∂u ⎪ ⎪ ∂t = aΔu − μ∇ · [u∇χ] + f (1 − uv) + g(x) ⎪ ⎪ ⎪ ∂v ⎪ ⎪ ⎪ ∂t = bΔv + ν∇ · [v∇χ] + f (1 − uv) + g(x) ⎪ ⎪ ⎪ ⎨0 = cΔχ − u + v + h(x) ⎪ u=v=χ =0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂u = ∂v = ∂χ = 0 ⎪ ∂n ∂n ∂n ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x)

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞), on ΓD × (0, ∞),

(8.1)

on ΓN × (0, ∞), in Ω,

in a two-dimensional bounded domain Ω with Lipschitz boundary ∂Ω. We make the following assumptions. The boundary ∂Ω of Ω is split into two parts ΓD and ΓN in such a way that (2.41) and (2.42) are satisfied. In particular, ΓD is a nonempty open set of ∂Ω. On ΓD , the Dirichlet boundary conditions are imposed on u, v, and χ ; on the other hand, on ΓN , the Neumann boundary conditions are imposed. At x ∈ ΓN , n(x) denotes the outer normal vector of the boundary. The mobilities μ and ν of drift-diffusions are both positive constants (> 0). Physically these mobilities are determined by the scattering process of electrons and holes by the crystal lattices, the impurities, and so on, and may depend on u, v and ∇u, ∇v. In this chapter, however, we are concerned with the simplest case that μ and ν are constants. From a physical point of view, this means that the densities of electrons and holes are not so high. The rates of generation and recombination of electrons and holes are nonnegative constants, and the two rates coincide. These rates are denoted by f ≥ 0. If the two rates are positive constants, then it is always possible to make them equal by suitable normalization. The function g(x) is a given nonnegative function such that 0 ≤ g ∈ L2 (Ω),

(8.2)

and h(x) is a given real function such that h ∈ L∞ (Ω; R).

(8.3)

1.2 Some Preliminary Results Let H˚ D1 (Ω) be the space given by (2.25). We consider the sesquilinear form ∇u · ∇v dx, u, v ∈ H˚ D1 (Ω). a(u, v) = Ω

It is easy to see that this form satisfies (2.6) and (2.7) on H˚ D1 (Ω). For (2.7), we may recall Remark 2.3. Let H˚ D1 (Ω) ⊂ L2 (Ω) ⊂ H˚ D1 (Ω)∗ = HD−1 (Ω) be a triplet of spaces, and let Λ be the associated sectorial operator of HD−1 (Ω) which is an isomorphism from H˚ D1 (Ω) onto HD−1 (Ω) and whose part in L2 (Ω) is a positive

1 Prototype Semiconductor Equations

347

definite self-adjoint operator. As noticed by Theorem 2.5, Λ is considered as a realization of the Laplace operator −Δ in HD−1 (Ω) under the Dirichlet boundary conditions on ΓD and the Neumann boundary conditions on ΓN . 1 By Theorem 2.35, the domain of the square root Λ 2 coincides with L2 (Ω), i.e., 1 D Λ 2 = L2 (Ω). Moreover, for

1 2

(8.4)

≤ θ ≤ 1, it holds that

D(Λθ ) = L2 (Ω), H˚ D1 (Ω) 2θ −1 ⊂ [L2 (Ω), H 1 (Ω)]2θ−1 = H 2θ−1 (Ω).

(8.5)

The part Λ|L2 may not have the optimal shift property that Λu ∈ L2 (Ω) always implies u ∈ H 2 (Ω). According to Theorem 2.10, however, it holds that D(Λ|L2 ) ⊂ Wp10 (Ω) with the estimate u Wp1 ≤ C Λu L2 , 0

u ∈ D(Λ|L2 ),

(8.6)

where p0 is a certain fixed number such that p0 > 2.

2

We finally extend the definition of the drift term ∇ · [u∇χ] for all u ∈ H p (Ω) and all χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω), provided that 2 < p < ∞. Proposition 8.1 Let 2 < p < ∞ be arbitrarily fixed. The correspondence (u, χ) → ∇ · [u∇χ] is continuous from H 2 (Ω) × [D(Λ|L2 ) ∩ Wp1 (Ω)] into L2 (Ω). It is possi2

ble to extend this correspondence continuously from H p (Ω) × [D(Λ|L2 ) ∩ Wp1 (Ω)] into HD−1 (Ω). Proof Let u ∈ H 2 (Ω) and χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω). Then, we observe directly that ∇ · [u∇χ] = ∇u · ∇χ − uΛχ ∈ L2 (Ω) in the sense of distribution. In addition, by the definition of Λ, ∇χ · ∇(uv) dx = ∇χ · (v∇u + u∇v) dx, (Λχ, uv)L2 = Ω

Ω

v ∈ H˚ D1 (Ω),

since u ∈ H 2 (Ω) and v ∈ H˚ D1 (Ω) imply that uv ∈ H˚ D1 (Ω). Noting (1.26), we obtain that u∇χ · ∇v dx, v ∈ H˚ D1 (Ω).

∇ · [u∇χ], vH −1 ×H˚ 1 = (∇ · [u∇χ], v)L2 = − D

D

Ω

2

For u ∈ H p (Ω) and χ ∈ D(ΛL2 ) ∩ Wp1 (Ω), it is observed (due to (1.86)) that |u∇χ∇v| dx ≤ u L Ω

2p p−2

∇χ Lp ∇v L2 ≤ Cp u

2

Hp

χ Wp1 v H˚ 1 . D

348

8 Semiconductor Models 2

Since H 2 (Ω) is a dense subspace of H p (Ω), we are led to extend ∇ · [u∇χ] ∈ 2

HD−1 (Ω) continuously for all u ∈ H p (Ω) and χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω) by the formula

∇ · [u∇χ], vH −1 ×H˚ 1 = − u∇χ · ∇v dx, v ∈ H˚ D1 (Ω). (8.7) D

D

Ω

We also have the same estimate ∇ · [u∇χ] H −1 ≤ C u D

2

Hp

χ Wp1 ,

2

u ∈ H p (Ω), χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω).

(8.8)

Of course, this proposition is available with p = p0 . However, in this case it holds that D(Λ|L2 ) ∩ Wp10 (Ω) = D(Λ|L2 ) due to (8.6). We choose as an underlying space for dealing with (8.1) the product space of HD−1 (Ω), i.e., ϕ X= ; ϕ ∈ HD−1 (Ω) and ψ ∈ HD−1 (Ω) . (8.9) ψ For electrons and holes, the initial densities are taken in the space u0 ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ v0 ∈ L2 (Ω) . K= v0

(8.10)

2 Local Solutions 2.1 Construction of Local Solutions We formulate problem (8.1) as the Cauchy problem for an abstract equation dU 0 < t < ∞, dt + AU = F (U ), (8.11) U (0) = U0 , in the space X. Here, A is a diagonal matrix operator A = diag{aΛ, bΛ} in X with domain u 1 1 ˚ ˚ D(A) = ; u ∈ HD (Ω) and v ∈ HD (Ω) , v where Λ is the operator introduced above. From (8.4) and (8.5) we have 1 D(A 2 ) = L2 (Ω), D(Aθ ) ⊂ H2θ−1 (Ω), 12 < θ ≤ 1

(8.12)

2 Local Solutions

349

(see (1.84) and (1.85)). The nonlinear operator F is given by F (U ) =

−μ∇ · [u∇(cΛ)−1 (−u + v + h(x))] + f (1 − uv) + g(x) ν∇ · [v∇(cΛ)−1 (−u + v + h(x))] + f (1 − uv) + g(x)

with domain D(F ) =

2 2 u ; u ∈ H p0 (Ω) and v ∈ H p0 (Ω) , v

where p0 > 2 is the number for which (8.6) holds. In view of (8.7), F (U ) is well defined. Let η = 12 + p10 < 1. Then, by (8.12) we have D(Aη ) ⊂ D(F ). The initial value U0 is taken in K. We are going to apply Theorem 4.1 with β = 12 and η = 12 + p10 . By virtue of (8.6) and (8.8),

∇ · [u∇Λ−1 (−u + v + h(x))] H −1 ≤ C u H 2/p0 − u + v + h L2 , D

∇ · [v∇Λ−1 (−u + v + h(x))] H −1 ≤ C v H 2/p0 − u + v + h L2 . D

Similarly, uv H −1 = D

sup uvw dx ≤ C

w H˚ 1 ≤1

Ω

D

sup

w H˚ 1 ≤1 D

u L

2p0 p0 −2

v L2 w Lp0

≤ C u H 2/p0 v L2 . Therefore, we verify by (8.12) that 1 1 F (U ) − F (V ) X ≤C A 2 U X + A 2 V X + 1 Aη (U − V ) X 1

+ ( Aη U X + Aη V X + 1)A 2 (U − V )X , U, V ∈ D(Aη ). This shows that (4.2) is fulfilled with β = 12 . Consequently, Theorem 4.1 is available to (8.11). We thus conclude that, for any U0 ∈ K, there exists a unique local solution U to (8.11) in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; X), where TU0 > 0 is determined by the norm U0 L2 only. Moreover, √ t AU (t) X + U (t) L2 ≤ CU0 ,

0 < t ≤ TU 0 ,

where CU0 > 0 is also determined by the norm U0 L2 only.

(8.13)

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8 Semiconductor Models

2.2 Nonnegativity of Solutions For U0 ∈ K, let U = t (u, v) be the local solution obtained above in the function space (8.13). We will prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . Let us first verify that U (t) is real valued. Indeed, it is clear that the complex conjugate U (t) of U (t) is also a local solution of (8.11) with the same initial value; so, by the uniqueness of solution, U (t) = U (t); hence, U (t) is real valued. Let us next introduce auxiliary linear problems. It is not difficult to construct sequences {uk }, {vk } of functions such that uk , vk ∈ C0,1 ([0, TU0 ]; L∞ (Ω))

(8.14)

and that, as k → ∞, uk → u and vk → v in C([0, TU0 ]; L2 (Ω)). (For example, we cut off the values of |u(t)| ≥ k and |v(t)| ≥ k and use the mollifier concerning the variable t.) Using uk and vk , we formulate the linear problem ⎧ ⎨ d U˜ k + AU˜ = B (t)U˜ + F, 0 < t ≤ T , k k k U0 dt (8.15) ⎩U˜ k (0) = U0 , in X. Here, Bk (t) is the family of closed linear operators given by

−μ∇ · [u˜ k ∇χk (t)] − u˜ k vk (t) u˜ Bk (t)U˜ k = , U˜ k = k ∈ D(Bk (t)), v˜k ν∇ · [v˜k ∇χk (t)] − uk (t)v˜k where χk (t) = (cΛ)−1 [−uk (t) + vk (t) + h], with domains D(Bk (t)) ≡ D(Aη ), and F is the vector of X given by F = [f + g(x)]e, where e = t (1, 1). The operators Bk (t) are weaker linear operators than A in the sense that D(A) ⊂ D(Aη ) ≡ D(Bk (t)) for all 0 < t ≤ TU0 . In addition, they are seen from (8.7) to have regularity [Bk (t) − Bk (s)]A−η L(X) ≤ Ck |t − s|,

0 ≤ s, t ≤ TU0 ,

due to (8.14). Then, we can apply Theorem 3.14 with μ˜ = 1 − η and ν˜ = 1 to this problem. It is then deduced that problem (8.15) possesses a unique solution U˜ k in the function space: U˜ k ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; X). Furthermore, by direct calculations (cf. Theorem 4.3), it is possible to verify that t η Aη [U˜ k (t) − U (t)] X ≤ C

1 sup A 2 [Uk (t) − U (t)]X ,

0≤t≤TU0

that is, as k → ∞, U˜ k (t) converges to U (t) for every t ∈ (0, TU0 ] in D(Aη ). Therefore, if we show that u˜ k (t) ≥ 0 and v˜k (t) ≥ 0, then it follows that u(t) ≥ 0 and v(t) ≥ 0.

2 Local Solutions

351

Our goal is therefore to prove that u˜ k (t) ≥ 0 and v˜k (t) ≥ 0 for every 0 < t ≤ TU0 . To this end, we use the truncation method. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. Since u˜ k ∈ C (0, TU0 ]; H˚ D1 (Ω) ∩ C1 (0, TU0 ]; HD−1 (Ω) , we get by (1.101) that the function ψk (t) = Ω H (u˜ k (x, t)) dx is continuously differentiable with the derivative ψk (t) = H (u˜ k ), −aΛu˜ k − μ∇ · [u˜ k ∇χk ]H˚ 1 ×H −1 D D dt + (H (u˜ k ), −u˜ k vk + f + g(x))L2 . By property (1.96), we have

H (u˜ k ), −aΛu˜ k H˚ 1 ×H −1 = −a D

D

∇H (u˜ k ) · ∇ u˜ k dx

Ω

|∇H (u˜ k )|2 dx ≤ 0.

= −a Ω

Similarly, by (8.7) and Lemma 8.1 presented below, u˜ k ∇H (u˜ k ) · ∇χk dx

H (u˜ k ), −μ∇ · [u˜ k ∇χk ]H˚ 1 ×H −1 = μ D

D

Ω

H (u˜ k )∇H (u˜ k ) · ∇χk dx

=μ Ω

=

μ 2c

H (u˜ k )2 (−uk + vk + h) dx.

Ω

Finally, due to (8.2) and f ≥ 0, H (u˜ k )[−u˜ k vk + f + g(x)] dx ≤ − vk H (u˜ k )2 dx. Ω

Ω

Thus, in view of (8.3) and (8.14), ψk (t) ≤ Ck ψk (t); consequently, ψk (t) ≤ ψk (0)eCk t . Therefore, ψk (0) = 0 implies ψk (t) ≡ 0, that is, u˜ k (t) ≥ 0 for every 0 < t ≤ TU0 . It is the same for v˜k (t) ≥ 0 for every 0 < t ≤ TU0 . Lemma 8.1 For u ∈ H˚ D1 (Ω) and χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω), where 2 < p < ∞, we have 1 u∇u · ∇χ dx = u2 Λχ dx. 2 Ω Ω Proof If u ∈ H˚ D1 (Ω) ∩ L∞ (Ω), then u2 ∈ H˚ D1 (Ω). Therefore, the result is verified directly noting that u∇u = 12 ∇u2 . For general u ∈ H˚ D1 (Ω), it is sufficient to approximate u by cutoff functions Ψk (u) ∈ H˚ D1 (Ω) ∩ L∞ (Ω), where Ψk (u) ≡ −k for

352

8 Semiconductor Models

u ≤ −k, Ψk (u) = u for |u| ≤ kd, and Ψk (u) ≡ k for u ≥ k. Owing to (1.96), we verify that Ψ (uk ) → u in H˚ D1 (Ω) as k → ∞. Meanwhile, note by (1.79) that it follows from u ∈ H˚ D1 (Ω) ⊂ H 1 (Ω) that u ∈ L 2p (Ω). p−2

3 Global Solutions 3.1 A Priori Estimates for Local Solutions For U0 ∈ K, let U = t (u, v) denote a local solution of (8.11) on [0, TU ] in the function space: 0 ≤ u ∈ C((0, TU ]; H˚ D1 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; HD−1 (Ω)), 0 ≤ v ∈ C((0, TU ]; H˚ D1 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; HD−1 (Ω)). (8.16) Proposition 8.2 There exists a constant C > 0 such that, for any U0 ∈ K and any local solution of (8.11) in the function space (8.16), it holds that U (t) L2 ≤ C( U0 L2 + 1),

0 ≤ t ≤ TU , U0 ∈ K.

(8.17)

Proof Consider the duality product of HD−1 (Ω) × H˚ D1 (Ω) between the first equation of (8.11) and u. Then, 1 d 2 2 u dx + a |∇u| dx − μ u∇u · ∇χ dx + f u2 v dx 2 dt Ω Ω Ω Ω = [f + g(x)]u dx, Ω

where χ = (cΛ)−1 [−u + v + h(x)]. Similarly, by the second equation of (8.11), 1 d 2 2 v dx + b |∇v| dx + ν v∇v · ∇χ dx + f uv 2 dx 2 dt Ω Ω Ω Ω = [f + g(x)]v dx. Ω

We sum up these two equalities after multiplying the first one by ν and the second one by μ, respectively. Then, since Lemma 8.1 yields that μν μν (−u∇u · ∇χ + v∇v · ∇χ) dx = (v 2 − u2 )Λχ dx 2 Ω Ω μν = (u − v)2 (u + v) dx 2c Ω μν + h(x)(v 2 − u2 ) dx, 2c Ω

3 Global Solutions

353

we have 1 d 2 dt

(νu + μv ) dx + 2

2

Ω

aν|∇u|2 + bμ|∇v|2 dx

Ω

μν + (u − v)2 (u + v) dx + f (νu + μv)uv dx 2c Ω Ω μν = [f + g(x)](νu + μv) dx + h(x)(u2 − v 2 ) dx. 2c Ω Ω

Furthermore, by (8.2), [f + g(x)](νu + μv) dx ≤ f + g L2 μu + νv L2 Ω

≤ f + g L2 ζ u 2L2 + v 2L2 + Cζ

with an arbitrarily small parameter ζ > 0. Similarly, by (8.3),

ζ (u − v)2 (u + v) + Cζ (u + v) dx h(x)(u2 − v 2 ) dx ≤ h L∞ Ω

Ω

≤ h L∞

ζ (u − v)2 (u + v) + ζ (u2 + v 2 ) + Cζ dx.

Ω

Meanwhile, by Poincaré’s inequality (see Remark 2.3), 2 2 aν|∇u| + bμ|∇v| dx ≥ α (u2 + v 2 ) dx Ω

Ω

with some positive number α > 0. Therefore, taking ζ > 0 sufficiently small, we obtain that α 1 d 2 2 (νu + μv ) dx + (u2 + v 2 ) dx ≤ C 2 dt Ω 2 Ω with some constant C > 0. Hence (cf. (1.58)),

u(t) 2L2 + v(t) 2L2 ≤ C e−δt u0 2L2 + v0 2L2 + 1 , with some positive exponent δ > 0 and constant C > 0.

0 ≤ t ≤ TU ,

(8.18)

3.2 Global Solutions As the a priori estimates (8.17) are verified for local solutions, we can apply Corollary 4.1 to conclude the global existence of solutions. Thus, for any U0 ∈ K, (8.11)

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8 Semiconductor Models

possesses a unique global solution in the function space: 0 ≤ u ∈ C((0, ∞); H˚ D1 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); HD−1 (Ω)), 0 ≤ v ∈ C((0, ∞); H˚ D1 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); HD−1 (Ω)). (8.19)

3.3 Global Norm Estimates For U0 ∈ K, let U (t; U0 ) be the global solution of (8.11) with the initial value U0 in the function space (8.19). By (8.18), there exist an exponent δ > 0 and a constant C˜ > 0 such that, for any U0 ∈ K, it holds that ˜ −δt U0 L2 + 1), U (t; U0 ) L2 ≤ C(e

0 ≤ t < ∞, U0 ∈ K.

(8.20)

4 Dynamical System 4.1 Construction of Dynamical System We are now in a position to construct a dynamical system determined from problem (8.11) in the universal space X by following the general methods described in Chap. 6, Sect. 5, where the exponent β will be taken as β = 12 . We already know that, for any U0 ∈ K, (8.11) and hence (8.1) possesses a unique global solution U (t; U0 ) in the function space (8.19). By S(t)U0 = U (t; U0 ), we define a nonlinear semigroup S(t) acting on K. Estimate (8.20) shows that (6.56) is fulfilled. So, (8.11) defines a dynamical system (S(t), K, L2 (Ω)). Recall that 1 D(A 2 ) = L2 (Ω).

4.2 Exponential Attractors We shall construct exponential attractors for the dynamical system (S(t), K, L2 (Ω)). To this end, however, it suffices to verify (6.59) and (6.60). The space condition (6.59) is clear. Meanwhile, (8.20) shows that the dissipative condition (6.60) is also fulfilled. Hence, (S(t), K, L2 (Ω)) enjoys a family of exponential attractors.

5 Three-Dimensional Problem In this section, we want to discuss the three-dimensional problem. Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary ∂Ω which is split into two parts ΓD and ΓN satisfying (2.41) and (2.42).

Notes and Further Researches

355

As a matter of fact, we can argue in a similar way as in the two-dimensional case. However, in order to extend definition of the drift term ∇ · [u∇χ] as (8.7), we have to assume the stronger shift property that Λu ∈ L2 (Ω) implies u ∈ Wp11 (Ω) with some p1 > 3 and the estimate u Wp1 ≤ C Λu L2 , 1

u ∈ D(Λ|L2 ).

(8.21)

We set the same underlying space X as (8.9). Three-dimensional problem (8.1) is then written as an abstract problem of the form (8.8), where A and F are the same linear and nonlinear operators, respectively, as before. The only difference is that F has the domain 3 3 u D(F ) = ; u ∈ H p1 (Ω) and v ∈ H p1 (Ω) , v where p1 > 3 is the number for which (8.21) is valid. Then, D(Aη ) ⊂ D(F ) with η = 12 + 2p3 1 < 1 due to (8.12) (which is independent of the dimensions), and F is

shown to fulfill (4.2) with β = 12 and η = 12 + 2p3 1 . We can then repeat the same arguments as in the preceding sections to obtain the local existence of solutions, nonnegativity of local solutions, and the a priori estimates of local solutions. Hence, under (8.21), we can deduce the uniqueness and global existence of solutions for any pair (u0 , v0 ) of initial functions in K (see (8.10)).

Notes and Further Researches The semiconductor model (8.1) was presented by Shockley to describe the flows of electrons and holes in a semiconductor in the middle of the last century. In his historic book [Sho50] published in 1950, we may find the origin of the macroscopic studies of physical systems which consist of many individuals possessing some uniform properties by the analytical methods. For the physical backgrounds of the semiconductor model and the details of modeling, we refer to the references Sze [Sze81], Selberherr [Sel83], and Gajewski–Gröger [GG90]. Many authors have already contributed to the study of semiconductor equations. The evolutional problems have been studied by many papers including [FI95a, FI95b, FI95c, Gaj85, GG86, Jer87, Moc74, Moc75, ST85]. In particular, the asymptotic behavior of solutions was studied in [FI95c, Gaj85, GG86, Moc74]. Mock [Moc74] first proved, in a simple case, that every solution converges to the stationary solution at an exponential rate. Gajewski [Gaj85] and Gajewski–Gröger [GG86] generalized this result, but they still assumed some conditions which guarantee the unique solvability of the stationary problem. In the general case where the stationary solutions are not unique, Fang–Ito [FI95c] constructed a global attractor for the dynamical system determined from (8.1). More precisely, they constructed a global attractor with finite Hausdorff dimension under the spectral gap condition for the operator Λ. The stationary problems have been studied, e.g., by [FN94, FN96, Jer85,

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8 Semiconductor Models

Moc72, Mar86] and [BF02, Chap. 6]. The existence of stationary solutions to (8.1) was shown in various situations. On the contrary, Bensoussan–Frehse [BF02, Theorem 6.2] gave some special conditions which imply the uniqueness of stationary solutions. The same results as in this chapter was already obtained in Favini–Lorenzi– Yagi [FLY06]. However, in [FLY06], the abstract result of [OY02] was used to construct local solutions. In addition, the result due to [EMZ00] was used to construct exponential attractors. As shown in Sect. 4.2, we were able to construct a finite-dimensional global attractor without any spectral gap condition as assumed by Fang–Ito [FI95c]. For constructing (local) solutions to (8.11), we indeed made an essential use of the shift property (8.6) which was established in Bensoussan– Frehse [BF02] (cf. also Lorenzi [Lor78]). In the three-dimensional case, however, no general conditions are known for ensuring such a shift property (8.21). Recently, mathematical models incorporating quantum effects are attracting interest of researchers. Ancona–Tiersten [AnT87] and Ancona–Iafrate [AI89] presented the following quantum-drift-diffusion model: √ ⎧ Δ u

⎨ ∂u √ + F, = aΔu − ∇ · [μu∇χ] − ∇ · αu∇ ∂t u √

⎩ ∂v = bΔv + ∇ · [νv∇χ] − ∇ · βv∇ Δ√ v + F ∂t v (cf. also Gardner [Gar94]). Here, u = u(x, t) and v = v(x, t) denote as before the densities√of electrons and holes, respectively, in a semiconductor device. The terms √ Δ u Δ v √ √ −αu∇ u and −βv v denote quantum corrections for the flows of u and v, respectively, α > 0 and β > 0 being some positive coefficients, and F denotes the rate of generation and recombination of electrons and holes. Problem of constructing the dynamical system determined from such a quantum model remains entirely to be studied.

Chapter 9

Activator–Inhibitor Models

Let us consider the activator–inhibitor system. The general model equation is given by the reaction–diffusion system ∂u ∂t = aΔu + γf (u, v) in Ω × (0, ∞), (9.1) ∂v ∂t = bΔv + γ g(u, v) in Ω × (0, ∞), in a domain Ω ⊂ Rd . Here, u denotes the concentration of the activator, and v the concentration of the inhibitor in Ω, respectively. The functions f (u, v) and g(u, v) denote the kinetics and are real smooth functions defined for u ≥ 0 and v ≥ 0. The activator and the inhibitor diffuse in Ω with diffusion rates a > 0 and b > 0, respectively. The constant γ > 0 denotes the reaction rate. Species of the activator or inhibitor must be specified in each activator–inhibitor system under consideration. The form of kinetic functions f (u, v) and g(u, v) must also be fixed in an adequate manner. Let the system possess a positive homogeneous stationary solution (u, v), i.e., f (u, v) = 0 and g(u, v) = 0. The partial derivatives of the kinetics at (u, v) are denoted by f u = fu (u, v), f v = fv (u, v), g u = gu (u, v), and g v = gv (u, v). We assume that f u > 0, that is, the activator is created autocatalystically, and that g v < 0 with the relation f u + g v < 0, that is, the inhibitor degrades faster than the activator grows. We assume also that f v < 0 and g u > 0; therefore, the inhibitor restrains growth of the activator, and oppositely, the activator produces the inhibitor, too. The relation f u g v > f v g u is also assumed. Hence, the partial derivatives are assumed to satisfy the following inequalities: f u > 0,

f u + g v < 0,

It then holds that f u + g v < 0,

f ugv > f v gu,

and f u g v > f v g u .

(9.2)

and ag v + bf u < 2 ab(f u g v − f v g u ) (9.3)

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_9, © Springer-Verlag Berlin Heidelberg 2010

357

358

9 Activator–Inhibitor Models

if the diffusion constants a and b coincide, i.e., a = b. According to the arguments in Chap. 6, Sect. 6.4 (since (6.82) is valid), this condition indicates that the homogeneous solution (u, v) is a stable stationary solution. Thus, under (9.2), the coincidence a = b provides the stability of the homogeneous solution (u, v). On the contrary, if the diffusion rate b of the inhibitor is greater than the a of the activator and their difference is large enough, then the last inequality in (9.3) becomes false, that is, (6.82) in Chap. 6, Sect. 6.4 is no longer valid. This means that the stationary solution (u, v) loses stability. Thus, under the same condition as (9.2), the relation b > a with sufficiently large difference can provide the instability of the homogeneous stationary solution. Such a principle is called the diffusion-driven instability and was originally proposed by Turing in 1952. This chapter is devoted to handling a typical model, choosing suitable functions f (u, v) and g(u, v), and to showing rigorously that the Turing instability principle actually takes place.

