Research on Evolution Equation Compendium, Volume 1

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RESEARCH ON EVOLUTION EQUATION COMPENDIUM. VOLUME 1 No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

RESEARCH ON EVOLUTION EQUATION COMPENDIUM. VOLUME 1

GASTON M. N'GUÉRÉKATA EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

ISBN: 978-1-61209-404-5 (E-Book)

Published by Nova Science Publishers, Inc. Ô  New York

CONTENTS Preface

ix

Double Scale Convergence and Homogenization of Quasilinear Parabolic Equations Xuming Xie

1

Second Grade Fluids with Enhanced Viscosity as Dynamical Systems Ahmed S. Bonfoh

11

Periodic Solutions of Some Evolution Equations with Infinite Delay Khalil Ezzinbi and James H.Liu

19

Internal Pollution and Discriminating Sentinel in Population Dynamics Problem O. Nakoulima and S. Sawadogo

29

Center Manifold and Stability in Critical Cases for Some Partial Functional Differential Equations Mostafa Adimy,Khalil Ezzinbi and Jianhong Wu

47

Stability Radii of Positive Linear Functional Differential Systems in Banach Spaces Pham Huu Anh Ngoc, Nguyen Van Minh and Toshiki Naito

75

Effects of Nonlocality and Phase Shift Definitions in Generalizing Levinson’s Theorem S.B.Qadri, B. Mulligan, M.F. Mahmood and J. Y. Al-Khal

99

Asymptotic Behavior of the Fitzhugh-Nagumo System Weishi Liu and Bixiang Wang A Partial Differential Equation with Nonautonomous Past Delay in L1-Phase Space S. Boulite, G.Fragnelli, M. Halloumi and L. Maniar Solving the Hyperbolic Problem Obtained by Transmutation Operator Hikmet Koyunbakan

129

165 183

vi

Contents

Elliptic Operators with Variable Coefficients Generating Fractional Resolvent Families Miao Li, Fu-Bo Li and Quan Zheng

195

Existence Results for Pseudo Almost Periodic Differential, Functional, and Neutral Integral Equations Toka Diagana

205

An Asymptotic Model of a Nonlinear Kelvin-Voigt Viscoelastic Plate Robert P.Gilbert and Robert J. Ronkese Mathematical Analysis of a Bilateral Obstacle Problem for a Class of Second-Order Operators Laurent Lévi and Guy Vallet

235

255

Application of the Picard Operators to Second Order ODE’s Alexandru Bica, Loredana Galea and Sorin Muresan

277

Doubly Nonlinear Degenerate Parabolic Equations on Carnot Groups Junqiang Han and Pengcheng Niu

287

Some Scalar Conservation Laws with Discontinuous Flux Julien Jimenez

297

A New Exact Solution to the Delayed Diffusion Equation P. M. Jordan

317

Level Sets for Reaction Diffusion Equations Amin Boumenir

327

On the Almost Periodicity of the Superposition of Functions Stanislaw Stoiński

333

Time Periodic Solutions for Quasigeostrophic Motion and Their Stability Mei-Qin Zhan Jacobian Feedback Loops Analysis II: Stability and Instability Bourama Toni Existence of Oscillating Solution for Nonlinear State-Dependent Delay Differential Equation Zhixian Yu and Rong Yuan Semilinear Abstract Differential Equations with Deviated Argument Ciprian G. Gal Singular Solutions of a Semi-Linear Elliptic Equation on Nonsmooth Domains Lotfi Riahi

343 355

367 381

387

Contents

vii

Existence of Weighted Pseudo Almost Periodic Solutions to Some NonAutonomous Differential Equations Toka Diagana

397

A Krasnoselskii-Type Fixed Point Theorem for Multifunctions Defined on a Hyperconvex Space Marcin Borowski

411

Index

417

PREFACE Chapter 1 - The behavior of inhomogeneous material, where the inhomogeneities are on a small scale, is of considerable interest in material sciences, mechanical engineering and in many technological applications, especially those involving composite materials. The theory of homogenization was created to model and predict the behavior of such material, when the inhomogeneities are on such a scale much smaller than the linear dimension of the system. In these situations, continuum models, in which rapid oscillation of the material properties is averaged, are sufficient to describe the behavior of the system. Moreover, they have the advantage of avoiding extensive numerical computations needed when the small scale behavior is taken into account. There have been numerous publications on homogenization theory and its applications in the past decades, the authors refer to the books by Bensoussan et al., Sanchez-Palencia and Jikov et al. For homogenization of nonlinear equations, a standard approach is using energy methods by Tartar and Murat; however, in some cases, it is difficult to work out energy method sin was first introduced by Nguetseng, later elaborated by Allaire; the advantage of this method is that the homogenized equations and proof of convergence are obtained in a single step. Recently Caffarelli proposed an alternative approach in which the homogenization be considered as a viscosity process. Following the idea by Caffarelli, Garcia-Azorero et al. proved that the uniform limit of the solutions at scale e to quasilinear parabolic equations is a viscosity solution to the homogenized equation. The use of viscosity solution methods was first considered by Evansor divergence structure PDE. In this paper, the authors shall use the method of double-scale convergence to derive the homogerized equation of the quasilinear parabolic equations considered in the references they site. Chapter 2 - The authors consider a modified 2d second grade incompressible fluid with a higher order viscosity term, and show that this nonlinear evolution system possesses an exponential attractor. This object gives a more precise description of the asymptotic behaviour of dynamical systems than the global attractor. Chapter 3 - In this work, the authors study the existence of periodic solutions for some partial functional differential equations with infinite delay. The authors assume that the linear part is not necessarily densely defined and satisfies the Hille-Yosida condition, and the phase space is chosen to be C for some decreasing function g from (-¥,0] to [1,¥). The authors also present a related Massera type result, namely the existence of a bounded solution on R+ implies the existence of a periodic solution.

x

Gaston M. N’Guerekata

Chapter 4 - The so called Lions sentinel method is applied to some time varying system with two time scale variables and with missing data. As an example, some population dynamics problem with missing data is considered. The two time scale variables represent the running time variable and the age of individuals. This insight seems to be new. Building the object sentinel leads to some null-controllability problem with restricted control variables and two time scale variables. Chapter 5 - In this work, the authors prove the existence of a center manifold for some partial functional differential equations, whose linear part is not necessarily densely defined but satisfies the Hille-Yosida condition. The attractiveness of the center manifold is also shown when the unstable space is reduced to zero. The authors prove that the flow on the center manifold is completely determined by an ordinary differential equation in a finite dimensional space. In some critical cases, when the exponential stability is not possible, the authors prove that the uniform asymptotic stability of the equilibrium is completely determined by the uniform asymptotic stability of the reduced system on the center manifold. Chapter 6 - In this paper the authors study stability radii of positive linear functional differential systems in Banach spaces under multi-perturbations and multi-affine perturbations. The authors prove that for the class of positive systems, complex stability radius, real stability radius and positive stability radius of positive systems under multiperturbations (or multi-affine perturbations) coincide and they are computed via a simple formula. The authors illustrated the obtained results by an example. Chapter 7 - This paper considers Levinson’s Theorem for nonlocal potentials. The effect of continuum bound states is clarified, and the presence of spurious states is introduced. Differences between definitions of the phase shift are discussed, with special attention given to discontinuities in the phase shift. Examples from nuclear physics emphasize the utility of this theorem and illustrate some of the more delicate aspects of the derivation. A summary is provided which analyzes certain crucial points in the discussion as they relate to previousattempts to obtain a generalized Levinson’s theorem. NUCLEAR REACTIONS Levinson’s Theorem, nonlocal potentialscontinuum bound states, spurious states, Jost functions, Fredholm determinants, definition of the phase shift. Chapter 8 - For the FitzHugh-Nagumo system defined on IR, the authors prove the existence of a compact global attractor in a weighted Sobolev space which contains bounded solutions, in particular, traveling wave and spatially periodic solutions. The authors also study the behavior of the global attractors as a parameter goes to zero. Although the limiting system for =0does not possess a bounded attracting set, the authors show that there exists a constant 0 such that global attractors for 0 < = are all contained in a compact subset of the phase space. Furthermore, the authors construct a compact local attractor for the limiting system and establish the upper semicontinuity of the global attractors of perturbed system and the local attractor of the limiting system. Chapter 9 - In this paper, the authors consider partial differential equations with nonautonomous past delay in L1-phase space. Here, the authors consider general delay operators given by Stieltjes integrals. Using the Miyadera-Voigt perturbation result, the authors show the wellposedness of these equations. The authors study also the robustness of some asymptotic properties, as asymptotic stability, almost periodicity and almost automorphy, under the nonautonomous past delay effect. As an application, the authors give a dynamical population equation.

Preface

xi

Chapter 10 - In this paper, the authors solve the problem of constructing the kernels of transmutation operator and give the generalized transmutation operator for a singular problem. Chapter 11 - In this paper, the authors show that under very general conditions, elliptic operator with zero boundary condition can generate fractional resolvent families. Chapter 12 - This paper is a survey devoted to the author’s recent results related to the existence and uniqueness of pseudo almost periodic solutions to some abstract differential equations and partial differential equations. Some of those results will be slightly modified and applied to some concrete problems. As main tools, the authors will make extensive use of the method of analytic semigroups, and both the Banach and Zima’s fixed-point principles. Applications include the study of pseudo almost periodic solutions to some boundary-value problems such as the heat equation with delay as well as the logistic differential equation. Chapter 13 - In the 1970s, Cowin and Hegedus introduced an adaptive elasticity model for bone deposition and reabsorption (remodeling). In 1998, Figueiredo and Trabucho published a paper on an asymptotic model of an adaptive elastic rod. Afterwards, Monnier and Trabucho proved the existence and uniqueness of a solution for this model; Trabucho himself extended this result to a model that contains non-linear strain terms along with linear strain terms in a rate remodeling equation. Here, it is proposed to introduce a Kelvin-Voigt model for the plate that includes non-linear terms rate remodelling terms with respect to strain and the time derivative of strain. As many biomechanists consider cancellous bone to be structure consisting of both rode and plates, adding the study of the plate permitsre a more realistic modeling of the trabeculae of spongy bone. Recent studies have suggested that the dissipation of acoustic energy in cancellous bone interrogation via ultrasound is due to the viscous properties of the trabeculae, rather than that of interstitial blood and marrow. This implies use of a Kelvin-Voigt model for both the plate-like and rod-like trabeculae. Chapter 14 – The authors investigate some inner bilateral obstacle problems for a class of strongly degenerate parabolic-hyperbolic quasilinear operators associated with homogeneous Dirichlet data in a multidimensional bounded domain. The authors first introduce the concept of an entropy process solution, more convenient and generalizing the notion of an entropy solution. Moreover, the boundary conditions are expressed by using the background of Divergence Measure Fields. The authors ensure that proposed definition warrants uniqueness. The existence of an entropy process solution is obtained through the vanishing viscosity and penalization methods. Chapter 15 - Using the Perov’s fixed point theorem and the fiber contractions theorem are obtained sufficient conditions for the smooth dependence by the end points of the solution of the two point boundary value problem for second order ODE’s. Finally, an application in the metabolism control is presented. Chapter 16 - In this paper the authors use variational methods to study the nonexistence of positive solutions for the following doubly nonlinear degenerate parabolic equations on Carnot groups:

xii

Gaston M. N’Guerekata

Here Ω is a bounded domain with smooth boundary in a Carnot group GVG is the horizontal gradient on G, T > 0, V ∈ L1loc (Ω), u0 is not identically zero, m ∈ R, 1< p < Q and m + p – 2 > 0. Chapter 17 - The authors deal with the scalar conservation law in a one dimensional bounded domain Ω: ∂tu+ ∂x(k( x) g(u)) = 0, associated with a bounded initial value u0. The function k is supposed to be bounded, discontinuous at { x0= 0} , and with bounded variation. A weak entropy formulation for the Cauchy problem has been introduced by J.D Towers. The existence and the uniqueness is proved by N. Seguin and J. Vovelle through a regularization of the function k. The authors generalize the definition of J.D Towers and the authors adapt the method developed in to establish an existence and uniqueness property in the case of the homogeneous Dirichlet boundary conditions. Chapter 18 - A new analytical solution to the one-dimensional delayed diffusion equation is derived in terms of the Lambert W-function. A new critical value of the thermal lag time parameter is also noted. The effects of varying this parameter are examined using numerical methods. Chapter 19 - The authors are concerned with the shape of the level sets of solution of reaction diffusion equations. Using the maximum principle the authors find sufficient conditions for their concavity. Chapter 20 - In this note the authors present some theorems on the superposition of a function which satisfies the Hölder condition and an (NSp)-almost periodic (a.p.) function, as well an Sp-a.p. function. Moreover, the authors prove a theorem on the superposition of a continuous function and an (NH)-a.p. function. In the following, the authors give a theorem on the superposition of a differentiable function and a Vp-a.p. function. Finally, the authors prove a theorem on the superposition of a differentiable function and an (NC(1))-a.p. function. Chapter 21 - In this article, the authors study the quasi geostrophic equation, which is a prototypical geophysical fluid model. The authors will show the existence of time-periodic solutions for any the Coriolis parameter and the Ekman dissipation constant with nonhomogeneous boundary conditions. Chapter 22 - The authors investigate the loop stability conditions of differential systems, that is, the conditions of invariance of the Jacobian spectrum under any variation of entries that leaves unchanged the loop structure. The authors use the dynamical properties of Jacobian Loops described by the products of the matrix entries under cyclic permutations of the indices. It appears that all k-order Feedback Loops given by the union of disjoint simple loops involving k variables must be positive for any asymptotic stable behavior. The authors also conjecture that the Loop structure requires a negative Feedback Loop of the system dimension for the onset of chaotic behavior. Chapter 23 - In this paper sufficient conditions for oscillation of all solutions are given for the equations

Preface

xiii

Chapter 24 - In this paper the authors prove local and global existence results for semilinear differential equations with deviated argument in Banach spaces. Chapter 25 - Under general conditions on the signed Radon measure µ the authors prove the existence of positive singular solutions for the problem Δu+µup=0, p >1 on bounded NTA domains in Rn,n ≥ 2. These results extend the recent ones proved by some authors to more general classes of potentials and domains. A new proof based on a simple fixed point argument is also given. Chapter 26 - The paper studies the so-called weighted pseudo almost periodic functions recently introduced by the author. Properties of those weighted pseudo almost periodic functions are discussed including a composition result of weighted pseudo almost periodic functions. The obtained results, subsequently, are utilized to study the existence and uniqueness of a weighted pseudo almost periodic solution to some non-autonomous abstract differential equations. Chapter 27 - The authors present a fixed point theorem for a sum of two convex-valued multi functions acting on a weakly compact, hyper convex subset of a normed space. The theorem is a multivalued version of a result of D. Bugajewski.

All the articles have been previously published in Journal of Evolution Equations, Volume 2 by Nova Science Publishers. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research.

In: Research on Evolution Equation Compendium. Volume 1 ISBN: 978-1-61209-404-5 Editor: Gaston M. N’Guerekata © 2009 Nova Science Publishers, Inc.

D OUBLE S CALE C ONVERGENCE AND H OMOGENIZATION OF Q UASILINEAR PARABOLIC E QUATIONS Xuming Xie∗ Department of Mathematics, Morgan State University 1700 E. Cold Spring Lane, Baltimore, MD 21251

Abstract We study the homogenization of the nonlinear parabolic equation x t (uε )t − div x (F(Dx uε , , 2 )) = f (t, x) ε ε where F(ξ, y) = F(ξ, y + m, s + k) for any integer vectors k, m ∈ Zn . We shall use double-scale convergence to derive the homogenized equations and prove a corrector result.

AMS Subject Classification: 35A35 Key Words: Nonlinear parabolic equations; Homogenization; Double-scale convergence; Monotonicity

1

Introduction

The behavior of inhomogeneous material, where the inhomogeneities are on a small scale, is of considerable interest in material sciences, mechanical engineering and in many technological applications, especially those involving composite materials. The theory of homogenization was created to model and predict the behavior of such material, when the inhomogeneities are on such a scale much smaller than the linear dimension of the system. In these situations, continuum models, in which rapid oscillation of the material properties is averaged, are sufficient to describe the behavior of the system. Moreover, they have the advantage of avoiding extensive numerical computations needed when the small scale behavior is taken into account. There have been numerous publications on homogenization theory and its applications in the past decades, we refer to the books by Bensoussan et al [2], Sanchez-Palencia [11] and Jikov et al [7]. For homogenization of nonlinear equations, a standard approach is using energy methods by Tartar and Murat; however, in some cases, it is difficult to work out energy method since the construction of test functions could be tricky. A two scale convergence method ∗ E-mail

address: [email protected]

2

Xuming Xie

was first introduced by Nguetseng [10], later elaborated by Allaire [1]; the advantage of this method is that the homogenized equations and proof of convergence are obtained in a single step. Recently Caffarelli [3] proposed an alternative approach in which the homogenization be considered as a viscosity process. Following the idea by Caffarelli, Garcia-Azorero et al [5] proved that the uniform limit of the solutions at scale ε to quasilinear parabolic equations is a viscosity solution to the homogenized equation. The use of viscosity solution methods was first considered by Evans [4] for divergence structure PDE. In this paper, we shall use the method of double-scale convergence to derive the homogerized equation of the quasilinear parabolic equations considered in [5]. Double-scale convergence was introduced in [6] and [8]. It is useful for homogenization of linear parabolic equations when the scale of time variable is different from that of space variable. Since it is a generalization of two scale convergence method, the double convergence method has the same advantage as two scale convergence method and is simpler than viscosity process used in [5]. We obtain that solutions to the microscopic equations doublescale converges to solutions of the same homogenized equations as in [5]; we also obtain a corrector result.