1 Prototype Reaction–Diffusion Model We take the kinetic functions f (u, v) = k1 − k2 u +

k3 u2 , v

g(u, v) = k4 u2 − k5 v,

presented by Gierer–Meinhardt [GM72] and Meinhardt [Mei82] (cf. also Meinhardt [Mei92] and Murray [Mur03, Chap. 2]), where ki are positive constants for i = 1, . . . , 5. After nondimensionalization (see [Mur03, p. 78]), the equations are written by ∂u u2 ∂t = aΔu + γ (d − cu + v ) in Ω × (0, ∞), ∂v ∂t

= bΔv + γ (u2 − v)

in Ω × (0, ∞).

So, we consider the initial-boundary-value problem ⎧ ∂u 2 = aΔu + γ (d − cu + uv ) ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂v 2 ∂t = bΔv + γ (u − v) ⎪ ∂u ⎪ = ∂v = 0 ⎪ ⎪ ⎩ ∂n ∂n v(x, 0) = v0 (x) u(x, 0) = u0 (x),

in Ω × (0, ∞), in Ω × (0, ∞), on ∂Ω × (0, ∞), in Ω,

(9.4)

where 0 < a ≤ b are diffusion constants of the activator and inhibitor, respectively, and where c, d, and γ are positive constants (> 0). We consider this problem in a three-dimensional C2 or convex, bounded domain Ω. Let from now on f (u, v) and g(u, v) denote the functions f (u, v) = d − cu +

u2 , v

g(u, v) = u2 − v,

(9.5)

2 Local Solutions

359

respectively. Then, fu (u, v) = −c + gu (u, v) = 2u,

2u u2 , fv (u, v) = − 2 , v v gv (u, v) − 1.

It is easy to verify that (9.4) possesses a unique homogeneous stationary solution (u, v), which is given by d +1 u= c

and v =

d +1 c

2 (9.6)

.

Obviously, f u = fu (u, v) =

c(1 − d) , d +1

f v = fv (u, v) = −

g u = gu (u, v) =

2(d + 1) , c

g v = gv (u, v) = −1.

c2 , (d + 1)2

We then easily verify that, at least under the assumptions that 0 < d < 1 and 0 < c

0 . (9.8) Ω v0 As the equation of (9.4) on u has a singularity at v = 0, we will impose the condition ess infΩ v0 > 0 on the initial functions v0 . We introduce a diagonal matrix operator A = diag{A1 , A2 } of X with domain D(A) = H2N (Ω), where A1 and A2 are realizations of the operators −aΔ + γ c

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9 Activator–Inhibitor Models

and −bΔ + γ in L2 (Ω), respectively, under the homogeneous Neumann boundary conditions on ∂Ω. On account of Theorem 1.25, A1 and A2 are positive definite self-adjoint operators of L2 (Ω) with domains HN2 (Ω) (see (2.34)), H2N (Ω) being the product space of HN2 (Ω). Consequently, A is a positive definite self-adjoint operator of X. Furthermore, on account of Theorems 16.7 and 16.9 in Chap. 16, it is verified that 3 D(Aθ ) = H2θ (Ω) if 0 ≤ θ < , 4 3 < θ ≤ 1. D(Aθ ) = H2θ N (Ω) if 4

(9.9) (9.10)

Let ε > 0 be an arbitrarily fixed positive number. We introduce a nonlinear operator defined as follows. Let η be a fixed exponent such that 34 < η < 1. Let Fε : D(Aη ) → X be given by γ d + γ u2 /χε (Re v) u (9.11) , U = ∈ D(Aη ). Fε (U ) = v γ u2 Here, χε (ξ ) is a cutoff function such that χε (ξ ) = ξ for ε ≤ ξ < ∞ and χε (ξ ) ≡ ε for −∞ < ξ ≤ ε; obviously, χε (ξ ) is a Lipschitz continuous function defined for ξ ∈ R. In view of (1.76) and (9.10), it holds true that D(Aη ) ⊂ H2η (Ω) ⊂ L∞ (Ω), which shows that Fε certainly maps D(Aη ) into X. Problem (9.4), at least for U0 ∈ K satisfying ess infΩ v0 > ε, is now formulated as the Cauchy problem for a semilinear abstract equation dU dt + AU = Fε (U ), 0 < t < ∞, (9.12) U (0) = U0 , in X. Let us apply the general results in Chap. 4 to construct local solutions. Our nonlinear operator Fε fulfills (4.2). In fact, we have γ u2

2 γ u˜ 2 ˜ 2 |χε (Re v) − χε (Re v)| ˜ + |u2 − u˜ 2 | ≤ Cε |u| + |u| χ (Re v) − χ (Re v) ˜ ε ε ≤ Cε |u|2 + |u| ˜ 2 + 1 (|v − v| ˜ + |u − u|). ˜ Therefore, ˜ L∞ + u − u ˜ L∞ ) Fε (U ) − Fε (U˜ ) X ≤ Cε ( u2 L2 + u˜ 2 L2 + 1)( v − v ≤ Cε u 2L4 + u ˜ 2L4 + 1 ( v − v ˜ L∞ + u − u ˜ L∞ ). 3

3

Noting that D(A 8 ) = H 4 (Ω) ⊂ L4 (Ω), we obtain that 3 2 3 2 Fε (U ) − Fε (U˜ ) X ≤ Cε A 8 U X + A 8 U˜ X + 1 × Aη (U − U˜ ) X ,

U, U˜ ∈ D(Aη ).

(9.13)

2 Local Solutions

361

This means that (4.2) is satisfied by Fε with β = 38 and the η fixed above. We can now apply Theorem 4.1. As a consequence, for any U0 ∈ K, (9.12) possesses a unique local solution in the function space: 3 U ∈ C((0, TU0 ]; D(A)) ∩ C [0, TU0 ]; D A 8 ∩ C1 ((0, TU0 ]; X),

(9.14)

3

where TU0 > 0 is determined by the norm A 8 U0 X only.

2.2 Nonnegativity of Local Solutions For U0 ∈ K, let U (t) = t (u(t), v(t)) be the local solution constructed above. We want to prove that u(t) ≥ 0 and v(t) > 0 for every 0 < t ≤ TU0 by using the truncation method. Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (9.12) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function given by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ϕ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative ϕ (t) = a

H (u)Δu dx + γ Ω

u2 H (u) d − cu + dx. χε (v) Ω

Property (1.96) provides that H (u)Δu dx = − ∇H (u) · ∇u dx = − |∇H (u)|2 dx ≤ 0. Ω

Ω

Ω

In addition, since H (u) ≤ 0 and H (u)u ≥ 0, it is seen that u2 H (u) d − cu + dx ≤ 0. χε (v) Ω Hence, ϕ (t) ≤ 0; consequently, ϕ(t) ≤ ϕ(0) for every 0 < t ≤ TU0 . Thus, ϕ(0) = 0 implies ϕ(t) ≡ 0, that is, u(t) ≥ 0 for 0 < t ≤ TU0 . As for v(t), we consider ψ(t) = Ω H (v(t) − e−γ t ε0 ) dx, where ε0 = ess infΩ v0 > 0. Then, from ψ (t) = b H (v − e−γ t ε0 )Δ[v − e−γ t ε0 ] dx Ω

−γ Ω

H (v − e−γ t ε0 )[v − e−γ t ε0 ] dx + γ

H (v − e−γ t ε0 )u2 dx

Ω

it follows that ψ (t) ≤ 0. Hence, v(t) ≥ e−γ t ε0 for 0 < t ≤ TU0 .

362

9 Activator–Inhibitor Models

This shows that, if ess infΩ v0 = ε0 > ε, then the local solution of (9.12) constructed above is a solution to (9.4), too, at least for t such that (v(t) ≥)e−γ t ε0 ≥ ε. The lower estimate for v(t) obtained in this subsection will be improved below by a uniform one.

3 Global Solutions 3.1 A Priori Estimates from Below Let U0 = t (u0 , v0 ) be an initial value in K with ess infx∈Ω v0 (x) = ε0 > 0. Let U = t (u, v) be any local solution of (9.12) on [0, T ] in the function space: U

3

0 ≤ u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; H 4 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 3

0 < v ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; H 4 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (9.15) Then, it holds that u(t) ≥ (d/c)(1 − e−γ ct ), v(t) ≥ ε0 e

−γ t

+ e(t),

0 ≤ t ≤ TU ,

0 ≤ t ≤ TU .

(9.16) (9.17)

t Here, e(t) denotes the function e(t) = γ (d/c)2 e−γ t 0 eγ s (1 − e−γ cs )2 ds; clearly, e(t) > 0 for t > 0 and limt→∞ e(t) = (d/c)2 . Note that these lower estimates are independent of the fixed number ε > 0 in (9.12). In fact, these estimates are verified by a quite analogous method to that used in Sect. 2.2. We consider the function u(t) ˜ = u(t) − (d/c)(1 − e−γ ct ), By (1.100), ϕ(t) ˜ =

Ω

H (u(t)) ˜ dx is continuously differentiable with the derivative

d u˜ H (u) ˜ dx = ϕ˜ (t) = dt Ω

0 ≤ t ≤ TU .

γ u2 −γ ct H (u) ˜ aΔu + γ (d − cu) + − γ de dx. χε (v) Ω

By the same arguments as before (note that ∇ u˜ = ∇u), we conclude that ϕ˜ (t) ≤ 0. Hence, ϕ(t) ˜ ≡ 0 for 0 ≤ t ≤ TU . This shows that (9.16) is valid. Consider next the function v(t) ˜ = v(t) − [ε0 e−γ t + e(t)], ˜ We put ψ(t) =

Ω

H (v(t)) ˜ dx. Then,

0 ≤ t ≤ TU .

3 Global Solutions

363

d v˜ dx dt Ω

= H (v) ˜ bΔv + γ (u2 − v) + ε0 γ e−γ t

ψ˜ (t) =

H (v) ˜

Ω

+ γ e(t) − γ (d/c)2 (1 − e−γ ct )2 dx. Since (9.16) is already known, it follows that 2 ˜ ψ (t) ≤ −b H (v)|∇v| ˜ dx − γ H (v) ˜ v˜ dx ≤ 0. Ω

Ω

˜ Hence, ψ(t) ≡ 0 for 0 ≤ t ≤ TU . We have thus verified (9.17).

3.2 A Priori Estimates for Sobolev Norms Let U0 ∈ K be an initial value satisfying ess infΩ v0 (x) = ε0 > 0. Let ε be such that 0 < ε ≤ min {e−γ t ε0 + e(t)}. 0≤t 0 being a constant independent of U0 and TU . Proof Let 2 < p < ∞ and 0 < q < ∞ be exponents to be fixed below. We put p w(t) = u(t) v(t)q . Then, d dt

qup ∂v pup−1 ∂u − dx v q ∂t v q+1 ∂t Ω p−1 qup pu aΔu − bΔv dx = vq v q+1 Ω

w dx = Ω

364

9 Activator–Inhibitor Models

+γ Ω

pup−1 u2 qup 2 d − cu + − (u − v) dx vq v v q+1

=I + II. By integration by parts, the first integral I is rewritten in the form up−2 up−1 ap(p − 1) q |∇u|2 − (a + b)pq q+1 ∇u · ∇v I =− v v Ω p u + bq(q + 1) q+2 |∇v|2 dx v up−2 =− ap(p − 1)|v∇u|2 − (a + b)pq(v∇u) · (u∇v) q+2 Ω v + bq(q + 1)|u∇v|2 dx. We here assume that p and q satisfy √ √ pq 2 ab . ≤ √ (p − 1)(q + 1) a + b

(9.20)

Then, it is easily observed that I ≤ 0. In the meantime, the second integral is equal to −q p−1 II = γ v du − (pc − q)up + v −(q+1) (pup+1 − qup+2 ) dx. Ω

Here, we notice that dup−1 ≤ ζ up + Cp,ζ p+1

pu

p+2

≤ζ u

for all u ≥ 0,

+ Cp,ζ

for all u ≥ 0,

with arbitrarily small numbers ζ > 0 and ζ > 0, provided that Cp,ζ > 0 and Cp,ζ > 0 are chosen in a suitable way. We assume in addition to (9.20) that q

0 and ζ > 0 sufficiently small and noting that v(t) ≥ ε (due to (9.18)), we obtain that γ cp w dx + Cε,p,q . II ≤ − 2 Ω γ cp d In this way, we have dt Ω w dx ≤ − 2 Ω w dx + Cε,p,q and conclude that w(t) L1 ≤ e−

γ cp 2 t

w(0) L1 + Cε,p,q ,

0 ≤ t ≤ TU .

(9.22)

3 Global Solutions

365

In view of (9.20) and (9.21), we can take p = 4, provided that q > 0 is taken sufficiently small. Let us indeed fix p = 4 and q such that 0 < q ≤ 2. Then, since 4 4 uv 2 L1 ≤ ε q−2 vuq L1 , (9.22) yields the desired estimate (9.19). The equation on u is written as the abstract equation du u(t)2 + A1 u = d + , dt v(t)

0 < t ≤ TU ,

in L2 (Ω). Therefore, u(t) = e−tA1 u0 +

0

t

u(s)2 ds, e−(t−s)A1 d + v(s)

where e−tA1 is the analytic semigroup on L2 (Ω) generated by −A1 and satisfies the estimate e−tA1 L2 ≤ e−γ ct . Furthermore, t 3 3 3 u(s)2 ds. A18 u(t) = e−tA1 A18 u0 + A18 e−(t−s)A1 d + v(s) 0 Using this expression, we immediately derive from (9.19) that 38 A u(t) 1

L2

3

≤ Cε e−δt A18 u0 L + 1 , 2

0 ≤ t ≤ TU ,

(9.23)

with some positive exponent δ > 0 and a constant Cε > 0 independent of U0 and TU . As well, 3

u(t) L4 ≤ Cε e−δt A18 u0 L + 1 , 0 ≤ t ≤ TU . 2

We can repeat the same argument for v(t) to conclude that 38 A v(t) 2

L2

3

3 ≤ Cε e−δt A18 u0 L + A28 v0 L + 1 , 2

2

0 ≤ t ≤ TU .

Hence, this, together with (9.23), provides the final estimate 3

A 8 U (t) ≤ Cε e−δt A 38 U0 + 1 , X X

0 ≤ t ≤ TU .

(9.24)

3.3 Global Solution Let U0 ∈ K be an initial value with ess infΩ v0 (x) = ε0 > 0. Taking ε > 0 small enough to satisfy min {ε0 e−γ t + e(t)} ≥ ε,

0≤t 0, this function U is not only a solution of (9.12) but also that of the original equation (9.4).

3.4 Global Norm Estimates For U0 ∈ K, let U (t) = U (t; U0 ) be the global of (9.12) with the initial value U0 in the function space (9.25). Of course, estimate (9.24) holds for the global solution, i.e., 3

A 8 U (t; U0 ) ≤ Cε e−δt A 38 U0 + 1 , 0 ≤ t < ∞. (9.26) X X

4 Dynamical System 4.1 Construction of Dynamical System For U0 ∈ K, let U (t; U0 ) denote the global solution of (9.4) with the initial value U0 . From (9.17) we notice that, if U (t; U0 ) = t (u(t), v(t)), then v(t) ≥ e(t) for every 0 ≤ t < ∞. This lower estimate is independent of the initial value. As we are concerned with asymptotic behavior of U (t; U0 ) for U0 ∈ K, without loss of generality we assume that all the initial functions v0 uniformly satisfy the estimate v0 (x) ≥ ε0 = e(1). Furthermore, v(t) ≥ ε for every 0 ≤ t < ∞, where ε > 0 is given by ε = min {e(1)e−γ t + e(t)}. 0≤t 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R ; U0 ∈ Kθ,R .

Meanwhile, by Proposition 6.1, 3 Aθ S(t)U0 X ≤ Cθ,R 1 + t −(θ− 8 ) ,

0 < t < ∞; U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞; U0 ∈ Kθ,R .

Hence, (6.56) is fulfilled with the exponent dynamical system in Dθ .

3 8

< θ < 1. So, (S(t), Kθ , Dθ ) defines a

4.2 Exponential Attractors Let us construct exponential attractors for (S(t), K, D 3 ). To this end, we can fol8 low the methods established in Chap. 6, Sect. 5, namely, it suffices to verify (6.59) and (6.60). It is clear that the space condition (6.59) is valid. Meanwhile, (9.26) shows that the dissipative condition (6.60) is also valid. So, it is concluded that (S(t), K, D 3 ) has exponential attractors. 8

For any exponent 38 ≤ θ < 1, we can equally construct exponential attractors of the dynamical system (S(t), Kθ , Dθ ), too.

4.3 Squeezing Property of S(t) As in Chap. 6, Sect. 5.2, we can construct a subset X of K which is a compact set of X, is a bounded set of D(A), and is an absorbing and invariant set of S(t). Then, (S(t), X, X) is a dynamical system in the space X. It is clear that (6.62) is valid. Then, S(t) enjoys the squeezing property (6.45)–(6.46). Hence, by Theorem 6.16, (S(t), X, X) has exponential attractors whose fractal dimensions are precisely estimated by (6.50).

5 Instability of Homogeneous Stationary Solution As mentioned in the beginning of this chapter, our next objective is to show that the homogeneous stationary solution U of (9.4) becomes unstable, provided that b is

368

9 Activator–Inhibitor Models

sufficiently large. For this purpose, we intend to construct a local unstable manifold of U employing the method described in Chap. 6, Sect. 6. Since we are concerned with only the local unstable manifold, we introduce a localized problem of (9.4) in a neighborhood of U .

5.1 Localized Problem We introduce the following initial-boundary-value problem ⎧ 2 ∂u ⎪ = aΔu + γ [d − cu + ϕ(u) ] in Ω × (0, ∞), ⎪ ⎪ ∂t ψ(v) ⎪ ⎪ ⎨ ∂v 2 in Ω × (0, ∞), ∂t = bΔv + γ [ϕ(u) − v] ⎪ ∂u ∂v ⎪ on ∂Ω × (0, ∞), ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v0 (x) in Ω. 0

(9.28)

Here, ϕ(u) is a cutoff function of u in a complex neighborhood of u such that u−u ϕ(u) ≡ u for |u − u| < 1 and ϕ(u) = |u−u| + u for |u − u| ≥ 1. Clearly, ϕ(u) is a bounded Lipschitz continuous function in C. Meanwhile, ψ(v) ≡ v for |v − v| < r v−v r + v for |v − v| ≥ r, where r > 0 is taken in such a way that and ψ(v) = |v−v|

r < v2 . Then, ψ(v) is a bounded Lipschitz continuous function in C with the estimate |ψ(v)| ≥ v2 for all v ∈ C. We can then repeat the same arguments as in Sects. 3 and 5 to construct local solutions and global solutions to (9.28) for all initial values U0 ∈ D(Aθ ) in the function space: U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); D(Aθ )) ∩ C1 ((0, ∞); X)

for any exponent θ such that 38 ≤ θ < 1. The homogeneous stationary solution U of (9.4) or of (9.12) is clearly a stationary solution to (9.28), too.

5.2 Complexified Dynamical System ˜ acting on The localized problem (9.28) introduced above defines a semigroup S(t) 3 θ ˜ Dθ = D(A ), 8 ≤ θ < 1, and defines a dynamical system (S(t), Dθ , Dθ ) which we call a comlexified dynamical system. We are concerned with the case 34 < θ < 1 because of the embedding Dθ = 2θ HN (Ω) ⊂ C(Ω). In this case, any solution of problem (9.12) is a solution of (9.28) in a small neighborhood of U in Dθ . Naturally, in such a neighborhood, any trajec˜ Dθ , Dθ ). In particular, U is an equilibrium tory of (S(t), Kθ , Dθ ) is that of (S(t), ˜ ˜ Dθ , Dθ ), then it is the of (S(t), Dθ , Dθ ). If U is stable as an equilibrium of (S(t),

5 Instability of Homogeneous Stationary Solution

369

same as that of (S(t), Kθ , Dθ ). On the other hand, even if U is unstable as an equi˜ librium of (S(t), Dθ , Dθ ), U is not automatically unstable as that of (S(t), Kθ , Dθ ). ˜ Nevertheless, as will be explained below, the instability of U in (S(t), Dθ , Dθ ) provides us an information which indicates its instability even in the original dynamical system (S(t), Kθ , Dθ ).

5.3 Unstable Manifold of U Problem (9.28) is now formulated in X as the Cauchy problem of the form (9.12) in which the nonlinear operator F˜ : D(Aη ) → X is given by u γ d + γ ϕ(u)2 /ψ(v) , U = ∈ D(Aη ), F˜ (U ) = v γ ϕ(u)2 where 34 < θ < η < 1. In the complex neighborhoods |u − u| < 1 of u and |v − v| < r of v, we have ϕ(u) ≡ u and ψ(v) ≡ v. So, in a small neighborhood of U in D(Aη ) ⊂ D(Aθ ) ⊂ C(Ω), F˜ is Fréchet differentiable with the derivative 2γ u γ u2 h1 − 2 ˜ v v F (U ) h = , h2 2γ u 0 η h u U= ∈ B D(A ) (U ; r), h = 1 ∈ D(Aη ). h2 v Then, it is possible to apply the results obtained in Chap. 6, Sect. 6. The derivative obviously fulfills conditions (6.68) and (6.69). In order to verify the spectrum separation condition (6.76), it suffices to check condition (6.83). To this end, we have to make the assumption that abμ2k − γ (ag v + bf u )μk + γ 2 (f u g v − f v g u ) = 0 for all μk ,

(9.29)

where 0 = μ0 < μ1 ≤ μ2 ≤ · · · are the eigenvalues of the Laplace operator −Δ under the homogeneous Neumann boundary conditions on ∂Ω. Since f u + g v < 0 (due to (9.2)), we have μk = (f u + g v )/(a + b) for all μk . Therefore, under (9.29), (6.83) and hence (6.76) is fulfilled. Under (9.7) (and hence under (9.2)), if a and b are such that ag v + bf u > 2 ab(f u g v − f v g u ), (9.30) then there may exist eigenvalues μk such that abμ2k − γ (ag v + bf u )μk + γ 2 (f u g v − f v g u ) < 0. Note that (9.30) is actually satisfied if a is fixed and b is sufficiently large because of ˜ + (U ; O) which f u > 0. In such a case, U has a nontrivial local unstable manifold W

370

9 Activator–Inhibitor Models

is tangential to the space U + X− at U , where X− is the sum of the eigenspaces of the linearized operator A = A − F˜ (U ) corresponding to the negative eigenvalues. ˜ + (U ; O) = dim X− = N (U ), where We already know that dim W (9.31) N(U ) = # μk ; abμ2k − γ (ag v + bf u )μk + γ 2 (f u g v − f v g u ) < 0 . ˜ + (U ; O) with dimenAs a consequence, U possesses a local unstable manifold W ˜ sion N(U ) in the dynamical system (S(t), Dθ , Dθ ). ˜ + (U ; O) by using (9.31). For Let us furthermore estimate the dimension of M simplicity, let Ω be a smooth bounded domain in R3 . According to [Tri78, Theorem 5.6.2], we have N (μ) ≡

0≤μk ≤μ

3 1 = O μ2

as μ → ∞.

In the meantime, let 0 < μ1 < μ2 < ∞ be the solution to the equation abμ2 − γ (ag v + bf u )μ + γ 2 (f u g v − f v g u ) = 0 for μ. Clearly, γ (ag v + bf u ) ∓ γ (ag v + bf u )2 − 4ab(f u g v − f v g u ) μ 1 , μ2 = . 2ab Then, we have N(U ) = N (μ2 ) − N (μ1 ). Remark 9.1 Let us consider some specific case where the diffusion constants are such that a = γ −1 a and b = γ b with a > 0 and b > 0. Fixing all other constants in (9.4), we let γ tend to infinity. It is easy to see that μ1 (γ ) ≤ C1 and μ2 (γ ) ≥ c2 γ 2 as γ → ∞ with some two positive constants C1 and c2 (> 0). Consequently, N(μ2 (γ )) − N (μ1 (γ )) ≥ Cγ 3 ; this means that, as γ → ∞, ˜ + (U ; O) = N (U ) ≥ Cγ 3 → ∞. dim W Let us consider finally the instability of U as an equilibrium of the original dynamical system (S(t), Kθ , Dθ ). We already know that U is an unstable equilib˜ + (U ; O) with dimen˜ rium of (S(t), Dθ , Dθ ) having a local unstable manifold W sion dim X− which is tangential to U + X− at U . Furthermore, the sum X− of the eigenspaces of A corresponding to the negative eigenvalues possesses an orthogonal basis which consists of real-valued functions (cf. Corollary 2.1). These facts then indicate that the local unstable manifold W+ (U ; O) in (S(t), Kθ , Dθ ) is neither trivial, that is, U is an unstable equilibrium of (S(t), Kθ , Dθ ).

Notes and Further Researches After the Turing’s pioneering work [Tur52], his instabilty principle was applied extensively. It is known that various pattern formations exhibited by physical, chemical, or biological systems can be reproduced by making use of activator–inhibitor

Notes and Further Researches

371

models. Accordingly, there are a certain number of forms for the kinetic functions on the activator and inhibitor, see [Mei92] and [Mur02, Chap. 6]. Gierer–Meinhardt [GM72] and Meihardt [Mei82] presented the functions k3 u2 , v

f (u, v) = k1 − k2 u +

g(u, v) = k4 u2 − k5 v.

(9.32)

The system (9.1) having these kinetic functions is called the Gierer–Meinhardt system. More generally, when the system has the kinetic functions f (u, v) = k1 − k2 u +

k3 up , vq

g(u, v) =

k4 ur − k5 v, vs

(9.33)

it is often called the generalized Gierer–Meinhardt system. When the kinetic functions are k2 up k3 ur g(u, v) = s − k4 v (9.34) f (u, v) = −k1 u + q , v v and when one imposes the Dirichlet boundary conditions on u and v, the problem is called the singular Gierer–Meinhardt system, see [CM00, CM03]. A system consisting of the diffusion equation on u and an ordinary differential equation on v which is obtained by letting b → ∞ and replacing g(u, v) by the integral Ω g(u, v) dx is called a shadow system of the Gierer–Meinhardt system. Murray [Mur88] (cf. also [Mur03, Chap. 3]) used the functions f (u, v) = k1 − k2 u − h(u, v), h(u, v) =

g(u, v) = k3 − k4 v − h(u, v),

k5 uv k6 + k7 u + k8 u2

in order to reproduce the animal coat patterns. A large number of papers have already been published for the Gierer–Meinhardt system. For the case of (9.32), the global existence was first obtained by Rothe [Rot84]. Masuda–Takahashi [MaT87] proved the global existence of solutions for the general case of (9.33), but they had to assume some restrictive condition on the exponents p, q, r, and s which excludes the case of (9.32). Afterward, Li–Chen– Qin [LCQ95] and Jiang [Jia06] got rid of such a restriction and proved the global q existence in the case where 0 < p − 1 < r and p−1 r < s+1 (which coves the case where p = 2, q = 1, r = 2, and s = 0). Wu–Huang [WH00] studied the asymptotic behavior of the global solutions. Evolutional problems for the shadow system were studied by Miyamoto [Miy06]. Exponential attractors are newly constructed for the Gierer–Meinhardt system. The results in Sect. 5 are also new. More articles have been published to contribute to the stationary problems of the Gierer–Meinhardt system than those to the evolutional problems. However, we here quote only pioneering papers Haken–Olbrich [HO78] and Takagi [Tak79] and recent papers Jiang–Ni [JN07] and Iron–Rumsey [IR07].

Chapter 10

Belousov–Zhabotinskii Reaction Models

This chapter is devoted to studying reaction–diffusion models for the Belousov– Zhabotinskii reaction. The reaction which consists of more than ten elementary chemical reactions arising simultaneously and which does not tend to any chemical equilibrium is known as one of typical self-organization phenomena in chemistry. In 1974, Field–Noyes presented a simple mathematical model for describing the complicated mechanism from a global point of view. Their reaction–diffusion model of dimensionless form is given by ⎧ ∂u −1 2 ⎪ ⎨ ∂t = aΔu + ε (qw − uw + u − u ) in Ω × (0, ∞), ∂v in Ω × (0, ∞), ∂t = bΔv + u − v ⎪ ⎩ ∂w −1 in Ω × (0, ∞). ∂t = dΔw + δ (−qw − uw + cv) Here, u denotes the concentration of HBrO2 , v the concentration of Ce4+ , and w the concentration of Br− , respectively, in a vessel represented by Ω. The positive constants δ, ε, q, and c are chemical parameters, where especially δ, ε, and q are considered to be small. In 1986, the model was simplified more by Keener–Tyson by assuming that δ cv is sufficiently small so that w is in quasi-equilibrium, i.e., w = u+q , and by using some scaling in the spatial variable so that the diffusions of u and v are εa and εb, respectively. Their model is then given by ∂u ∂t ∂v ∂t

= εaΔu +

1 ε

u(1 − u) − cv u−q in Ω × (0, ∞), u+q

= εbΔv + u − v

in Ω × (0, ∞).

(10.1)

In this chapter, we consider the initial-boundary-value problems of these equations under the Neumann boundary conditions. We show the global existence of solutions, construct a dynamical system, and construct exponential attractors. Especially, for the Keener–Tyson model, we show that, if ε and q are sufficiently small in a specific sense, then the problem has no stable homogeneous stationary solutions. In addition, in such a case, the fractal dimension of exponential attractor is estimated from below by some quantity which tends to infinity as ε → 0. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_10, © Springer-Verlag Berlin Heidelberg 2010

373

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10 Belousov–Zhabotinskii Reaction Models

1 Field–Noyes Model Let us first consider the Field–Noyes model. We consider the initial-boundary-value problem ⎧ ∂u −1 2 ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu + ε (qw − uw + u − u ) ⎪ ⎪ ⎪ ∂v ⎪ ⎪ = bΔv + u − v in Ω × (0, ∞), ⎪ ⎨ ∂t ∂w −1 in Ω × (0, ∞), ∂t = dΔw + δ (−qw − uw + cv) ⎪ ⎪ ⎪ ∂u ∂v ∂w ⎪ on ∂Ω × (0, ∞), ⎪ ∂n = ∂n = ∂n = 0 ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), w(x, 0) = w (x) in Ω, 0 0 0 (10.2) in a three-dimensional C2 or convex, bounded domain Ω. Here, δ, ε, q, and c are positive constants (>0), and a, b, and d are positive diffusion rates. The three unknown functions u, v and w are assumed to satisfy the homogeneous Neumann boundary conditions on ∂Ω.

2 Local Solutions 2.1 Construction of Local Solutions We construct local solutions by formulating (10.2) as the Cauchy problem for a semilinear abstract equation. We set the underlying space X as the product ⎧⎛ ⎞ ⎫ ⎨ f ⎬ (10.3) X = L2 (Ω) = ⎝ g ⎠ ; f ∈ L2 (Ω), g ∈ L2 (Ω), h ∈ L2 (Ω) . ⎩ ⎭ h We introduce a diagonal matrix operator A = diag {A1 , A2 , A3 } of X, where A1 , A2 and A3 are realizations of −aΔ + ε −1 , −bΔ + 1 and −dΔ + δ −1 q, respectively, in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. They are positive definite self-adjoint operators of L2 (Ω) with domains HN2 (Ω), see Theorems 2.6 and 2.7. Therefore, A is a positive definite self-adjoint operator of X. Furthermore, by Theorems 16.7 and 16.9 in Chap. 16, the domains of its fractional powers are given by 3 D(Aθ ) = H2θ (Ω) if 0 ≤ θ < , 4 3 < θ ≤ 1. D(Aθ ) = H2θ N (Ω) if 4

2 Local Solutions

375

We also define the nonlinear operator F : D(Aη ) → X by ⎛ −1 ⎞ ε (qw − uw + 2u − u2 ) ⎠ , U ∈ D(Aη ), u F (U ) = ⎝ δ −1 (−uw + cv)

(10.4) 2η

where the exponent η is fixed in such a way that 34 < η < 1. Since HN (Ω) ⊂ H 2η (Ω) ⊂ C(Ω) due to (1.76), it follows that D(Aη ) ⊂ C(Ω). We also set the space of initial values by ⎧⎛ ⎞ ⎫ ⎨ u0 ⎬ K = ⎝ v0 ⎠ ; 0 ≤ u0 ∈ L2 (Ω), 0 ≤ v0 ∈ L2 (Ω), 0 ≤ w0 ∈ L2 (Ω) . ⎩ ⎭ w0 We can then formulate (10.2) as the Cauchy problem dU 0 < t < ∞, dt + AU = F (U ),

(10.5)

U (0) = U0 ,

in X. Applications of the general results in Chap. 4 to (10.5) are quite easy. The structural assumption (4.21) on F is immediately verified. Indeed, from |u2 − u˜ 2 | ≤ (|u| + |u|)|u ˜ − u| ˜ it follows that ˜ L∞ )u − u ˜ L2 u2 − u˜ 2 L2 ≤ (uL∞ + u ≤ (uH 2η + u ˜ H 2η )u − u ˜ L2 ,

2η

u, u˜ ∈ HN (Ω).

Similarly, ˜ L2 + u ˜ H 2η w − w ˜ L2 ), uw − u˜ w ˜ L2 ≤ C(wH 2η u − u 2η

u, w, u, ˜ w˜ ∈ HN (Ω). η

η

η

2η

Therefore, in view of D(A1 ) = D(A2 ) = D(A3 ) = HN (Ω), we deduce that F (U ) − F (U˜ )X ≤ Cε (Aη U X + Aη U˜ X + 1)U − U˜ X ,

U, U˜ ∈ D(Aη ). (10.6)

Hence, F fulfills (4.21). Theorem 4.4 (with G(t) ≡ 0) thus provides that, for any initial value U0 ∈ X(⊃ K), (10.5) possesses a unique local solution in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; X) ∩ C1 ((0, TU0 ]; X),

(10.7)

where TU0 > 0 depends only on the norm U0 X . It also holds that tAU (t)X + U (t)X ≤ CU0 ,

0 < t ≤ TU 0 ,

(10.8)

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10 Belousov–Zhabotinskii Reaction Models

where CU0 > 0 also depends only on the norm U0 X .

2.2 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, v, w) be the local solution of (10.5) constructed above. We want to show that u(t) ≥ 0, v(t) ≥ 0, and w(t) ≥ 0 for every 0 < t ≤ TU0 . To this end, we have to use the modified nonlinear operator ⎞ ⎛ ⎞ ⎛ −1 u ε (qw − uw + 2u − u2 ) ⎠ , U = ⎝ v ⎠ ∈ D(Aη ), |u| F˜ (U ) = ⎝ w δ −1 (−uw + cv) to (10.4). And we consider an auxiliary problem ˜ dU ˜ ˜ ˜ dt + AU = F (U ), 0 ≤ t < ∞, U˜ (0) = U0 .

(10.9)

We can repeat the same arguments as in the preceding subsection to construct a unique local solution U˜ = t (u, ˜ v, ˜ w) ˜ to (10.9) on an interval [0, T˜U0 ] in the same ˜ solution space as (10.7), U also satisfying (10.8). The local solution U˜ (t) is then real valued, since the complex conjugate U˜ (t) of U˜ (t) is also a local solution of (10.9) with the same initial value, and the uniqueness of solution implies that U˜ (t) = U˜ (t). Furthermore, as is shown below, it holds that u(t) ˜ ≥ 0, v(t) ˜ ≥ 0 and w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . Let us verify first that v(t) ˜ ≥ 0 by the truncation method. Let H (v) ˜ be a C1,1 cut1 2 ˜ ≡0 off function for −∞ < v˜ < ∞ given by H (v) ˜ = 2v˜ for −∞ < v˜ < 0 and H (v) ˜ dx is continuously diffor 0 ≤ v˜ < ∞. By (1.100), the function ϕ(t) = Ω H (v(t)) ferentiable with the derivative H (v)Δ ˜ v˜ dx + H (v)(| ˜ u| ˜ − v) ˜ dx. ϕ (t) = b Ω

Ω

By property (1.96), H (v)Δ ˜ v˜ dx = − ∇H (v) ˜ · ∇ v˜ dx = − |∇H (v)| ˜ 2 dx ≤ 0. Ω

Ω

Ω

In addition, since H (v) ˜ ≤ 0 and H (v) ˜ v˜ ≥ 0, it follows that ϕ (t) ≤ 0. Conse˜ quently, ϕ(t) ≤ ϕ(0) for 0 < t ≤ TU0 . Thus, ϕ(0) = 0 implies ϕ(t) ≡ 0, that is, v(t) ˜ ≥ 0 for 0 ≤ t ≤ T˜U0 . ˜ By Let us next verify that w(t) ˜ ≥ 0. We use the function ψ(t) = Ω H (w(t))dx. the same reason as for ϕ(t), we have −1 H (w)Δ ˜ w˜ dx + δ H (w)(−q ˜ w˜ − u˜ w˜ + cv) ˜ dx. ψ (t) = d Ω

Ω

3 Global Solutions

377

Similarly, by property (1.96), Ω H (w)Δ ˜ w˜ dx ≤ 0. Since H (w) ˜ ≤ 0, 0 ≤ H (w) ˜ w˜ ≤ 2H (w) ˜ and v(t) ˜ ≥ 0, it is seen that H (w)(−q ˜ w˜ − u˜ w˜ + cv) ˜ dx ≤ 2u(t) ˜ H (w) ˜ dx. L∞ Ω

Ω

t ˜ ˜ Hence, ψ (t) ≤ 2u(t) L∞ ψ(t); consequently, ψ(t) ≤ ψ(0) exp[2 0 u(s) L∞ ds] η ˜ ˜ ˜ for 0 < t ≤ TU0 . We here know that u(s) L∞ ≤ CA U (s)X and, in view of (10.8), that Aη U (s)X ≤ CU0 s −η . Thus, ψ(0) = 0 implies ψ(t) ≡ 0, that is, w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . ˜ dx. Using the known reLet us finally consider the function φ(t) = Ω H (u(t)) −η , we can repeat the same ˜ sults that w(t) ˜ ≥ 0 and u(t) ˜ L∞ + w(t) L∞ ≤ CU0 t arguments as for w(t) ˜ to verify that u(t) ˜ ≥ 0. In this way, we have shown that u(t) ˜ ≥ 0, v(t) ˜ ≥ 0, and w(t) ˜ ≥ 0 for 0 ≤ t ≤ T˜U0 . The componentwise nonnegativity of U˜ (t) now implies that F˜ (U˜ (t)) = F (U˜ (t)) for 0 ≤ t ≤ T˜U0 ; this then means that U˜ (t) is a local solution of (10.5), too. By the uniqueness of a solution of (10.5), we must have U˜ (t) = U (t) for 0 ≤ t ≤ T˜U0 . Hence, if T˜U0 ≥ TU0 , we finish the proof. Otherwise, we put T0 = sup{0 < T ≤ TU0 ; u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for every 0 < t ≤ T }. Since Ω H (u(T0 )) dx = limtT0 Ω H (u(t)) dx = 0, we see that u(T0 ) ≥ 0. By a similar reason, we have v(T0 ) ≥ 0 and w(T0 ) ≥ 0. So, if T0 = TU0 , we finish the proof. If T0 < TU0 , we will again consider problem (10.9) but with initial time T0 and initial value U (T0 ). Then, we conclude that there is τ > 0 such that u(t) ≥ 0, v(t) ≥ 0, and w(t) ≥ 0 for every t such that T0 ≤ t ≤ T0 + τ . This is a contradiction; hence, T0 = TU0 .

3 Global Solutions 3.1 A Priori Estimates of Local Solutions For U0 ∈ K, let U = t (u, v, w) denote a local solution of (10.5) on [0, TU ] in the function space: 0 ≤ U ∈ C (0, TU ]; H2N (Ω) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (10.10) Consider the inner product of the equation of (10.5) and U in X. From the equation on u we have 1 d 2 2 −1 u dx + a |∇u| dx + ε u2 dx 2 dt Ω Ω Ω = ε −1 (quw − u2 w + 2u2 − u3 ) dx

Ω

≤ Ω

ε −1 (4−1 q 2 w + 2u2 − u3 ) dx,

(10.11)

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10 Belousov–Zhabotinskii Reaction Models

where we used the inequality qu ≤ 1 d 2 dt

q2 4

+ u2 . By the equation on v,

1 v dx + b |∇v| dx + 2 Ω Ω 2

(uv − 2−1 v 2 ) dx

v dx =

2

2

Ω

Ω

(u2 − 4−1 v 2 ) dx.

≤

(10.12)

Ω

Finally, by the equation on w, 1 d 2 dt

w 2 dx + d

Ω

=

|∇w|2 dx + Ω

δ −1 q 2

w 2 dx Ω

δ −1 (−2−1 qw 2 − uw 2 + cvw) dx

Ω

≤

δ −1 (−4−1 qw 2 + c2 q −1 v 2 ) dx.

(10.13)

Ω

After multiplying (10.12) by a parameter ξ > 0, we sum up (10.11), (10.12), and (10.13) to obtain that 1 1 d (u2 + ξ v 2 + w 2 ) dx + (2ε −1 u2 + ξ v 2 + δ −1 qw 2 ) dx 2 dt Ω 2 Ω −1 ≤ (2ε + ξ − ε−1 u)u2 + (c2 q −1 δ −1 − 4−1 ξ )v 2 Ω

+ 4−1 q(ε −1 q − δ −1 w)w dx. Let us fix the parameter ξ so that ξ = 4c2 q −1 δ −1 and let us notice the following two estimates (2ε−1 + ξ )u2 ≤ ε −1 u3 +

4 (2ε −1 + ξ )3 ε 2 27

ε −1 qw ≤ δ −1 w 2 + 4−1 δ 2 ε −2 q 2 Then, d dt

for all 0 ≤ u < ∞,

for all 0 ≤ w < ∞.

(u2 + ξ v 2 + w 2 ) dx + μ Ω

(u2 + ξ v 2 + w 2 ) dx ≤ C Ω

with μ = min{2ε −1 , 1, δ −1 q} > 0 and some constant C > 0. Solving this differential inequality (cf. (1.58)), we obtain that u(t)2L2 + ξ v(t)2L2 + w(t)2L2 ≤ e−μt u0 2L2 + ξ v0 2L2 + w0 2L2 + Cμ−1 ,

0 ≤ t ≤ TU . (10.14)

4 Dynamical System

379

3.2 Global Existence Let U0 ∈ K. For any local solution U of (10.5) in the function space (10.10), from (10.14) it follows that U (t)X ≤ C U0 X + 1 , 0 ≤ t ≤ TU . (10.15) Then, we can apply Corollary 4.3 to conclude the global existence of solution. Indeed, for any U0 ∈ K, (10.5) possesses a unique global solution in the function space: 0 ≤ U ∈ C (0, ∞); H2N (Ω) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (10.16)

3.3 Global Norm Estimate For U0 ∈ K, the global solution U (t; U0 ) of (10.5) in (10.16) with initial value U0 satisfies the estimate ˜ U (t; U0 )X ≤ C[e−μt U0 X + 1],

0 ≤ t < ∞, U0 ∈ K,

(10.17)

with some exponent μ˜ > 0 and constant C > 0. Indeed, this is verified directly from (10.14).

4 Dynamical System 4.1 Construction of Dynamical System We can now construct a dynamical system determined from (10.5). We take the universal space X given by (10.3) and follow the general methods described in Chap. 6, Sect. 5. We already know that, for any U0 ∈ K, (10.5) possesses a unique global solution U (t; U0 ) in the function space (10.16). By S(t)U0 = U (t; U0 ) we define a nonlinear semigroup S(t) acting on K. Estimate (10.17) shows that (6.56) is fulfilled. So, (10.5) determines a dynamical system (S(t), K, X).

4.2 Exponential Attractors Let us construct exponential attractors for the dynamical system (S(t), K, X). Thanks to the general results, it suffices to verify (6.59) and (6.60). But, the space condition (6.59) is clear. Meanwhile, (10.17) shows that the dissipative condition (6.60) is also fulfilled. Hence, there exists an absorbing and invariant set X of

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10 Belousov–Zhabotinskii Reaction Models

(S(t), K, X) such that X is a compact subset of X and is a bounded subset of D(A). Furthermore, the dynamical system (S(t), X, X) enjoys a family of exponential attractors M. It is clear in the present case that (6.62) is valid. Therefore, S(t) enjoys the squeezing property (6.45)–(6.46). Hence, (S(t), X, X) has exponential attractors whose fractal dimensions are estimated precisely by (6.50).

5 Keener–Tyson Model We now consider the initial-boundary-value problem ⎧ u−q

∂u 1 ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu + ε 2 u(1 − u) − cv u+q ⎪ ⎪ ⎪ ∂v 1 ⎨ = bΔv + (u − v) in Ω × (0, ∞), ∂t ε ∂v ⎪ on ∂Ω × (0, ∞), ⎪ ∂u ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩ v(x, 0) = v0 (x) in Ω, u(x, 0) = u0 (x),

(10.18)

in a three-dimensional C2 or convex, bounded domain Ω. Note that we have rewritten the equations for u and v in the original Keener–Tyson model (10.1) by scaling the temporal variable as τ = ε −1 t . The new temporal variable τ is again denoted by t as in (10.18). We fix the coefficients a > 0,

b > 0,

c > 0,

0 < q < 1.

(10.19)

In the meantime, 0 < ε ≤ 1 is considered as a control parameter of the system. We use the notation f (u, v) = u(1 − u) − cv u−q u+q , (10.20) g(u, v) = u − v. For u ≥ 0 and v ≥ 0, the equations f (u, v) = 0 and g(u, v) = 0 obviously have two solutions (0, 0) and (u, v). Here, u is a unique positive solution to the quadratic equation (u + q)(1 − u) = c(u − q),

(10.21)

and v = u. More precisely,

u = v = 1 − c − q + (c + q − 1)2 + 4q(c + 1) /2.

(10.22)

It is easy to see from (10.21) that q < u < 1. Thus, problem (10.18) has two homogeneous stationary solutions (u, v) ≡ (0, 0) and (u, v).

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381

We are then concerned with the stability and instability of these homogeneous stationary solutions. Clearly, we have fu (u, v) = 1 − 2u − gu (u, v) = 1,

2cqv , (u + q)2

fv (u, v) = −c

u−q , u+q

gv (u, v) = −1.

In particular, fu (0, 0) = 1,

fv (0, 0) = c,

gu (0, 0) = 1,

gv (0, 0) = −1.

These imply that the relation fu (0, 0)gv (0, 0) − fv (0, 0)gu (0, 0) = −(1 + c) < 0 always takes place, which shows on account of (6.84) that (0, 0) is always unstable. We denote the partial differential coefficients as f u = fu (u, v) = 1 − 2u − g u = gu (u, v) = 1,

2cqu , (u + q)2

f v = fv (u, v) = u − 1,

g v = gv (u, v) = −1.

It is clear that f u < 1. This estimate is actually optimal when the coefficients of equations are taken as in (10.19); indeed, when c = 1, it is seen from (10.22) that √ √ u = O( q), and hence f u = 1 − O( q) when q is small. Since f u − (1 − u) = 2cqu −u − (u+q) 2 < 0, it is seen that the differential coefficients always satisfy the relation f u g v > f v g u . According to (6.82), a sufficient condition in order that (u, v) satisfies the stability condition (6.67) is read as 1 f + g v < 0, ε u

f ugv > f v gu

and εag v + bf u < 2 εab(f u g v − f v g u ).

(10.23) When f u ≤ 0, all these inequalities are fulfilled automatically whatever ε > 0 is; hence, in this case, (u, v) is stable. On the other hand, when 0 < f u < 1, these inequalities may fail depending on the control parameter ε > 0. Indeed, if ε > max

2

b2 f u

, fu , ( ab(f u g v − f v g u ) + −abf v g u )2

then the inequalities in (10.23) are still valid; therefore, (u, v) is stable. But, if ε > 0 is small enough, then some of inequalities in (10.23) fail; so, (u, v) is no longer stable.

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10 Belousov–Zhabotinskii Reaction Models

6 Local Solutions 6.1 Construction of Local Solutions It is carried out in an analogous way to the Field–Noyes model by formulating (10.18) as the Cauchy problem of the form (10.5) in the product space X = L2 (Ω). We introduce a diagonal matrix operator A = diag{A1 , A2 } of X, where A1 and A2 are realizations of −aΔ + 1 and −bΔ + 1, respectively, in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. They are positive definite selfadjoint operators of L2 (Ω) with domains HN2 (Ω). Therefore, A is a positive definite self-adjoint operator of X. Furthermore, the domains of its fractional powers are 3 given by D(Aθ ) = H2θ (Ω) if 0 ≤ θ < 34 and D(Aθ ) = H2θ N (Ω) if 4 < θ ≤ 1. We η introduce also a nonlinear operator F : D(A ) → X given by u + ε −2 [u(1 − u) − cv(u − q)(|u| + q)−1 ] F (U ) = , U ∈ D(Aη ), (10.24) ε −1 u + (1 − ε−1 )v 2η

where the exponent η is fixed as 34 < η < 1. Since HN (Ω) ⊂ H 2η (Ω) ⊂ C(Ω) due to (1.76), it follows that D(Aη ) ⊂ C(Ω). The space of initial values is given by u0 K= (10.25) ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ v0 ∈ L2 (Ω) . v0 Problem (10.18) is then formulated as the Cauchy problem for an abstract evolution equation dU 0 < t < ∞, dt + AU = F (U ), (10.26) U (0) = U0 , of the same form as (10.5) in X. Applications of the general results in Chap. 4 are quite direct as before. The Lipschitz condition (4.21) on F is easily verified. Indeed, from uv u˜ v˜ ≤ C[|v − v| ˜ + (|v| + |v|)|u ˜ − u|] ˜ |u| + q − |u| ˜ +q it follows that uv u˜ v˜ ≤ C[v − v ˜ L2 + (vH 2η + v ˜ H 2η )u − u ˜ L2 ], |u| + q − |u| ˜ + q L2 2η

2η

u, v ∈ HN (Ω); u, ˜ v˜ ∈ HN (Ω). η

η

2η

Therefore, in view of D(A1 ) = D(A2 ) = HN (Ω), we deduce that F (U ) − F (U˜ )X ≤ Cε (Aη U X + Aη U˜ X + 1)U − U˜ X , U, U˜ ∈ D(Aη ),

(10.27)

7 Global Solutions

383

hence, F fulfills (4.21). We thus conclude that, for any initial value U0 ∈ K, (10.26) possesses a unique local solution in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; X) ∩ C1 ((0, TU0 ]; X),

(10.28)

where TU0 > 0 depends only on the norm U0 X .

6.2 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, v) be the local solution of (10.26) constructed above. It is then shown by the same methods as in Sect. 2.2 that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ T U0 . We introduce the modified nonlinear operator u u + ε−1 [|u|(1 − u) − c|v|(u − q)(|u| + q)−1 ] ˜ , U= ∈ D(Aη ), F (U ) = v ε −1 |u| + (1 − ε−1 )v to (10.24) and consider the same auxiliary problem as (10.9). Let U˜ (t) denote the local solution of the auxiliary problem with initial value U0 ∈ K on an interval [0, T˜U0 ]. Employing the truncation method as before, we conclude that U˜ (t) ≥ 0 componentwise for every 0 < t ≤ T˜U0 . Consequently, F˜ (U˜ (t)) = F (U˜ (t)) for 0 ≤ t ≤ T˜U0 . This means that U˜ (t) is a local solution of (10.26), too. By the uniqueness of local solution, we deduce that U˜ (t) = U (t) for 0 ≤ t ≤ T˜U0 . Therefore, U (t) ≥ 0 for 0 ≤ t ≤ T˜U0 . We can repeat the similar arguments until we obtain that U (t) ≥ 0 for every 0 < t ≤ TU0 . Now, since u(t) ≥ 0, we have |u(t)| = u(t). This means that the local solution U of (10.26) is a local solution of the original problem (10.18), too.

7 Global Solutions 7.1 A Priori Estimates of Local Solutions For U0 ∈ K, let U = t (u, v) denote a local solution of (10.26) on [0, TU ] in the function space: 0 ≤ U ∈ C (0, TU ]; H2N (Ω) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (10.29) Consider the inner product of the equation of (10.26) and U in X. From the equation on u we have 1 1 d 2 2 u dx + a |∇u| dx + u2 dx 2 dt Ω 2 Ω Ω

384

10 Belousov–Zhabotinskii Reaction Models

=

[(2

−1

+ε

−2

)u − ε 2

−2 3

u ] dx − ε

Ω

−2

c

uv Ω

u−q dx. u+q

u−q u−q If u > q, then −ε−2 cuv u+q ≤ 0. And, if 0 ≤ u ≤ q, then −ε −2 cuv u+q ≤ ε −2 cqv. So, 1 d 1 u2 dx + a |∇u|2 dx + u2 dx 2 dt Ω 2 Ω Ω −1 −2 2 −2 3 −2 ≤ [(2 + ε )u − ε u ] dx + ε cq v dx. Ω

Ω

In the meantime, from the equation on v, we have 1 d 1 v 2 dx + b |∇v|2 dx + v 2 dx = [ε −1 uv + (2−1 − ε −1 )v 2 ] dx 2 dt Ω 2 Ω Ω Ω 3 −2 2 −1 −ε ε u + v 2 dx. ≤ 4 Ω Therefore, it follows that 1 1 d (u2 + v 2 ) dx + (u2 + v 2 ) dx 2 dt Ω 2 Ω 3 −1 −2 2 −2 3 −2 −1 −ε ≤ [(2 + 2ε )u − ε u ] + ε cqv + v 2 dx. 4 Ω Here, we notice that 4 (2−1 + 2ε −2 )u2 ≤ ε −2 u3 + (2 + 2−1 ε 2 )3 ε −2 for all 0 ≤ u < ∞, 27 3 2 −2 −1 v + (4 − 3ε)−1 c2 q 2 ε −3 for all 0 ≤ v < ∞. ε cqv ≤ ε − 4 Hence, d dt

(u2 + v 2 ) dx + Ω

(u2 + v 2 ) dx ≤ Cε−3 .