2

Formulation of the Problem

We consider the following nonlinear parabolic equation x t (uε )t − div x (F(Dx uε , , 2 )) = f (t, x) in U ε ε

(2.1)

where F(ξ, y, s) is a vector field in Rn × Rn × R satisfying the following conditions: (1) Periodicity: F(ξ, y, s) is 1-periodic in y and s, i.e., F(ξ, y + m, s + k) = F(ξ, y, s) for all m ∈ Zn , k ∈ Z. (2) Monotonicity: λ|ξ − η|2 ≤< F(ξ, y, s) − F(η, y, s), ξ − η >≤ Λ|ξ − η|2

(2.2)

for some positive constants λ and Λ which are independent of ε, for all ξ, η ∈ Rn and for a.e. y and s. (3) Growth: There exists C > 0 such that |F(ξ, y, s)| ≤ C|ξ|

(2.3)

where C is positive and independent of ε. (4) regularity: F(ξ, y, s) is continuous in ξ and measurable in y and s. Remark 2.1. Conditions (1) - (3) are the same as in [5], the regularity condition (4) is the well known Caratheodory condition, which is weaker than the regularity condition required in [5]. In [5], F(ξ, y, s) is required to satisfy Lipschitz condition in ξ for a.e (y, s) ∈ Rn × R.

Double Scale Convergence and Homogenization of Quasilinear ...

3

Lemma 2.2. For f (t, x) ∈ L∞ ([0, T ], L2 (U)), there exists a unique solution uε ∈ L2 ([0, T ], H01 (U))∩C([0, T ], L2 (U)), ∂t (uε ) ∈ L2 ([0, T ], H −1 (U)) to equation (2.1) with initial and boundary condition uε |t=0 = 0,

uε |∂U = 0.

(2.4)

Furthermore, we have following estimates kuε kL2 ([0,T ]×U) ≤ C,

k∇x ukL2 ([0,T ]×U) ≤ C,

x t kF(∇x uε , , 2 )kL2 ([0,T ]×U) ≤ C, ε ε

(2.5) (2.6)

where C is a positive constant and independent of ε. Proof. The existence and uniqueness and (2.5) follow from the standard Galerkin Method, see Theorem 30.A of [12] with evolution triple V = H01 (U), H = L2 (U),V ∗ = H −1 (U). (2.6) follows from (2.5) and the growth condition (2.3). Our goal in this paper is to investigate the behavior of the sequence uε (t, x) of solutions to (2.1) as ε → 0. We expect that uε (t, x) converges to a function u(t, x) in a double scale sense , which will be explained shortly. We are going to derive the effective equations that the function u(t, x) will satisfy. We are also going to estimate how fast that uε (t, x) will converge to u(t, x), i.e the so called corrector results.

3

Double Scale Convergence and the Main Theorem

To investigate the behavior of the sequence uε (t, x) as ε → 0, we are going to use the notion of double-scale convergence which is a generalization of two-scale convergence. It was introduced by Holmbom [6] and Goudon and Poupaud [8] to study linear parabolic equations. Let C#∞ (Y ) denote the space of those C∞ functions periodic on Y , where Y = [0, 1] × [0, 1]n . Let dZ = dt dx ds dy. Definition 3.1. {uε (t, x)} ⊂ L2 ([0, T ] × U) double-scale converges to u(t, x, s, y) ∈ L2 ([0, T ] × U ×Y ) iff for any φ(t, x, s, y) ∈ C∞ ([0, T ] × U,C#∞ (Y )), T

lim

ε→0 0

U

uε (t, x) · φ(t, x,

t x , )dxdt = ε2 ε

T 0

U Y

u(t, x, s, y) · φ(t, x, s, y)dZ.

Remark 3.2. The above definition uses C∞ test functions φ; however as pointed out in [1, 6], test function φ(t, x, s, y) could be any ”admissible” test function. Functions in spaces L2 ([0, T ] × U,C# (Y )), or L#2 (Y,C([0, T ] × U)) are admissible. Recall the space W21 ([0, T ], H01 (U), L2 (U)) = {u(t, x) : u ∈ L2 ([0, T ], H01 (U)), ∂t u ∈ The following compactness theorem ([6],Theorem 3.1) will be essential:

L2 ([0, T ], H −1 (U))}.

4

Xuming Xie

Theorem 3.3. (i) If {uε (t, x)} is a bounded sequence in L2 ([0, T ] × U)), then there exists u(t, x, s, y) ∈ L2 ([0, T ] × U, L#2 (Y )) such that a subsequence of {uε (t, x)} double-scale converges to u(t, x, s, y) in the sense of Definition 3.1. (ii) If {uε (t, x)} is a bounded sequence in W21 ([0, T ], H01 (U), L2 (U)), then there exist u(t, x) ∈ L2 ([0, T ], H 1 (U)) and u1 (t, x, s, y) ∈ L2 ([0, T ] × U, L2 ([0, 1], H#1 ([0, 1]n ))) such that a subsequence of {uε (t, x)} double-scale converges to u(t, x) and a subsequence of ∇x uε doublescale converges to ∇x u + ∇y u1 in the sense of Definition 3.1. Proof. See Theorem 3.1 in [6]. Remark 3.4. In [8], double-scale convergence is defined for a sequence of measures; a compact theorem is also obtained (see Proposition 4.4 in [8]). The lemma below is essential in the proof of the main homogenization theorem. Lemma 3.5. Assume that {uε } is a bounded sequence in W21 ([0, T ]; H01 (U), L2 (U)), and u(t, x) and u1 (t, x, s, y) are as in Theorem 3.3, then T

lim

ε→0 0

U

t x 1 ε (u (t, x) − u(t, x))ψ(t, x, 2 , )dt dx = ε ε ε

T 0

U Y

u1 (t, x, s, y)ψ(t, x, s, y)dZ

(3.1) for any ψ(t, x, s, y) = b(t)c(x)p(y)q(s), where b(t) ∈ C0∞ ([0, T ]), c(x) ∈ C0∞ (U), p(y) ∈ L#2 ([0, 1]n ), q(s) ∈ L#2 ([0, 1]). Proof. See Corollary 3.3 in [6]. Now we are ready to prove the following main theorem: Theorem 3.6. The solution sequence uε (t, x) as in Lemma 2.2 double-scale converges to a function u(t, x) in L2 ([0, T ], H01 (U)), and ∇x uε double-scale converges to 2 1 ∇x u(t, x) + ∇y u1 (t, x, s, y) where (u, u1 ) is the unique solution in L ([0, T ], H0 (U)) × 2 2 1 n L [0, T ] ×U; L ([0, 1], H# ([0, 1] )) of the homogenized problem ut − div x F(∇x u(t, x) + ∇y u1 , y, s)dyds = f in U, (3.2) Y

(u1 )s − div y F(∇x u(t, x) + ∇y u1 , y, s) = 0

in U ×Y,

(3.3)

u|t=0 = 0, u|∂U = 0

(3.4)

u1 (t, x, s, y) Y-periodic in (s, y).

(3.5)

Proof. The varational formulation of (2.1) is following T U

uε (T, x)φ(T, x)dx −

0

U

uε ∂t φ dx dt T

+ 0

for φ(t, x) ∈ W21 ([0, T ], H01 (U), L2 (U)).

x t F(∇uε , , 2 ) · ∇φ dx dt = ε ε U

T 0

U

f φ dx dt (3.6)


5

From Lemma 2.2 and Theorem 3.3, up to a subsequence, there are u(t, x), u1 (t, x, s, y) and F0 (t, x, y, s) such that uε (t, x) double-scale converges to u(t, x) ∇x uε (t, x) double-scale converges to ∇x u(t, x) + ∇y u1 (t, x, s, y) x t F(∇x uε , , 2 ) double-scale converges to F0 (t, x, y, s). ε ε

(3.7)

Let x t φ(t, x) = b(t)ψ1 (x) + εb(t)φ1 (x, , 2 ), ε ε

(3.8)

where φ1 (x, y, s) = c(x)p(y)q(s), b(t) ∈ C0∞ ([0, T ]), ψ1 (x), c(x) ∈ H01 (U)), p(y) ∈ C#∞ ([0, 1]n ), q(s) ∈ C#∞ ([0, 1]). Plugging (3.8) into (3.6) T

x t uε ∂t [b(t)ψ1 (x) + εb(t)φ1 (x, , 2 )]dxdt ε ε U T x t x t F(∇uε , , 2 ) · [b(t)∇x ψ1 + b(t)∇y φ1 (x, , 2 )]dx dt + ε ε ε ε U 0 T x t f [b(t)ψ1 (x) + εb(t)φ1 (x, , 2 )]dx dt. (3.9) = ε ε 0 U

− 0

Note T

x t uε ∂t [εb(t)φ1 (x, , 2 )]dxdt ε ε U 0 T x t 1 uε b(t)(φ1 )s (x, , 2 )dxdt + = ε ε U ε 0 T x t 1 (uε − u)b(t)(φ1 )s (x, , 2 )dxdt + = ε ε U ε 0

T

= 0

U

T

x t εuε b0 (t)φ1 (x, , 2 )dxdt ε ε U 0 T x t 1 u(x,t)b(t)(φ1 )s (x, , 2 )dxdt ε ε U ε 0 T x t + εuε b0 (t)φ1 (x, , 2 )dxdt ε ε 0 U T x t 1 x t (uε − u)b(t)(φ1 )s (x, , 2 )dxdt + ε∂t [b(t)u(x,t)]φ1 (x, , 2 )dxdt ε ε ε ε ε U 0 T x t εuε b0 (t)φ1 (x, , 2 )dxdt. (3.10) + ε ε U 0

Taking limit ε → 0 and using Lemma 3.5, we have T

lim

ε→0 0

T

x t uε ∂t [εb(t)φ1 (x, , 2 )]dxdt = lim ε→0 ε ε U

U

0

1 x t (uε − u)b(t)(φ1 )s (x, , 2 )dxdt ε ε ε

T

= 0

U

u1 (x,t, y, s)b(t)(φ1 )s (x, y, s)dZ. (3.11)

6

Xuming Xie Taking double-scale limit in (3.9) and using (3.11), we have T

T

− 0 T

U

u∂t bψ1 (x) −

+ U Y

0

0

U

u1 (x,t, y, s)b(t)(φ1 )s (x, y, s)dZ T

F0 (t, x, y, s) · (b(t)∇ψ1 + b(t)∇y φ1 (t, x, y, s))dZ =

0

U

f b(t)ψ1 (t, x)dtdx. (3.12)

In above equation, let φ1 = 0 , we obtain: T

T

− 0

U

u∂t bψ1 (x) +

0

U Y

T

b(t)F0 (t, x, y, s) · ∇ψ1 dZ =

0

U

f b(t)ψ1 (t, x)dtdx. (3.13)

Letting ψ1 = 0 in (3.12), we have T

T

− 0

U

u1 (x,t, y, s)b(t)(φ1 )s (x, y, s)dZ +

U Y

0

b(t)F0 (t, x, y, s) · ∇y φ1 (x, y, s)dZ = 0.

(3.14) Now the question is to identify F0 (t, x, y, s) in terms of u, u1 and F. To this end, for τ > 0 , Φ(t, x, y) ∈ C0∞ (U ×Y × [0, T ]) and functions uk1 (t, x, s, y) ∈ C0∞ (U ×Y × [0, T ]) such that uk1 (t, x, s, y) → u1 (t, x, s, y) strongly in L2 ([0, T ] ×U, L2 ([0, 1], H#1 ([0, 1]n ))), let

(3.15)

x t x vkε (t, x) = ∇x [u(t, x) + εuk1 (t, x, , 2 )] + τΦ(t, x, ). ε ε ε Then vkε (t, x) double-scale converges to ∇x u(t, x) + ∇y uk1 (t, x, y, s) + τΦ(t, x, y) ≡ vk (t, x, y). (3.16) From monotonicity condition T 0

h x t i x t F(∇uε , , 2 ) − F(vkε , , 2 ) · (∇x uε − vkε ) ≥ 0. ε ε ε ε U

(3.17)

Using equation (3.6) with φ = uε , T 0

1 x t F(∇uε , , 2 ) · ∇uε dxdt = − ε ε 2 U

U

u2ε (T )dx +

T 0

U

f uε dx dt.

(3.18)

Combining (3.17) and (3.18), −

1 2

U

T

− 0

T

T

x t F(vkε , , 2 ) · ∇uε dxdt ε ε U U 0 0 T x t x t F(∇uε , , 2 ) · vkε dxdt + F(vkε , , 2 ) · vkε dxdt ≥ 0. (3.19) ε ε ε ε U U 0

u2ε (T, x) dx +

f uε dxdt −

Since uε (T, x) , up to a subsequence, weakly converges to u(T, x) in L2 (U), it follows that lim

ε→0 U

u2ε (T, x) dx ≥

u2 (T, x) dx. U

(3.20)


7

Taking the double scale limit in (3.19) and using (3.20), we have −

1 2

T

u2 (T, x) dx +

T

f u dxdt − 0

U

− 0

0

U

T U Y

U Y

F(vk , y, s) · [∇x u + ∇y u1 ] dZ T

F0 (t, x, y, s) · vk dZ +

0

F(vk , y, s) · vk dZ ≥ 0. (3.21) U Y

By growth, regularity conditions and Lebesgue’s dominated convergence theorem, and (3.15) T

lim

k→∞ 0

U Y

F(vk , y, s) · [∇x u + ∇y u1 ]dZ T

= 0

T

F(∇x u + ∇y u1 + τΦ, y, s) · [∇x u + ∇y u1 ] dZ, (3.22)

F(vk , y, s) · vk dZ

lim

k→∞ 0

U Y

U Y T

= 0

T

lim

k→∞ 0

U Y

U Y

F(∇u + ∇y u1 + τΦ, y, s) · [∇u + ∇y u1 + sΦ] dZ, (3.23)

F0 (t, x, y, s) · vk dZ T

= U Y

0

F0 (t, x, y) · [∇u + ∇y u1 + τΦ] dZ. (3.24)

Leting k → ∞ in (3.21) and using (3.22)-(3.24), we have 1 − 2

T

2

T

u (T )dx +

f u dxdt − 0

U

0

U T

− 0

+τ

T U Y

0

U Y

U Y

F0 (t, x, y, s) · ∇x u dZ

F0 (t, x, y, s) · ∇y u1 dZ

[F(∇u + ∇y u1 + τΦ, y, s) − F0 (t, x, y, s)] · Φ dZ ≥ 0. (3.25)

Replacing φ in (3.6) by u(t, x), we have T

− U

uε (T, x) u(T, x)dx +

0

U

T

uε ∂t u dxdt + T

− 0

f u dxdt 0

U

x t F(∇uε , , 2 ) · ∇x u dydxdt = 0. (3.26) ε ε U

Taking the double-scale limit in the above T

−

T

u(T, x)u(T, x)dx+ U

0

U

u∂t u dxdt +

T

f udxdt − 0

U

0

U Y

F0 (t, x, y, s)·∇x udZ = 0, (3.27)

8

Xuming Xie

which is equivalent to −

1 2

T

u2 (T )dx +

T

f u dxdt −

U

0

U

0

U Y

F0 (t, x, y, s) · ∇x u dZ = 0.

(3.28)

Consider a sequence of functions of b(t)φ1 (x, y, s) that converges strongly to u1 (t, x, s, y) in L2 ([0, T ] ×U, L2 ([0, 1], H#1 ([0, 1]n ))). Passing limit in (3.14), we have T U Y

0

F0 (t, x, y, s) · ∇y u1 dZ =

T

u1 ∂s u1 dZ = 0.

(3.29)

[F(∇u + ∇y u1 + τΦ, y, s) − F0 (t, x, y, s)] · Φ dZ ≥ 0.

(3.30)

0

U Y n+1

From (3.28) and above , (3.25) becomes T 0

U Y

Letting τ → 0 we have T 0

U Y

[F(∇u + ∇y u1 , y, s) − F0 (t, x, y, s)] · Φ dZ ≥ 0,

(3.31)

F0 (t, x, y, s) = F(∇x u + ∇y u1 , y, s).

(3.32)

which implies

Finally, (3.2) follows from (3.13) and (3.32); and (3.3) follows from (3.14) and (3.32).

Remark 3.7. Our homogenized equation (3.2) is the same as (II.7) in [5], (3.3) is the same as (II.3) in [5]. Theorem 3.8. Assume that u1 (t, x, s, y) ∈ L2 ([0, T ] ×U,C#1 (Y )), then lim kuε (t, x) − u(t, x) − εu1 (x,t,

ε→0

t x , )k 2 = 0. 1 ε2 ε L ([0,T ],H (U))

Proof. We consider h x t i vε = ∇x u(t, x) + εu1 (x,t, , 2 ) , ε ε which double-scale converges to ∇u + ∇y u1 . By monotonicity condition: T 0

h x t x t i F(∇uε , , 2 ) − F(vε , , 2 ) · (∇uε − vε )dtdx ≥ λ ε ε ε ε U

T 0

U

|∇uε − vε |2 dxdt. (3.33)

As in the proof ofTheorem 3.6 , the left hand side of (3.33) goes to zero, which implies n that the sequence ∇x uε − u(t, x) − εu1 (x,t, xε , εt2 ) converges to zero in L2 ([0, T ] ×U) .


9

References [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23.6 (1992) 1482-1518. [2] A. Bensoussan, J.L. Lions, G. Papanicolau. Asymptotic Analysis for Periodic Structures. (1978) Amsterdam, North Holland. [3] Caffarelli, L. A. A note on nonlinear homogenization. Comm. Pure. Appl. Math. LII (1999) 829-838. [4] L.C. Evans. Periodic homogenization of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinbeugh sect. A, 120 (1992) 245-265. [5] J. Garcia-Azorero, C.E. Gutierrez, I. Peral, Homogenization of quasilinear parabolic equations in periodic media, Comm. Partial Diff. Equa. vol 28 (2003) 1887-1910. [6] A. Holmbom, Homogenization of parabolic equations an alternative approach and some corrector-type results. Applications of Mathematics 42(1997) 321-342. [7] V. Jikov, S. Koslov, O. Oleinik, Homogenization of Differential Operators and Integral Functions, (1994) Berlin, Springer. [8] T. Goudon, F. Poupaud, Homogenization of transport equations: weak mean field approximation. SIAM J Math. Anal. 36(2004) 856-881. [9] F. Murat, L. Tartar. H-convergence, topics in the mathematical modeling of composite materials, L. Cherkaev, R.V. Kohn ed., Progress in nonlinear differential equations and their applications, Birkhauser, Boston, 1998, 21-43. [10] G, Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989) 608-623. [11] E. Sanchez-Palencia, Non-homogenous Media and Vibration theory. Lecture notes in physics 127 Berlin, Springer. [12] E. Zeidler, Nonlinear Functional Analysis and Its Application, IIB (1990), Berlin, Springer. [13] X. Xie, A note on homogenization of quasilinear parabolic equations, preprint, 2004.


S ECOND G RADE F LUIDS WITH E NHANCED V ISCOSITY AS DYNAMICAL S YSTEMS Ahmed S. Bonfoh∗ The Abdus Salam International Centre for Theoretical Physics, Mathematics Section, Strada Costiera 11, 34014 Trieste, Italy

Abstract We consider a modified 2d second grade incompressible fluid with a higher order viscosity term, and show that this nonlinear evolution system possesses an exponential attractor. This object gives a more precise description of the asymptotic behaviour of dynamical systems than the global attractor.

Keywords: modified second grade fluids, global attractor, exponential attractors. 2000 AMS MS Classifications: 35A05, 35B45, 35B40

1

Introduction

The study of the longtime behaviour of solutions to nonlinear evolution equations arising from mechanics and physics is a capital issue, as it is important, to understand and predict their evolution. Many dissipative systems possess a global attractor, which is a compact set lying in the phase space, and which attracts uniformly the trajectories starting from bounded sets when time goes to infinity. In other words, the global attractor is the unique largest bounded invariant set which embodies all the permanent regimes corresponding to all possible initial data. However, the global attractor may present some major defaults for practical purposes. Indeed, the rate of attraction of the trajectories may be small and very sensitive to perturbations. In order to give a more precise description of the longtime behaviour of such systems, C. Foias et al. have introduced a new object called an exponential attractor. This object is a compact and positively invariant set which contains the global attractor, has finite fractal dimension and attracts all the trajectories starting from bounded sets at a uniform exponential rate. Moreover, exponential attractors are more robust than global attractors with respect to approximations and perturbations (see [6]). In this paper, we consider a modified 2d fluid of second grade and investigate the existence of exponential attractors. ∗ E-mail

address: [email protected], Phone: +390402240220, Fax: +39040224163

12

2

Ahmed S. Bonfoh

Setting of the Problem

We consider a modified second grade incompressible fluid in two-dimensional space by adding the higher order viscosity term µ∆2 u, that is, ∂ 2 ∂t (u − α∆u) + µ∆ u − ν∆u + curl (u − α∆u) × u

div u = 0, u|∂Ω = 0, u|t=0 = u0 .

= f + ∇p, (2.1)

The function u = u(x,t) is the velocity, p = p(x,t) is the pressure, the density of the fluid is supposed to be ρ = 1, f = f (x) is the given external body force, and the parameters α, µ and ν are given strictly positive constants (ν is the kinematic viscosity). The case µ = 0 corresponds to the usual fluids of second grade, which has been studied in [3], [4], [5] and in the references therein. For both cases µ = 0 and α = 0, (2.1) reduces to the Navier-Stokes equations. Let us remark that a superviscosity term is often added for the numerical solution of the turbulent Navier-Stokes equations (see [7]). We assume that the fluid fills a simply-connected bounded open set Ω, with smooth and connected boundary ∂Ω of class C 4,1 . We denote by n the unit normal vector to ∂Ω, directed outside Ω. We denote by k.k and (., .) the usual norm and scalar product in L2 (Ω) (and also in 2 L (Ω)2 ). We define the following functional sets: V = {u ∈ C 0∞ (Ω)2 , div u = 0}, H = the closure of V in L2 (Ω)2 , and V = the closure of V in H01 (Ω)2 . V is a Hilbert space with the scalar product (u, v)V = (u, v) + α(∇u, ∇v). We also define the Hilbert space W = {u ∈ V, curl(u − α∆u) ∈ L2 (Ω)} endowed with the scalar product (u, v)W = (u, v)V + (curl(u − α∆u), curl(v − α∆v)). Note that, in two spatial dimension, the operator curl is a scalar. ∂u1 ∂u2 ∂u1 2 That is, for u = (u1 , u2 ), we have curl u = ∂u ∂x1 − ∂x2 and ×u = (− ∂x1 , ∂x2 ). It has been shown in [3] that W = {u ∈ [H 3 (Ω) ∩ H01 (Ω)]2 , div u = 0} and also the norms kqkH 3 (Ω)2 and k curl(q − α∆q)k are equivalent on W . Throughout this paper, the same letter c (and sometimes ci , i = 0, 1, 2, ...) denotes positive constants which may change from line to line.

3

Existence of the Global Attractor

The existence and uniqueness of solutions of problem (2.1) with µ = 0 have been investigated, for instance, in [3] and [4]. These results allowed I. Moise et al. to consider the semigroup {S(t)}t≥0 of operators S(t) : W → W, u0 7→ u(t), u(t) being the solution at time t of (2.1) with µ = 0. They proved that this semigroup is asymptotically compact (it is not compact), weakly continuous and possesses the global attractor in W via energy equations (cf. [9]). Now, when µ > 0, we can show that the semigroup {S(t)}t≥0 generated by (2.1) is uniformly compact, continuous, Lipschitz on bounded sets of H 4 (Ω)2 , and possesses the global attractor in W , which is bounded in H 4 (Ω)2 if f ∈ H 2 (Ω)2 (see, e.g., [1] and [10] for more details about the theory of global attractors). We state the following results.

Second Grade Fluids with Enhanced Viscosity as Dynamical Systems

13

Theorem 3.1 If u0 ∈ W and f ∈ H 1 (Ω)2 , then there exists a unique solution u to problem ∞ + (2.1) satisfying u ∈ L∞ (R+ ;W )∩L2 (0, T ;W ∩H 4 (Ω)2 ), with ∂u ∂t ∈ L (R ;V ), for any T > 0. Theorem 3.2 We assume that f ∈ H 1 (Ω)2 . Then, the semigroup {S(t)}t≥0 possesses the global attractor A in W . Furthermore, if f ∈ H 2 (Ω)2 , then the attractor A is bounded in H 4 (Ω)2 . The proofs of Theorem 3.1 and 3.2 are based on showing, firstly, that curl(u − α∆u) ∈ L∞ (R+ ; L2 (Ω)) ∩ L2 (0, T ; H 1 (Ω)), and then using the equivalence of norms. We can note that the norms k curl(u − α∆u)kH 1 (Ω) and kukH 4 (Ω)2 are also equivalent on W ∩ H 4 (Ω)2 . Here, a new property is the boundedness of the attractor A in H 4 (Ω)2 . This result comes from the existence of a bounded absorbing set B2 for {S(t)}t≥0 in H 4 (Ω)2 . This is a consequence of showing that, for any u satisfying Theorem 3.1, curl(u − α∆u) belongs to a bounded absorbing set of H 1 (Ω), for all time t ≥ t2 , for some t2 > 0. Let us give a proof. We set b = curl(u−α∆u) and remark that curl(b×u) = u.∇b, for all u ∈ W ∩H 4 (Ω)2 . If u is the solution of (2.1) satisfying Theorem 3.1, then b is the unique function in L∞ (R+ ; L2 (Ω)) ∩L2 (0, T ; H 1 (Ω)), for all T > 0, solution of the following system (see [4]): ∂b ∂t

− αµ ∆b + u.∇b + ( αµ − ν)∆ curl u = curl f , b|t=0 = curl(u0 − α∆u0 ).

(3.1)

We consider a special basis {w j } j≥1 of H 1 (Ω), which is orthogonal in L2 (Ω), where w j , j ∈ N? , is the solution of the homogeneous Neumann problem: w j − αµ ∆w j = λ j w j , ∂w j ∂n = 0,

(3.2)

and where 0 < λ1 ≤ λ2 ≤ λ3 ..., λ j → +∞ for j → +∞, {λ j } j≥1 being the family of eigenvalues associated with the family of eigenfunctions {w j } j≥1 . We can note that w j is also the solution of the following variational problem: (w j , v) + αµ (∇w j , ∇v) = λ j (w j , v), ∀v ∈ H 1 (Ω).

(3.3)

Since ∂Ω is regular enough, w j belongs to H 2 (Ω), and, the norms kw − αµ ∆wk and kwkH 2 (Ω) are equivalent on H 2 (Ω). We now implement a Galerkin method with the basis {w j } j≥1 . Noting that (u.∇b, b) = 0 (see [3]), we formally obtain µ µ µ 1 d 2 2 2 2 dt (kbk + α k∇bk ) + kb − α ∆bk + α (∇(u.∇b), ∇b) +( αµ − ν)[ αµ (∇∆ curl u, ∇b) + (∆ curl u, b)] = kbk2 + αµ k∇bk2 + αµ (∇ curl f , ∇b) + (curl f , b).

(3.4)

µ µ d 2 2 2 2 2 dt (kbk + α k∇bk ) + c1 kbkH 2 (Ω) ≤ c2 (kbk + α k∇bk ) +τk∇uk2L∞ (Ω) kbk2H 2 (Ω) + c3 kuk2H 4 (Ω)2 + c4 k f k2H 2 (Ω)2 ,

(3.5)

Therefore,

14

Ahmed S. Bonfoh

for any τ > 0. If u0 ∈ W and f ∈ H 1 (Ω)2 , then u satisfies Theorem 3.1, and ku(t)kL∞ (Ω) ≤ c1 k∇u(t)kL∞ (Ω) ≤ c2 ku(t)kW ≤ c, ∀t ≥ 0. Using this latter fact together with a proper choice of τ, equation (3.5) reduces to µ d 2 2 2 dt (kbk + α k∇bk ) + c1 kbkH 2 (Ω) ≤ c2 (kbk2 + αµ k∇bk2 ) + c3 k f k2H 2 (Ω)2 .

(3.6)

If f ∈ H 2 (Ω)2 , then we deduce the existence of a bounded absorbing set B2 for {S(t)}t≥0 in H 4 (Ω)2 , by applying the uniform Gronwall lemma to (3.6), and using the equivalence of norms.

4

Existence of Exponential Attractors

Let E be a metric space, X a compact subset of E and consider a continuous semigroup {S(t)}t≥0 , on E, mapping X into X. We recall the definition of an exponential attractor for {S(t)}t≥0 . Definition 4.1 A compact set M is called an exponential attractor for {S(t)}t≥0 for the topology of E if: - M contains the global attractor; that is, A ⊂ M ; - M is positively invariant under S(t); that is, S(t)M ⊂ M , ∀t ≥ 0; - the fractal dimension of M is finite; - there exists a constant c0 > 0 such that, for every bounded subset B ⊂ X, there exists a constant c1 (B) > 0 such that distE (S(t)B, M ) ≤ c1 e−c0t , ∀t ≥ 0, where distE is the Hausdorff semi-distance with respect to the metric of E: distE (A, B) = sup inf ka − bkE . a∈A b∈B

Sufficient conditions ensuring the existence of exponential attractors in Hilbert spaces are given in [6]. It depends on a dichotomy principle called the squeezing property. Definition 4.2 The semigroup {S(t)}t≥0 verifies the squeezing property on X if, for a real number η belonging to [0, 41 [, there exists a projection PN ? : E → E, with finite rank N ? (η), and a time t ? such that ∀ (φ, ψ) ∈ X 2 , if

k(I − PN ? )(S(t ? )φ − S(t ? )ψ)kE ≥ kPN ? (S(t ? )φ − S(t ? )ψ)kE , then kS(t ? )φ − S(t ? )ψkE ≤ ηkφ − ψkE .

(4.1)

Proposition 4.1 If {S(t)}t≥0 satisfies the squeezing property on X and if S(t ? ) is Lipschitz on X with Lipschitz constant L, then there exists an exponential attractor M for {S(t)}t≥0 on X such that the fractal dimension of M is bounded as follows: ) . dF (M ) ≤ N ? max(1, ln(16L+1) ln 2

(4.2)


15

In the order to verify the squeezing property, Babin and Nicolaenko introduced in [2] a method based on a decomposition of the difference of two trajectories. A consequence of their result is stated as follows (see [8]). Proposition 4.2 Let E and V be two Hilbert spaces such that the inclusion V ⊂ E is compact. Let X ⊂ E be a closed set and S(t) : X → X be a semigroup. Let us furthermore assume that there exists a projection PN ? with finite rank N ? such that k(I − PN ? )ykE ≤ c(N ? )kykV , ∀y ∈ V , where c(N ? ) → 0 as N ? → +∞. If there exists ϕ1 , ϕ2 satisfying S(t)φ − S(t)ψ = ϕ1 (t) + ϕ2 (t) and kϕ1 (t)k2E ≤ d(t)kφ − ψk2E , kϕ2 (t)kV2 ≤ h(t)kφ − ψk2E , ∀φ, ψ ∈ X, where d(t) is continuous and satisfies limt→+∞ d(t) = 0 and h(t) is continuous, then {S(t)}t≥0 enjoys the squeezing property on X for the topology of E. We now want to apply this result to show the existence of an exponential attractor on the W set X = t≥t1 S(t)B2 , where B2 is a bounded absorbing set for {S(t)}t≥0 in H 4 (Ω)2 and t1 is such that S(t)B2 ⊂ B2 , ∀t ≥ t1 . The set X is compact and positively invariant by S(t). Let u1 and u2 be two solutions of (2.1) satisfying Theorem 3.1. We then write u = u1 − u2 as sum of two functions u = u1 + u2 , where u1 and u2 are the solutions of the following problems: µ ∂ 1 1 1 1 ∂t (u − α∆u ) − α ∆(u − α∆u ) div u1 = 0,

= 0,

u1 |∂Ω = 0, u1 |t=0 = u0 ,

(4.3)

and µ ∂ 2 2 2 2 ∂t (u − α∆u ) − α ∆(u − α∆u ) + curl(u1 − α∆u1 ) × u1 −curl(u2 − α∆u2 ) × u2 + ( αµ − ν)∆u = 0, div u2 = 0,

(4.4)

u2 |∂Ω = 0, u2 |t=0 = 0, respectively. Similarly to the decomposition of u, we consider the difference b = b1 − b2 of the two corresponding solutions of (3.1) and write b = b1 + b2 , where µ ∂b1 1 1 ∂t + b − α ∆b = 0, b1 |t=0 = curl(u0 − α∆u0 ),

(4.5)

and ∂b2 ∂t

+ b2 − αµ ∆b2 + u1 .∇b1 − u2 .∇b2 + ( αµ − ν)∆ curl u − b = 0, b2 |t=0 = 0.

(4.6)

We formally take the L2 −scalar product of (4.5) with b1 , and obtain 2µ d 1 2 1 2 1 2 dt kb k + 2kb k + α k∇b k

= 0.

(4.7)

Therefore, d 1 2 1 2 dt kb k + ckb k

≤ 0,

(4.8)

16

Ahmed S. Bonfoh

and kb1 (t)k ≤ c1 e−ct kb0 k, 0t kb1 k2H 1 (Ω) dτ ≤ ckb0 k2 , ∀t ≥ 0.

(4.9)

Finally, we deduce from the equivalence of norms previously noticed that 2 , ∀t ≥ 0. ku1 (t)kW ≤ c1 e−ct ku0 kW , 0t ku1 k2H 4 (Ω)2 dτ ≤ c2 ku0 kW

(4.10)

We now note that u1 .∇b1 − u2 .∇b2 = u1 .∇b + u.∇b2 , and equation (4.6) becomes ∂b2 ∂t

+ b2 − αµ ∆b2 + u1 .∇b + u.∇b2 + ( αµ − ν)∆ curl u − b = 0, b2 |t=0 = 0.

(4.11)

Still working with the basis {w j } j≥1 considered in Section 3, we obtain, by proceeding as for (3.4), µ µ µ 1 d 2 2 2 2 2 2 2 2 2 dt (kb k + α k∇b k ) + kb − α ∆b k + α (∇(u1 .∇b), ∇b ) µ 2 2 2 +(u1 .∇b, b ) + α (∇(u.∇b2 ), ∇b ) + (u.∇b2 , b ) +( αµ − ν)[ αµ (∇∆ curl u, ∇b2 ) + (∆ curl u, b2 )] = αµ (∇b, ∇b2 ) + (b, b2 ).