Ω

Solving this differential inequality (cf. (1.58)), we obtain that u(t)2L2 + v(t)2L2 ≤ e−t u0 2L2 + v0 2L2 + Cε −3 , 0 ≤ t ≤ TU .

(10.30)

7.2 Global Existence Let U0 ∈ K. For any local solution U of (10.26) in (10.29), from (10.30) it follows that U (t)X ≤ U0 2X + Cε −3 , 0 ≤ t ≤ TU . (10.31)

8 Dynamical System

385

Then, we can use Corollary 4.3 to conclude the global existence of solution. Indeed, for any U0 ∈ K, (10.26) possesses a unique global solution in the function space: 0 ≤ U ∈ C (0, ∞); H2N (Ω) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (10.32)

7.3 Global Norm Estimate For U0 ∈ K, let U (t; U0 ) be the global solution of (10.26) in (10.32) with the initial value U0 . Then, (10.30) yields that U (t; U0 )2X ≤ e−t U0 2X + Cε −3 ,

0 ≤ t < ∞, U0 ∈ K.

(10.33)

8 Dynamical System 8.1 Construction of Dynamical System Let us now construct a dynamical system determined from (10.26). As before, we can follow the general methods described in Chap. 6, Sect. 5. We already know that, for any U0 ∈ K, (10.26) and hence (10.18) possesses a unique global solution U (t; U0 ) in the function space (10.32). By S(t)U0 = U (t; U0 ), we define a nonlinear semigroup S(t) acting on K. Estimate (10.33) shows that (6.56) is fulfilled. So, (10.26) determines a dynamical system (S(t), K, X). For any exponent 0 < θ < 1, (10.26) equally determines a dynamical system in the universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (10.24) fulfills (4.21) and a fortiori (4.2) with β = θ . D Let Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

Meanwhile, by Proposition 6.1, Aθ S(t)U0 X ≤ CAS(t)U0 θX S(t)U0 1−θ X ≤ Cθ,R (1 + t −θ ),

0 < t < ∞, U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that the global norm estimate (6.56) is valid in the space Dθ , too. Then, by the similar arguments as in the proof of Proposition 6.2, S(t)U0 is seen to be continuous from [0, ∞) × Kθ to Kθ . So, (S(t), Kθ , Dθ ) becomes a dynamical system in Dθ .

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10 Belousov–Zhabotinskii Reaction Models

8.2 Exponential Attractors We construct exponential attractors for (S(t), K, X). Thanks to the general results, it suffices to verify (6.59) and (6.60). However, the space condition (6.59) is clear. Meanwhile, (10.33) shows that the dissipative condition (6.60) is also fulfilled. Hence, there exists an absorbing and invariant set X of (S(t), K, X) such that X is a compact subset of X and is a bounded subset of D(A). In addition, (S(t), X, X) has a family of exponential attractors M. For any exponent 0 < θ < 1, these exponential attractors M keep their properties as exponential attractors in the universal space Dθ , too. Hence, the exponential attractors M of (S(t), K, X) are exponential attractors of (S(t), Kθ , Dθ ). As before, we can verify that S(t) enjoys the squeezing property (6.45)–(6.46). Hence, by Theorem 6.16, (S(t), X, X) has exponential attractors whose fractal dimensions are estimated by (6.50).

9 Instability of Homogeneous Stationary Solution 9.1 Localized Problem Let U = t (u, v) (actually, u = v) be the positive homogeneous stationary solution to (10.18) given by (10.22). We intend to investigate its stability and instability. For this purpose, we begin with introducing the localized problem of (10.18) in a neighborhood of U ⎧ ∂u ⎪ ∂t = aΔu − u + χ(u) ⎪ ⎪

⎪ ⎪ ⎪ + ε−2 χ(u)(1 − χ(u)) − cχ(v) χ(u)−q ⎪ χ(u)+q ⎨ ∂v −1 −1 ∂t = bΔv − v + ε χ(u) + (1 − ε )χ(v) ⎪ ⎪ ⎪ ⎪ ∂u = ∂v = 0 ⎪ ⎪ ⎪ ⎩ ∂n ∂n v(x, 0) = v0 (x) u(x, 0) = u0 (x),

in Ω × (0, ∞), in Ω × (0, ∞),

(10.34)

on ∂Ω × (0, ∞), in Ω.

Here, χ(u) is a cutoff function of u in a complex neighborhood of u such that u−u u + u for |u − u| ≥ u. It is clear χ(u) ≡ u for |u − u| < u and χ(u) = |u−u| that |χ(u) + q| ≥ q for u ∈ C, |χ(u)| ≤ 2u for u ∈ C, and |χ(u1 ) − χ(u2 )| ≤ |u1 − u2 | for u1 , u2 ∈ C. Since u = v, χ(v) also is a cutoff function of v in the same neighborhood of v. We can then repeat the same arguments as in Sects. 6 and 7 to construct local solutions and global solutions to (10.34) for any initial value U0 ∈ D(Aθ ), where 0 ≤ θ < 1, in the function space: U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); D(Aθ )) ∩ C1 ((0, ∞); X).

9 Instability of Homogeneous Stationary Solution

387

9.2 Complexified Dynamical System ˜ The localized problem (10.34) defines a semigroup S(t) acting on Dθ = D(Aθ ). ˜ Therefore, (10.34) defines a complexified dynamical system (S(t), Dθ , Dθ ) for any 0 ≤ θ < 1. Let us fix 34 < θ < 1. Then, Dθ ⊂ C(Ω). In a suitable neighborhood of U in Dθ , any solution to the original problem (10.18) is a solution of (10.34), too. In such ˜ a neighborhood, any trajectory of (S(t), Kθ , Dθ ) is that of (S(t), Dθ , Dθ ). In par˜ ticular, U is an equilibrium of (S(t), Dθ , Dθ ). If U is stable as an equilibrium of ˜ (S(t), Dθ , Dθ ), then it is equally stable as that of (S(t), Kθ , Dθ ). However, even if ˜ U is unstable as an equilibrium of (S(t), Dθ , Dθ ), we cannot say that U is automatically unstable as that of (S(t), Kθ , Dθ ). The situation is quite similar as for the ˜ activator–inhibitor model handled in Chap. 9. The instability of U in (S(t), Dθ , Dθ ) provides us an information which indicates the instability in the original dynamical system (S(t), Kθ , Dθ ), too.

9.3 Case where fu (u, v) ≤ 0 Problem (10.34) is formulated as the Cauchy problem of an abstract equation in X of the form (10.26) in which the nonlinear operator F˜ : D(Aη ) → X, where 34 < θ < η < 1, is given by χ(u) + ε −2 [χ(u)(1 − χ(u)) − cχ(v)(χ(u) − q)(χ(u) + q)−1 ] ˜ F (U ) = . ε −1 χ(u) + (1 − ε−1 )χ(v) In the complex neighborhood |u − u| < u and |v − v| < v of (u, v), we have χ(u) ≡ u and χ(v) ≡ v. So, in a suitable neighborhood of U in D(Aη ) ⊂ D(Aβ ) ⊂ C(Ω), F˜ is Fréchet differentiable with the derivative 1 + ε−2 [(1 − 2u) − 2cqv(u + q)−2 ] −ε −2 c(u − q)(u + q)−1 F˜ (U ) h = 1 1 − ε −1 η h h u × 1 , U= ∈ B D(A ) (U ; r), h = 1 ∈ D(Aη ). h2 h2 v It is now possible to apply the general results obtained in Chap. 6, Sect. 6. It is easy to check that F˜ fulfills conditions (6.68) and (6.69), and a fortiori (6.64). Let us consider in the present subsection the case where fu (u, v) ≤ 0.

(10.35)

Since the inequalities in (10.23) are all satisfied (whatever ε is), condition (6.82) and hence condition (6.67) is fulfilled. We already know that (6.64) and (6.67) provide the exponential stability. Hence, under (10.35), U is an exponentially stable ˜ equilibrium of (S(t), Dθ , Dθ ).

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10 Belousov–Zhabotinskii Reaction Models

9.4 Case where 0 < fu (u, v) < 1 Let us next consider the case where 0 < fu (u, v) < 1. We can still follow the arguments of Chap. 6, Sect. 6. If ε > 0 is large enough to satisfy ε > max

2

b2 f u

, fu , ( ab(f u g v − f v g u ) + −abf v g u )2

then the inequalities in (10.23) are all valid. Hence, by the same reason as before, ˜ U is an exponentially stable equilibrium of (S(t), Dθ , Dθ ). For the remaining ε, we make the assumption that

abμ2k − ε −2 (εag v + bf u )μk + ε −3 (f u g v − f v g u ) = 0 −εag v + bf u > (a + b) −εf v g u .

for all μk , and

(10.36) Here, 0 = μ0 < μ1 ≤ μ2 ≤ · · · are the eigenvalues of the Laplace operator −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. This assumption means that condition (6.83) is fulfilled. We already know that (6.83) implies the spectrum separation condition (6.76). Hence, (6.68), (6.69), and (6.76) are fulfilled. We have thus shown that, under (10.36), U has a smooth local un˜ + (U ; O) which is tangential to the space U + X− at U , where stable manifold W X− is the eigenspace of A = A − F˜ (U ) for the negative eigenvalues. Furthermore, ˜ + (U ; O) = dim X− = N (U ), where dim W N (U ) = # μk ; abμ2k − ε −2 (εag v + bf u )μk + ε −3 (f u g v − f v g u ) < 0 . (10.37) We remark that if ε is sufficiently small, then U has a nontrivial local unstable manifold. Indeed, if ε is sufficiently small, then we have εag v + bf u > 2 εab(f u g v − f v g u ), which actually implies N (U ) ≥ 1. We remark also that, if ε → 0 (the other parameters in (10.34) are fixed), then ˜ + (U , O) tends to infinity. For simplicity, assume that Ω is a smooth bounded dim W domain in R3 . Then, according to [Tri78, Theorem 5.6.2], N (μ) ≡ 0≤μk ≤μ 1 = 3

3

3

O(μ 2 ). Therefore, N (U ) = N (μ2 (ε)) − N (μ1 (ε)) = O(μ2 (ε) 2 ) − O(μ1 (ε) 2 ), where 0 < μ1 (ε) < μ2 (ε) < ∞ are the solutions to the equation abμ2 −ε −2 (εag v + bf u )μ + ε −3 (f u g v − f v g u ) = 0 on μ. From μ1 (ε), μ2 (ε) =

(εag v + bf u )2 ∓

(εag v + bf u )2 − 4εab(f u g v − f v g u ) 2ε 2 ab

Notes and Further Researches

389

we find two positive constants C1 and c2 such that μ1 (ε) ≤ C1 ε −1 and μ2 (ε) ≥ c2 ε −2 , respectively. Hence, as ε → 0, ˜ + (U ; O) = N (U ) ≥ Cε −3 → ∞. dim W Let us finally consider the instability of U as an equilibrium of (S(t), Kθ , Dθ ). ˜ As is proved above, U has in (S(t), Dθ , Dθ ) a smooth local unstable manifold ˜ W+ (U ; O) with dimension N (U ) which is tangential to U + X− at U . Furthermore, X− is the eigenspace which corresponds to negative eigenvalues of A and possesses an orthogonal basis which consists of real-valued functions (cf. Corollary 2.1). These facts indicate that U is an unstable equilibrium of (S(t), Kθ , Dθ ), too.

Notes and Further Researches The Field–Noyes model was introduced in the papers [FN74a, FN74b], cf. also [TF80]. The model has already attracted interest of many mathematicians. Traveling wave solutions for one-dimensional problem were first constructed by Murray [Mur76] and were succeeded by the papers [FT79, Gib80, GC95, LS96, Tro77, WXY94, YW87]. The stability of traveling wave solutions was discussed by KL90]. In the meantime, there are only a few papers on the asymptotic behavior of solutions of the Field and Noyes model. Wang–Wu WW95] constructed for the d-dimensional problem under the Dirichlet boundary conditions a finite-dimensional global attractor. (But they handle the case where the parameter q vanishes.) They succeeded also in constructing an inertial manifold when Ω is a square domain. Pao Pao88] proved in some specific cases that solutions satisfying the Robin boundary conditions converge to the homogeneous stationary solutions as t → ∞. The stationary problems were studied by BK93, PW04]. For the numerical study, we quote MM82]. The results obtained in Sect. 4 are new. The Keener–Tyson model (10.18) was presented in the paper KT86]. The results on the Keener–Tyson model in this chapter were announced in YOS] without full proof. In YOS], the continuous dependence of exponential attractors M with respect to the parameter ε is also proved by employing Theorem 6.17.

Chapter 11

Forest Kinematic Model

In this chapter, we handle a forest kinematic model. In 1994, Kuznetsov, Antonovsky, Biktashev, and Aponina presented a system of parabolic-ordinary differential equations for three unknown functions u, v, and w for describing kinetics of a simplified forest ecosystem. The unknown function w (density of seeds) satisfies a linear diffusion equation in a domain Ω ⊂ R2 , v (density of old age class trees) satisfies a linear ordinary differential equation at each point of Ω, and u (density of young age class trees) does a nonlinear ordinary differential equation at each point of Ω with a cubic kinetic function on u and v. We treat this system as a degenerate parabolic system; “degenerate” means that the ordinary differential equations for u and v are regarded as a diffusion equation with vanishing diffusion rate. We may lose of course spatial smoothing of solutions for u and v. What is worse, at least in the level of numerical simulation (see Sect. 4.5), u and v can tend to discontinuous functions of Ω as t → ∞ even if they start from continuous functions at t = 0. These evidences make us be careful for choosing underlying function spaces X1 and X2 for treating the equations on u and v, respectively. First, Xi (i = 1, 2) must be closed in the usual product of functions, because the equations contain a cubic kinetic function on u and v and the solution u, v has no smoothing. Secondly, Xi must contain discontinuous functions in Ω. We are thus led to choose L∞ (Ω) as the spaces Xi (i = 1, 2).

1 Model Equations Consider the initial-boundary-value problem ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = βδw − γ (v)u − f u ⎪ ⎪ ⎪ ∂v ⎪ ⎪ = f u − hv in Ω × (0, ∞), ⎪ ⎨ ∂t ∂w in Ω × (0, ∞), ∂t = dΔw − βw + αv ⎪ ⎪ ⎪ ⎪ w=0 on ∂Ω × (0, ∞), ⎪ ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), w(x, 0) = w (x) in Ω, 0 0 0 (11.1) A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_11, © Springer-Verlag Berlin Heidelberg 2010

391

392

11 Forest Kinematic Model

in a two-dimensional C2 or convex, bounded domain Ω. In Ω a prototype forest ecosystem which simply consists of mono-species is considered. We suppose that age generations of trees are divided only into two classes, namely, the young age class and old age class. The unknown functions u(x, t) and v(x, t) denote the tree densities of the young and old age classes in Ω at time t, respectively. The third unknown function w(x, t) denotes the density of seeds in the air over Ω at time t. The third equation describes the kinetics of seeds; d is a diffusion rate in the air, and α and β are the seed production and seed deposition rates, respectively. While the first and second equations describe the growth of the young and old age class trees, respectively, δ is the seed establishment rate (0 < δ ≤ 1), γ (v) is the mortality of young trees, which is allowed to depend on the old-tree density v, f is the aging rate, and h is the mortality of old generation trees. On w, we impose the homogeneous Dirichlet conditions on ∂Ω. The initial functions u0 (x), v0 (x), and w0 (x) are given nonnegative functions in Ω. We assume that the mortality γ (v) is given by a square function of the form γ (v) = a(v − b)2 + c,

(11.2)

which means that the mortality hits its minimum when the old-tree density is a specific optimal value b. We assume also that a, b, c, d, f, h, α, and β are positive constants (> 0) and that δ is a constant such that 0 < δ ≤ 1. We shall first prove the global existence of solutions and then construct a dynamical system determined from (11.1). Fortunately, our dynamical system has a Lyapunov function given by (11.31). Thanks to this Lyapunov function, we can investigate the asymptotic behavior of solutions, and indeed they are shown to tend to some stationary solutions in a certain weak sense. Furthermore, we see that the structure of stationary solutions changes drastically depending on the parameters in the system, especially on the parameter h.

2 Local Solutions 2.1 Abstract Formulation We will handle equations on u and v in L∞ (Ω) and that on w in L2 (Ω). The underlying space is therefore set by ⎛ u ⎞ X = ⎝ v ⎠ ; u, v ∈ L∞ (Ω), w ∈ L2 (Ω) . (11.3) w In addition, the space of initial functions is taken as ⎛ u ⎞ ⎝ ⎠ v ; 0 ≤ u, v ∈ L∞ (Ω), 0 ≤ w ∈ L2 (Ω) . K= w

(11.4)

2 Local Solutions

393

We formulate problem (11.1) as the Cauchy problem for an abstract semilinear equation dU dt + AU = F (U ), 0 < t < ∞, (11.5) U (0) = U0 , in X. Here, A = diag{f, h, Λ} is a diagonal matrix operator of X with domain ⎛ u ⎞ 2 D(A) = ⎝ v ⎠ ; u, v ∈ L∞ (Ω), w ∈ HD (Ω) , w where Λ is a realization of the operator −dΔ + β in L2 (Ω) under the homogeneous Dirichlet boundary conditions and is a positive definite self-adjoint operator of L2 (Ω) with D(Λ) = HD2 (Ω) (see (2.31)). By Theorems 16.12 and 16.13 in Chap. 16, the domains of Λθ , 0 < θ < 1, are characterized by D(Λθ ) = H 2θ (Ω) for 0 ≤ θ < 14 and D(Λθ ) = HD2θ (Ω) for 14 < θ ≤ 1, θ = 34 . Then, A is a sectorial operator of X. For 0 < θ < 1, its fractional powers are given by Aθ = diag{f θ , hθ , Λθ } with ⎛ u ⎞ D(Aθ ) = ⎝ v ⎠ ; u, v ∈ L∞ (Ω), w ∈ D(Λθ ) . w

The nonlinear operator F is given by ⎛ ⎞ βδw − γ (v)u ⎠, fu F (U ) = ⎝ αv

⎛ ⎞ u U = ⎝ v ⎠ ∈ D(Aη ), w

(11.6)

where η is an arbitrarily fixed exponent such that 12 < η < 1. As D(Λη ) ⊂ L∞ (Ω) due to (1.76) and consequently as D(Aη ) ⊂ L∞ (Ω), F is certainly an operator from D(Aη ) into X.

2.2 Construction of Local Solutions It is possible to apply Theorem 4.4 to (11.5). Indeed, after some direct calculations, we get

F (U ) − F (U˜ )X ≤ C U 2X + U˜ 2X + 1 Aη (U − U˜ )X , U, U˜ ∈ D(Aη ), (11.7) with some constant C > 0. This shows that (4.21) is fulfilled. Therefore, for any U0 ∈ K, (11.5) possesses a unique local solution in the function space: u, v ∈ C([0, TU0 ]; L∞ (Ω)) ∩ C1 ((0, TU0 ]; L∞ (Ω)), w ∈ C((0, TU0 ]; HD2 (Ω)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)). (11.8)

394

11 Forest Kinematic Model

Here, TU0 > 0 is determined by the norm U0 X only. Moreover, the local solution satisfies the estimate tAU (t)X + U (t)X ≤ CU0 ,

0 < t ≤ TU 0 ,

(11.9)

with some constant CU0 > 0 depending only on U0 X . We observe also that w(t)L∞ ≤ Cw(t)H 2η ≤ CU0 t −η ,

0 < t ≤ TU 0 .

(11.10)

Then, thanks to (1.104), we can verify the representation formulas u(t) = e

−

t

0 [γ (v(s))+f ] ds

v(t) = e−ht v0 + f

u0 + βδ

t

e−

t s

[γ (v(τ ))+f ] dτ

w(s) ds,

(11.11)

0

t

e−h(t−s) u(s) ds.

(11.12)

0

In verifying (11.11), (11.10) was used.

2.3 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, v, w) be the local solution of (11.5) constructed above. Our goal is to show that u(t) ≥ 0, v(t) ≥ 0, and w(t) ≥ 0 for every 0 < t ≤ TU0 . For this purpose, however, we have to introduce the modified nonlinear operator F˜ (U ) = t (βδw − γ (v)u, f u, αχ(Re v)),

U ∈ D(Aη ),

where χ(v) is a cutoff function given by χ(v) ≡ 0 for −∞ < v < 0 and χ(v) = v for 0 ≤ v < ∞. We also have to consider the auxiliary problem

d U˜ dt

+ AU˜ = F˜ (U˜ ), ˜ U (0) = U0 .

0 < t < ∞,

(11.13)

We can in fact repeat the same arguments as in the preceding subsection to de˜ v, ˜ w) ˜ duce that, for any U0 ∈ K, (11.13) possesses a unique local solution U˜ = t (u, on an interval [0, T˜U0 ] in the function space (11.8)–(11.9) (note that χ(Re v1 ) − χ(Re v2 )L2 ≤ v1 − v2 L2 for v1 , v2 ∈ L2 (Ω)). Before showing the nonnegativity of u, v, w, we will show that of u, ˜ v, ˜ w. ˜ Let us first notice that U˜ is real valued. Indeed, if U˜ is a local solution of (11.13), then its complex conjugate is also a local solution with the same initial value; so, by the uniqueness of solution, they must coincide; this implies that U˜ is real valued. Let us secondly verify the nonnegativity of w. ˜ We use another cutoff function H (w) ˜ which is the C1,1 function given by H (w) ˜ = 12 w˜ 2 for −∞ < w˜ < 0

3 Global Solutions

395

and H (w) ˜ ≡ 0 for 0 ≤ w˜ < ∞. According to (1.100), the function ψ(t) = H ( w(t)) ˜ dx is continuously differentiable with the derivative Ω ψ (t) = d H (w)Δ ˜ w˜ dx + H (w)[−β ˜ w˜ + αχ(v)] ˜ dx. Ω

Ω

By property (1.96), H (w)Δ ˜ w˜ dx = − ∇H (w) ˜ · ∇ w˜ dx = − |∇H (w)| ˜ 2 dx ≤ 0. Ω

Ω

Ω

H (w)[−β ˜ w˜

In addition, it is clear that Ω + αχ(v)] ˜ dx ≤ 0. Hence, ψ (t) ≤ 0; consequently, ψ(t) ≤ ψ(0) for 0 < t ≤ T˜U0 . Thus, ψ(0) = 0 implies ψ(t) ≡ 0, i.e., w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . As for u˜ and v, ˜ we use representations (11.11) and (11.12), which hold equally for U˜ . Therefore, we conclude that u(t) ˜ ≥ 0 and v(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . We now notice that v(t) ˜ ≥ 0 implies that χ(v(t)) ˜ = v(t); ˜ in other words, U˜ is a local solution of the original problem (11.5), too. The uniqueness of solution then yields that U (t) ≡ U˜ (t). Hence, u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for 0 < t ≤ T˜U0 . If T˜U0 ≥ TU0 , we finish the proof. If not, we put T 0 = sup{0 < T ≤ ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for every 0 < t ≤ T }. Since TU0 ; u(t) Ω H (u(T0 )) dx = limtT0 Ω H (u(t)) dx = 0, we see that u(T0 ) ≥ 0. By similar reasons, we have v(T0 ) ≥ 0, w(T0 ) ≥ 0. So, if T0 = TU0 , we finish the proof. If T0 < TU0 , we will consider again problem (11.13) but with the initial time T0 and the initial value U (T0 ). Repeating the same arguments above, we conclude that there is τ > 0 such that u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for every t such that TU0 ≤ t ≤ TU0 + τ . This is a contradiction; hence, T0 = TU0 .

3 Global Solutions 3.1 A Priori Estimates for Local Solutions For U0 ∈ K, let U = t (u, v, w) be a local solution of (11.5) on [0, TU ] in the function space: 0 ≤ u, v ∈ C([0, TU ]; L∞ (Ω)) ∩ C1 ((0, TU ]; L∞ (Ω)), 0 ≤ w ∈ C([0, TU ]; L2 (Ω)) ∩ C((0, TU ]; HD2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (11.14) We establish the following a priori estimates for local solutions. Proposition 11.1 There exist an exponent ρ > 0 and a constant C > 0 such that the estimate U (t)X ≤ C[e−ρt U0 X + 1],

0 ≤ t ≤ TU ,

holds for any local solution U in the function space (11.14).

(11.15)

396

11 Forest Kinematic Model

Proof In the proof, C, C1 , C2 , . . . , stand for some constants which are determined only by Ω and by the initial constants in the equations of (11.1). Similarly, ρ stands for some positive exponent determined in the same manner. Step 1. Let us first estimate the norm U (t)L2 . Multiply the first equation of (11.1) by u and integrate the product in Ω. Then, 1 d 2 2 u dx + f u dx = βδ wu dx − γ (v)u2 dx 2 dt Ω Ω Ω Ω f u2 dx + C1 w 2 dx − γ (v)u2 dx. ≤ 2 Ω Ω Ω (11.16) Multiply the third equation of (11.1) by w and integrate the product in Ω. Then, 1 d 2 2 w dx + d |∇w| dx + β w 2 dx 2 dt Ω Ω Ω β vw dx ≤ w 2 dx + C2 v 2 dx. =α 2 Ω Ω Ω Let C3 > 0 be a constant such that C1 C3 ≤ β4 . Multiply (11.16) by C3 and add the product to the above inequality. Then, we obtain that C3 d 1 d C3 f β 2 2 2 u dx + w dx + u dx + w 2 dx 2 dt Ω 2 dt Ω 2 Ω 4 Ω v 2 dx − C3 γ (v)u2 dx. (11.17) ≤ C2 Ω

Ω

Multiply next the second equation of (11.1) by v and integrate the product in Ω. Then, 1 d 2 2 v dx + h v dx = f uv dx. 2 dt Ω Ω Ω Let C4 > 0 be a constant such that C4 h ≥ 2C2 . Multiply the present equality by C4 and add the product to inequality (11.17) to obtain that C3 d C4 d 1 d C3 f u2 dx + v 2 dx + w 2 dx + u2 dx 2 dt Ω 2 dt Ω 2 dt Ω 2 Ω β 2 2 v dx + w dx ≤ C4 f uv dx − C3 γ (v)u2 dx. + C2 4 Ω Ω Ω Ω We here notice that C2f 2 C4 f uv − C3 γ (v)u2 = − C3 a(v − b)2 u2 − C4 f (v − b)u + 4 4C3 a

3 Global Solutions

397

C 2 f 2 b2 C 2 f 2 1 b2 − C3 cu2 − C4 f bu + 4 + 4 + 4C3 c 4C3 a c C 2 f 2 1 b2 ≤ 4 + . 4C3 a c Therefore, d dt

(C3 u + C4 v + w ) dx + ρ 2

2

(C3 u2 + C4 v 2 + w 2 ) dx ≤ C.