(4.12)

µ µ d 2 2 2 2 2 2 2 2 2 2 dt (kb k + α k∇b k ) + c1 kb kH 2 (Ω) ≤ c2 (kb k + α k∇b k ) +σ1 (ku1 k2L∞ (Ω) + k∇u1 k2L∞ (Ω) )kbk2H 2 (Ω) +σ2 (kuk2L∞ (Ω) + k∇uk2L∞ (Ω) )kb2 k2H 2 (Ω) +σ3 kuk2H 4 (Ω)2 + σ4 (k∇bk2 + kbk2 ),

(4.13)

Therefore,

for any σ1 , σ2 , σ3 , σ4 > 0. We note that kuk2H 4 (Ω)2 ≤ ckbk2H 1 (Ω) , and kbkY2 ≤ kb1 kY2 + kb2 kY2 , where Y = H 1 (Ω) or H 2 (Ω) . According to the results in Section 3, if u01 , u02 ∈ X, then u1 (t), u2 (t) ∈ X, ∀t ≥ 0. We recall that X is a bounded set of H 4 (Ω)2 . This is equivalent to the fact that b1 (t) and b2 (t) belong to a bounded set of H 1 (Ω). We also have 0t kbi k2H 2 (Ω) dt ≤ c(t), i = 1, 2, for all t ≥ 0. Using these boundedness and choosing properly constants σ1 , σ2 , σ3 and σ4 , we get µ d 2 2 2 2 2 2 dt (kb k + α k∇b k ) + c1 kb kH 2 (Ω) ≤ c2 (1 + kb2 k2H 2 (Ω) )(kb2 k2 + αµ k∇b2 k2 ) +c3 kb1 k2 kb2 k2H 2 (Ω) + c4 kb1 k2H 2 (Ω) .

(4.14)

Finally, we obtain by applying the Gronwall lemma that c2

t (1+kb2 k2

H 2 (Ω) kb2 (t)k2H 1 (Ω) ≤ c1 e 0 + 0t kb1 k2H 2 (Ω) ds],

)dτ

[ 0t kb1 k2 kb2 k2H 2 (Ω) ds ∀t > 0.

(4.15)

Noting (4.9) and the equivalence of norms, we get c2

ku2 (t)kH 4 (Ω)2 ≤ c1 e

0t (1+kb2 k2H 2 (Ω) )dτ ku k , ∀t > 0. 0 W

(4.16)

Both estimates (4.10) and (4.16) allow us to say that {S(t)}t≥0 enjoys the squeezing property on X. Moreover, S(t) is Lipschitz on X; and we consider PN ∗ as the projection on the space generated by the N ∗ first eigenfunctions of the operator −∆ on V ∩ H 2 (Ω)2 . We then have the following result. Theorem 4.1 The semigroup {S(t)}t≥0 possesses an exponential attractor M on X.


17

Acknowledgments I am currently an Abdus Salam International Centre for Theoretical Physics Post-Doctoral Research Fellow, and I would like to thank the Centre for all its support.

References [1] A.V. Babin and M.I. Vishik, Attractors of evolution equations, North-Holland, Amsterdam, London, New York, Tokyo, 1992. [2] A. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dyn. Diff. Equ. 7 (1995), 567-589. [3] D. Cioranescu and E.H. Ouazar, Existence and uniqueness for fluids of second grade, Nonlin. Part. Diff. Equ., Collège de France, Pitman 109 (1984), 178-197. [4] D. Cioranescu and V. Girault, Weak and classical solutions of a family of second grade fluids, Int. J. Non-Linear Mechanics 32(2) (1997), 317-335. [5] J.E. Dunn and R.L. Fosdick, Thermodynamics, stability and boundedness of fluids of complexity two and fluids of second grade, Arch. Rat. Mech. Anal. 56(3) (1974), 191-252. [6] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Masson, Paris, 1994. [7] J.L. Lions, Quelques Méthodes de Résolutions des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [8] A. Miranville, Exponential attractors for a class of evolution equations by a decomposition method, C. R. Acad. Sci. Paris, 328(1) (1999), 145-150. [9] I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity 11(7) (1998), 1369-1393. [10] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, (2nd Edition) Springer-Verlag, Berlin, Heidelberg, New York, 1997.


P ERIODIC S OLUTIONS OF S OME E VOLUTION E QUATIONS WITH I NFINITE D ELAY∗ Khalil Ezzinbi 1†and James H. Liu 2‡ 1 Facult´ e des Sciences Semlalia, D e´ partement de Mathématiques, B.P. 2390, Marrakesh, Morocco 2 Department of Mathematics and Statistics James Madison University, Harrisonburg, VA 22807, USA

Abstract In this work, we study the existence of periodic solutions for some partial functional differential equations with infinite delay. We assume that the linear part is not necessarily densely defined and satisfies the Hille-Yosida condition, and the phase space is chosen to be Cg for some decreasing function g from (−∞, 0] to [1, ∞). We also present a related Massera type result, namely the existence of a bounded solution on R+ implies the existence of a periodic solution.

Keywords and phrases : Periodic solutions, infinite delay, Kuratowski’s measure of noncompactness, condensing map.

1 Introduction The purpose of this work is to study the existence of periodic solutions for the following partial functional differential equation with infinite delay 0 u (t) = Au(t) + f (t, ut ), t > 0, (1.1) u(s) = φ(s), s ≤ 0, where A is an unbounded linear operator on a Banach space X. Here, the operator A is not necessarily densely defined on X, and satisfies the Hille-Yosida condition: there exist M0 ≥ 1 and ω0 ∈ R such that (ω0 , +∞) ⊂ ρ(A) and |R (λ, A)n | ≤

M0 , for n ∈ N and λ > ω0 , (λ − ω0 )n

where ρ(A) is the resolvent set of A and R (λ, A) = (λ − A)−1 . The initial data φ belongs to some phase space Cg to be defined later. ∗ This

research is supported by a TWAS Research Grant under contract: No 03-030 RG/MATHS/AF/AC address: [email protected] ‡ E-mail address: [email protected] † E-mail

20

Khalil Ezzinbi and James H. Liu

The function f : R+ ×Cg → X is continuous and T –periodic in t. As usual, the history function ut ∈ Cg is defined by ut (s) = u(t + s), s ≤ 0. Periodic solutions are derived for Eq.(1.1) by [5] in a general phase space with axioms when A is densely defined; by [6, 7] in Cg when A is time dependent and densely defined; and by [4] in a general phase space with axioms when A is non-densely defined. In this work we use the techniques developed in [4, 5, 6, 7] to present an extension of [4, 5, 6, 7] by studying Eq.(1.1) in Cg when A is non-densely defined. This work is organized as follows: In Section 2, we recall some results about the existence, uniqueness, and estimations of solutions. In Section 3, we prove that the Poincaré map in condensing, which is then used to derive the existence of periodic solutions by using certain boundedness of solutions. In section 4, we prove a Massera type theorem when f is linear with respect to the second argument, showing that the existence of a bounded solution on IR+ is sufficient for the existence of periodic solutions.

2

Solutions and Their Estimations

First, we make the following assumptions: (H1 ) A satisfies the Hille-Yosida condition. (H2 ) f is continuous, and Lipschitzian with respect to the second argument. Lemma 2.1. [3] Let A0 be the part of the operator A in D(A), which is defined by D(A0 ) = {x ∈ D(A) : Ax ∈ D(A)}, A0 x = Ax. Then A0 generates a C0 -semigroup (T0 (t))t≥0 on D(A). Now, we define the phase space Cg for Eq.(1.1) as in [6]. Lemma 2.2. [6] There exists an integer K0 > 1 such that 1 ( )K0 −1 M0 < 1, 2 where M0 = supt∈[0,T ] |T0 (t)|. Next, let w0 = KT0 , then there exists a function g on (−∞, 0] such that g(0) = 1, g(−∞) = ∞, g is decreasing on (−∞, 0], and for d ≥ w0 one has sup s≤0

1 g(s) ≤ . g(s − d) 2

For the function g given in the above lemma, we define the continuous functions space Cg n o |φ(θ)| Cg = φ : φ ∈ C((−∞, 0]; X) and lim =0 . θ→−∞ g(θ)

Periodic Solutions of Some Evolution Equations with Infinite Delay

21

If we provide Cg with the norm |φ|g = sup s≤0

|φ(s)| , φ ∈ Cg , g(s)

then Cg is a Banach space. Next, we recall some results concerning existence and uniqueness of integral solutions for Eq.(1.1). Definition 2.3. [1] Let φ ∈ Cg . A function u : R → X is called an integral solution of Eq.(1.1) if the following conditions hold: (i) u is continuous on [0, ∞), (ii) u0 = φ, (iii) 0t u (s) ds ∈ D (A), for t ≥ 0, (iv) u (t) = φ (0) + A 0t u (s) ds + 0t f (s, us )ds, for t ≥ 0. Theorem 2.4. [2] Assume that (H1 ) and (H2 ) hold. Then, for all φ ∈ Cg such that φ (0) ∈ D (A), Eq.(1.1) has a unique integral solution u. Moreover, let u and v be two integral solutions of Eq.(1.1) on (−∞, L], L > 0. Then for t ∈ [0, L], |ut − vt |g ≤ M1 |u0 − v0 |g ek1t , where M1 and k1 are some positive constants. The integral solution of Eq.(1.1) is given by the following variation of constants formula  t  T0 (t)φ (0) + lim T0 (t − s) Bλ f (s, us )ds , t ≥ 0, u (t) = λ→+∞ 0  φ(t), t ≤ 0,

where Bλ = λ(λ − A)−1 , and u(t) ∈ D(A), t ≥ 0. If D(A) = X, then the integral solutions coincide with the known mild solutions given in [5] and references therein. Later on, we will follow [4, 5, 6] and call “integral solutions” as “solutions”. As an immediate consequence of the above theorem, we conclude that solutions of Eq.(1.1) are locally bounded in t and φ. We can get more information on the estimation of solutions. Lemma 2.5. [6] Let u be a continuous function on (−∞, T ] such that u0 ∈ Cg . Then for any 0 ≤ h < r ≤ T with r − h ≥ w0 (w0 is from Lemma 2.2), one has ! 1 |ur |g ≤ max sup |u(s)| , |uh |g . 2 s∈[h,r] In the sequel, we examine the Kuratowski’s measure of non-compactness, which will be used in the next section to study periodic solutions via fixed points of a condensing operator. The Kuratowski’s measure of non-compactness (or the α measure) for a bounded set H of a Banach space Y with norm | · |Y is defined as n o α(H) = inf d > 0 : H has a finite cover of diameter < d . The following are some basic properties of the α measure of non-compactness.

22


Lemma 2.6. Let B1 and B2 be bounded sets of a Banach space Y . Then α(B1 ) ≤ dia(B1 ). (dia(B1 ) = sup{|x − y|Y : x, y ∈ B1 }.) α(B1 ) = 0 if and only if B1 is precompact. α(λB1 ) = |λ|α(B1 ), λ ∈ ℜ. (λB1 = {λx : x ∈ B1 }) α(B1 ∪ B2 ) = max{α(B1 ), α(B2 )}. α(B1 + B2 ) ≤ α(B1 ) + α(B2 ). (B1 + B2 = {x + y : x ∈ B1 , y ∈ B2 }) α(B1 ) ≤ α(B2 ) if B1 ⊆ B2 . In the sequel, we assume that (H3 ) T0 (t) is compact on D(A) whenever t > 0. Next, define Cg0 = {φ ∈ Cg : φ(0) ∈ D(A)}. For D ⊂ Cg0 and u(φ) the unique solution with u0 (φ) = φ, we define Wl (D) = {ul (φ) : φ ∈ D} and W[h,r] (D) = {u[h,r] (φ) : φ ∈ D}, where u[h,r] means the restriction of u on [h, r]. Lemma 2.7. [6] Assume that (H1 ), (H2 ) and (H3 ) are satisfied and let D ⊂ Cg0 . Then for any 0 ≤ h < r ≤ T with r − h ≥ w0 (w0 is from Lemma 2.2), one has n o 1 α(Wr (D)) ≤ max α(W[h,r] (D)), α(Wh (D)) . 2

Lemma 2.8. Let D ⊂ Cg0 be bounded. Then α(W[h,r] (D)) = 0 for any 0 < h < r ≤ T . Proof. Let 0 < h < r ≤ T. For φ ∈ D and t ∈ [h, r] , one has u(t, φ) = T0 (t)φ(0) + lim

t

λ→+∞ 0

T0 (t − s)Bλ f (s, us (·, φ))ds.

Define w by w(t, φ) = lim

t

λ→+∞ 0

T0 (t − s)Bλ f (s, us (·, φ))ds, for φ ∈ D for t ∈ [h, r] .

Then t

lim

λ→+∞ 0

T0 (t − s)Bλ f (s, us (·, φ))ds = T0 (ε) lim

t−ε

λ→+∞ 0 t

+ lim

λ→+∞ t−ε

T0 (t − ε − s)Bλ f (s, us (·, φ))ds

T0 (t − s)Bλ f (s, us (·, φ))ds.

Since T0 (ε) is compact, there exists a compact set Wε such that t−ε T0 (ε) lim T0 (t − ε − s)Bλ f (s, us (·, φ))ds : φ ∈ D ⊆ Wε . λ→+∞ 0

From the boundedness of f , there exists a positive constant a such that t lim T0 (t − s)Bλ f (s, us (·, φ))ds ≤ aε, uniformly in φ ∈ D. λ→+∞ t−ε


23

We deduce that the set {w(t, φ) : φ ∈ D} is totally bounded and therefore is relatively compact in X. To prove the equicontinuity, let 0 < h ≤ t0 < t ≤ r. Then t |w(t, φ) − w(t0 , φ)| ≤ lim T0 (t − s)Bλ f (s, us (·, φ))ds λ→+∞ t0 t 0 + (T0 (t − t0 ) − I) lim T0 (t0 − s)Bλ f (s, us (·, φ))ds . λ→+∞ 0

There exists a positive constant b such that t lim λ→+∞ t T0 (t − s)Bλ f (s, us (·, φ))ds ≤ b(t − t0 ), uniformly in φ ∈ D. 0

Moreover, W0 =

t 0

lim

λ→+∞ 0

and it’s well known that

T0 (t0 − s)Bλ f (s, us (·, φ))ds : φ ∈ D is relatively compact

lim (T0 (h) − I)u = 0, uniformly in u ∈ W0 .

h→0+

On the other hand, T0 (t)φ(0) = T0 (t − h)T0 (h)φ(0).

(2.1)

Since T0 (h) is compact, using (2.1) we get that the family {T0 (·)φ(0) : φ ∈ D} is equicontinuous in C([h, r] ; X). Consequently, we deduce that lim |u(t, φ) − u(t0 , φ)| = 0, uniformly in φ ∈ D.

t−t0 →0+

n o It follows that u(·, φ)|[h,r] : φ ∈ D is equicontinuous. By Arzéla-Ascoli’s theorem, we n o deduce that u(·, φ)|[h,r] : φ ∈ D is relatively compact in C([h, r] ; X). Hence the result is true by using Lemma 2.6.

3

Existence of Periodic Solutions

In this section, we study periodic solutions of Eq.(1.1). For this purpose, we use the Poincaré map P defined by P(φ) = uT (·, φ), φ ∈ Cg0 , where u(·, φ) denotes the (integral) solution of Eq.(1.1) with u0 (·, φ) = φ. In the sequel, we will show that P has a fixed point. We note that a fixed point of P gives rise to a periodic solution. First, we prove that the operator P is condensing. Theorem 3.1. Assume that (H1 ), (H2 ) and (H3 ) hold. Then the Poincaré map P is condensing in Cg0 .

24


Proof. Since |P(φ1 ) − P(φ2 )|g = |uT (φ1 ) − uT (φ2 )|g ≤ M1 ek1 T |φ1 − φ2 |g , it implies that P is continuous and takes bounded sets into bounded sets. Let D ⊂ Cg0 be bounded with α(D) > 0. Then, using lemmas 2.7 and 2.8 (w0 is from Lemma 2.2), n o 1 α(P(D)) = α(WT (D)) ≤ max α(W[T −w0 ,T ] (D)), α(WT −w0 (D)) 2 1 = α(WT −w0 (D)) 2 n o 1 1 ≤ max α(W[T −2w0 ,T −w0 ] (D)), α(WT −2w0 (D)) 2 2 1 2 = ( ) α(WT −2w0 (D)) 2 n o 1 2 1 ≤ ( ) max α(W[T −3w0 ,T −2w0 ] (D)), α(WT −3w0 (D)) 2 2 1 3 = ( ) α(WT −3w0 (D)) 2 ...... n o 1 1 ≤ ( )K0 −1 max α(W[0,w0 ] (D)), α(D) . (3.1) 2 2 Next, for · ∈ [0, w0 ], · T0 (· − h)Bλ f (h, uh (φ))dh : φ ∈ D . W[0,w0 ] (D) ⊆ {T0 (·)φ(0) : φ ∈ D} + lim λ→+∞ 0

And for t ∈ [0, w0 ], |T0 (t)φ1 (0) − T0 (t)φ2 (0)| ≤ M0 |φ1 (0) − φ2 (0)| ≤ M0 |φ1 − φ2 |g ,

(3.2)

where M0 = supt∈[0,T ] |T0 (t)|. Consequently, using an argument similar to that in [6], we see that for · ∈ [0, w0 ], one has α{T0 (·)φ(0) : φ ∈ D} ≤ M0 α(D). Also, we can see that for · ∈ [0, w0 ], α{ lim

·

λ→+∞ 0

T0 (· − h)Bλ f (h, uh (φ))dh : φ ∈ D} = 0.

Therefore we have that α(W[0,w0 ] (D)) ≤ M0 α(D). Thus, we have n o 1 1 α(P(D)) ≤ ( )K0 −1 max M0 α(D), α(D) 2 2 1 K0 −1 ≤ ( ) M0 α(D) < α(D). 2 This completes the proof. To derive periodic solutions, we list the following fixed point theorems.

(3.3)


25

Theorem 3.2 (Sadovskii’s fixed point theorem). [8] Let P be a condensing operator on a Banach space Y , i.e., P is continuous and takes bounded sets into bounded sets, and α(P(B)) < α(B) for every bounded set B of Y with α(B) > 0. If P(H) ⊆ H for a convex, closed and bounded set H of Y , then P has a fixed point in H. Theorem 3.3. [7] Suppose S0 ⊆ S1 ⊆ S2 are convex bounded subsets of a Banach space Y , S0 and S2 are closed, and S1 is open in S2 , and suppose P is a condensing operator in Y . If P j (S1 ) ⊆ S2 , j ≥ 0, and there is a number N(S1 ) such that Pk (S1 ) ⊆ S0 , k ≥ N(S1 ), then P has a fixed point. Note that the Sadovskii’s fixed point theorem requires that the Poincaré operator maps a bounded set into itself. Thus some kind of boundedness of the solutions is required. As for Theorem 3.3, some asymptotic boundedness of the solutions is required. Thus we list the following definitions. Definition 3.4. [6] The solutions of Eq.(1.1) are said to be locally strictly bounded if there exists a constant B > 0 such that |φ|g ≤ B implies that its solution satisfies ku(t, φ)k ≤ B for t ∈ [0, T ]. Definition 3.5. [7] The solutions of Eq.(1.1) are said to be ultimate bounded if there is a bound B > 0, such that for each B3 > 0, there is a K > 0, such that |φ|g ≤ B3 and t ≥ K imply that its solution satisfies ku(t, φ)k < B. We now study the relationship between the boundedness and the periodicity of solutions of Eq.(1.1). By using Theorem 3.2 and Definition 3.4, we have Theorem 3.6. Let (H1 ), (H2 ) and (H3 ) be satisfied. If the solutions of Eq.(1.1) are locally strictly bounded (or assume that solutions are non-increasing in norm k · k on [0, T ]), then Eq.(1.1) has a T periodic solution. Proof. Let H = {φ ∈ Cg0 : |φ|g ≤ B} with B from Definition 3.4. Then H is convex, closed and bounded in Cg0 . Next, for u(·) = u(·, φ) with φ ∈ H, the locally strictly boundedness implies that ku(t)k ≤ B for t ∈ [0, T ]. Then we obtain from Lemma 2.5 that |P(φ)|g = |uT (φ)|g ≤ max

n

sup ku(s)k, s∈[0,T ]

n 1 o ≤ max B, B = B. 2

o 1 |φ|g 2 (3.4)

Thus the result is true by using Theorem 3.2. By using Theorem 3.3 and Definition 3.5, we have the following result, whose proof is omitted here since it is similar to one in [7]. Theorem 3.7. Let (H1 ), (H2 ) and (H3 ) be satisfied. If the solutions of Eq.(1.1) are ultimate bounded, then Eq.(1.1) has a T periodic solution.

26

4


A Massera Type Theorem

In this section, we study the existence of a periodic solution for the following nonhomogeneous partial functional differential equation ( d x(t) = Ax(t) + L(t, xt ) + h(t), t > 0, dt x0 = φ,

(4.1)

where L is a continuous function from IR+ × Cg into X, linear with respect to the second argument and T -periodic in t; h is a continuous T -periodic function with values in X. We will obtain a Massera type theorem for Eq. (4.1). More precisely, we have Theorem 4.1. Let (H1 ) and (H3 ) be satisfied. If Eq. (4.1) has a bounded solution y (with y(0) ∈ D(A)) on IR+ in the sense that sup |y(t)| < +∞, then it has a T -periodic solution. t∈IR+

Proof. Let y be a bounded solution of Eq. (4.1) on IR+ . From the definition of Cg , we know that {yt : t ≥ 0} is also bounded in Cg . Let D := co {ynT : n ∈ IN} ,

where co denotes the closure of the convex hull. Then D is a nonempty bounded closed convex subset of Cg0 . Let ψ ∈ D . Then there exists a sequence (ψk )k ⊂ co {ynT : n ∈ IN} such that nk nk ψk = Σ αki ynk T , αki ≥ 0, Σ αki = 1, and ψ − ψk g → 0 as k → ∞. i=1

i

i=1

Then, one has

nk

xT (·, ψk ) = Σ αki xT (·, ynk T ) i

i=1 nk

= Σ αki ynk T +T . i=1

i

Thus, xT (·, ψk ) ∈ D . Since xT (·, ψ) − xT (·, ψk ) g → 0 as k → +∞, and D is closed, we deduce that P(ψ) = xT (·, ψ) ∈ D for all ψ ∈ D , which implies that P(D ) ⊂ D . By Theorem 3.2, we conclude that Eq. (4.1) has a T -periodic solution.

References [1] M. Adimy, H. Bouzahir and K. Ezzinbi, Existence for a class of partial functional differential equations with infinite delay, Nonlin. Anal., 46(2001), 91-112, [2] M. Adimy, H. Bouzahir and K. Ezzinbi, Local existence and stability for a class of partial functional differential equations with infinite delay, Nonlin. Anal., 48(2002), 323-348.


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[3] W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics. Vol. 96, Birkhauser Verlag, 2001. [4] R. Benkhalti and H. Bouzahir and K. Ezzinbi, Existence of Periodic solutions for some partial functional differential equations with infinite delay, J. Math. Anal. Appl., 256(2001), 257-280. [5] H. Henriquez, Periodic solutions of quasi-linear partial functional differential equations with unbounded delay, Funkcial. Ekvac., 37(1994), 329-343. [6] J. Liu, Periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 247(2000), 627-644. [7] J. Liu, T. Naito and N. Minh, Bounded and Periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 286(2003), 705-712. [8] B. Sadovskii, On a fixed point principle, Funct. Anal. Appl., 2(1967), 151-153.


I NTERNAL P OLLUTION AND D ISCRIMINATING S ENTINEL IN P OPULATION DYNAMICS P ROBLEM O. Nakoulima 1∗and S. Sawadogo 2† 1 Universit´ e des Antilles et de la Guyane, Facult e´ des Sciences Exactes et Naturelles, D e´ partement de Math e´ matiques/Informatiques, Campus de Fouillole, 97159 Pointe a` Pitre Cedex, France. 2 Universit´ e de Ouagadougou, UFR/Sciences Exactes et Appliqu e´ es, Département de Math e´ matiques/Informatiques, 03 BP 7021 Ouagadougou 03-Burkina Faso

Abstract The so called Lions sentinel method is applied to some time varying system with two time scale variables and with missing data. As an example, some population dynamics problem with missing data is considered. The two time scale variables represent the running time variable and the age of individuals. This insight seems to be new. Building the object sentinel leads to some null-controllability problem with restricted control variables and two time scale variables.

AMS Subject Classification: 93B05, 92D25, 35B37, 35K05 Key Words: population dynamics, controllability, sentinels.

1 Introduction 1.1 Population Dynamics Problem with Missing Data The unknown real function y depends on variables t, a, x where t ∈ (T0 , T ) stands for the running time, a ∈ (0, A) for the age of individuals and x ∈ Ω ⊂ RN for the space variable. The number y (t, a, x) is the distribution of a − year old individuals at time t at the point x.

For relevant notation we agree on the following. If some function φ depends on several variables (x1 , x2 , ..xn ) then we denote by φ (xk ) or φ (xk , xl )the function of the other variables. Possible confusion is avoided in precising the definition domain. The function y has to satisfy the following two time scale varying equation  ∂y ∂y + − ∆y + µy = f + λ fb in Q = (T0 , T ) × (0, A) × Ω = U × Ω    ∂t ∂a y = 0 in Σ = (T0 , T ) × (0, A) × ∂Ω = U × Ω (1) y (T0 ) = y0 + τb y0 in QA = (0, A) × Ω    y (0) = 0A β (a) y (a) da in QT = (T0 , T ) × Ω ∗ E-mail † E-mail

address: [email protected] address: [email protected]

30

O. Nakoulima and S. Sawadogo

It is assumed that Ω is open and bounded with C 2 boundary Γ = ∂Ω and µ (t, a, x) ≥ 0, β (t, a, x) ≥ 0. The parameters of the problem have the following sense : the bound T > 0 is the horizon of the problem (the lower bound of running time scale is T0 in order to avoid confusion with the lower bound of age time scale which is actually 0), the bound A is the expectation of life, the weight β is the natural fertility rate, the function µ = µ (t, x, a) is the natural death rate of a−year old individuals at time t > 0 and in the position x, the function f corresponds to external flow and y0 = y0 (a, x) is the initial distribution of individuals. Convenient assumptions for mesurability and integrability of functions are made. In particular f ∈ L2 (Q). The data of the state equation (1) are incomplete in the following sense : (a) fb ∈ L2 (Q) and yb0 ∈ L2 (QA ) are unknown functions with

b

f 2 ≤ 1, yb0 L2 (QA ) ≤ 1 L (Q)

(b) λ ∈ R and τ ∈ R are unknown numbers and small enough.

In the model (1), the term τb y0 represents missing data where as λ fb stands for pollution term. The general question we want to address is for any given extra observation of the system (1), (H0 ) can we evaluate λ fb, without any attempt at computing τb y0 ?

Lions[8] first investigated parabolic patterns. Here we construct sentinels when the supports of the observation function and of the control function are included in two different open subsets of RN . This point of view has already been proposed by Nakoulima [10].

The sentinels theory relies on three features: the state equation, the observation system and some particular evaluation function : the sentinel itself. State equation: The equation (1) governs the state y. We assume the following [1] : ∞ ((0, A) × (T , T ) × Ω), β ∈ L+ 0 (H1) ∃δ ∈ (0, A) s.t. β(a) = 0 for a ∈ (δ, A); ∞ (H2) µ ∈ Lloc ([0, A); L∞ ((T0 , T ) × Ω)), µ ≥ 0 a.e. in QA ; ( ∀t, T0 < t < T0 + A, ∀x ∈ Ω, lim 0t µ(ι, a − t + ι, x)dι = +∞, a−→A (H3) ∀t, A < t < T, ∀x ∈ Ω, lim 0a µ(t − a + α, α, x)dα = +∞. a−→A

For the biological comments about the model and for the basic existence of the solution to (1) we refer to [2, 7, 11]. For the sake of simplicity, we use indifferently y, y(t, a, x; λ, τ) or y(λ, τ) to denote the unique solution of (1). It is relevant since λ and τ are fixed parameters. Observation system: the observation is the knowledge, along some time period, of some function yobs which is defined on some strip over some nonempty open subset O ⊂ Ω, called observatory, as follows M

yobs = m0 + ∑ βi mi . i=1

(2)

Sentinels for Population Dynamics

31

where the functions m0, m1 , ..., mM are given in L2 (U × O) but where the real coefficients βi are unknown. We assume the coefficients βi are ”small”. We refer to the terms βi mi as the interference terms. We can assume without loss of generality that the functions mi are linearly independent.

(3)

h0 ∈ L2 (U × O)

(4)

Sentinel. Let and let ω ⊂ Ω, open and nonempty and set G =U × ω. For any control function w ∈ L2 (G ), set S(λ, τ) = h0 y(t, a, x; λ, τ)dtdadx + wy(t, a, x; λ, τ)dtdadx. U

Choose now w ∈

O

L2 (G )

U

ω

such that the following holds:

(i). S is stationary with respect to the missing terms τb y0 , i.e. ∂S (0, 0) = 0 ∀b y0 ; ∂τ

(6)

(ii). S is stationary with respect to interference terms βi mi ,i.e.

U

O

h0 mi dtdadx +

U

ω

wmi dtdadx = 0, 1 ≤ i ≤ M;

(7)

and (iii). the norm kwk 2 is minimal among control functions in L2 (G ) which L (U×ω) satisfy the above conditions kwkL2 (G ) = minimum. (8) Remark 1.1 According to (8), the function S if it exists, is unique. We refer to function S as the sentinel. Remark 1.2 If the functions mi are null functions, no information comes from (7). The sentinel S is defined only by (6) and (8). If mi 6= 0, the sentinel S is defined by (6), (7) and (8) and it is called a discriminating sentinel. Remark 1.3 The support sup p (mi ) of functions mi is assumed to be included in O. Suppose ω ∩ O = 0/ then U ω wmi dtdadx = 0. There are then no contraints on the control w. Therefore, it suffices to choose h0 such that h0 is orthogonal to each mi . So, without loss of generality, it may be assumed that ω ⊂ O. (9)

1.2

Equivalence to Null-Controllability with Constraints on the Control

Here it will be shown that the existence of such a control function w dealing (6)-(8) is equivalent to null-controllability property for the system with constrained control. Consider the function yτ = ∂y ∂τ where y corresponds to parameter values λ = 0, τ = 0. The function yτ is the solution of the problem

32

O. Nakoulima and S. Sawadogo       

∂yτ ∂yτ ∂t + ∂a − ∆yτ + µyτ = 0 yτ = 0 on Σ; yτ (T0 ) = yb0 in QA ; yτ (0) = 0A β(a)yτ (a)da

in Q; (10) in QT .

Under the assumptions (H1 ) − (H3 ) the linear problem (10) gets one only solution yτ such that yτ (t, A, x) = 0. For the details of the proof we refer to [2, 7, 11]. We now consider the stationary condition (6). The stationary condition holds if and only if

U

O

h0 yτ dtdadx +

U

ω

wyτ dtdadx = 0, ∀ yb0 , yb0 L2 (QA ) ≤ 1.

(11)

In order to transform the equation (11), introduce now the classical adjoint state. More precisely, consider the solution q = q(t, a, x) of the linear problem  ∂q ∂q − − − ∆q + µq = βq(0) + h0 χO + wχω in Q,    ∂t ∂a q=0 in Σ, (12)  q(T ) = 0 in QA ,   q(A) = 0 in QT ,

where χO and χω are indicator functions for the open sets O and ω respectively. There is one only solution in L2 (Q) as some consequence of the fixed point theorem for contracting mapping [2, 3]. The so called adjoint state q depends on the unknown w and its utility comes from the following process. First, multiply both members of the differential equation in (12) by yτ , and integrate by parts over Q A

U

O

h0 yτ dtdadx +

U

ω

wyτ dtdadx =

0

Ω

q(0, a, x)b y0 dadx ,

(13)

∀b y0 ∈ L2 (QA ) yb0 L2 (QA ) ≤ 1.

Thus, the condition (6) (or (11) ) holds if and only if

q(0, a, x) = 0 a.e (a, x) ∈ (0, A) × Ω.

(14)

Then, consider the constraints (7). Let K ⊂ L2 (U × ω) be the subspace generated by the family {mi χω }1≤i≤M K =span {mi χω |1 ≤ i ≤ M} . (15) There is one unique k0 ∈ K such that

U

Denoting by K if and only if

⊥

O

h0 mi dtdadx +

U

ω

k0 mi dtdadx = 0, 1 ≤ i ≤ M.

(16)

the orthogonal subspace of K in L2 (U × ω) , the condition (7) holds w − k0 = v ∈ K ⊥ .

(17)


33

The above considerations show that finding the control w such that the pair (w, S) satisfies (6)–(8) comes to find the control v such that the pair (v, q) satisfies the following system v ∈ K ⊥ , q ∈ L2 (Q) (18)  ∂q ∂q − − − ∆q + µq = βq(0) + h0 χO + k0 χω + vχω in Q,    ∂t ∂a q = 0 on Σ, (19)  q(T ) = 0 in QA ,   q(A) = 0 in QT , q(T0 ) = 0 in QA ,

(20)

kvkL2 (U×ω) = minimum.

(21)

Problem (18)-(21) stands for some null-controllability problem with constrained control variable v.

1.3

Controllability Problem

For the problem (18)-(21), two matters are considered. The first one consists in solving the null-controllability problem, and the second one consists in characterizing the optimal solution (21) by some optimality system. The problem (18)-(21) is solved when K ={0} (i.e. setting without constraints or free constraints) in several issues by various methods [1], [4]. In the present paper both points are considered in the general setting K 6= {0}.

2

Null-Controllability with Constraints on the Control

Consider the following general null-controllability problem. Find some pair (v, q) such that v ∈ K ⊥ , q ∈ L2 (Q);

 ∂q ∂q − − − ∆q + µq    ∂t ∂a q  q(T )   q(A)

= = = =

βq(0) + h + vχω in Q, 0 in Σ, 0 in QA , 0 in QT ,

(22)

(23)

q(T0 ) = 0 in QA ,

(24)

kvkL2 (U×ω) = minimum,

(25)

where h ∈ L2 (Q) (for h = h0 χO + k0 χω we recognize (18)-(21)).

2.1

Observability Inequality Adapted to Constraints

The observability inequality we are looking for is some consequence of Carleman’s inequality. In order to state Carleman’s inequality, introduce now some objects and notations. Choose first some auxiliary function ψ ∈ C2 (Ω) which satisfies the following conditions : ψ(x) > 0 ∀x ∈ Ω, ψ(x) = 0 ∀x ∈ Γ, |∇ψ(x)| 6= 0 ∀x ∈ Ω − ω0 ,

34


where ω0 denotes any open set such that ω0 ⊂ ω (for example ω0 can be some small enough open ball). Such a function ψ exists according to A. Fursikov and O. Yu. Imanuvilov [6]. For any positive parameter value λ define then the following weight functions eλψ(x) e2λkψk∞ − eλψ(x) , α(t, a, x) = , at(T − t) at(T − t)

ϕ(t, a, x) =

(26)

and adopt the following notations  ∂ − ∆ + µI,  L = ∂t∂ + ∂a ∂ L∗ = − ∂t∂ − ∂a − ∆ + µI,  V = ρ ∈ C∞ (Q), ρ = 0 on Σ .