2

Ω

Ω

Hence (cf. (1.58)), we conclude that U (t)2L2 ≤ C e−ρt U0 2L2 + 1 ,

0 ≤ t ≤ TU .

(11.18)

Step 2. We will estimate the norm w(t)L∞ . We have t η − t−s Λ − t−s Λ βt βt Λη w(t) = Λη e− 2 Λ e− 2 Λ w0 + α Λ e 2 e 2 v(s) ds. 0

Then, βt

w(t)H 2η ≤ C(1 + t −η )e− 2 w0 L2 + C e−tΛ

t

0

β 1 + (t − s)−η e− 2 (t−s) v(s)L2 ds,

≤ e−βt

where we used the estimate for t ≥ 0 (recall that Λ ≥ β). FurtherL2 more, by (11.18), t β 1 + (t − s)−η e− 2 (t−s) v(s)L2 ds 0

t

β 1 + (t − s)−η e− 2 (t−s) ds

≤C 0

+ Ce

−ρ t

0

≤ C[e

−ρ t

t

β 1 + (t − s)−η e−( 2 −ρ )(t−s) e−(ρ−ρ )s dsU0 L2

U0 L2 + 1],

where 0 < ρ < min{ β2 , ρ}. Replacing ρ by new one, we thus conclude that w(t)L∞ ≤ Cw(t)H 2η ≤ C[(1 + t −η )e−ρt U0 L2 + 1],

0 < t ≤ TU . (11.19)

Step 3. Let us finally estimate the norms u(t)L∞ and v(t)L∞ by using (11.11) and (11.12). In view of (11.19), we have t −f t e−f (t−s) [(1 + s −η )e−ρs U0 L2 + 1] ds. u(t)L∞ ≤ e u0 L∞ + C 0

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Therefore, replacing ρ by new one, we conclude that u(t)L∞ ≤ C[e−ρt U0 X + 1],

0 ≤ t ≤ TU .

In a similarly way, by (11.12), v(t)L∞ ≤ C[e−ρt U0 X + 1],

0 ≤ t ≤ TU .

These, together with (11.18), finally yield the desired estimates (11.15).

3.2 Global Solutions As the a priori estimates have been established for local solutions of (11.5) with U0 ∈ K, we can apply Corollary 4.3 to conclude the global existence of solutions. Indeed, for any U0 ∈ K, there exists a unique global solution to (11.5) in the function space: 0 ≤ u, v ∈ C([0, ∞); L∞ (Ω)) ∩ C1 ((0, ∞); L∞ (Ω)), 0 ≤ w ∈ C((0, ∞); HD2 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (11.20)

3.3 Global Norm Estimates For U0 ∈ K, let U (t) = U (t; U0 ) be the global solution of (11.5) with the initial value U0 in the function space (11.20). From (11.15) it immediately follows that U (t; U0 )X ≤ C[e−ρt U0 X + 1],

0 ≤ t < ∞, U0 ∈ K.

(11.21)

In particular, U (t; U0 )X ≤ C[U0 X + 1],

0 ≤ t < ∞, U0 ∈ K.

This, together with (11.9), yields that AU (t; U0 ) ≤ (1 + t −1 )p0 (U0 X ),

0 < t < ∞, U0 ∈ K,

(11.22)

with a suitable continuous increasing function p0 (·). In particular, w(t)H 2 ≤ (1 + t −1 )p0 (U0 X ),

0 < t < ∞, U0 ∈ K.

As a consequence, w(t)L∞ ≤ (1 + t −η )p0 (U0 X ),

0 < t < ∞, U0 ∈ K.

We can further prove the following norm estimates.

(11.23)

3 Global Solutions

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Proposition 11.2 For the derivative U (t), it holds that u (t)L∞ ≤ (1 + t −η )p1 (U0 X ),

v (t)L∞ ≤ p1 (U0 X ),

0 < t < ∞, U0 ∈ K,

0 < t < ∞, U0 ∈ K,

w (t)L2 ≤ (1 + t −1 )p1 (U0 X ),

(11.24) (11.25)

0 < t < ∞, U0 ∈ K,

(11.26)

where p1 (·) is an appropriate continuous increasing function. Proof In view of (11.23), (11.24) is verified directly from the equation of u. In the meantime, (11.25) is obvious by (11.21). The last estimate (11.26) is an immediate consequence of (11.22). Proposition 11.3 For the second order derivative U (t), it holds that u (t)L∞ ≤ (1 + t −1−η )p2 (U0 X ),

v (t)L∞ ≤ (1 + t

−η

0 < t < ∞, U0 ∈ K,

(11.27)

)p2 (U0 X ),

0 < t < ∞, U0 ∈ K,

(11.28)

w (t)L2 ≤ (1 + t −2 )p2 (U0 X ),

0 < t < ∞, U0 ∈ K,

(11.29)

where p2 (·) is an appropriate continuous increasing function. Proof By the equation of v in (11.5), v (t) = f u (t) − hv (t),

0 < t < ∞.

Then, v ∈ C2 ((0, ∞); L∞ (Ω)), and estimate (11.28) is verified by (11.24) and (11.25). We can next use the regularity property of solutions of linear evolution equations, regarding the equation of w as a linear equation with the unknown function w and given force function v. Since v belongs to C2 ([τ, ∞); L2 (Ω)), we conclude by Theorem 3.6 that w ∈ C2 ((τ, ∞); L2 (Ω)) with the estimate w (t)L2 + w (t)H 2 ≤ C 1 + (t − τ )−1 w (τ )L2 ,

τ < t < ∞,

where τ > 0 is any positive time. So, taking τ = 2t , we verify (11.29) and the estimate w (t)H 2 ≤ (1 + t −2 )p2 (U0 X ),

0 < t < ∞.

As a consequence, we verify that w (t)L∞ ≤ Cw (t)H 2η ≤ (1 + t −1−η )p2 (U0 X ),

0 < t < ∞.

Then, (11.27) is observed by differentiating the equation of u in t .

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4 Dynamical System We now construct a dynamical system determined from (11.5). It is possible to follow the general strategy described in Chap. 6, Sect. 5.1 by taking β = 0. Estimate (11.21) indeed shows that (6.56) is fulfilled; therefore, it is immediate to conclude that (11.5) defines a dynamical system (S(t), K, X). For any exponent 0 < θ < 1, (11.5) equally defines a dynamical system in the universal space Dθ = D(Aθ ) with the phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (11.6) fulfills (4.21) and a fortiori (4.2) with β = θ . D Let Kθ,R = K ∩ B θ (0; R). Applying Theorem 3.12, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

This, together with (11.22), then yields that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that (6.56) is valid in the space Dθ , too. Therefore, for any 0 < θ < 1, (S(t), Kθ , Dθ ) defines a dynamical system. X Estimate (11.15) means that, for any KR = K ∩ B (0; R), there exists tR > 0 such that ˜ =K ˜ S(t)KR ⊂ K ∩ B X (0; 2C) 2C

for any tR ≤ t < ∞.

˜ Thus, we Moreover, let B = S(1)K2C˜ . Then, by (11.22), B ⊂ B D(A) (0; 2p(2C)). have observed that (S(t), K, X) possesses an absorbing set B (⊂ K) which is a bounded set of D(A).

4.1 Lyapunov Function We can build up a Lyapunov function Ψ (U ) for (S(t), K, X). Let U0 ∈ K, and let S(t)U0 = U (t). Put ϕ(t) = f u(t) − hv(t), 0 ≤ t < ∞. From the equations of u and v in (11.5) it is easily obtained that ∂ϕ = f {βδw − [γ (v) + f ]u} − h(f u − hv) ∂t = fβδw − [γ (v) + f + h]ϕ − h[γ (v) + f ]v. Multiply this by ϕ(t) = ∂v ∂t and integrate the product in Ω. Then, ∂v d 1 d ϕ 2 dx + h Γ (v) dx − fβδ w dx 2 dt Ω dt Ω Ω ∂t 2 ∂v = − [γ (v) + f + h] dx, ∂t Ω v where Γ (v) = 0 [γ (v)v + f v] dv.

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401

Meanwhile, multiply the equation of w in (11.5) by ∂w ∂t and integrate the product in Ω. Then, d d ∂w 2 β d ∂w 2 2 |∇w| dx + w dx − α v dx. dx = − 2 dt Ω 2 dt Ω Ω ∂t Ω ∂t From these two energy equalities we obtain that α 2 dfβδ d fβ 2 δ 2 2 ϕ + |∇w| + hαΓ (v) + w − (f αβδ)vw dx dt Ω 2 2 2 2 ∂v ∂w 2 α[γ (v) + f + h] dx ≤ 0. (11.30) =− + fβδ ∂t ∂t Ω Note that hαΓ (v) +

fβ 2 δ 2 w − (f αβδ)vw ≥ −C 2

for all v, w ≥ 0,

with some constant C > 0. So, the functional given by α dfβδ (f u − hv)2 + |∇w|2 + hαΓ (v) Ψ (U ) = 2 2 Ω 1 fβ 2 δ 2 w − (f αβδ)vw dx, U ∈ D A 2 , + 2

(11.31)

becomes a Lyapunov function for the dynamical system (S(t), K, X). The Lyapunov function provides the following two propositions. Proposition 11.4 For any trajectory S(t)U0 = U (t), it holds that ∞ dU 2 dt (t) dt < ∞. 1 L2 Proof Integrate both sides of (11.30) on an interval [1, T ]. Then, 2 T ∂v ∂w 2 α[γ (v) + f + h] dx dt + fβδ ∂t ∂t Ω 1 α dfβδ ≤ ϕ(1)2 + |∇w(1)|2 + hαΓ (v(1)) 2 Ω 2 fβ 2 δ 2 w(1) + f αβδv(T )w(T ) dx. + 2 By (11.21) it is observed that 2 ∞ ∂v ∂w 2 α[γ (v) + f + h] dx dt < ∞. + fβδ ∂t ∂t 1 Ω

(11.32)

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Differentiating the equation of u in (11.1) with respect to t, we have ∂u ∂w ∂v ∂ 2u − γ (v)u − [γ (v) + f ] . = βδ ∂t ∂t ∂t ∂t 2 ∂u ∂t

Multiply this by 1 d 2 dt

Ω

∂u ∂t

and integrate the product in Ω. Then,

2

2 ∂w ∂v ∂u ∂u − γ (v)u dx − (γ (v) + f ) dx ∂t ∂t ∂t ∂t Ω Ω 2 ∂v ∂w 2 dx ≤ C(U0 X + 1) + ∂t ∂t Ω 2 f ∂u − dx. 2 Ω ∂t

dx =

βδ

Integrating this inequality on [1, T ], we obtain that f 2

T 1

Ω

∂u ∂t

2 dx dt ≤

2 ∂u (1) dx Ω ∂t + C(U0 X + 1)

1 2

T

1

On account of (11.32), we conclude that sired integrability has been verified.

f 2

Ω

∞ 1

∂v ∂t

2

∂u 2 Ω ( ∂t ) dx dt

+

∂w ∂t

2 dx dt.

< ∞. Hence, the de

Proposition 11.5 For any trajectory S(t)U0 = U (t), the derivative 0 as t → ∞ in the L2 norm.

dU dt (t)

tends to

d dU dt (t)2L2 | ≤ CU0 for 1 ≤ t < ∞ Proof Propositions 11.2 and 11.3 yield that | dt with some constant CU0 > 0 depending on U0 X . This, together with integrability dU 2 of dU dt (t)L2 in (1, ∞), implies the desired convergence dt (t)L2 → 0.

We finally verify the following fact. 1

Theorem 11.1 The Lyapunov function Ψ : D(A 2 ) → C is Fréchet differentiable, and U ∈ D(A) is a stationary solution of (11.1) if and only if Ψ (U ) = 0. Proof By direct calculations it is observed that Ψ is Fréchet differentiable, and its derivative is given by Ψ (U )U = α(f u − hv)(f u − hv) + dfβδ∇w · ∇w + hαΓ (v)v Ω

+ fβ 2 δww − f αβδ(vw + wv) dx,

4 Dynamical System

403

⎛ ⎞ ⎛ ⎞ u u 1 U = ⎝v ⎠, U = ⎝v ⎠ ∈ D A2 . w w Consequently, Ψ (U )U =

f α(f u − hv)u + α[h2 v + hγ (v)v − fβδw]v

Ω

+ fβδ[d∇w · ∇w + (βw − αv)w] dx. Therefore, Ψ (U ) = 0 if and only if f u − hv = 0, h2 v + hγ (v)v − fβδw = 0 and [d∇w · ∇w + (βw − αv)w] dx = 0 for all w ∈ H˚ 1 (Ω). Ω

Furthermore, Ψ (U ) = 0 if and only if f u − hv = 0, hu + γ (v)u − βδw = 0, and −dΔw + βw − αv = 0 in H −1 (Ω). Hence, Ψ (U ) = 0 if and only if U is a stationary solution of (11.1).

4.2 ω-limit Sets For each U0 ∈ K, the ω-limit set of the trajectory S(t)U0 is defined by ω(U0 ) =

{S(τ )U0 ; t ≤ τ < ∞} (closure in X),

t≥0

see Chap. 6, Sect. 1.2. However, we have the following observation. Remark 11.1 As will be seen in Sect. 4.5, numerical examples suggest that there exists a trajectory which starts from an initial value U0 ∈ K consisting of a continuous function of Ω componentwise but, as t → ∞, converges to a discontinuous stationary solution U = t (u, v, w). If this is the case, then any sequence of the form S(tn )U0 cannot converge to U in the topology of X. (Indeed, on account of the topology of X, u and v must be continuous functions; in the meantime, since w ∈ H 2 (Ω), w is also a continuous function.) Therefore, for U0 ∈ K, it must hold that ω(U0 ) = ∅. This suggests also that the dynamical system (S(t), K, X) cannot have a global attractor. In view of this observation, we are led to introduce other types of ω-limit sets. A sequence Un = t (un , vn , wn ) in X is said to be L2 convergent to U = t (u, v, w) ∈ X as n → ∞ if u → u in L (Ω), v → v in L (Ω), and w → w n 2 n 2 n in L2 (Ω), respectively. Then, using this topology, we define the L2 -ω-limit set of

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S(t)U0 , U0 ∈ K, by L2 -ω(U0 ) =

{S(τ )U0 ; t ≤ τ < ∞}

(closure in the L2 topology of X).

t≥0

In the meantime, a sequence Un = t (un , vn , wn ) in X is said to be weak* convergent to U = t (u, v, w) ∈ X as n → ∞ if un → u weak* in L∞ (Ω), vn → v weak* in L∞ (Ω), and wn → w in L2 (Ω), respectively. Using this topology, we define the w*-ω-limit set of S(t)U0 , U0 ∈ K, by w*-ω(U0 ) =

{S(τ )U0 ; t ≤ τ < ∞} (closure in the weak* topology of X).

t≥0

Concerning these ω-limit sets, we have the following results. Theorem 11.2 For each U0 ∈ K, w*-ω(U0 ) is a nonempty set. Proof Let U0 ∈ K and S(t)U0 = U (t). On account of (11.22), we notice that the Banach–Alaoglu theorem of sequentially weak* compact version (see Brezis [Bre83, IV. 3. C]) and the compact embedding theorem, i.e., H 2 (Ω) ⊂ L2 (Ω), are available to conclude that there exists a time sequence {tn } tending to ∞ such that u(tn ) → u weak* in L∞ (Ω), v(tn ) → v weak* in L∞ (Ω), and w(tn ) → w in L2 (Ω), that is, U (tn ) → U in the weak* topology of X. Theorem 11.3 For each U0 ∈ K, ω(U0 ) ⊂ L2 -ω(U0 ) ⊂ w*-ω(U0 ). Proof The first relation ω(U0 ) ⊂ L2 -ω(U0 ) is obvious by the definition. Let U ∈ L2 -ω(U0 ). Then, there exists a sequence {tn } ∞ such that S(tn )U0 = Un (t) → U in the L2 topology of X. Let ϕ ∈ L1 (Ω). For any f ∈ L2 (Ω), ϕ[u(tn ) − u] dx ≤ ϕ − f L u(tn ) − uL + f [u(tn ) − u] dx . ∞ 1 Ω

Ω

Since L2 (Ω) is dense in L1 (Ω) and since (11.21) is true, we verify that, as tn → ∞, | Ω ϕ[u(tn ) − u] dx| → 0. Hence, u(tn ) → u in the weak* topology of L∞ (Ω). It is the same for the weak* convergence of v(tn ) to v. Thus, we have U ∈ w*-ω(U0 ). Theorem 11.4 For U0 ∈ K, let there exist a sequence {tn } tending to ∞ such that the sequence of functions S(tn )U0 converges to U ∈ X componentwise almost everywhere in Ω. Then, U ∈ L2 -ω(U0 ). Proof By (11.21) and (11.23), the almost everywhere convergence implies the L2 convergence of the sequences of u(tn ), v(tn ), and w(tn ). Hence, U ∈ L2 -ω(U0 ).

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405

4.3 Constituents of L2 -ω-limit Sets Generally speaking, if there exists a Lyapunov function, then one can prove that the ω-limit set consists of equilibria of a dynamical system. However, in the present case, we do not know that S(t) is continuous with respect to the weak* topology of X. We can merely prove the assertion only for the L2 -ω-limit set. Theorem 11.5 If L2 -ω(U0 ) is nonempty, then the set consists of equilibria of (S(t), K, X). Proof We first notice that S(t) : K → X is continuous with respect to the L2 topology of X. Indeed, by similar techniques as in Step 1 of the proof of Proposition 11.1, we can establish the local Lipschitz condition S(t)U0 − S(t)U˜ 0 L2 ≤ CT ,R U0 − U˜ 0 L2 , 0 ≤ t ≤ T ; U0 , U˜ 0 ∈ K with U0 L2 + U˜ 0 L2 ≤ R. The detailed proof is however left to the reader. Let U ∈ L2 -ω(U0 ). Then, there exists a sequence tn → ∞ such that S(tn )U0 = U (tn ) → U in the L2 topology. By (11.22), we can take a subsequence {w(tn )} of {w(tn )} such that w(tn ) → w in H 1 (Ω). It is then easy to see that w = w . Meanwhile, u(tn ) → u and v(tn ) → v in any Lp topology with any finite 2 ≤ p < ∞. By these facts we conclude that the value Ψ (U (tn )) of the Lyapunov function is convergent to Ψ (U ) as tn → ∞. That is, Ψ (U ) = lim Ψ (U (tn )) = inf Ψ (S(t)U0 ) ≡ Ψ∞ . n →∞

0≤t 0. 3f

We here notice that

2 (c + f )2 u2 + (αδv)2 + (βδw)2 − (c + f )αδuv − αβδ 2 vw − (c + f )βδwu + 3εv 2

2

(c + f )αδu 2 2 αβδ 2 w 2 − α δ +ε v + = √ α2 δ2 + ε v − √ α2δ2 + ε α2δ2 + ε (c + f )2 u2 (βδw)2 + v2 + 2 2 . + [βδw − (c + f )u]2 + ε 2 2 α δ +ε α δ +ε

4 Dynamical System

407

Therefore, with an appropriate exponent ρ > 0, d (C1 u2 + C2 v 2 + C3 w 2 ) dx + ρ (C1 u2 + C2 v 2 + C3 w 2 ) dx ≤ 0, dt Ω Ω where Ci > 0 (i = 1, 2, 3) are some positive constants. Thus, U (t)L2 ≤ Ce−ρt U0 L2 ,

0 < t < ∞, U0 ∈ K.

This shows that, as t → ∞, S(t)U0 converges to O exponentially in the L2 topology. We can then repeat similar calculations as in Steps 2 and 3 of the proof of Proposition 11.1 to derive the exponential decay of S(t)U0 in the X norm. In particular, the assertion of theorem is verified. We secondly consider the case where ab2 < 3(c + f ). We notice in this case that the cubic function Q(v) ≡

h [γ (v) + f ]v, fβδ

−∞ < v < ∞,

(11.36)

has the following properties. Lemma 11.1 When ab2 < 3(c + f ), Q (v) > 0 for all −∞ < v < ∞. Therefore, w = Q(v) is an increasing function, and its inverse v = Q−1 (w) is a single-valued smooth function for −∞ < w < ∞; moreover, the inverse function is uniformly Lipschitz continuous, i.e., |Q−1 (w) − Q−1 (w)| ˜ ≤ L|w − w|, ˜ −∞ < w, w˜ < ∞. On the contrary, when ab2 ≥ 3(c + f ), Q (v) has real roots 0 < v1 ≤ v2 < ∞ (when ab2 = 3(c + f ), v1 = v2 ). Proof We obviously see from (11.2) that h (3av 2 − 4abv + ab2 + c + f ) fβδ 2b 2 ab2 − 3(c + f ) h 3a v − . − = fβδ 3 3

Q (v) =

Therefore, all the assertions of lemma are clear.

Using these properties of Q(v), we prove the following theorem. Theorem 11.7 Let ab2 < 3(c + f ). Then, L2 -ω(U0 ) = w*-ω(U0 ) for every U0 ∈ K. Proof Let S(t)U0 = t (u(t), v(t), w(t)). Consider any time sequence {tn } which tends to ∞ as n → ∞. By (11.22), w(tn )H 2 is a bounded sequence; so, we can choose a subsequence {tn } for which {w(tn )} is convergent to w in H 1+ε (Ω) and

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11 Forest Kinematic Model

hence in L∞ (Ω). From the first and second equations of (11.1), it is easily observed that 1 du γ (v(tn )) + f dv Q(v(tn )) = w(tn ) − (tn ) − (tn ). βδ dt fβδ dt So, by (11.36), 1 du γ (v(tn )) + f dv v(tn ) = Q−1 w(tn ) − (tn ) − (tn ) . βδ dt fβδ dt Since w(tn ) → w in L∞ (Ω) and since Proposition 11.5 is valid, we conclude that v(tn ) converges to v = Q−1 (w) in L2 (Ω). Since Proposition 11.5 provides in particular that, as t → ∞, f u(t)−hv(t) → 0 in L2 (Ω), we conclude also that u(tn ) converges to fh v in L2 (Ω). Thus, we have shown that t (u(tn ), v(tn ), w(tn )) → t (u, v, w) in L (Ω). 2 We now know that any sequence t (u(tn ), v(tn ), w(tn )) has a subsequence which converges to some vector of X in the L2 topology. In particular, the relation w*-ω(U0 ) ⊂ L2 -ω(U0 ) must be true.

4.5 Numerical Examples We here present some numerical example. Let Ω = [0, 1] × [0, 1] be a quadratic domain. Set the parameters as α = β = δ = 1.0, f = g = 1.0, a = 1.0, b = 3.0, c = 0.2, and d = 0.05. We treat h as a control parameter and put h = 0.0488 in order that the parameters satisfy the relation 0 0 is the minimal eigenvalue of the operator −Δ in L2 (Ω) equipped with the homogeneous Dirichlet boundary conditions.

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11 Forest Kinematic Model

5.1 Localized Problem in a Neighborhood of O For investigating the stability and instability of O, we can follow the general strategy in Chap. 6, Sect. 6. Let χ(u) be a cutoff function defined on the complex plane C such that χ(u) = u for u : |u| < 1 and χ(u) = u/|u| for u : |u| ≥ 1. The localized problem in a neighborhood of O is given by ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = βδχ(w) − γ (χ(v))χ(u) − f u ⎪ ⎪ ⎪ ∂v ⎪ ⎪ in Ω × (0, ∞), ⎨ ∂t = f χ(u) − hv ∂w ⎪ ∂t = dΔw − βw + αχ(v) ⎪ ⎪ ⎪ w =0 ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), 0 0

in Ω × (0, ∞), on ∂Ω × (0, ∞), w(x, 0) = w0 (x) in Ω. (11.38) We can then repeat the same arguments as in Sect. 1 to construct local solutions and global solutions for all initial values from X in the function space: U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); X) ∩ C1 ((0, ∞); X). ˜ acting Therefore, the localized problem (11.38) defines a continuous semigroup S(t) ˜ on X. Furthermore, it is verified that (S(t), X, X) defines a dynamical system which is a complexified dynamical system of (S(t), K, X). It is also verified that, if U0 ∈ D(Aθ ) with 0 < θ < 1, then U belongs to: U ∈ C((0, ∞); D(A)) ∩ C([0, X); D(Aθ )) ∩ C1 ((0, ∞); D(Aθ )). ˜ From this we can equally verify that, for any 0 < θ < 1, (S(t), Dθ , Dθ ) is a dynamθ ical system in Dθ , where Dθ = D(A ). We now fix 12 < θ < 1 in order to have Dθ ⊂ L∞ (Ω). Then, in a suitable neighborhood of O in the space Dθ , any trajectory of (S(t), Kθ , Dθ ) is that of ˜ ˜ (S(t), Dθ , Dθ ). In particular, O is an equilibrium of (S(t), Dθ , Dθ ), too. Therefore, ˜ if O is stable as an equilibrium of (S(t), Dθ , Dθ ), then it is the same as that of (S(t), Kθ , Dθ ). However, we cannot say automatically that, even if O is unstable ˜ in (S(t), Dθ , Dθ ), it is the same as that of (S(t), Kθ , Dθ ). Nevertheless, the insta˜ bility of O in (S(t), Dθ , Dθ ) provides an information which indicates its instability in the original system (S(t), Kθ , Dθ ). In fact, that shows existence of a smooth lo˜ cal unstable manifold W˜ + (O; O) in (S(t), Dθ , Dθ ), which is tangential to the space X− at O, where X− is a finite- or infinite-dimensional eigenspace corresponding to the negative eigenvalues of the linearized operator A − F˜ (O) given by (11.39). Furthermore, X− has a basis consisting of vectors which are componentwise realvalued functions in Ω. ˜ Let us next verify the Fréchet differentiability of S(t) in a neighborhood of O. ˜ is determined by the Cauchy problem for a semilinear equation of We know that S(t)

5 Homogeneous Stationary Solution

411

the form (11.5) in which the nonlinear operator is replaced by the following operator F˜ (U ) : D(Aη ) → X, where 12 < θ < η < 1: ⎛ ⎞ ⎛ ⎞ βδχ(w) − γ (χ(v))χ(u) u ⎠ , U = ⎝ v ⎠ ∈ D(Aη ). f χ(u) F˜ (U ) = ⎝ αχ(v) w In a neighborhood of O in D(Aη ) ⊂ D(Aθ ) ⊂ L∞ (Ω), F˜ is Fréchet differentiable with the derivative ⎛ ⎞ βδr − γ (v)p − γ (v)uq ⎠, fp F˜ (U ) · H = ⎝ αq ⎛ ⎞ p η U ∈ B D(A ) (O; r), H = ⎝ q ⎠ ∈ D(Aη ). r Therefore, F˜ is of class C1,1 and satisfies conditions (6.68) and (6.69). So, the gen˜ : Dθ → eral results obtained in Chap. 6, Sect. 6 are available to conclude that S(t) Dθ is of class C1,1 in a neighborhood of O in Dθ for 0 ≤ t ≤ T , where T > 0. In ˜ is Fréchet differentiable at O for any 0 ≤ t < ∞ with the differential particular, S(t) ˜ O = e−tA , where e−tA is the analytic semigroup on X generated by S(t) ⎛ ⎞ m 0 −βδ h 0 ⎠ (11.39) −A = −(A − F˜ (O)) = ⎝−f 0 −α Λ with m = γ (0) + f = ab2 + c + f .