(27)

Now the inequality can be formulated. There exist numbers λ0 > 1 and s0 > 1 and there exists some number C > 0 which only depends on the open sets Ω and ω, such that for any λ > λ0 , for any s ≥ s0 = s0 (µ, T, λ) and for any ρ ∈ V the following holds:

e−2sα |ρt + ρa |2 + |∆ρ|2 dtdadx + sλ2 ϕe−2sα |∇ρ|2 dtdadx Q sϕ Q

≤C

+ s3 λ4 ϕ3 e−2sα |ρ|2 dtdadx Q

−2sα

e

Q

2

|Lρ| dtdadx +

T

A

T0

0

ω

s λ ϕ e

3 4 3 −2sα

2

|ρ| dtdadx . (28)

The above inequality is the global Carleman’s inequality for which we refer to [1] and [12]. We are now concerned with our own inequality. Since ϕ does not vanish, set √ esα 1 θ = √ or = ϕ ϕe−sα . ϕ ϕ θ Then θ ∈ C2 (Q) and

1 θ

(29)

is bounded. By substitution in (28) the following inequality holds

1 1 1 |ρ|2 dtdadx ≤ C( 2 3 3 4 |Lρ|2 dtdadx + 2 |ρ|2 dtdadx). 2 θ θ ϕ s λ θ Q Q U ω

(30)

As a consequence of the boundeness of θ1 and ϕ3 s13 λ4 , some first observability inequality comes 1 |ρ|2 dtdadx ≤ C( |Lρ|2 dtdadx + |ρ|2 dtdadx). (31) 2 Qθ Q U ω Another observability inequality follows which is adapted to the constraints setting using the subspace K (15). Recall first that K is finite dimensional. (32) Denote by P = the orthogonal projection operator from L2 (G ) on K . The following lemma is the key for our results

(33)

Sentinels for Population Dynamics Lemma 2.1 Assume any function k ∈ K ∩ L2 (U, H 1 (ω)) such that ∂k ∂k ∂t + ∂a − ∆k + µk = 0 in G is identically zero in G .

35

(34)

Then there exists some positive constant C > 0 such that :

1 |ρ|2 dtdadx ≤ C( |Lρ|2 dtdadx + |ρ − Pρ|2 dtdadx) ∀ρ ∈ V . 2 Qθ Q G

(35)

Remark 2.2 The above assumption has been already set by Lions [8] for the parabolic system. Proof 2.1 Suppose that (35) does not hold. Then  1 2   ∀ j ∈ N∗ , ∃ρ j ∈ V , 2 ρ j dtdadx = 1, U Ωθ 2 2 1 1   Lρ j dtdadx ≤ and (ρ j − Pρ j ) χω dtdadx ≤ . j j U Ω G

(36)

The forthcomming proof consists in extracting some subsequence (ρ jk )k such that the following contradiction holds lim k

1 2 ρ jk dtdadx → 0. 2 Qθ

Denote by (h|g)L2 (G ) the natural scalar product in the Hibert space L2 (G ) . Let {k1 , k2 , ..., kM } be some orthonormal basis of K . 1) We show first that for any i = 1, 2, ..., M the numerical sequence ((ρ j |ki )L2 (G ) ) j∈N∗ is

2 is bounded bounded or equivalently that the sequence Pρ j 2 L (G )

j

Start with the norm inequality

G

2 1 dtdadx Pρ χ j ω θ2

Since

1 θ2

12

≤

G

21 2 1 ρ j χω dtdadx θ2 21 2 1 + 2 (ρ j − Pρ j ) χω dtdadx . (37) G θ

is bounded and by (36) it follows that there is some number γ ∀ j ∈ N,

G

2 1 Pρ j χω dtdadx ≤ γ. 2 θ

Since K is finite dimensional, norms are equivalent. Particularly the mappings k 7−→ |k|2 dtdadx and k 7−→ G

G

1 2 |k| dtdadx θ2

are equivalent norms on K . There is then some number γ0 2 ∀ j ∈ N, Pρ j χω dtdadx ≤ γ0 . G

(38)

36

O. Nakoulima and S. Sawadogo The relation ((ρ j − Pρ j ) χω ∈ K ⊥ , ∀ j ∈ N∗ means the following ((ρ j − Pρ j ) χω |ki )L2 (G ) = 0 ∀i, 1 ≤ i ≤ M, ∀ j ∈ N∗ . Thus

M

M

i=1

i=1

Pρ j χω = ∑ (Pρ j χω |ki )L2 (G ) ki = ∑ (ρ j |ki )L2 (G ) ki .

(39)

(40)

and from orthonormality

G

thus

2 M

Pρ j χω 2 dtdadx = ∑ (ρ j |ki )L2 (G ) = Pρ j 2 2 i=1

L (G )

Pρ j 2 2 ≤ γ0 . L (G )

(41)

(43)

2) Since (Pρ j χω ) j∈N is bounded and

(ρ j − Pρ j ) χω 2 2 = (ρ j − Pρ j ) χω 2 dtdadx → 0, L (G ) G

then the sequence (ρ j χω ) j∈N∗ is bounded. There is some weakly convergent subsequence still denoted by (ρ j χω ) j∈N∗ such that ρ j χω * g weakly in L2 (G ) .

(45)

Since subsequences have the same limit as convergent sequence ρ j χω − Pρ j χω → 0 strongly in L2 (G ) .

(46)

Since K finite dimensional, there is some possibly sub-subsequence still denoted by (ρ j χω ) j∈N∗ such that Pρ j χω → ζ strongly in L2 (G ) . (47) As a consequence of (46) and (47),(45) ρ j χω −→ g = ζ strongly in L2 (G ) . Then (46) Pρ j χω → g = ζ strongly in L2 (G ) . Since P is some compact operator and by (45) Pρ j χω → Pg strongly in L2 (G ) . Therefore Pg = g and so g ∈ K . 3) There is however some better knowledge about the function g. Actually g = 0. From (36) Lρ j → 0 strongly in L2 (Q) . Then Lρ j χω −→ 0 strongly in L2 (G ) . Thus Lρ j χω * 0 weakly in D 0 (G ) and so Lg = 0. Following (34) g = 0 on G .


37

4) The two preceding steps deal the conclusion ρ j χω → 0 strongly in L2 (G ) . Since ρ j ∈ V , the observability inequality (31) applies as follows: 2 2 1 2 ∀ j ∈ N , 2 ρ j dtdadx ≤ C Lρ j dtdadx + ρ j dtdadx . Qθ Q G It is clear then that

lim j

The proof is now completed.

2.2

(48)

1 2 ρ j dtdadx = 0. 2 Qθ

Existence of Optimal Control Variable

Consider now the following symetric bilinear form ρdtdadx + (ρ − Pρ)(b ρ − Pb ρ)dtdadx. ∀ρ ∈ V ,∀b ρ ∈ V , a(ρ, b ρ) = LρLb Q

(49)

G

According to Lemma 3, this symetric bilinear form is a scalar product on V . Let V be the completion of V with respect to the related norm : p ρ 7−→ kρkV = a(ρ, ρ). (50)

The closure of V is the Hilbert space V. Remark that the norm k.kV is related with the right member of the inequality (35). Similarly, the left member of (35) leads to the norm

1 ∀ρ ∈ V , |ρ|θ = 2 |ρ|2 dtdadx θ Q

21

.

The completion of V is the weighted Hilbert space usually denoted by L21 . The inequalθ ity (35) has the following meaning ∀ρ ∈ V , |ρ|θ ≤ C kρkV .

(51)

This inequality extends to ρ ∈ V . This shows that V is continuously imbedded in L21 . θ We can state the following Lemma 2.3 Let us consider h such that h ∈ L2 (Q) and θh ∈ L2 (Q). The linear form defined by ρ ∈ V 7→ hρdtdadx Q

is continuous.

(52)

38


Proof 2.2 From Cauchy-Schwarz inequality in L2 (Q) 1 ∀ρ ∈ V , hρdtdadx ≤ ( θ2 |h|2 dtdadx) 2 × ( Q

∀ρ ∈ V ,

Q

1 1 |ρ|2 dtdadx) 2 2 Qθ

(53)

hρdtdadx ≤ kθhk 2 |ρ| ≤ C kθhk 2 kρk . L L V Q θ

Since V = V and the function is linear V , the inequality extends to V : ∀ρ ∈ V, hρdtdadx ≤ C kρkV .

(54)

Q

that was to be proved.

By the Lax-Milgram theorem [5], for any function h as above, there exists one and only one solution ρθ ∈ V of the variational equation: a(ρθ , ρ) = hρdtdadx ∀ρ ∈ V.

(55)

vθ = −(ρθ − Pρθ χω )χω

(56)

qθ = Lρθ .

(57)

Q

Set then

Remark 2.4 In the statement of the null-controllability problem, there are boundary and initial or end conditions. These conditions concern the values of the control or state functions at the points of the boudary for example. The solutions dealt by means of functionnal analysis are not functions but elements of function spaces which are equivalence classes. As a consequence boundary or initial or end values of the solutions have to be considered in function spaces. Such a question has been adressed by Lions-Magenes. We refer to [9] for deriving the following trace theorems inregular opens sets Ω. Assume q ∈ L2 (U × Ω) ' L2 U, L2 (Ω) and ∆q ∈ H −1 (U, L2 (Ω)). Then 1

q|U×Γ ∈ H −1 (U, H − 2 (Γ)).

The meaning of q|Σ , the trace of q on Σ, is clear. ∂q 2 −2 Assume q ∈ L2 (U × Ω) ' L2 [T0 , T ] × [0, A] , L2 (Ω) and ∂q ∂t + ∂a ∈ L (U, H (Ω)). Then q ∈ C ([0, A], L2 ([T0 , T ] , H −2 (Ω))) ∩ C ([T0 , T ], L2 ([0, A] , H −2 (Ω))). That means there exists some function qe : [T0 , T ] × [0, A] → L2 ([T0 , T ] , H −2 (Ω)) standing for q ∈ L2 (U × Ω) which is separately continuous so that the following values in L2 ([T0 , T ] , H −2 (Ω)) get sense ∀ (t, a) ∈ U, q (t, a) = qe (t, a)


39

and q(T ) ∈ L2 ([0, A] , H −2 (Ω))

q(0) ∈ L2 ([T0 , T ] , H −2 (Ω))

q(A) ∈ L2 ([T0 , T ] , H −2 (Ω)) Proposition 2.1 Keep the notations in the statement of the null-controllability problem. Assume (34) and the function h as in(52). Then there exists some couple (v, q) is such that (22)-(24) hold. Proof 2.3 The couple (ν, q) = (νθ , qθ ) defined above from ρθ stands for the solution. Since ρθ ∈ V then vθ ∈ L2 (G ) and qθ ∈ L2 (Q). Since Pρθ χω ∈ K then vθ = −(ρθ − Pρθ χω )χω ∈ K ⊥ . By direct substitution in the formulas (49), (55) and (57) it follows ∀ρ ∈ V, qθ Lρdtdadx + (ρθ − Pρθ )(ρ − Pρ)dtdadx = hρdtdadx. Q

(58)

Q

G

Because Pρχω ∈ K , it reduces to ∀ρ ∈ V, qθ Lρdtdadx = hρdtdadx − (ρθ − Pρθ )ρdtdadx,

(59)

∀ρ ∈ V, qθ Lρdtdadx = hρdtdadx + vθ ρdtdadx.

(60)

Q

Q

G

i.e. Q

Q

G

In the duality frame D (Q),D 0 (Q) (60) means that L∗ qθ = h + vθ χω in D 0 (Q).

(61)

Besides h + vθ χω ∈ L2 (Q), then L∗ qθ ∈ L2 (Q). Since qθ ∈ L2 (Q) and ∆qθ ∈ H −1 (U, L2 (Ω)) and by the above remark qθ|U×Γ ∈ 1 2 −2 θ H −1 (U, H − 2 (Γ)). Similarly, since qθ ∈ L2 (Q)) and ∂q∂tθ + ∂q ∂a ∈ L (U, H (Ω)) qθ (0) ∈ L2 ([0, A] , H −2 (Ω)), qθ (T ) ∈ L2 ([0, A] , H −2 (Ω)); qθ (0) ∈ L2 ([T0 , T ] , H −2 (Ω)) and qθ (A) ∈ L2 ([T0 , T ] , H −2 (Ω)). Taking into account (61), integrate by parts ∂ρ ∀ρ ∈ V qθ Lρdtdadx + dtda qθ , ∂ν H − 21 (Γ),H 12 (Γ) Q U T

+ 0 A

[hqθ (t, 0), ρ(t, 0)iH −2 (Ω),H 2 (Ω) − hqθ (t, A), ρ(t, A)iH −2 (Ω),H 2 (Ω) ]dt

+ [hqθ (T0 , a), ρ(T0 , a)iH −2 (Ω),H 2 (Ω) − hqθ (T, a), ρ(T, a)iH −2 (Ω),H 2 (Ω) ]da 0

= (h + vθ χω )ρdtdadx. (62) Q

40

O. Nakoulima and S. Sawadogo By (60) since V ⊂ V , it follows ∂ρ ∀ρ ∈ V , dtda qθ , ∂ν H − 21 (Γ),H 12 (Γ) U T

+ 0 A

[hqθ (t, 0), ρ(t, 0)iH −2 (Ω),H 2 (Ω) − hqθ (t, A), ρ(t, A)iH −2 (Ω),H 2 (Ω) ]dt

+ [hqθ (T0 , a), ρ(T0 , a)iH −2 (Ω),H 2 (Ω) − hqθ (T, a), ρ(T, a)iH −2 (Ω),H 2 (Ω) ]da 0

= 0. (63) Then, successively, we get qθ = 0 on Σ, qθ (T0 ) = 0 and qθ (T ) = 0 in QA , qθ (0) = 0 and qθ (A) = 0 in QT . Since qθ (0) = 0 we have L∗ qθ = βqθ (0) + h + vθ χω . Hence the proof is completed. Remark 2.5 Adapted observability inequality (35) shows that the choice of the scalar product on V is not unique. Thus, there exists an infinity of control functions v such that (22)-(24) hold. Consider the set of control variables v such that (22)-(24) hold. By Proposition 4 this set is nonempty and it is clearly convex and closed in L2 (G ) . Therefore, there exists a unique control variable vbθ of minimal norm in L2 (G ) . We state then the following :

Theorem 2.6 Assume (34). For every h satisfying (52), there exists some control variable v such that (22)-(24) hold. Moreover, we can get a unique control vbθ such that (25) holds.

3

Now, we are concerned with the optimality system for vbθ .

Optimality System for the Optimal Control Variable

Let qbθ be the solution of system (23) submitted to the optimal control vbθ . A classical way to derive the optimality system for the couple (b vθ , qbθ ) is the penalization method by Lions [8]. Describe now the method. Let ε > 0 . Define the functional 1 Jε (v, q) = kvk2 2 2 L (G )

2 (64)

∂q ∂q 1

+ − − − ∆q + µq − βq(t, 0, x) − h − vχω

2 2ε ∂t ∂a L (Q) for any couple (v, q) such that  v ∈ K ⊥ , q ∈ L2 (Q),    ∂q ∂q − ∂t − ∂a − ∆q + µq − βq(0) ∈ L2 (Q),  q = 0 on Σ; q(T ) = 0 in QA ; q(A) = 0 in QT ;   q(T0 ) = 0 in QA .

(65)

Consider the minimization problem

inf{Jε (v, q) | (v, q) subject to (65)}.

(66)


41

Proposition 3.1 Under the assumptions of Theorem 8, the minimization problem has an optimal solution. There exists (vε , qε ) such that Jε (vε , qε ) = min{Jε (v, q) | (v, q) subject to (65)}.

(67)

Proof 3.1 Let (vn , qn ) be a minimizing sequence satisfying (65). The sequence (Jε (vn , qn ))n is bounded from above Jε (vn , qn ) ≤ γ (ε) (68) then

  kvn kL2 (G ) ≤ C(ε)

∂q

∂q  − ∂tn − ∂an − ∆qn + µqn − βqn (0) − h − vn χω

L2 (Q)

≤

√ εC(ε).

(69)

There is some subsequence of (vn )n , still denoted by (vn )n , such that vn * vε weakly in L2 (G ).

(70)

As a consequence (65) the (sub)sequence (qn )n is bounded kqn kL2 (Q) ≤ C.

(71)

There is some subsequence of (qn )n , still denoted by (qn )n such that qn * qε weakly in L2 (Q).

(72)

lim inf Jε (vn , qn ) ≥ Jε (vε , qε ).

(73)

Then We deduce that (vε , qε ) is a unique optimal control, from the strict convexity of Jε . Now, we study the convergence of (vε , qε )ε . Proposition 3.2 Let ((vε , qε ))ε be the family of solutions of (67). Then

limε→0 vε = vbθ weakly in L2 (U × ω) limε→0 qε * qbθ weakly in L2 (Q).

(74)

Proof 3.2 (b vθ , qbθ ) satisfies (22)-(24), then from the structure (64) of Jε (v, q), we have of course   kvε kL2 (G ) ≤ C

√ (75)

∂q

∂q  − ∂tε − ∂aε − ∆qε + µqε − βqε (t, 0, x) − h − vε χω 2 ≤ C ε. L (Q)

Where C’ s are various constants independent of ε. qε verifies (65) (75) then kqε kL2 (Q) ≤ C.

(76)

42


There are a subseqence of (vε , qε )ε , again denoted by (vε , qε )ε , v0 ∈ L2 (G ) and q0 ∈ L2 (Q) such that vε * v0 weakly in L2 (G ), v0 ∈ K ⊥ ; qε * q0 weakly in L2 (Q).

(77)

q0 (T0 ) = 0 in QA .

(79)

kvε k2L2 (G ) ≤ Jε (vε , qε )

(80)

kv0 k2L2 (G ) ≤ lim inf Jε (vε , qε ).

(81)

Then qε * q0 weakly in D 0 (Q) and by the weak continuity of the operator L∗ in D 0 (Q) it follows L∗ qε * L∗ q0 weakly in D 0 (Q). Moreover the traces functions are continuous, then the pair (v0 , q0 ) satisfies the system  ∂q 0 − ∂t0 − ∂q   ∂a − ∆q0 + µq0 = βq0 (0) + h + v0 χω in Q,  q0 = 0 on Σ, (78)  q (T ) = 0 in QA ,   0 q0 (A ) = 0 in QT . From the following estimate

we get

Since the pair (b vθ , qbθ ) satisfies (22)-(24)

lim inf Jε (vε , qε ) ≤ kb vθ k2L2 (G ) .

Thus kv0 kL2 (G ) ≤ kb vθ kL2 (G )

and then

kv0 kL2 (G ) = kb vθ kL2 (G ) .

Hence v0 = vbθ . Since (78) has a unique solution, it follows q0 = qbθ .

Let us express the optimality conditions satisfied by (vε , qε ), as optimal solution of (67). Proposition 3.3 The assumptions are as in Theorem 8. The couple (vε , qε ) is optimal solution of problem (67) if and only if there is one function ρε such that the triplet {vε , qε , ρε } satisfies the following so called optimality system  ∂q ∂qε ε   − ∂t − ∂a − ∆qε + µqε = βqε (0) + h + vε χω + ερε in Q,  qε = 0 in Σ, (82)  q (T ) = 0 in QA ,   ε qε (A) = 0 in QT ,   

qε (T0 ) = 0 in QA ,

∂ρε ∂ρε ∂t + ∂a − ∆ρε + µρε = 0 in Q, ρε = 0 in Σ, ρε (0) = 0A β(a)ρε (a)da in QT ,

vε = −(ρε − Pρε χω )χω .

(82)

(83) (84)


43

Proof 3.3 Express the Euler-Lagrange optimality conditions which characterize (vε , qε ). For any (v, ϕ) such that (65) the following holds

G

vε vdtdadx+ ∂qε ∂qε 1 (− − − ∆qε + µqε − βqε (0) − h − vε χω ) ε Q ∂t ∂a ∂ϕ ∂ϕ − ∆ϕ + µϕ − βϕ(0) − vχω )dtdadx = 0. (85) × (− − ∂t ∂a

Define the adjoint state 1 ∂qε ∂qε ρε = − − − − ∆qε + µqε − βqε (0) − h − vε χω . ε ∂t ∂a

(86)

Then (82) holds . For any (v, ϕ) such that (65), (85) becomes

G

and

vε vdtdadx + ρε (− Q

∂ϕ ∂ϕ − − ∆ϕ + µϕ − βϕ(0) − vχω )dtdadx = 0. ∂t ∂a

(87)

Integrate by parts in (87). As a consequence the couple (vε , ρε ) is shown to satisfy  ∂ρε ∂ρε  ∂t + ∂a − ∆ρε + µρε = 0 in Q (88) ρ = 0 on Σ  ε ρε (0) = 0A β(a)ρε (a)da

G

(vε + ρε ) vdtdadx = 0, ∀v ∈ K ⊥ .

(89)

Hence vε + ρε χω ∈ K . Since vε ∈ K ⊥ then vε + ρε χω = P (vε + ρε χω ) = Pρε χω and thus vε = − (ρε − Pρε χω ) χω . (90) Hence the assertion follows. Remark 3.1 There is no available information concerning ρε (A) in QT , ρε (0) in QA , ρε (T ) in QA . A priori estimates for approximate adjoint state (ρε )ε are now looked for. This is the essential point. From (90) and (75) k(ρε − Pρε χω ) χω kL2 (G ) ≤ C. (91) Since Lρε = 0, then kρε kV ≤ C.

(92)

ρε * b ρθ weakly in V.

(93)

Therefore there are a subseqence (ρε )ε still denoted by (ρε )ε and b ρθ ∈ V such that

44


Following the lines of the proof of Lemma 3, we conclude with the weak convergence ρε * b ρθ in L2 (G ). Thus Pρε χω → χ0 strongly in L2 (G ) (93) so that χ0 ∈ K . By (91) and (93) there is some χ1 ∈ K

⊥

ρε − Pρε χω * χ1 weakly in V so that b ρθ = χ0 + χ1 . Thus χ0 = Pb ρθ χω and

ρε χω − Pρε χω * b ρθ χω − Pb ρθ χω weakly in L2 (G ).

(94)

The above argument leads to the following statement :

Theorem 3.2 The assumptions are as in Theorem 8. The couple (b vθ , qbθ ) is the optimal b solution of problem (22)-(25) if and only if there is a function ρθ such that the triplet (b vθ , qbθ , b ρθ ) is solution of the following optimality system vbθ ∈ K ⊥ , qbθ ∈ L2 (Q), b ρθ ∈ V

 ∂bq qθ − ∂tθ − ∂b qθ + µb qθ = βb qθ (0) + h + vbθ χω in Q,   ∂a − ∆b  qbθ = 0 on Σ,  b q (T ) = 0 in QA , θ   qbθ (A) = 0 in QT ,   

(96)

qbθ (T0 ) = 0 in QA ,

(97)

vbθ = −(b ρθ − Pb ρθ χω )χω .

(99)

∂b ρθ ∂b ρθ ∂t + ∂a b ρθ = 0

− ∆b ρθ + µb ρθ = 0 in Q, on Σ,   b ρθ (a)da in QT , ρθ (0) = 0A β(a)b

4

(95)

(98)

Discriminating Sentinels

It is clear, when h = h0 χO + k0 χω that the problem (18)-(21) and the problem (22)-(25) are the same. The results concerning the former apply to the latter. Consider then the results obtained in the previous sections. More precisely, assume that K is defined as in (15) and h0 , k0 are such that (52) holds. Assume also (9). Let (b vθ , qbθ , b ρθ ) be defined as in Theorem 13. Then vbθ = −(b ρθ − Pb ρθ χω )χω . (100)

Definition - The function S defined by S(λ, τ) =

U

O

h0 y(λ, τ)dtdadx +

U

ω

(k0 − (b ρθ − Pb ρθ χω ))y(λ, τ)dtdadx

(101)

is called discriminating sentinel defined by the open subsets O and ω, and by the function h0 .


45

Return to the problem (H0 ). The above function S allows to lay down the relevant question. Because of (6) we can write S(λ, τ) = S(0, 0) + λ

∂S (0, 0) for λ and τ small, ∂λ

(102)

where S is defined by (101). On the other hand, by (2) and (7) S(λ, τ) =

U

O

h0 m0 dtdadx +

U

ω

So that

U

O

h0 (m0 − y0 )dtdadx +

U

We also have

where yλ is given by

ω

vbθ m0 dtdadx.

vbθ (m0 − y0 )dtdadx = λ

∂S (0, 0). ∂λ

∂S (0, 0) = h0 yλ dtdadx + vbθ yλ dtdadx ∂λ U O U ω    

∂yλ ∂yλ ∂t + ∂a − ∆yλ + µyλ yλ = 0 in Σ,

= fb in Q,

(103)

(104)

(105)

(106)

y (T ) = 0 in QA ,    λ 0 yλ (0) = 0A β(a)yλ (a)da in QT .

Let qbθ (h0 ) be the solution (uniquely defined by h0 ) of (96). Multiplying (96) by yλ it follows bθ yλ dtdadx = qbθ (h0 ) fbdtdadx. h0 yλ dtdadx + v (107) U

O

U

ω

Q

Therefore (104) becomes

U

O

h0 (m0 − y0 )dtdadx +

U

ω

vbθ (m0 − y0 )dtdadx = λ qbθ (h0 ) fbdtdadx.

(108)

Q

This equation allows to evaluate the right hand side of (108) and therefore it is an available information concerning λ fb. A perturbation (or a ”pollution”) fb will be stealthy for the sentinel defined by h0 if

Q

5

Conclusion

qbθ (h0 ) fbdtdadx = 0.

(109)

As a conclusion we announce two lines in prolongation of our work. The first one is concerned with a generalization of the idea of sentinel which tackles new observation or control designs. Indeed it has been shown above that sentinel by J. L. Lions can hoock on some two-time scale varying system with both interior observation and control. But several other designs are realic. For example, only one part of the boundary

46


may be observed, nd the control may be constrained and/or restricted to the boundary. Different mixing are possible which could be solved by means of some setting so called Carmeman inequality. This the aim of some forthcomming paper. The second line in process is concerned with furtivity as it is defined at the end of above section.

Acknowledgments Authors wish to express theirs gratitudes to Robert Janin for his many helpful suggestions.

References [1] B.Ainseba, 2002, Exact and approximate controllability of age and space population dynamics structured model, J. Math.Anal.Appl. [2] B.Ainseba and M.Langlais, 2000, On a population dynamics control problem with age dependence and spatial structure, J. Math. Anal. Appl. 248,455–474. [3] B. Ainseba et M.Langlais, 1996, Sur un problème de contrôle d’une population structurée en aˆ ge et en espace.C.R.Acad.Sci.Paris,t.323,serie I, 269–274. [4] B.Ainseba and S.Anita, 2001, Local exact controllability of the age-dependent population dynamics with diffusion.Abstract Appl.Anal.,6, 357-368. [5] H.Brezis: Analyse fonctionnelle.Théorie et application. Masson, Paris 1983. [6] A.Fursikov, O.Imanuvilov,1996, Controllability of evolution equation.Lecture Notes Series 34,RIM-GARC, Seoul National University. [7] M.Giovanna and M.Langlais, 1982, Age-dependent population diffusion with external constraint., J.Math.Biology, 14, 7-94. [8] J.L.Lions: Sentinelles pour les systèmes distribués a` données incomplètes.Masson, Paris 1992. [9] J.L. Lions et M. Magenes . (1968) Problèmes aux limites non homogènes et applications. Paris, Dunod,Vol. 1 et 2. [10] O. Nakoulima, 2004, Contrôlabilité a` zéro avec contraintes sur le contrôle. C. R. Acad. Sci. Paris, Ser.I 339/6 405-410. [11] A. Ouédraogo, O. Traoré, 2003, Sur un problème de dynamique des populations, Imhotep. Vol 4 n◦ 1. [12] S.Sawadogo, 2005, Thèse unique.Université de Ouagadougou.


C ENTER M ANIFOLD AND S TABILITY IN C RITICAL C ASES FOR S OME PARTIAL F UNCTIONAL D IFFERENTIAL E QUATIONS∗ Mostafa Adimy1†, Khalil Ezzinbi2‡ and Jianhong Wu 3§ de Pau et des Pays de l’Adour, Laboratoire de Mathématiques Appliquées CNRS UMR 5142 Avenue de l’université 64000, Pau, France 2 Universit´ e Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P. 2390, Marrakesh, Morocco 3 York University, Department of Mathematics and Statistics, Faculty of Pure and Applied Sciences, 4700 Keele Street North York, Ontario, Canada, M3J 1P3 1 Universit´ e

Abstract In this work, we prove the existence of a center manifold for some partial functional differential equations, whose linear part is not necessarily densely defined but satisfies the Hille-Yosida condition. The attractiveness of the center manifold is also shown when the unstable space is reduced to zero. We prove that the flow on the center manifold is completely determined by an ordinary differential equation in a finite dimensional space. In some critical cases, when the exponential stability is not possible, we prove that the uniform asymptotic stability of the equilibrium is completely determined by the uniform asymptotic stability of the reduced system on the center manifold.

Key Words: Hille-Yosida operator, integral solution, semigroup, variation of constants formula, center manifold, attractiveness, reduced system, critical case, asymptotic stability, approximation. 2000 Mathematical Subject Classification: 34K17, 34K19, 34K20, 34K30, 34G20, 47D06. ∗ This

research is supported by Grant from CNCPRST (Morocco) and CNRS(France) Ref. SPM 17769, by TWAS Grant under contract Ref. 03-030 RG/MATHS/AF/AC, by the Canada Research Chairs Program, by Natural Sciences and Engineering Research Council of Canada, and by Mathematics for Information Technology and Complex Systems. † E-mail address: [email protected] ‡ E-mail address: [email protected]: to whom all correspondence should be sent § E-mail address: [email protected]

48

1

Mostafa Adimy, Khalil Ezzinbi and Jianhong Wu

Introduction

The aim of this paper is to study the existence of a center manifold and stability in some critical cases for the following partial functional differential equation ( d u(t) = Au(t) + L(ut ) + g(ut ), t ≥ 0 dt u0 = ϕ ∈ C := C ([−r, 0] ; E) ,

(1.1)

where A is not necessarily densely defined linear operator on a Banach space E and C is the space of continuous functions from [−r, 0] to E endowed with the uniform norm topology. For every t ≥ 0 and for a continuous u : [−r, +∞) −→ E, the function ut ∈ C is defined by ut (θ) = u(t + θ) for θ ∈ [−r, 0] . L is a bounded linear operator from C into E and g is a Lipschitz continuous function from C to E with g(0) = 0. In this work, we assume that A is a Hille-Yosida operator: there exist ω ∈ lR and M0 ≥ 1 such that (ω, ∞) ⊂ ρ(A) and (λI − A)−n ≤

M0 for λ ≥ ω and n ∈ N, (λ − ω)n

where ρ(A) is the resolvent set of A. In [21], the authors proved the existence, regularity and stability of solutions of (1.1) when A generates a strongly continuous semigroup, which is equivalent by Hille-Yosida Theorem to that A is a Hille-Yosida operator and D(A) = E. In [3], the authors used the integrated semigroup approach to prove the existence and regularity of solutions of (1.1) when A is only a Hille-Yosida operator. Moreover, it was shown that the phase space of equation (1.1) is given by n o Y := ϕ ∈ C : ϕ(0) ∈ D(A) . Assume that the function g is differentiable at 0 with g0 (0) = 0. Then the linearized equation of (1.1) around the equilibrium zero is given by ( d v(t) = Av(t) + L(vt ), t ≥ 0 dt v0 = ϕ ∈ C.

(1.2)

If all characteristic values (see section 2) of equation (1.2) have negative real part, then the zero equilibrium of (1.1) is uniformly asymptotically stable. However, if there exists at least one characteristic value with a positive real part, then the zero solution of (1.1) is unstable. In the critical case, when exponential stability is not possible and there exists a characteristic value with zero real part, the situation is more complicated since either stability or instability may hold. The subject of the center manifold is to study the stability in this critical case. For differential equations, the center manifold theory has been extensively studied, we refer to [6], [7], [8], [12], [13], [14], [15], [16], [19], [20] and [24]. In [17] and [22], the authors proved the existence of a center manifold when D(A) = E.

Center Manifold and Stability in Critical Cases for Some Partial Functional ...

49

They established the attractiveness of this manifold when the unstable space is reduced to zero. In [11], the authors proved the existence of a center manifold for a given map. Their approach was applied to show the existence of a center manifold for partial functional differential equations in Banach spaces in the case when the linear part generates a compact strongly continuous semigroup. Recently, in [18], the authors studied the existence of invariant manifolds for an evolutionary process in Banach spaces and in particularly for some partial functional differential equations. For more details about the center manifold theory and its applications in the context of partial functional differential equations, we refer to the monograph [27]. Here we consider equation (1.1) when the domain D(A) is not necessarily dense in E. The nondensity occurs, in many situations, from restrictions made on the space where the equations are considered (for example, periodic continuous solutions, H"older continuous functions) or from boundary conditions ( the space C1 with null value on the boundary is not dense in the space of continuous functions). For more details, we refer to [1], [2], [3], [4] and [5]. The organization of this work is as follows: in section 2, we recall some results of integral solutions and the semigroup solution and we describe the variation of constants formula for the associated non-homogeneous problem of (1.2). We also give some results on the spectral analysis of the linear equation (1.2). In section 3, we prove the existence of a global center manifold. In section 4, we prove that this center manifold is exponentially attractive when the unstable space is reduced to zero. In section 5, we prove that the flow on the center manifold is governed by an ordinary differential equation in a finite dimensional space. In section 6, we prove a result on the stability of the equilibrium in the critical case. We also establish a new reduction principal for equation (1.1). In section 7, we study the existence of a local center manifold when g is only defined and C1 -function in a neighborhood of zero. In the last section, we propose a result on the stability when zero is a simple characteristic value and no characteristic value lies on the imaginary axis.

2

Spectral Analysis and Variation of Constants Formula

In the following we assume (H1 ) A is a Hille-Yosida operator. Definition 2.1. A continuous function u : [−r, +∞) → E is called an integral solution of equation (1.1) if t

u(s)ds ∈ D(A) for t ≥ 0, t u(s)ds + L ii) u(t) = ϕ(0) + A

i)

0

iii) u0 = ϕ.

0

t 0

us ds +

t 0

g(us )ds for t ≥ 0,

We will call, without causing any confusion, the integral solution the function ut , for t ≥ 0. Let A0 be the part of the operator A in D(A) which is defined by n o ( D(A0 ) = x ∈ D(A) : Ax ∈ D(A) A0 x = Ax for x ∈ D(A0 ).

50

Mostafa Adimy, Khalil Ezzinbi and Jianhong Wu The following result is well known (see [3]).