˜ O 5.2 Spectrum of S(t) Let us characterize in this subsection the spectrum of A in X, i.e., σ X (A). To this end, let us find λ ∈ C such that, for any H = t (p, q, r) ∈ X, the vector equation (λ − A)U = H has a unique solution U = t (u, v, w) ∈ D(A). Obviously, this problem is equivalent to ⎧ ⎪ ⎨(λ − m)u + βδw = p, f u + (λ − h)v = q, ⎪ ⎩ αv + (λ − Λ)w = r. It then follows that [(λ − m)(λ − h)(λ − Λ) + f αβδ]w = f αp − α(λ − m)q + (λ − m)(λ − h)r.

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11 Forest Kinematic Model

Therefore, if λ is a solution of the equation (λ − m)(λ − h) = 0, i.e., λ = m or λ = h, then w cannot belong to H 2 (Ω) in general. This means that m and h do not belong to ρ X (A), i.e., m, h ∈ σ X (A). So, let λ be neither m nor h. Then, we verify that λ ∈ ρ X (A)

⇔

λ+

f αβδ ∈ ρ L2 (Λ). (λ − m)(λ − h)

λ ∈ σ X (A)

⇔

λ+

f αβδ = dμn + β (λ − m)(λ − h)

In other words,

for some n = 0, 1, 2, . . . , where (0 0. The construction of a dynamical system for (11.1) in the underlying space X was done by Chuan–Yagi [CY06] and Shirai–Chuan–Yagi [SCY07a]. The structure of the ω-limit set of trajectories was studied by Chuan–Tsujikawa–Yagi [CTY06] and Shirai–Chuan–Yagi [SCY07b]. Although we know the existence of the Lyapunov function given by (11.31), there remain many essential questions unanswered for the ω-limit sets. For example, does the limit set w*-ω(U0 ) consist only of stationary solutions of (11.1)? In any case, is the limit set L2 -ω(U0 ) nonempty? When do the

Notes and Further Researches

415

two limit sets w*-ω(U0 ) and L2 -ω(U0 ) coincide? Can we utilize the Simon and Łojasiewicz method (cf. [Sim83] and [Loj63, Loj65]) in order to show that the ω-limit sets are singletons? The stability and instability of the homogeneous stationary solution O were studied by Shirai–Chuan–Yagi [SCY08]. In Chuan–Tsujikawa–Yagi [CTY09], the existence of an infinite number of discontinuous stationary solutions was indicated but under the homogeneous Neumann boundary conditions on w. In the Dirichlet case, however, there is no such a rigorous result. Mola–Yagi [MY09] treated a forest kinematic model with memory.

Chapter 12

Chemotaxis Models

Budrene–Berg discovered [BB91, BB95] a remarkable aggregation pattern generated by chemotactic bacteria. In 1995, Woodward–Tyson–Myerscough–Murray– Budrene–Berg presented the mathematical model ⎧ u ⎨ ∂u = aΔu − μ∇ · ∇ρ + cu 1 − us in Ω × (0, ∞), ∂t (1+αρ)2 (12.1) rρ p ∂ρ u su ⎩ = bΔρ + in Ω × (0, ∞), ∂t 1+βuq − 1+γρ for describing the process of pattern formation. Here, Ω is a two-dimensional domain where bacteria are bred. The function u(x, t) denotes the population density of bacteria in Ω at time t . The function ρ(x, t) denotes the concentration of chemical attractant in Ω at time t . The movement of biological individuals consists of two effects: one is a random walk, and the other is a directed movement in a sense that they have a tendency to move toward a higher concentration direction of the chemical substance. The latter directed movement is called chemotaxis. In 1970, Keller–Segel introduced a mathematical formulation for chemotaxis by the diffusion–advection equation ∂u = aΔu − μ∇ · [u∇χ(ρ)], ∂t where χ(ρ) is a sensitivity function of bacteria toward the chemoattractant, and μ > 0 is an attraction rate. Several forms are proposed as sensitivity functions. The ρ typical ones are such that χ(ρ) = ρ, ρ 2 , log(ρ + 1), log ρ, ρ+1 , etc. In (12.1), the ρ authors took the function χ(ρ) = 1+αρ . In (12.1), the term cu(1 − us ) denotes the proliferation of bacteria by the logistic law, where s denotes the concentration of the substrate. In this model, s is assumed sup to be spatially homogeneous and constant. The term 1+βu q denotes the production of chemoattractant by bacteria, where p, q are some nonnegative exponents and ur ρ denotes the consumption of chemoattractant β ≥ 0 is a constant. The term − 1+γρ by bacteria, where r is some nonnegative exponent, and γ ≥ 0 is a constant. The bacteria and the chemical substance diffuse with rates a > 0 and b > 0, respectively. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_12, © Springer-Verlag Berlin Heidelberg 2010

417

418

12 Chemotaxis Models

We will consider in this chapter the following simplified model: ∂u 2 in Ω × (0, ∞), ∂t = aΔu − μ∇ · [u∇χ(ρ)] + cu − γ u ∂ρ ∂t

= bΔρ − dρ + νu

in Ω × (0, ∞).

(12.2)

The sensitivity function will not be specified but will be assumed to satisfy condiρ tion (12.4). Actually, χ(ρ) = ρ, log(ρ +1), and ρ+1 are allowed. For the production term, we take p = 1 and β = 0. We ignore the consumption of chemical substance by bacteria, i.e., r = 0; instead we consider the unconstrained resolution which is denoted by −dρ with rate d > 0. First, we construct a dynamical system for the initial-boundary-value problem of (12.2) imposing the homogeneous Neumann boundary conditions for u and ρ and show that the dynamical system possesses exponential attractors. Secondly, we investigate stability and instability of the homogeneous stationary solution cνthe (u, ρ) = γc , dγ . In fact, we show that, when the product μν is small, (u, ρ) is stable and, on the other hand, that, when μν is large enough, (u, ρ) is no longer stable. Moreover, we show that the degree of instability tends to infinity as μν → ∞ and consequently the fractal dimensions of exponential attractors also tend to infinity.

1 Chemotaxis Model Without Proliferation Before studying (12.2), let us consider a chemotaxis model without proliferation. We are concerned with the following problem: ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu − μ∇ · [u∇χ(ρ)] ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu in Ω × (0, ∞), ∂t (12.3) ∂ρ ∂u ⎪ = ∂n = 0 on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ⎪ ⎪ ⎩u(x, 0) = u (x), ρ(x, 0) = ρ (x) in Ω. 0 0 As mentioned above, such a chemotaxis model was founded by Keller–Segel [KS70]. The form of the equations of (12.3) is of some particular case of the general ones presented in the paper. Here, Ω is a two-dimensional C2 or convex, bounded domain. We assume that the sensitivity function χ(ρ) is a real smooth function for 0 ≤ ρ < ∞ satisfying the condition i d χ sup i (ρ) < ∞ for i = 1, 2. (12.4) dρ 0≤ρ0). First, we construct local solutions for problem (12.3). Secondly, in the case where χ(ρ) is linear (= ρ), we establish a priori estimates of local solutions for the initial functions u0 (x) ≥ 0 and ρ0 (x) ≥ 0 with sufficiently small norm u0 L1 . Consequently, for such initial functions, we can show the global existence of solutions.

1 Chemotaxis Model Without Proliferation

419

However, we must notice the fact due to Herrero–Velázquez [HV96] that the local solutions to (12.3) blow up in finite time, provided that the norms u0 L1 of their initial functions are large.

1.1 Abstract Formulation Let us formulate problem (12.3) as the Cauchy problem for an abstract evolution equation. It will be spontaneous to try to handle the first equation of (12.3) on u in the space L2 (Ω). Then, we are naturally led to handle the second one in the Sobolev space H 2 (Ω). As will be observed in establishing a priori estimates for (12.3) (see Propositions 12.2 and 12.3), we have to couple the norms u(t)2L2 and ρ(t)2H 2 . This means that it is natural to handle (12.3) in the product space of u ∈ L2 (Ω) and ρ ∈ H 2 (Ω). By this reason, our underlying space is taken as u X= ; u ∈ L2 (Ω) and ρ ∈ HN2 (Ω) . (12.5) ρ Since ρ satisfies the homogeneous Neumann boundary conditions, the space for ρ is taken as HN2 (Ω). In the underlying space X, the chemotactic term ∇ · [u∇χ(ρ)] can be treated as a lower term. In other words, we are able to formulate problem (12.3) as a semilinear problem of the form (4.1), although the first diffusion equation of (12.3) on u is quasilinear in form. This is the merit of taking an underlying space of the form (12.5). In this way, we rewrite (12.3) by dU 0 < t < ∞, dt + AU = F (U ), (12.6) U (0) = U0 , in X, where U = t (u, ρ). Here, the linear operator A is given by

A1 0 . A= −ν A2

(12.7)

The two operators A1 = −aΔ + 1 and A2 = −bΔ + d are positive definite selfadjoint operators of L2 (Ω) with domains D(A1 ) = D(A2 ) = HN2 (Ω). However, A2 4 (Ω) = D(A2 ) = {ρ ∈ H 2 (Ω); Δρ ∈ H 2 (Ω)} is regarded as an operator from HN 2 N N 2 2 into D(A2 ) = HN (Ω). (Note that, as Ω is not assumed to be of C4 class, we cannot 4 (Ω) ⊂ H 4 (Ω).) So, the domain of A is given by expect that HN 2 u 2 4 D(A) = ; u ∈ HN (Ω) and ρ ∈ HN 2 (Ω) . ρ The nonlinear operator F is given by

−μ∇ · [u∇χ(Re ρ)] + u F (U ) = . 0

(12.8)

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12 Chemotaxis Models

Since ∇ · [u∇χ] = ∇u · ∇χ + uΔχ , it is possible to see that ∇ · [u∇χ]L2 ≤ CuH 1+ε χH 2 ,

u ∈ H 1+ε (Ω), χ ∈ HN2 (Ω)

(12.9)

with an arbitrary fixed exponent 0 < ε < 12 . Indeed, note that H 1+ε (Ω) ⊂ L∞ (Ω) due to (1.76) and H 1+ε (Ω) ⊂ Hp1 (Ω) for p such that 2 < p < 2/(1 − ε) due to (1.74). We then observe that F (U ) is an operator from D(Aη ) into X, where the exponent η is given by η = 1+ε 2 . Note also by (1.93) that ρ → χ(Re ρ) is a bounded operator from HN2 (Ω) into itself.

1.2 Construction of Local Solutions In order to show the existence of local solutions to (12.6), let us verify that A and F fulfill the structural assumptions made in Chap. 4, Sect. 1. Denote by B the multiplicative operator by the number −ν. Then, A is obviously written by

A1 0 A= . (12.10) B A2 We regard B as a linear operator from D(B) ⊂ L2 (Ω) to HN2 (Ω); clearly, D(B) = HN2 (Ω), and B is a bounded operator from D(A1 ) onto HN2 (Ω). Hence, it is an immediate consequence of Theorem 2.16 that the operator A is a sectorial operator of the product space X. Let us next investigate the domains of fractional powers Aθ for 0 ≤ θ ≤ 1. According to Theorems 16.7 and 16.7 in Chap. 16, the domains D(Aθ1 ) and D(Aθ2 ) are characterized by D Aθ1 = D Aθ2 = H 2θ (Ω), D Aθ1 = D Aθ2 = HN2θ (Ω),

3 0≤θ < , 4 3 < θ ≤ 1. 4

Consequently, we have 2θ (Ω) = ρ ∈ HN2 (Ω); Δρ ∈ H 2(θ−1) (Ω) , D Aθ2 = HN 2(θ−1) 2θ 2 (Ω) D(Aθ2 ) = HN 2 (Ω) = ρ ∈ HN (Ω); Δρ ∈ HN

7 1≤θ < , 4 7 < θ ≤ 2. 4

Consider a diagonal operator AD = diag {A1 , A2 } acting in X. Then, AD is a selfadjoint operator of X; therefore, AθD = diag {Aθ1 , Aθ2 }. More precisely, θ u 3 2(θ+1) 2θ ; u ∈ H (Ω) and ρ ∈ HN (Ω) , 0 < θ < , D AD = ρ 4 u 3 2(θ+1) D AθD = ; u ∈ HN2θ (Ω) and ρ ∈ HN 2 (Ω) , < θ ≤ 1. ρ 4

1 Chemotaxis Model Without Proliferation

421

We then verify the following property. Proposition 12.1 The domains of fractional powers of A are given by D(Aθ ) = D AθD , 0 ≤ θ ≤ 1, with the norm equivalence C −1 AθD ·X ≤ Aθ · X ≤ C AθD ·X ,

(12.11)

0 ≤ θ ≤ 1.

Proof We will apply Theorem 16.4 to A. It is clear that the analogous statement to Theorem 16.4 is true for matrix operators of the form (12.10). The multiplicative operator B is clearly an operator from L2 (Ω) to HN2 (Ω) = D(A2 ) with the domain D(B) = HN2 (Ω). Therefore, B is a bounded operator from D(A1 ) into HN2 (Ω). In the meantime, u ∈ HN2 (Ω) belongs to D(B ∗ ) if and only if the functional (u, B·)H 2 is continuous with respect to the L2 topology. Since N

(u, Bv)H 2 = (A2 u, −A2 νv)L2 = −ν(A2 u, A2 v)L2 , u ∈ D(A22 ) implies u ∈ D(B ∗ ). N

This indeed means that D(A22 ) ⊂ D(B ∗ ) and B ∗ is a bounded operator from D(A22 ) into L2 (Ω). Hence, we have verified that Theorem 16.4 is available to A. We now know that D(Aθ ) = [D(A), X]θ . From this we deduce that D(Aθ ) = [D(A), X]θ = [D(AD ), X]θ = D(AθD ) because of the coincidence D(A) = D(AD ). The norm equivalence is also verified by Theorem 16.4. Note that ρ → χ(Re ρ) is a continuous and bounded operator from HN2 (Ω) into itself due to (1.93). Furthermore, by (12.9) and (12.11) (θ = η), we observe that F satisfies the Lipschitz condition (4.21). We therefore know that Theorem 4.4 is available to problem (12.6). As (4.21) is valid in the present case, we are accordingly led to set the space of initial values by u0 (12.12) ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ ρ0 ∈ HN2 (Ω) . K= ρ0 We have thus proved that, for each U0 ∈ K, (12.6) (and hence (12.3)) possesses a unique local solution U = t (u, ρ) in the function space: u ∈ C((0, TU0 ]; HN2 (Ω)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)). ρ ∈ C((0, TU0 ]; HN U0 U0 2 N N (12.13) In addition, it holds for U that dU + tAU (t)X + U (t)X ≤ CU , 0 < t ≤ TU . t (12.14) (t) 0 0 dt X Here, the positive time TU0 > 0 and the constant CU0 > 0 are determined by the norm U0 X only.

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12 Chemotaxis Models

1.3 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, ρ) be the local solution of (12.6) constructed above. Our goal is to prove that u(t) ≥ 0 and ρ(t) ≥ 0 for every 0 < t ≤ TU0 by the truncation method. Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (12.6) with the same initial value. Therefore, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative ψ (t) = (H (u), aΔu − μ∇ · [u∇χ(ρ)])L2 . Applying (1.96) to H (u), we observe that

(H (u), aΔu)L2 = −a

∇H (u) · ∇u dx = −a Ω

|∇H (u)|2 dx.

Ω

In the meantime, by (1.96) again,

−(H (u), μ∇ · [u∇χ(ρ)])L2 = μ

u∇H (u) · ∇χ(ρ) dx

Ω

H (u)∇H (u) · ∇χ(ρ) dx

=μ Ω

=−

μ 2

H (u)2 Δχ(ρ) dx. Ω

Since Δχ(ρ) = χ (ρ)Δρ + χ (ρ)|∇ρ|2 ,

(12.15)

(12.13) shows that Δχ(ρ)L2 ≤ CU for 0 ≤ t ≤ TU0 . Therefore, −(H (u), μ∇ · [u∇χ(ρ)])L2 ≤

μ H (u)2L4 Δχ(ρ)L2 2

≤ CU H (u)H 1 H (u)L2 a ≤ H (u)2H 1 + CU H (u)2L2 . 2 Hence, ψ (t) ≤ CU ψ(t); consequently, ψ(t) ≤ ψ(0)eCU t . Then, ψ(0) = 0 implies ψ(t) ≡ 0, i.e., u(t) ≥ 0 for 0 < t ≤ TU0 . It is similar for the proof of ρ(t) ≥ 0 for 0 < t ≤ TU 0 .

2 Case where χ (ρ) = ρ

423

2 Case where χ(ρ) = ρ In this section, let us consider problem (12.6) in a particular case where the sensitivity function is given by χ(ρ) = ρ,

0 ≤ ρ < ∞.

(12.16)

We can establish a priori estimates for sufficiently small initial functions u0 , and consequently we obtain the global existence of solutions.

2.1 Global Solutions for Small Initial Functions Let U0 ∈ K, and let U be any local solution of (12.6) on [0, TU ] in the function space: 0 ≤ u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)). 0 ≤ ρ ∈ C((0, TU ]; HN U U 2 N N (12.17) Proposition 12.2 There exists a positive number r > 0 and a continuous increasing function p(·) such that the estimate u(t)L2 + ρ(t)H 2 ≤ p(u0 L2 + ρ0 H 2 ),

0 ≤ t ≤ TU ,

(12.18)

holds for any local solution U = t (u, ρ) of (12.6) in (12.17) with initial value U0 ∈ K satisfying the condition u0 L1 ≤ r. Proof In the proof, the notation p(·) stands for some continuous increasing functions which are determined only by Ω and by the initial constants in the equations of (12.3). Similarly, the notation C stands for some constants, which are determined in the same way as p(·). Step 1. Integrate the first equation of (12.6) in Ω. Then, obviously, d u dx = 0. dt Ω Since u(t) ≥ 0, it follows that u(t)L1 = u0 L1 ,

0 ≤ t ≤ TU .

Step 2. We here introduce the quantity u log(u + 1) dx, N1,log (u) =

0 ≤ u ∈ L2 (Ω).

Ω

The purpose of this step is to estimate N1,log (u(t)) for the local solution U .

(12.19)

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12 Chemotaxis Models

Multiply the first equation of (12.6) by log(u + 1) and integrate the product in Ω. Then, since

log(u + 1)Δu dx = − Ω

Ω

1 |∇(u + 1)|2 dx = −4 u+1

√ |∇ u + 1|2 dx

Ω

and

log(u + 1)∇ · [u∇ρ] dx = Ω

∇[(u + 1) log(u + 1) − u] · ∇ρ dx Ω

u log(u + 1)Δρ dx =

+

[u − log(u + 1)]Δρ dx,

Ω

Ω

it follows that d dt

[(u + 1) log(u + 1) − u] dx + 4a Ω

√ |∇ u + 1|2 dx

Ω

=μ Ω

[log(u + 1) − u]Δρ dx ≤ ζ Δρ2L2 + Cζ u2L2

(12.20)

with any ζ > 0. In the meantime, from the second equation of (12.6) we obtain that 1 d 2 dt

ρ 2 dx + b Ω

|∇ρ|2 dx + d Ω

ρ 2 dx = ν Ω

uρ dx ≤ Ω

d ρ2L2 + Cu2L2 2

and 1 d 2 dt

=ν

|∇ρ| dx + b Ω

|Δρ| dx + d

2

|∇ρ|2 dx

2

Ω

Ω

b uΔρ dx ≤ Δρ2L2 + Cu2L2 . 4 Ω

Fixing the parameter ζ as ζ = b4 , we sum up these three differential inequalities to obtain that √ d ψ(t) + dψ(t) + 4a |∇ u + 1|2 dx + b |Δρ|2 dx ≤ C u2L2 + 1 , dt Ω Ω where ψ(t) = (u + 1) log(u + 1) − uL1 + ρ2H 1 . We notice the fact that 1/3 2/3 1/3 √ 4/3 uL2 ≤ uL1 uL4 ≤ uL1 u + 1L8 √ √ 1/3 √ 1/3 ≤ CuL1 u + 1L2 (∇ u + 1L2 + u + 1L2 ),

0 ≤ u ∈ H 1 (Ω).

2 Case where χ (ρ) = ρ

425

Hence, on account of (12.19), d ψ(t) + dψ(t) + 4a dt

√

|∇ u + 1| dx + b

|Δρ|2 dx

2

Ω

Ω

2 √ ≤ u0 L3 1 p(u0 L1 ) ∇ u + 12L2 + 1 . 2

We now fix a radius r > 0 small enough to satisfy that r 3 p(r) ≤ 3a. Then, if u0 L1 ≤ r, it holds that √ d a|∇ u + 1|2 + b|Δρ|2 dx ≤ C. ψ(t) + dψ(t) + dt Ω As a consequence (cf. (1.58)), ψ(t) ≤ e−dt ψ(0) + C,

0 ≤ t ≤ TU .

In particular, N1,log (u(t)) + ρ(t)2H 1 ≤ C e−dt N1,log (u0 ) + ρ0 2H 1 + 1 ,

0 ≤ t ≤ TU . (12.21)

Proposition 1.5 is also available to conclude that t u(τ ) + 12H 1 + Δρ(τ )2L2 dτ s

≤ C (t − s) + N1,log (u0 ) + ρ0 2H 1 + 1 ,

0 ≤ s < t ≤ TU .

Step 3. The present step is devoted to showing the estimate u(t)2L2 + ρ(t)2H 2 ≤ p(N1,log (u0 ) + ρ0 H 1 ) × e−δt u0 2L2 + ρ0 2H 2 + 1 ,

0 ≤ t ≤ TU ,

with some exponent δ > 0. Now, when estimate (12.21) is established, the techniques of its proof are quite analogous to that of the proof (Step 4) of Proposition 12.3 below, which establishes a similar a priori estimate for a general sensitivity function including the case of (12.16). So, the continued proof will be given there. It is now possible to apply Corollary 4.3 to conclude the global existence of solutions. In fact, there is a number r > 0 such that, for any U0 ∈ K satisfying u0 L1 ≤ r, (12.3) possesses a unique global solution in the function space: 0 ≤ u ∈ C((0, ∞); HN2 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)), 4 (Ω)) ∩ C([0, ∞); H 2 (Ω)) ∩ C1 ((0, ∞); H 2 (Ω)). 0 ≤ ρ ∈ C((0, ∞); HN 2 N N (12.22)

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12 Chemotaxis Models

2.2 Lyapunov Function When (12.16) holds, we can construct a Lyapunov function. Let U = t (u, ρ) denote a local or global solution of (12.6) in the function space (12.17). Let ε > 0 be a ∂uε positive parameter and put uε = u + ε. It is clear that ∂u ∂t = ∂t and that aΔu − μ∇ · [u∇ρ] = aΔuε − μ∇ · [uε ∇ρ] + εμΔρ. Therefore, the first equation of (12.6) is written as ∂uε = ∇ · [uε ∇(a log uε − μρ)] + εμΔρ. ∂t Multiply this equation by (a log uε − μρ) and integrate the product in Ω. Then, ∂uε ∂u uε |∇(a log uε − μρ)|2 dx + Rε , a log uε dx − μ ρ dx = − Ω ∂t Ω ∂t Ω where Rε = εμ Ω (a log uε − μρ)Δρ dx. In the meantime, multiplying the equation of ρ by ∂ρ ∂t and integrating the product in Ω, we have Ω

∂ρ ∂t

2

b d dx = − 2 dt

d d |∇ρ| dx − 2 dt Ω

ρ dx + ν

2

2

Ω

u Ω

∂ρ dx. ∂t

Therefore, d dt

bμ dμ 2 |∇ρ|2 + ρ − μνuρ dx 2 2 Ω

2 ∂ρ νuε |∇(a log uε − μρ)|2 + μ dx + νRε ≤ νRε . =− ∂t Ω aν(uε log uε − uε ) +

Let 0 < s < t ≤ TU . Integrating this inequality in [s, t], we obtain that bμ dμ aν(uε (τ ) log uε (τ ) − uε (τ )) + |∇ρ(τ )|2 + ρ(τ )2 2 2 Ω τ =t t − μνu(τ )ρ(τ ) dx ≤ νRε (τ ) dτ. τ =s

It is not difficult to verify that limε→0

s

t s

Rε (τ ) dτ = 0. Hence, if we set

bμ 2 dμ 2 aν(u log u−u)+ |∇ρ| + ρ −μνuρ dx, Φ(U ) = 2 2 Ω

U ∈ X, (12.23)

then Φ(U (t)) ≤ Φ(U (s)). This means that Φ(U ) plays a role of the Lyapunov function for (12.3). Note that the function u log u can be extended for all 0 ≤ u < ∞ as a continuous function.

3 Chemotaxis Model with Proliferation

427

2.3 Blowup of Solutions Herrero–Velázquez [HV96, HV97] proved that, for suitable initial functions (at least, when u0 L1 are sufficiently large), the local solutions to (12.6) blow up in finite time. Theorem 12.1 There are initial values U0 ∈ K for which the local solutions to (12.6) satisfy lim supt→T U (t)X = ∞ with finite time T < ∞. Hence, (12.6) with the initial values U0 have no global solutions.

3 Chemotaxis Model with Proliferation Let us consider the initial-boundary-value problem for the chemotaxis model with proliferation ⎧ ∂u 2 in Ω × (0, ∞), ⎪ ⎪ ∂t = aΔu − μ∇ · [u∇χ(ρ)] + cu − γ u ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu in Ω × (0, ∞), ∂t (12.24) ∂ρ ∂u ⎪ = =0 on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ∂n ⎪ ⎪ ⎩u(x, 0) = u (x), ρ(x, 0) = ρ (x) in Ω, 0

0

in a two-dimensional C2 or convex, bounded domain Ω. We still assume that χ(ρ) satisfies (12.4) and assume that γ > 0 is a positive constant. As before, we handle problem (12.24) in the product space X given by (12.5), and, as a space of initial functions, we take the space K given by (12.12).

3.1 Local Solutions In order to construct local solutions to (12.24), we can argue in a quite analogous way as for problem (12.3). In fact, we rewrite (12.24) as an abstract problem of the form (12.6) in X. The linear operator A is given in a similar way by (12.7). However, now, the nonlinear operator F is given (instead of (12.8)) by

−μ∇ · [u∇χ(Re ρ)] + (c + 1)u − γ u2 . (12.25) F (U ) = 0 Let 0 < ε < 12 be arbitrarily fixed and put η = 1+ε 2 . Then, as before (by (12.9) and (12.11)), F is an operator from D(Aη ) into X. It is clear that u2 L2 ≤ CuL2 uH 1+ε ,

u ∈ H 1+ε (Ω).

Then, this, together with (12.9), implies that F fulfills the Lipschitz condition (4.21).

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For each U0 ∈ K, we can therefore construct a unique local solution to (12.24). Furthermore, by the truncation method, we can verify the nonnegativity of the local solution. In this way, for any initial value U0 ∈ K, problem (12.24) possesses a unique local solution in the function space: 0 ≤ u ∈ C((0, TU0 ]; HN2 (Ω)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)), 0 ≤ ρ ∈ C((0, TU0 ]; HN U0 U0 2 N N where the positive time TU0 > 0 is determined by the norm U0 X only. In addition, estimate (12.14) holds for the present local solution, too.

3.2 A Priori Estimates of Local Solutions For U0 ∈ K, let U = t (u, ρ) denote a local solution of (12.24) on [0, TU ] in the function space: 0 ≤ u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)). 0 ≤ ρ ∈ C((0, TU ]; HN U U 2 N N (12.26) We can establish the following a priori estimates for local solutions. Proposition 12.3 There exists a continuous increasing function p(·) such that the estimate U (t)X ≤ p(U0 X ),

0 ≤ t ≤ TU ,

(12.27)

holds for any local solution U of (12.24) in the function space (12.26). Proof We put f (u) = cu − γ u2 ,

0 ≤ u < ∞.

The notation p(·) stands for some continuous increasing functions which are determined only by χ(·) and Ω and by the initial constants in the equations of (12.24). Similarly, the notation C stands for some constants which are determined in the same way as p(·). Step 1. Integration of the first equation of (12.24) in Ω gives (c + 1)2 d 2 u dx = (cu − γ u ) dx ≤ dx. −u + dt Ω 4γ Ω Ω Solving this differential inequality (cf. (1.58)), we see that Ω

u dx ≤ e−t u0 L1 +

(c + 1)2 |Ω|. 4γ

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429

Since u(t) ≥ 0, we conclude that u(t)L1 ≤ e−t u0 L1 +

(c + 1)2 |Ω|, 4γ

0 ≤ t < TU .

(12.28)

Furthermore, for any 0 ≤ s < t ≤ TU , we have u(t)L1 − u(s)L1 = So,

γ s

t

u(τ )2L2

t s

t

dτ ≤ c s

(cu − γ u2 ) dx dτ.

Ω

u(τ )L1 dτ + u(s)L1 ;

hence, on account of (12.28),

t s

(c + 1)2 u(τ )2L2 dτ ≤ γ −1 u0 L1 + |Ω| [c(t − s) + 1], 4γ

0 ≤ s < t ≤ TU . (12.29)

Step 2. Multiply the second equation of (12.24) by Δρ and integrate the product in Ω. Then, 1 d 2 2 |∇ρ| dx + b |Δρ| dx + d |∇ρ|2 dx 2 dt Ω Ω Ω b ν2 uΔρ dx ≤ Δρ2L2 + u2L2 . = −ν 2 2b Ω Similarly, multiplying the second equation of (12.24) by ρ, we obtain that 1 d 2 2 2 ρ dx + b |∇ρ| dx + d ρ dx = −ν uρ dx 2 dt Ω Ω Ω Ω ≤

d ν2 ρ2L2 + u2L2 . 2 2d

Summing up these differential inequalities, we obtain that d |∇ρ|2 + ρ 2 dx + b|Δρ|2 + 2(d + b)|∇ρ|2 + dρ 2 dx dt Ω Ω ≤ ν 2 (b + d)(bd)−1 u(t)2L2 .

(12.30)

As (12.29) holds for u(t), it is possible to solve this by applying Proposition 1.4. Consequently, we deduce that ρ(t)2H 1 ≤ e−dt ρ0 2H 1 + C(u0 L1 + 1),

0 ≤ t ≤ TU .

(12.31)

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At the same time, t Δρ(τ )2L2 dτ ≤ C(u0 L1 + 1)[(t − s) + ρ0 H 1 + 1],

0 ≤ s < t ≤ TU .

s

Furthermore, since it holds that ρH 2 ≤ C(ΔρL2 + ρL2 ),

ρ ∈ HN2 (Ω),

due to (2.33), it is also verified that t ρ(τ )2H 2 dτ ≤ C(u0 L1 + 1)[(t − s) + ρ0 H 1 + 1],

(12.32)

0 ≤ s < t ≤ TU .

s

(12.33)

Step 3. In this step, we use the notation p1 (U0 ) = p(u0 L1 + ρ0 H 1 ), where p(·) is some continuous increasing function. Multiply the first equation of (12.24) by log(u + 1) and integrate the product in Ω. Then, by the same calculation as for (12.20), we have √ d [(u + 1) log(u + 1) − u] dx + 4a |∇ u + 1|2 dx dt Ω Ω = μ [log(u + 1) − u]Δχ(ρ) dx + f (u) log(u + 1) dx. Ω

Ω

In addition, we observe that Δχ(ρ) = χ (ρ)Δρ + χ (ρ)|∇ρ|2 . Then, the first integral in the right-hand side can be estimated as follows. By (12.4), [log(u + 1) − u]χ (ρ)Δρ dx ≤ CuL2 ΔρL2 ≤ C u2L2 + Δρ2L2 , Ω

while, by (12.4), (12.31), and (12.32), [log(u + 1) − u]χ (ρ)|∇ρ|2 dx ≤ CuL2 ∇ρ2L4 ≤ CuL2 ρH 2 ρH 1 Ω

≤ CρH 1 u2L2 + ρ2H 2 ≤ p1 (U0 ) u2L2 + ρ2H 2 .

As for the second integral, it is easy to see that f (u) log(u + 1) ≤ cu2

for all 0 ≤ u < ∞.

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431

Similarly, 1 (u + 1) log(u + 1) ≤ u2 + u for all 0 ≤ u < ∞. 2 These estimates thus yield that

1 [f (u) + u + 1] log(u + 1) dx ≤ c + u2L2 + uL1 . 2 Ω Our inequality is then reduced to √ d [(u + 1) log(u + 1) − u] dx + 4a |∇ u + 1|2 dx dt Ω Ω + [(u + 1) log(u + 1) − u] ≤ p1 (U0 ) u(t)2L2 + ρ(t)2H 2 + 1 . Ω

As we already know that (12.29) and (12.33) are valid, it is possible to solve this inequality by using Proposition 1.4. We then deduce that N1,log (u(t)) ≤ e−t N1,log (u0 ) + p1 (U0 ),

0 ≤ t ≤ TU .

(12.34)

Step 4. In this step, we use the notation p2 (U0 ) = p(N1,log (u0 ) + ρ0 H 1 ). Multiply the first equation of (12.24) by u and integrate the product in Ω. Then, 1 d u2 dx + a |∇u|2 dx = μ u∇u · ∇χ(ρ) dx + f (u)u dx. 2 dt Ω Ω Ω Ω Here, u∇u · ∇χ(ρ) dx = − Ω

1 2

u2 Δχ(ρ) dx, Ω

Δχ(ρ) = χ (ρ)Δρ + χ (ρ)|∇ρ|2 . Furthermore, with the aid of (1.98) (p = 2, q = 1, and r = 3), we verify that 2

1

uL3 ≤ ζ uH3 1 N1,log (u) 3 + Cζ uL1 with any ζ > 0, and, by the moment inequality (2.117), we observe that 1 2 1 2 3 2 1 1 ΔρL3 ≤ CΔρH3 1 ΔρL3 2 ≤ CA2 ρH3 1 A2 ρL3 2 ≤ C A22 ρ L3 2 A22 ρ L3 2 .

Therefore, on account of (12.4), (12.31), and (12.34), 3 2 χ (ρ)u2 Δρ dx ≤ Cu2L3 ΔρL3 ≤ p2 (U0 ) ζ u2H 1 + A22 ρ L + Cζ . − Ω

2

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12 Chemotaxis Models

Similarly, − Ω

4

2

χ (ρ)u2 |∇ρ|2 dx ≤ Cu2L3 ∇ρ2L6 ≤ Cu2L3 ρH3 2 ρH3 1 3 2 ≤ p2 (U0 ) ζ u2H 1 + A22 ρ L + Cζ 2

with any ζ > 0. Thus, we obtain that 1 d 2 2 u dx + a |∇u| dx + γ u3 dx 2 dt Ω Ω Ω 3 2 ≤ p2 (U0 ) ζ u2H 1 + A22 ρ L + Cζ . 2

(12.35)

In this step, it is necessary to combine two inequalities for u and ρ. Multiply the second equation of (12.24) by A22 ρ and integrate the product in Ω. Then, 1 3 3 2 1 d |A2 ρ|2 dx + A22 ρ dx = ν A22 u · A22 ρ dx 2 dt Ω Ω Ω 3 1 2 ≤ A22 ρ L + Cu2H 1 , 2 2 i.e., d dt

|A2 ρ|2 dx + Ω

Ω

32 2 A ρ dx ≤ Cu(t)2 1 . 2 H

After multiplying a parameter 2ξ > 0 to (12.35), we add the product to this inequality to obtain that 3 2 2 d ξ u + |A2 ρ|2 dx + (2aξ − C)|∇u|2 + 2γ ξ u3 dx + A22 ρ dx dt Ω Ω Ω 3 2 ≤ p2 (U0 )ξ ζ u2H 1 + A22 ρ L + Cζ . 2

Now, fix the parameters ξ and ζ so that 2aξ − C ≥ 1 and p2 (U0 )ξ ζ ≤ 12 . Then, we arrive at the differential inequality 3 2 1 dψ + δψ + u(t)2H 1 + A22 ρ(t)L ≤ p2 (U0 ) 2 dt 2 for ψ(t) = ξ u(t)2L2 +A2 ρ(t)2L2 with some exponent δ > 0. We can now employ Proposition 1.4 to conclude that ξ u(t)2L2 + A2 ρ(t)2L2 ≤ e−δt ξ u0 2L2 + A2 ρ0 2H 2 + p2 (U0 ), 0 ≤ t ≤ TU , i.e., u(t)2L2 + ρ(t)2H 2 ≤ p2 (U0 ) e−δt u0 2L2 + ρ0 2H 2 + 1 ,

0 ≤ t ≤ TU . (12.36)

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433

As well, t 3 2 u(τ )2H 1 + A22 ρ(τ )L dτ ≤ p2 (U0 ) (t − s) + u0 2L2 + ρ0 2H 2 + 1 , 2

s

0 ≤ s < t ≤ TU . We have in this way established the desired a priori estimate (12.27).

3.3 Global Solutions We can now apply Corollary 4.3 to conclude the global existence of solutions. As a result, for any U0 ∈ K, (12.24) possesses a unique global solution in the function space: 0 ≤ u ∈ C((0, ∞); HN2 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)), 4 (Ω)) ∩ C([0, ∞); H 2 (Ω)) ∩ C1 ((0, ∞); H 2 (Ω)). 0 ≤ ρ ∈ C((0, ∞); HN 2 N N (12.37) For U0 ∈ K, let U (t; U0 ) be the global solution of (12.24) with the initial value U0 in the function space (12.37). Recalling that Proposition 12.3 was established by four steps, let us apply (12.28) to U (t; U0 ) in the interval 0, 4t , (12.31) in the interval 4t , 2t4 , (12.34) in the interval 2t4 , 3t4 , and (12.36) in the interval 3t4 , t , respectively. Then, we obtain the (more refined than (12.27)) estimate U (t; U0 )X ≤ p(e−δt p(U0 X ) + 1),

0 ≤ t < ∞, U0 ∈ K,

(12.38)

with some suitable exponent δ > 0.

3.4 Dynamical System Let us construct a dynamical system determined from problem (12.24) in the universal space X by following the general methods in Chap. 6, Sect. 5. Indeed, we can take β = 0. Furthermore, (12.38) shows that (6.56) is fulfilled. Consequently, it is deduced that problem (12.24) determines a dynamical system (S(t), K, X). For any exponent 0 < θ < 1, (12.24) equally determines a dynamical system in a universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (12.25) fulfills (4.21) and a fortiori (4.2) with β = θ . Let D Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

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12 Chemotaxis Models

Meanwhile, by Proposition 6.1, Aθ S(t)U0 X ≤ Cθ,R t −θ ,

0 < t < ∞, U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that (6.56) is valid in the space Dθ , too. So, (S(t), Kθ , Dθ ) defines a dynamical system.

3.5 Exponential Attractors Let us next construct exponential attractors for (S(t), K, X). To this end, it suffices to verify (6.59) and (6.60). However, the space condition (6.59) is fulfilled in the present case. Meanwhile, (12.38) shows that the dissipative condition (6.60) is also fulfilled. Hence, (S(t), K, X) possesses a family of exponential attractors. For any exponent 0 < θ < 1, these exponential attractors are seen to keep their properties as exponential attractors in the universal space Dθ , too. Remark 12.1 As the linear operator A given by (12.7) is not symmetric, A does not satisfy (6.62). So, it is unclear whether S(t) enjoys the squeezing property (6.45)– (6.46) or not. However, if Ω is of C3 class, then one can use the shift property (Δu ∈ H 1 (Ω), together with u ∈ HN2 (Ω), implies u ∈ H 3 (Ω)) to rewrite (12.24) into the form (12.6) in an underlying space different from (12.5) in which the linear operator is a self-adjoint operator satisfying (6.62). For details, see Osaki– Tsujikawa–Yagi–Mimura [OTYM02]. In such an abstract formulation, the semigroup enjoys the squeezing property, and therefore the dimension of exponential attractor is estimated precisely by (6.50).

3.6 Numerical Examples In this subsection, we present some numerical examples. Let Ω = [−8, 8] × [−8, 8] be a quadratic domain. We consider the linear sensitivity function χ(ρ) = ρ and the growth function f (u) = u2 (1 − u). More precisely, we consider the problem ⎧ ∂u 2 ⎪ ∂t = 0.0625Δu − μ∇ · [u∇ρ] + u (1 − u) in Ω × (0, ∞), ⎪ ⎪ ⎪ ⎨ ∂ρ = Δρ − 32ρ + u in Ω × (0, ∞), ∂t ∂ρ ⎪ ∂u ⎪ = =0 on ∂Ω × (0, ∞), ⎪ ⎪ ⎩ ∂n ∂n ρ(x, 0) = ρ0 (x) in Ω. u(x, 0) = u0 (x),

(12.39)

3 Chemotaxis Model with Proliferation

435

Fig. 1 Stationary patterns for μ = 6.2, μ = 7.2, and μ = 8.5

We treat μ > 0 as a control parameter. As the growth function in (12.39) is a cubic function, we cannot apply the results obtained in this section to (12.39) directly, but, as a matter of fact, we can repeat the same arguments in order to conclude for any 0 < μ < ∞ that (12.39) equally defines a dynamical system (S(t), K, X) which possesses exponential attractors, cf. Aida–Efendiev–Yagi [AEY05]. Note 1 ) whatever also that (12.39) has a homogeneous stationary solution (u, ρ) = (1, 32 0 < μ < ∞ is. 1 ), where ε(x) is a The initial functions (u0 (x), ρ0 (x)) are taken as (1 + ε(x), 32 small perturbation which vanishes outside of a small disk centered at the origin. So, (u0 , ρ0 ) is taken in a neighborhood of the stationary solution (u, ρ). When μ is small enough, the homogeneous stationary solution (u, ρ) is asymptotically stable (cf. (12.48)). The numerical solution is also observed to tend to the stationary solution (u, ρ). When μ = 6.2, the homogeneous stationary solution is no longer stable (cf. (12.49)), and the numerical solution is observed to tend to an inhomogeneous stationary solution with honeycomb structure as in Fig. 1(a). (In the figure, white indicates high concentration of biological individuals, and black oppositely shows the graph of u.) As μ increases from 6.2 to 7.2 and 8.5, the inhomogeneous stationary solution changes its types from honeycomb to stripe and perforated stripe as Fig. 1 shows. When μ still increases, such an inhomogeneous and ordered stationary solution is no longer found. That may lose its stability or completely vanish. Instead, the numerical solution is observed to tend to some moving pattern. When μ = 9.0, a moving perforated labyrinthine pattern is observed as in Fig. 2. When μ = 11.0, a chaotic spot pattern is observed as in Fig. 3. Each spot continues to move in a chaotic manner, a few spots are combined here and there in Ω; on the other hand, some new spots are generated to conserve the total number of spots at every moment. When μ becomes larger, moving patterns as in Figs. 2 and 3 disappear. Instead, the numerical solution is observed to tend to a stationary pattern again. A number of steady spots are regularly located in Ω as in Fig. 4, but these stationary patterns seem to be quite different from those in Fig. 1.

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Fig. 2 Moving perforated labyrinthine pattern for μ = 9.0

Fig. 3 Chaotic spot pattern for μ = 11.0

Fig. 4 Stationary patterns for μ = 25.0, μ = 40.0, and μ = 70.0

These numerical results seem to be very interesting when we interpret them in accordance with the analytical result that the dimension of exponential attractors of the dynamical system for (12.39) increases as the parameter μ increases (cf. (12.50)).

4 Instability of Homogeneous Stationary Solution of (12.24)

437

4 Instability of Homogeneous Stationary Solution of (12.24) We are concerned with homogeneous stationary solutions to (12.24). The homogeneous solution is given as a solution to cu − γ u2 = 0, −dρ + νu = 0, cν . We want to investigate the stability and instanamely, (u, ρ) = (0, 0) and γc , dγ cν . bility of the nonzero homogeneous stationary solution (u, ρ) = γc , dγ In this section, we make the new assumption on χ(ρ) that χ(ρ) is real analytic in a neighborhood of ρ.

(12.40)

4.1 Localized Problem From (12.40) we can extend the sensitivity function as an analytic function in a complex neighborhood |ρ − ρ| < R; furthermore, this analytic function can be extended over the whole complex plane in such a way that the extended function, denoted by χ˜ (ρ), is a C2 function with respect to the real variables (ρ , ρ ) ∈ R2 such that ρ = ρ + iρ ∈ C with uniformly bounded partial derivatives in C up to the second order. Similarly, let ϕ(u) be an extension of the function u such that ϕ(u) ≡ u in a complex neighborhood |u − u| < R and ϕ(u) is a C1 function with respect to the real variable (u , u ) ∈ R2 such that u = u + iu ∈ C with uniformly bounded partial derivatives up to the first order. Using these functions, we introduce the localized problem ⎧ ∂u 2 ⎪ ∂t = aΔu − u − μ∇ · [ϕ(u)∇ χ˜ (ρ)] + (c + 1)ϕ(u) − γ ϕ(u) ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu ∂t ∂ρ ⎪ ∂u ⎪ ∂n = ∂n =0 ⎪ ⎪ ⎩ ρ(x, 0) = ρ0 (x) u(x, 0) = u0 (x),

in Ω × (0, ∞), in Ω × (0, ∞),

on ∂Ω × (0, ∞), in Ω. (12.41) We can handle this localized problem in a quite analogous way as for (12.24). In fact, the problem is formulated as the Cauchy problem for an abstract equation in X of the form (12.6) in which the nonlinear operator F˜ : D(Aη ) → X, where 1+ε 1 2 ≤ η < 1 and 0 < ε < 2 , is given by

−μ∇ · [ϕ(u)∇ χ˜ (ρ)] + (c + 1)ϕ(u) − γ ϕ(u)2 ˜ , F (U ) = 0

u U= ∈ D(Aη ). ρ

Obviously, F˜ is a nonlinear operator from D(Aη ) into X. Let 0 ≤ θ < η. It is then possible to construct local solutions and global solutions to (12.41) for any initial

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value U0 ∈ D(Aθ ) in the function space: U ∈ C([0, ∞); D(Aθ )) ∩ C((0, ∞); D(A)) ∩ C 1 ((0, ∞); X).

4.2 Complexified Dynamical System ˜ Problem (12.41) then defines a nonlinear semigroup S(t) acting on Dθ = D(Aθ ), ˜ 0 ≤ θ < η. Furthermore, (12.41) defines a complexified dynamical system (S(t), Dθ , Dθ ) in Dθ . We are concerned with the case where 12 < θ < η. Since Dθ ⊂ C(Ω) for θ > 12 , any solution to (12.24) is a solution of (12.41) in a suitable neighborhood of U in Dθ . Naturally, in such a neighborhood, any trajectory of (S(t), Kθ , Dθ ) is that of ˜ ˜ Dθ , Dθ ). If U is stable (S(t), Dθ , Dθ ). In particular, U is an equilibrium of (S(t), ˜ as an equilibrium of (S(t), Dθ , Dθ ), then it is the same as that of (S(t), Kθ , Dθ ). ˜ Dθ , Dθ ), we On the other hand, even if U is unstable as an equilibrium of (S(t), cannot say that U is automatically unstable as that of (S(t), Kθ , Dθ ). However, as ˜ Dθ , Dθ ) provides us an important information before, the instability of U in (S(t), which indicates the instability even in the original system (S(t), Kθ , Dθ ), too. We can follow the general methods described in Chap. 6, Sect. 6. It is indeed sufficient to verify the differentiability conditions (6.68) and (6.69) for the nonlinear operator F˜ (U ) and the spectrum separation condition (6.76) for the linearized operator A − F˜ (U ).

4.3 Differentiability of F˜ (U ) If r > 0 is sufficiently small so that Aθ (U − U ) < r implies U − U C < R, then, θ in the open ball B D(A ) (U ; r), F˜ : D(Aη ) → X is seen to be Fréchet differentiable with the derivative

−μ∇ · [h∇χ(ρ)] − μ∇ · {u∇[χ (ρ)]} + (c + 1)h − 2γ uh ˜ , F (U )H = 0

u h η D(Aβ ) ∈ D(A ) ∩ B U= (U ; r), H = ∈ D(Aη ). (12.42) ρ Furthermore, it is immediate to check that F˜ (U ) fulfills conditions (6.68) and (6.69).

4.4 Spectrum Separation Condition of A For A = A − F˜ (U ), let us verify condition (6.76).

4 Instability of Homogeneous Stationary Solution of (12.24)

439

Let Λ be the realization of −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions. Let 0 = μ 0 < μ1 ≤ μ 2 ≤ · · · → ∞ 1

be the eigenvalues of Λ in L2 (Ω), and let φ0 = |Ω|− 2 , φ1 , φ2 , . . . , be the corresponding real-valued eigenfunctions which constitute an orthonormal basis of L2 (Ω) (cf. Corollary 2.1). Using Λ, we can characterize the space HN2 (Ω) by ∞ ∞ 2 2 2 HN (Ω) = D(Λ) = ρ = ηk φk ; (μk + 1) |ηk | < ∞ k=0

k=0

with the inner product (ρ, ρ )H 2 = N

∞ (μk + 1)2 ηk ηk ,

ρ=

k=0

∞

ηk φk , ρ =

k=0

∞

ηk φk .

k=0

Then, we notice that our underlying space X given by (12.5) is obtained as the infinite sum of two-dimensional orthogonal subspaces

φk 0 + ηk ; ξk , ηk ∈ C , k = 0, 1, 2, . . . , Xk = U = ξ k 0 φk equipped with the norm U 2Xk = |ξk |2 + (μk + 1)2 |ηk |2 . It is then obvious that Φk =

φk 0

and Ψk = (μk + 1)−1

0 φk

are orthonormal bases of Xk . According to (12.42), we have

c + 1 − 2γ u −μuχ (ρ)Δ . F˜ (U ) = 0 0 Note that u =

c γ

and put χ1 = χ (ρ) > 0. Since

aΛ + 1 0 A= −ν bΛ + d

−c + 1 ˜ and F (U ) = 0

μuχ1 Λ , 0

we observe that A = A − F˜ (U ) maps the subspace Xk into itself, that is, Xk is invariantunder A for every k. Consequently, the operator A can also be decomposed as A = ∞ k=0 Ak , where Ak is the part of A in Xk . Obviously, Ak Φk = (aμk + c)Φk − ν(μk + 1)Ψk ,

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12 Chemotaxis Models

Ak Ψk = −μuχ1 μk (μk + 1)−1 Φk + (bμk + d)Ψk . The transformation matrix M k of Ak with respect to the basis Φk , Ψk is then given by

aμk + c −ν(μk + 1) . Mk = bμk + d −μuχ1 μk (μk + 1)−1 Since |λ − M k | = [λ − (aμk + c)][λ − (bμk + d)] − μνuχ1 μk , the characteristic equation of M k is written by λ2 − [(a + b)μk + c + d]λ + (aμk + c)(bμk + d) − μνuχ1 μk = 0. So, the discriminant satisfies Dk = [(a − b)μk + c − d]2 + 4μνuχ1 μk ≥ 0. That is, for every k = 0, 1, 2, . . . , the matrix M k has two real eigenvalues λk ≤ λk . Furthermore, for λ = λk , λk , (λ − M k )−1 =

1 λ − (bμk + d) (λ − λk )(λ − λk ) −μuχ1 μk (μk + 1)−1

−ν(μk + 1) . λ − (aμk + c) (12.43)

For verifying (6.76), we here make the assumption that abμ2k + (ad + bc − μνuχ1 )μk + cd = 0 for all k = 0, 1, 2, . . . .

(12.44)

If (aμk + c)(bμk + d) − μνuχ1 μk = abμ2k + (ad + bc − μνuχ1 )μk + cd > 0, (12.45) then the eigenvalues of M k are both positive, i.e., 0 < λk ≤ λk . Furthermore, the following estimate abμ2k + (ad + bc − μνuχ1 )μk + cd < λk ≤ λk (a + b)μk + c + d < (a + b)μk + c + d −

abμ2k + (ad + bc − μνuχ1 )μk + cd (12.46) (a + b)μk + c + d

holds for the eigenvalues. In the meantime, if abμ2k + (ad + bc − μνuχ1 )μk + cd < 0,

(12.47)

then one of the eigenvalues is negative, and the other is positive, i.e., λk < 0 < λk .

4 Instability of Homogeneous Stationary Solution of (12.24)

441

Let λ ∈ iR. Consider the parts Ak on Xk . Let ψk : Xk → C2 be a natural isomorphism such that ψk (ξk Φk + ηk Ψk ) = t (ξk , ηk ) (ψk = ψk−1 = 1); then Ak is written as Ak = ψk−1 M k ψk . Consequently, (λ − Ak )−1 = ψk−1 (λ − M k )−1 ψk . Therefore, if k are sufficiently large so that (12.45) take place, then (12.43) provides, with the aid of (12.46) and the obvious inequalities |λ − λk | ≥ λk and |λ − λk | ≥ λk , that (λ − Ak )−1 ≤ Cλ , L(X ) k

Cλ > 0 being independent of k. This fact means that λ ∈ iR belongs to ρ(A) if and / σ (A) if and only only if λ ∈ ρ(Ak ) for every k = 0, 1, 2, . . . . In other words, λ ∈ λ∈ / σ (Ak ) = {λk , λk } for every k. We already know that, under (12.44), iR ∩ σ (Ak ) = ∅ for every k = 0, 1, 2, . . . . Hence, iR ∩ σ (A) = ∅, that is, the separation condition (6.76) is fulfilled.

4.5 Stability or Instability Conditions Assume the condition

μνuχ1

0) and that the parameters c, d, f, g, μ, ν, ζ , and K are positive (> 0) constants. It is easy to see that (13.1) has the unique homogeneous stationary solution Kf ν Kf νζ f , , . (13.2) U= c Kcd + f ν (Kcd + f ν)g We shall first prove the global existence of solutions to (13.1) and construct a dynamical system. We shall next show that the dynamical system possesses exponential attractors and that the homogeneous stationary solution U becomes unstable if μ and ζ are sufficiently large.

2 Local Solutions 2.1 Abstract Formulation Let us formulate problem (13.1) as the Cauchy problem for an abstract evolution equation. We want to handle the first equation of (13.1) on u in the L2 space. In order to rewrite (13.1) into a semilinear abstract equation, we will choose the H 1 Sobolev space for handling the third equation on ρ. It is then natural to handle the second equation on v in the same H 1 space, because the equation has no smoothing effect. So, our underlying space is set by ⎧⎛ ⎞ ⎫ ⎨ u ⎬ X = ⎝ v ⎠ ; u ∈ L2 (Ω), v ∈ H 1 (Ω), ρ ∈ H 1 (Ω) . (13.3) ⎩ ⎭ ρ We are then led to formulate (13.1) as the problem dU 0 < t < ∞, dt + AU = F (U ), U (0) = U0 ,

(13.4)

in X. Here, A = diag {A1 , d, A2 } is a diagonal matrix operator of X, where A1 = −aΔ + c and A2 = −bΔ + g are positive definite self-adjoint operators of L2 (Ω)

2 Local Solutions

447

with domains D(A1 ) = D(A2 ) = HN2 (Ω) (see (2.34)). But, A2 must be regarded 3

1

as an operator from D(A22 ) = {ρ ∈ HN2 (Ω); Δρ ∈ H 1 (Ω)} into D(A22 ) = H 1 (Ω). Note that, as Ω is of C3 class, Theorem 2.9 ensures the shift property that Δρ ∈ ∂ρ = 0 implies that ρ ∈ H 3 (Ω). Therefore, H 1 (Ω) with ∂n 32 ∂ρ 3 3 D A2 = HN (Ω) = ρ ∈ H (Ω); = 0 on ∂Ω . ∂n Consequently, the domain of A is given by ⎧⎛ ⎞ ⎫ ⎨ u ⎬ D(A) = ⎝ v ⎠ ; u ∈ HN2 (Ω), v ∈ H 1 (Ω), ρ ∈ HN3 (Ω) . ⎩ ⎭ ρ

(13.5)

Obviously, A is a positive definite self-adjoint operator of X. The nonlinear operator F is given by ⎛ ⎞ ⎞ ⎛ u −μ∇ · [u∇ρ] + f (13.6) F (U ) = ⎝ ν(1 − Kv )u ⎠ , U = ⎝ v ⎠ ∈ D(Aη ), ζv ρ 3 4

where η is a fixed exponent such that

< η < 1.

2.2 Construction of Local Solutions Let us first characterize the domains of fractional powers Aθ for 0 < θ < 1. Since A is diagonal, it is clear that Aθ = diag {Aθ1 , d θ , Aθ2 } for 0 < θ < 1, where Aθ2 is θ+ 1

1

an operator from D(A2 2 ) to D(A22 ). According to Theorems 16.7 and 16.9 in Chap. 16, it is known that D(Aθ1 ) = D(Aθ2 ) = H 2θ (Ω), 0 ≤ θ < 34 , D(Aθ1 ) = D(Aθ2 ) = HN2θ (Ω), 34 < θ ≤ 1. In addition, as Ω is of C3 class, it is seen that ∂ρ D(Aθ2 ) = HN2θ (Ω) = ρ ∈ H 2θ (Ω); = 0 on ∂Ω , ∂n

3 1≤θ ≤ . 2

As a consequence, it is deduced that ⎛u⎞ D(Aθ ) = ⎝ v ⎠ ; u ∈ H 2θ (Ω), v ∈ H 1 (Ω), ρ ∈ H 2θ+1 (Ω) , ρ

1 0≤θ < , 4 (13.7)

448

13 Termite Mound Building Model

⎛u⎞

D(Aθ ) = ⎝ v ⎠ ; u ∈ H 2θ (Ω), v ∈ H 1 (Ω), ρ ρ

∈ HN2θ+1 (Ω)

,

⎛u⎞ D(Aθ ) = ⎝ v ⎠ ; u ∈ HN2θ (Ω), v ∈ H 1 (Ω), ρ ∈ HN2θ+1 (Ω) , ρ

1 3 0 is determined by the norm A 2 U0 X only. In addition, 1 1 t 2 AU (t)X + A 2 U (t)X ≤ CU0 , 1

where CU0 > 0 is also determined by A 2 U0 X only.

0 < t ≤ TU 0 ,

(13.12)

2 Local Solutions

449

2.3 Nonnegativity of Local Solutions Let U (t) be a unique local solution of (13.4) in (13.11) for U0 ∈ K. We want to show that U (t) ∈ K for every 0 < t ≤ TU0 . The local solution U is real valued. Indeed, the complex conjugate U of U is clearly a solution of (13.4) with the same initial value; then, by the uniqueness of solution, it must hold that U (t) = U (t) for every 0 < t ≤ TU0 ; hence, U (t) is a real-valued function. Let us denote U = t (u, v, ρ). We first prove that u ≥ 0 by the truncation method. 2 Let H (u) be a C1,1 cutoff function such that H (u) = u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. Consider the function ϕ(t) = Ω H (u(t)) dx. By (1.100), ϕ is a continuously differentiable function with the derivative ϕ (t) = H (u){aΔu − μ∇ · [u∇ρ] − cu + f } dx. Ω

Applying (1.96) to H (u), it is observed that H (u)aΔu = −a ∇H (u) · ∇u dx = −a |∇H (u)|2 dx. Ω

Ω

Ω

In the meantime, − H (u)μ∇ · [u∇ρ] dx = μ u∇H (u) · ∇ρ dx = μ H (u)∇H (u) · ∇ρ dx Ω

Ω

μ = 2

Ω

μ ∇H (u)2 · ∇ρ dx = − 2 Ω

H (u)2 Δρ dx.

Ω

Since Δρ(t)L2 ≤ CU for 0 ≤ t ≤ TU0 , we have μ H (u)2 Δρ dx ≤ C H (u)2 L ΔρL2 ≤ CH (u)2L4 ΔρL2 − 2 2 Ω 3

1

≤ CU H (u)H2 1 H (u)L2 2 a ≤ H (u)2H 1 + CU H (u)2L2 . 2

Finally, it is clear that Ω H (u)[−cu + f ] dx ≤ 0. Hence, we proved that ϕ (t) ≤ CU ϕ(t) and that ϕ(t) ≤ eCU t ϕ(0). Consequently, ϕ(0) = 0 implies ϕ(t) ≡ 0, i.e., u(t) ≥ 0 for 0 < t ≤ TU0 . We next verify that 0 ≤ v(t) ≤ K for 0 < t ≤ TU0 . Since u ≥ 0 is already seen, the proof for v(t) ≥ 0 is similar or even easier. In order to prove that v(t) ≤ K, we rewrite the equation of v in the form ∂w νu =− d+ w + dK, ∂t K

450

13 Termite Mound Building Model

where w = K − v. Then, the property w0 = K − v0 ≥ 0 at t = 0 implies w(t) ≥ 0 for every 0 < t ≤ TU0 . In this way, we observe that 0 ≤ v(t) ≤ K for 0 < t ≤ TU0 . We finally have to prove that ρ(t) ≥ 0 for 0 < t ≤ TU0 . Since v ≥ 0 is already known, this is also seen by the truncation method employed above. We have thus proved that U0 ∈ K implies U (t) ∈ K for each 0 < t ≤ TU0 .

3 Global Solutions 3.1 A Priori Estimates for Local Solutions For U0 ∈ K ∩ D(A), let U (t) denote any local solution of (13.4) on [0, TU ] in the function space: 1 U ∈ C((0, TU ]; D(A)) ∩ C([0, TU ]; D(A 2 ) ∩ C1 ((0, TU ]; X), (13.13) U (t) ∈ K, 0 ≤ t ≤ TU . We intend to establish the a priori estimate for such a local solution. For this purpose, we have to employ a realization of −bΔ + g under the homogeneous Neumann boundary conditions in the space L6 (Ω). We know by Theorem 2.12 that the realization, denoted by A2,6 , is a sectorial operator of L6 (Ω) with angle < π2 and generates an analytic semigroup e−tA2,6 on L6 (Ω) which satisfies the estimate e−tA2,6 L6 ≤ Ce−δ1 t , 0 ≤ t < ∞, with some exponent δ1 > 0. Moreover, ∂ρ = 0 on ∂Ω}. Since we know by Theorem 2.15 that D(A2,6 ) = {ρ ∈ H62 (Ω); ∂n 3

HN3 (Ω) ⊂ D(A2,6 ), we see that ρ0 ∈ D(A22 ) implies ρ0 ∈ D(A2,6 ). Furthermore, for 0 < θ < 1, consider the fractional power Aθ2,6 . On account of Remark 16.6 in

Chap. 16, we verify that D(Aθ2,6 ) ⊂ H62θ (Ω) for any 0 < θ < θ . In particular, we 9

8

9 8 10 1 (Ω) due to (1.76). and θ = 10 ; then, D(A2,6 ) ⊂ H65 (Ω) ⊂ H∞ take θ = 10 We can state what we want to prove now.

Proposition 13.1 There exists a continuous increasing function p(·) such that the estimate 9 9 1 A 2 U (t) + A 10 ρ(t) ≤ p A 12 U0 + A 10 ρ0 , 2,6 2,6 X L X L

0 ≤ t ≤ TU , (13.14) holds for any local solution U of (13.4) with U0 ∈ K ∩ D(A) in the function space (13.13). 6

6

Proof Throughout the proof, the notation p(·) stands for some continuous increasing functions which are determined only by Ω and by the initial constants in the equations of (13.1). Similarly, the notation C stands for some constants which are determined only by Ω and by the initial constants in (13.1).

3 Global Solutions

451

Step 1. Integrate the first equation of (13.4) in Ω. Then, d u dx = −c u dx + f |Ω|. dt Ω Ω So,

Ω

u dx = e−ct u0 L1 + f |Ω|

t

e−c(t−s) ds.

0

Since u(t) ≥ 0, it follows that u(t)L1 ≤ e−ct u0 L1 +

f |Ω|, c

0 ≤ t ≤ TU .

(13.15)

Step 2. In this step, let us treat the third equation of (13.4) in L6 (Ω). From (13.13) v we see that dv dt L2 = − dv + ν(1 − K )uL2 ≤ C, so that v(t) − v(s)L2 ≤ 1

C|t − s|. This then implies v ∈ C 3 ([0, TU ]; L6 (Ω)) because of 2

1

v(t) − v(s)L6 ≤ v(t) − v(s)L3 ∞ v(t) − v(s)L3 2 ,

0 ≤ s < t ≤ TU .

Then, thanks to Theorem 3.4, ρ(t) is written by t e−(t−s)A2,6 ζ v(s) ds, ρ(t) = e−tA2,6 ρ0 + 0 9

10 where A2,6 is the realization of −bΔ + g in L6 (Ω) mentioned above. Operate A2,6 to the equality. Then, t 9 9 9 10 10 10 −(t−s)A2,6 A2,6 ρ(t) = e−A2,6 t A2,6 ρ0 + A2,6 e ζ v(s) ds.

0

On account of 0 ≤ v(s) ≤ K, we have 9 109 A ρ(t) ≤ Ce−δ1 t A 10 ρ0 + C 2,6 2,6 L L 6

6

t 0

≤ C e−δ1 t A2,6 ρ0 L + 1 , 9 10

9

δ1

(t − s)− 10 e− 2 (t−s) v(s)L6 ds 0 ≤ t ≤ TU .

6

In particular, −δ t 109 1 A ρ(t)H∞ 1 ≤C e 2,6 ρ0 L + 1 , 6

0 ≤ t ≤ TU ,

with some exponent δ1 > 0. Step 3. In this step, let p1 (U0 ) denote the quantity 109 p1 (U0 ) = p u0 L1 + A2,6 ρ0 L . 6

(13.16)

452

13 Termite Mound Building Model

Multiply the first equation of (13.4) by u and integrate the product in Ω. Then, 1 d u2 dx + a |∇u|2 dx + c u2 dx 2 dt Ω Ω Ω =μ u∇u · ∇ρ dx + f u dx Ω

Ω

7 a ≤ ∇u2L2 + C uL3 7 + ∇ρ14 L14 + f uL1 . 4 3 Here, by (1.77), we have 28

7

uL 7 ≤ CuH351 uL351 . 3

So, 7

28

7

uL3 7 ≤ CuH151 uL151 ≤ ζ˜ u2H 1 + Cζ˜ u7L1 3

with an arbitrary number ζ˜ > 0. In the meantime, it is clear that ∇ρL14 ≤ ρH∞ 1 . We take ζ˜ > 0 small enough to ensure that a c 1 d u2 dx + |∇u|2 dx + u2 dx ≤ p1 (U0 ). 2 dt Ω 2 Ω 2 Ω Hence, solving this inequality (see (1.57)), we conclude that u(t)2L2 ≤ e−ct u0 2L2 + p1 (U0 ),

0 < t ≤ TU .

(13.17)

In addition, by Proposition 1.5, we conclude that

t s

u(τ )2H 1 dτ ≤ C p1 (U0 )(t − s) + u0 2L2 ,

0 ≤ s < t ≤ TU .

(13.18)

Step 4. Differentiate the second equation of (13.4) by xi (i = 1, 2, 3). Then, νu v ∂ Di v = − d + Di v + ν 1 − Di u. ∂t K K Multiply this equation by Di v and integrate the product in Ω. Then, 1 d νu v 2 2 |Di v| dx + |Di v| dx = (Di u)(Di v) dx d+ ν− 2 dt Ω K K Ω Ω ≤

d Di v2L2 + CDi u2L2 . 2

3 Global Solutions

453

So, it follows that d dt

|Di v|2 dx + d Ω

Ω

|Di v|2 dx ≤ CDi u(t)2L2 .

Employing Proposition 1.4, in view of (13.18), we conclude that Di v(t)2L2 ≤ e−dt Di v0 2L2 + p1 (U0 ) + u0 2L2 ,

0 ≤ t ≤ TU .

We sum up these estimates for i = 1, 2, 3 to conclude that v(t)2H 1 ≤ C e−dt v0 2H 1 + p1 (U0 ) + u0 2L2 , 0 ≤ t ≤ TU .

(13.19)

Step 5. In view of (13.19), let p2 (U0 ) denote p2 (U0 ) = p(v0 H 1 + u0 L2 + p1 (U0 )). Let us use the triplet H 1 (Ω) ⊂ L2 (Ω) ⊂ H 1 (Ω)∗ . We remember that A2 can be extended as an operator from H 1 (Ω) to H 1 (Ω)∗ , see Theorem 2.4. On the other hand, A2 is an operator from HN3 (Ω) to H 1 (Ω). So, A22 maps HN3 (Ω) to H 1 (Ω)∗ . We consider the duality product of the third equation of (13.4), which has a meaning in H 1 (Ω), and the vector A22 ρ(t) ∈ H 1 (Ω)∗ . Then, since ρ1 , A22 ρ2 H 1 ×H 1∗ = A2 ρ1 , A2 ρ2 H 1 ×H 1∗ = (A2 ρ1 , A2 ρ2 )L2 we have 1 d 2 dt

32 2 A ρ dx = ζ

|A2 ρ|2 dx + Ω

for ρ1 ∈ HN2 (Ω), ρ2 ∈ HN3 (Ω),

Ω

2

1

Ω

3

A22 v · A22 ρ dx

1 3 2 ≤ A22 ρ L + Cv2H 1 . 2 2 On account of (13.19), we obtain that 3 2 d 2 |A2 ρ| dx + A22 ρ dx ≤ p2 (U0 ). dt Ω Ω −1

(13.20)

3

Since A2 ρL2 ≤ A2 2 L(L2 ) A22 ρL2 , it follows that 2 d 2 |A2 ρ| dx + δ2 A2 ρ dx ≤ p2 (U0 ), dt Ω Ω −1

where δ2 = A2 2 −1 L(L2 ) > 0. Solving this (cf. (1.58)), we conclude that ρ(t)2H 2 ≤ Ce−δ2 t ρ0 2H 2 + p2 (U0 ),

0 ≤ t ≤ TU .

(13.21)

454

13 Termite Mound Building Model

As well, by Proposition 1.5, we conclude that t 3 2 A ρ(τ )2 dt ≤ C[p2 (U0 )(t − s) + ρ0 2 ], H 2 L

0 ≤ s < t ≤ TU .

2

s

In other words, t ρ(τ )2 3 dτ ≤ C[p2 (U0 )(t − s) + ρ0 2 ], H H

0 ≤ s < t ≤ TU .

s

Step 6. On account of (13.21), let us denote p3 (U0 ) = p(ρ0 H 2 + p2 (U0 )). Multiply the first equation of (13.4) by Δu and integrate the product in Ω. Then, 1 d |∇u|2 dx + a |Δu|2 dx + c |∇u|2 dx = μ ∇ · [u∇ρ]Δu dx 2 dt Ω Ω Ω Ω a 2 ≤ |Δu| dx + C (|∇u · ∇ρ| + u|Δρ|)2 dx. 4 Ω Ω Here, Ω

|∇u|2 |∇ρ|2 dx ≤ C∇u2L2 ∇ρ2L∞ ≤ CuH 2 uL2 ρ2H 1

∞

≤ ζ˜ u2H 2 + Cζ˜ u2L2 ρ4H 1 ∞ 2 2 ≤ ζ˜ ΔuL2 + Cζ˜ uL2 ρ4H 1 + 1 , ∞

where ζ˜ > 0 is an arbitrary number. Similarly, using H (Ω) ⊂ L∞ (Ω) (due to (1.76)), we have 5 1 u2 |Δρ|2 dx ≤ u2L∞ Δρ2L2 ≤ u2 5 ρ2H 2 ≤ uH3 2 uL3 2 ρ2H 2 5 3

H3

Ω

≤ ζ˜ u2H 2 + Cζ˜ u2L2 ρ12 H2 ≤ ζ˜ Δu2L2 + Cζ˜ u2L2 ρ12 +1 , H2 where ζ˜ > 0 is an arbitrary number. Therefore, by (13.17) and (13.21), we obtain that d |∇u|2 dx + a |Δu|2 dx + c |∇u|2 dx ≤ p3 (U0 ). (13.22) dt Ω Ω Ω Solving this (cf. (1.58)), we conclude that ∇u(t)2L2 ≤ e−ct ∇u0 2L2 + p3 (U0 ),

0 ≤ t ≤ TU .

3 Global Solutions

455

This, together with (13.17), yields that u(t)2H 1 ≤ e−ct u0 2H 1 + p3 (U0 ),

0 ≤ t ≤ TU .

As well, by Proposition 1.5, t Δu(τ )2L2 dτ ≤ C p3 (U0 )(t − s) + u0 2H 1 ,

(13.23)

0 ≤ s ≤ t ≤ TU .

s

By (13.16), (13.19), (13.21), and (13.23), we have obtained the desired estimate.

3.2 Construction of Global Solutions Let U0 ∈ K. As shown in Sect. 2.2, there exists a unique local solution to (13.4) with the initial value U0 on an interval [0, TU0 ]. Let 0 < t1 < TU0 . Then, U1 = U (t1 ) ∈ D(A) ∩ K. We consider problem (13.4) but substituting t1 for the initial time 0 and U1 for the initial value U0 . Of course, the problem also possesses a unique local solution in (13.11). Since U0 ∈ D(A) ∩ K, we can apply Proposition 13.1 to any local solution of the problem with initial time t1 and initial value U1 . Then, Corollary 4.1 provides the existence of the global solution. In this way, we verify that, for any U0 ∈ K, (13.4) possesses a unique global solution in the function space: 1 U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); D(A 2 )) ∩ C1 ((0, ∞); X), (13.24) U (t) ∈ K, 0 < t < ∞.

3.3 Global Norm Estimates For U0 ∈ K, let U (t; U0 ) be the global solution of (13.4) with the initial value U0 in (13.24). We observe that 1 A 2 U (t; U0 ) ≤ p e−δt p A 12 U0 + 1 , 0 ≤ t < ∞, U0 ∈ K, (13.25) X X where p(·)’s are some continuous increasing functions, and δ is a positive exponent. In fact, let us first assume that U0 ∈ D(A) ∩ K. We will apply estimates (13.15), (13.16), (13.17), (13.19), (13.21), and (13.23) established above. Applying (13.23) on [ 4t5 , t] and using (13.14), we have u(t)2H 1

2 4t 4t ≤ e u + p3 U 5 H 1 5 c 4t ≤ e− 5 t p(AU0 X ) + p3 U . 5 − 5c t

456

13 Termite Mound Building Model

By the definition of p3 (·), we observe that 2 4t 4t 4t = p ρ . p3 U + p2 U 5 5 H 2 5 Applying (13.21) on

3t 5

, 4t5 and using (13.14), we have

2 4t δ ≤ Ce− 52 t ρ 3t + p2 U 3t ρ 5 H 2 5 H 2 5 δ2 3t . ≤ e− 5 t p(AU0 X ) + p2 U 5 Let us estimate p2 (U ( 3t5 )). It is the same for p2 (U ( 4t5 )). By the definition of p2 (·), we observe that 2 2 3t 3t 3t + p1 U 3t v p2 U = p u . + 5 5 L2 5 H 1 5 Applying (13.19) on

2t 5

, 3t5 and using (13.14), we have

2 2 2 3t v ≤ e− d5 t v 2t + u 2t + p1 U 2t 5 1 5 1 5 5 H

H

L2

2 2t 2t − d5 t . p(AU0 X ) + u + p1 U ≤e 5 5 L2

In addition, applying (13.17) on

t

2t

5, 5

, we have

2 2 2t u ≤ e− 5c t u t + p1 U t 5 5 L2 5 L2 c t ≤ e− 5 t p(AU0 X ) + p1 U . 5 (It is the same for the estimate u( 3t5 )2L2 .) Let us next estimate p1 (U ( 5t )). (It is the same for p1 (U ( 2t5 )).) By the definition of p1 (·), we observe that 9 t 10 t t A2,6 ρ p1 U + . = p u 5 5 L1 5 L6 Finally, we apply (13.15) and (13.16) on [0, 5t ] to obtain that t −ct u 5 ≤ C e 5 AU0 X + 1 , L1

4 Dynamical System

457

9 10 A ρ t 2,6 5

δ1 ≤ C e− 5 t AU0 X + 1 .

L6

We have thus verified that u(t)H 1 ≤ p(e−δt p(AU0 X ) + 1),

0 ≤ t < ∞,

with some exponent δ > 0. It is possible to estimate v(t)H 1 and ρ(t)H 2 in an analogous way. We therefore observe that 1 A 2 U (t) ≤ p(e−δt p(AU0 X ) + 1), 0 ≤ t < ∞, (13.26) X with some exponent δ > 0. Let now U0 ∈ K, and let us verify (13.25). By (13.12), there exists a time τ = 1 1 τ (A 2 U0 X ) > 0 depending only on the norm A 2 U0 X such that 1 1 1 t 2 AU (t)X + A 2 U (t)X ≤ p A 2 U0 X ,

0 < t ≤ τ.

(13.27)

For τ ≤ t < ∞, (13.26) yields that 1 A 2 U (t) ≤ p(e−δ(t−τ ) p(AU (τ )X ) + 1), X

τ ≤ t < ∞.

Hence, 1 A 2 U (t) ≤ p e−δt eδτ p τ − 12 p A 12 U0 +1 , X X

τ ≤ t < ∞.

This, together with (13.27), immediately implies (13.25).

4 Dynamical System 4.1 Construction of Dynamical System Let us construct a dynamical system determined from problem (13.4). We can entirely follow the general methods in Chap. 6, Sect. 5 by taking β = 12 . Indeed, (13.25) shows that (6.56) is fulfilled. Hence, problem (13.4) determines a dynamical 1 system (S(t), K, D 1 ), where D 1 = D(A 2 ). 2

2

For any exponent 12 < θ < 1, (13.4) equally determines a dynamical system in the universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (13.6) fulfills (4.2) with β = 12 and a fortiori with β = θ . D

Let Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

458

13 Termite Mound Building Model

Meanwhile, by Proposition 6.1, 1

Aθ S(t)U0 X ≤ Cθ,R t 2 −θ ,

0 < t < ∞, U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that (6.56) is valid in the space Dθ , too. So, (S(t), Kθ , Dθ ) is a dynamical system. Let us finally construct a dynamicalsystem in the universal space X. For any 0 < R < ∞, let KR = K 1 ,R . Let XR = 0≤t 0 for which the following assertion is valid. For any bounded subset B of K, there exists a time tB > 0 such that ˜ sup sup S(t)U0 D 1 ≤ C.

U0 ∈B t≥tB

(13.28)

2

D1

˜ is an absorbing set. This means that the closed ball B 2 (0; C) We can then follow the same arguments as in Chap. 6, Sect. 5.2 to construct an absorbing set X ⊂ K which is invariant under S(t), is a closed subset of X, and is a bounded subset of D(A). Let R > 0 be a radius large enough to satisfy X ⊂ KR ⊂ XR . As (S(t), XR , X) is already known to be a dynamical system in X, we conclude that (S(t), X, X) also defines a dynamical system in X. Furthermore, since X is an absorbing set of (S(t), K, D 1 ), the asymptotic behavior of trajectories 2 of (S(t), K, D 1 ) is reduced to that of (S(t), X, X). 2

4.3 Global Attractor We are concerned with constructing a global attractor for (S(t), X, X). We must however notice that the general methods described in Chap. 6, Sect. 5.3 are not available, because the space condition (6.59) fails now. The domain D(A) given by (13.3) is not compactly embedded in X. So, we have to construct it in a direct way.

4 Dynamical System

459

We set !

A=

(closure in X).

S(s)X

(13.29)

0≤t

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