Lemma 2.2. A0 generates a strongly continuous semigroup (T0 (t))t≥0 on D(A). For the existence and uniqueness of an integral solution of (1.1), we need the following condition. (H2 ) g : C −→ E is Lipschitz continuous. The following result can be found in [3]. Proposition 2.3. Assume that (H1 ) and (H2 ) hold. Then for ϕ ∈ Y, equation (1.1) has a unique global integral solution on [−r, ∞) which is given by the following formula  t  T0 (t)ϕ(0) + lim T0 (t − s)Bλ (L(us ) + g(us )) ds, t ≥ 0 u(t) = (2.1) λ→∞  ϕ(t), t ∈ [−r, 0] , 0

where Bλ = λ(λI − A)−1 for λ ≥ ω.

Assume that g is differentiable at zero with g0 (0) = 0. Then the linearized equation of (1.1) at zero is given by equation (1.2). Define the operator U(t) on Y by U(t)ϕ = vt (., ϕ), where v is the unique integral solution of equation (1.2) corresponding to the initial value ϕ. Then (U(t))t≥0 is a strongly continuous semigroup on Y . One has the following linearized principle. Theorem 2.4. Assume that (H1 ) and (H2 ) hold. If the zero equilibrium of (U(t))t≥0 is exponentially stable, in the sense that there exist N0 ≥ 1 and ε ≥ 0 such that |U(t)| ≤ N0 e−εt for t ≥ 0, then the zero equilibrium of equation (1.1) is locally exponentially stable, in the sense that there exist δ ≥ 0, µ ≥ 0 and k ≥ 1 such that |xt (., ϕ)| ≤ ke−µt |ϕ| for ϕ ∈ Y with |ϕ| ≤ δ and t ≥ 0, where xt (., ϕ) is the integral solution of equation (1.1) corresponding to initial value ϕ. Moreover, if Y can be decomposed as Y = Y1 ⊕Y2 where Yi are U-invariant subspaces of Y , 1 Y1 is a finite-dimensional space and with ω0 = lim log |U(h)|Y2 | we have h→∞ h inf {|λ| : λ ∈ σ (U(t)|Y1 )} ≥ eω0t for t ≥ 0, where σ (U(t)|Y1 ) is the spectrum of U(t)|Y1 , then the zero equilibrium of equation (1.1) is unstable, in the sense that there exist ε ≥ 0 and sequences (ϕn )n converging to 0 and (tn )n of positive real numbers such that |xtn (., ϕn )| ≥ ε.


51

The above theorem is a consequence of the following result. For more details on the proof, we refer to [3]. Theorem 2.5. [9] Let (V (t))t≥0 be a nonlinear strongly continuous semigroup on a subset Ω of a Banach space Z and assume that x0 ∈ Ω is an equilibrium of (V (t))t≥0 such that V (t) is differentiable at x0 , with W (t) the derivative at x0 of V (t) for each t ≥ 0. Then, (W (t))t≥0 is a strongly continuous semigroup of bounded linear operators on Z. If the zero equilibrium of (W (t))t≥0 is exponentially stable, then x0 is locally exponentially stable equilibrium of (V (t))t≥0 . Moreover, if Z can be decomposed as Z = Z1 ⊕ Z2 where Zi are W -invariant 1 subspaces of Z, Z1 is a finite-dimensional space and with ω1 = lim log |W (h)|Z2 | we have h→∞ h inf {|λ| : λ ∈ σ (W (t)|Z1 )} ≥ eω1t for t ≥ 0, then the equilibrium x0 is unstable in the sense that there exist ε ≥ 0 and sequences (yn )n converging to x0 and (tn )n of positive real numbers such that |V (tn )yn − x0 | ≥ ε. Some informations of the infinitesimal generator of (U(t))t≥0 can be found in [5]. For example, we know that Theorem 2.6. The infinitesimal generator AU of (U(t))t≥0 on Y is given by  

ϕ ∈ C1 ([−r, 0] ; E) : ϕ(0) ∈ D(A), ϕ0 (0) ∈ D(A) and D(AU ) = ϕ0 (0) = Aϕ(0) + L(ϕ)  0 AU ϕ = ϕ for ϕ ∈ D(AU ).

We now make the next assumption about the operator A.

(H3 ) The semigroup (T0 (t))t≥0 is compact on D(A) for t ≥ 0. Theorem 2.7. Assume that (H3 ) holds. Then, U(t) is a compact operator on Y for t ≥ r. e be a bounded subset of Y . We use Ascoli-Arzela’s theorem to show Proof. n Let t ≥ r andoD e is relatively compact in Y . Let ϕ ∈ D, e θ ∈ [−r, 0] and ε ≥ 0 such that that U(t)ϕ : ϕ ∈ D t + θ − ε ≥ 0. Then (U(t)ϕ) (θ) = T0 (t + θ)ϕ(0) + lim

λ→+∞ 0

t+θ

T0 (t + θ − s)Bλ L(U(s)ϕ)ds.

Note that

t+θ 0

T0 (t + θ − s)Bλ L(U(s)ϕ)ds t+θ−ε

= 0

T0 (t + θ − s)Bλ L(U(s)ϕ)ds +

t+θ t+θ−ε

T0 (t + θ − s)Bλ L(U(s)ϕ)ds

52


and t+θ−ε

t+θ−ε

T0 (t +θ−s)Bλ L(U(s)ϕ)ds = T0 (ε) lim

lim

λ→+∞ 0

λ→+∞ 0

T0 (t +θ−ε−s)Bλ L(U(s)ϕ)ds.

The assumption (H3 ) implies that T0 (ε)

t+θ−ε

lim

λ→+∞ 0

e T0 (t + θ − ε − s)Bλ L(U(s)ϕ)ds : ϕ ∈ D

is relatively compact in E. As the semigroup (U(t))t≥0 is exponentially bounded, then there exists a positive constant b1 such that t+θ lim ≤ b1 ε for ϕ ∈ D. e T (t + θ − s)B L(U(s)ϕ)ds 0 λ λ→+∞ t+θ−ε

Consequently, the set lim

t+θ

λ→+∞ t+θ−ε

e T0 (t + θ − s)Bλ L(U(s)ϕ)ds : ϕ ∈ D

n o e is relatively compact in E, is totally bounded in E. We deduce that (U(t)ϕ) (θ) : ϕ ∈ D for each θ ∈ [−r, 0]. For the completeness of the proof, we need to show the equicontinuity property. Let θ, θ0 ∈ [−r, 0] such that θ ≥ θ0 . Then (U(t)ϕ) (θ) − (U(t)ϕ) (θ0 ) = (T0 (t + θ) − T0 (t + θ0 )) ϕ(0) t+θ

+ lim

λ→+∞ 0


t+θ0

− lim

λ→+∞ 0

T0 (t + θ0 − s)Bλ L(U(s)ϕ)ds.

Furthermore, t+θ 0

T0 (t + θ − s)Bλ L(U(s)ϕ)ds =

t+θ0 0


t+θ

+ t+θ0

T0 (t + θ − s)Bλ L(U(s)ϕ)ds.

Consequently, |(U(t)ϕ) (θ) − (U(t)ϕ) (θ0 )| ≤ |T0 (t + θ) − T0 (t + θ0 )| |ϕ(0)| t+θ0 + lim (T0 (t + θ − s) − T0 (t + θ0 − s)) Bλ L(U(s)ϕ)ds λ→+∞ 0 t+θ + lim T0 (t + θ − s)Bλ L(U(s)ϕ)ds . λ→+∞ t+θ0


53

Assumption (H3 ) implies that the semigroup (T0 (t))t≥0 is uniformly continuous for t ≥ 0. Then lim |T0 (t + θ) − T0 (t + θ0 )| = 0. θ→θ0

The semigroup (U(t))t≥0 is exponentially bounded. Consequently, there exists a positive constant b2 such that t+θ lim T0 (t + θ − s)Bλ L(U(s)ϕ)ds ≤ b2 (θ − θ0 ) λ→+∞ t+θ0

and

t+θ0

lim

λ→+∞ 0

(T0 (t + θ − s) − T0 (t + θ0 − s)) Bλ L(U(s)ϕ)ds

= (T0 (θ − θ0 ) − I) lim

λ→+∞ 0

t+θ0

T0 (t + θ0 − s)Bλ L(U(s)ϕ)ds.

e0 in E such that We have proved that there exists a compact set K t+θ0

lim

λ→+∞ 0

e0 for ϕ ∈ D. e T0 (t + θ0 − s)Bλ L(U(s)ϕ)ds ∈ K

Using Banach-Steinhaus’s theorem, we obtain

e0 . lim (T0 (θ − θ0 ) − I)x = 0 uniformly in x ∈ K

θ→θ0

This implies that

e lim (U(t)ϕ) (θ) − (U(t)ϕ) (θ0 ) = 0 uniformly in ϕ ∈ D.

θ→θ+ 0

We can prove in similar way that

e lim (U(t)ϕ) (θ) − (U(t)ϕ) (θ0 ) = 0 uniformly in ϕ ∈ D.

θ→θ− 0

n o e is compact for t ≥ r. By Ascoli-Arzela’s theorem, we conclude that U(t)ϕ : ϕ ∈ D

Now, we consider the spectral properties of the infinitesimal generator AU . We denote by E, without causing confusion, the complexication of E. For each complex number λ, we define the linear operator ∆(λ) : D(A) → E by ∆(λ) = λI − A − L(eλ· I), where eλ· I : E → C is defined by eλ· x (θ) = eλθ x, x ∈ E and θ ∈ [−r, 0] .

(2.2)

Definition 2.8. We say that λ is a characteristic value of equation (1.2) if there exists x ∈ D(A)\{0} solving the characteristic equation ∆(λ)x = 0.

54


Since the operator U(t) is compact for t ≥ r, the spectrum σ(AU ) of AU is the point spectrum σ p (AU ). More precisely, we have Theorem 2.9. The spectrum σ(AU ) = σ p (AU ) = {λ ∈ C : ker ∆(λ) 6= {0}} . Proof. Let λ ∈ σ p (AU ). Then there exists ϕ ∈ D(AU )\{0} such that AU ϕ = λϕ, which is equivalent to ϕ(θ) = eλθ ϕ(0), for θ ∈ [−r, 0] and ϕ0 (0) = Aϕ(0) − L(ϕ) with ϕ(0) 6= 0. Consequently ∆(λ)ϕ(0) = 0. Conversely, let λ ∈ C such that ker ∆(λ) 6= {0} . Then there exists x ∈ D(A)\{0} such that ∆(λ)x = 0. If we define the function ϕ by ϕ(θ) = eλθ x for θ ∈ [−r, 0] , then ϕ ∈ D(AU ) and AU ϕ = λϕ, which implies that λ ∈ σ p (AU ). The growth bound ω0 (U) of the semigroup (U(t))t≥0 is defined by −κt ω0 (U) = inf κ ≥ 0 : sup e |U(t)| < ∞ . t≥0

The spectral bound s(AU ) of AU is defined by s(AU ) = sup {Re(λ) : λ ∈ σ p (AU )} . Since U(t) is compact for t ≥ r, then it is well known that ω0 (U) = s(AU ). Consequently, the asymptotic behavior of the solutions of the linear equation (1.2) is completely obtained by s(AU ). More precisely, we have the following result. Corollary 2.10. Assume that (H1 ), (H2 ) and (H3 ) hold. Then, the following properties hold, i) if s(AU ) < 0, then (U(t))t≥0 is exponentially stable and zero is locally exponentially stable for equation (1.1); ii) if s(AU ) = 0, then there exists ϕ ∈ Y such that |U(t)ϕ| = |ϕ| for t ≥ 0 and either stability or instability may hold; iii) if s(AU ) ≥ 0, then there exists ϕ ∈ Y such that |U(t)ϕ| → ∞ as t → ∞ and zero is unstable for equation (1.1). As a consequence of the compactness of the semigroup U(t) for t ≥ r and by Theorem 2.11, p.100, in [10], we get the following general spectral decomposition of the phase space Y. Theorem 2.11. There exist linear subspaces of Y denoted by Y− , Y0 and Y+ respectively with Y = Y− ⊕Y0 ⊕Y+ such that i) AU (Y− ) ⊂ Y− , AU (Y0 ) ⊂ Y0 , and AU (Y+ ) ⊂ Y+ ; ii) Y0 and Y+ are finite dimensional;


55

iii) σ(AU |Y0 ) = {λ ∈ σ(AU ) : Re λ = 0} , σ(AU |Y+ ) = {λ ∈ σ(AU ) : Re λ ≥ 0}; iv) U(t)Y− ⊂ Y− for t ≥ 0, U(t) can be extended for t ≤ 0 when restricted to Y0 ∪ Y+ and U(t)Y0 ⊂ Y0 ,U(t)Y+ ⊂ Y+ for t ∈ lR; v) for any 0 < γ < inf {|Re λ| : λ ∈ σ(AU ) and Re λ 6= 0} , there exists K ≥ 0 such that |U(t)P− ϕ| ≤ Ke−γt |P− ϕ| for t ≥ 0, γ |U(t)P0 ϕ| ≤ Ke 3 |t| |P0 ϕ| for t ∈ lR, |U(t)P+ ϕ| ≤ Keγt |P+ ϕ| for t ≤ 0, where P− , P0 and P+ are projections of Y into Y− ,Y0 and Y+ respectively. Y− ,Y0 and Y+ are called stable, center and unstable subspaces of the semigroup (U(t))t≥0 . The following result deals with the variation of constants formula for equation (1.1) which are taken from [5]. Let hX0 i be defined by hX0 i = {X0 c : c ∈ E} , where the function X0 c is defined by (X0 c) (θ) =

0 if θ ∈ [−r, 0) , c if θ = 0.

We introduce the space Y ⊕ hX0 i , endowed with the following norm |ϕ + X0 c| = |ϕ| + |c| . The following result is taken from [5]. f Theorem 2.12. The continuous extension A U of the operator AU defined on Y ⊕ hX0 i by: n o ( 1 0 f D(A U ) = ϕ ∈ C ([−r, 0] ; E) : ϕ(0) ∈ D(A) and ϕ (0) ∈ D(A) 0 0 f A U ϕ = ϕ + X0 (Aϕ(0) + L(ϕ) − ϕ (0)),

e0 ≥ 1 such that (ω e ∈ lR and M e , ∞) ⊂ is a Hille-Yosida operator on Y ⊕ hX0 i: there exist ω f ρ(AU ) and e0 M −n f e and n ∈ N, (λI − A ) for λ ≥ ω ≤ U e )n (λ − ω

f f with ρ(A U ) the resolvent set of AU . Moreover, the integral solution u of equation (1.1) is given for ϕ ∈ Y, by the following variation of constants formula t

ut = U(t)ϕ + lim

λ→∞ 0

−1 for λ ≥ ω f fλ = λ(λI − A e. where B U)

fλ (X0 g(us )) ds for t ≥ 0, U(t − s)B

(2.3)

e0 = 1. Otherwise, we can renorm Remarks. i) Without loss of generality, we assume that M e0 = 1. the space Y ⊕ hX0 i in order to get an equivalent norm for which M ii) For any locally integrable function ρ : lR → E, one can see that the following limit exists: t

lim

λ→∞ s

fλ X0 ρ(τ)dτ for t ≥ s. U(t − τ)B

56


3

Global Existence of the Center Manifold

Theorem 3.1. Assume that (H1 ) and (H3 ) hold. Then, there exists ε ≥ 0 such that if |g(ϕ1 ) − g(ϕ2 )| < ε, |ϕ1 − ϕ2 | ϕ1 6=ϕ2

Lip(g) = sup

then, there exists a bounded Lipschitz map hg : Y0 → Y− ⊕ Y+ such that hg (0) = 0 and the Lipschitz manifold Mg := {ϕ + hg (ϕ) : ϕ ∈ Y0 } is globally invariant under the flow of equation (1.1) on Y . Proof. Let B = B (Y0 ,Y− ⊕Y+ ) denote the Banach space of bounded maps h : Y0 → Y− ⊕Y+ endowed with the uniform norm topology. We define F = {h ∈ B : h is Lipschitz, h(0) = 0 and Lip(h) ≤ 1} .

Let h ∈ F and ϕ ∈ Y0 . Using the strict contraction principle, one can prove the existence of solution of the following equation ϕ vt

t

ϕ

vt = U(t)ϕ + lim

λ→∞ 0

0 fλ X0 g(vϕτ + h(vϕτ )) dτ, t ∈ lR. U(t − τ) B

(3.1)

We now introduce the mapping Tg : F → B by 0

Tg (h)ϕ = lim

λ→∞ −∞ 0

+ lim

− fλ X0 g(vϕτ + h(vϕτ )) dτ U(−τ) B

λ→∞ +∞

+ fλ X0 g(vϕτ + h(vϕτ )) dτ. U(−τ) B

The first step is to prove that Tg maps F into itself. Let ϕ1 , ϕ2 ∈ Y0 and t ∈ lR. Suppose that Lip(g) < ε. Then t ϕ1 γ γ vt − vtϕ2 ≤ Ke 3 |t| |ϕ1 − ϕ2 | + 2K |P0 | ε e 3 |t−τ| vϕτ 1 − vϕτ 2 dτ . 0 By Gronwall’s lemma, we get that

ϕ γ ϕ e− 3 |t| vt 1 − vt 2 ≤ K |ϕ1 − ϕ2 | e2K|P0 |ε|t|

and

ϕ1 γ vt − vtϕ2 ≤ K |ϕ1 − ϕ2 | e[ 3 +2K|P0 |ε]|t| for t ∈ lR.

If we choose ε such that then

γ 2K |P0 | ε < , 6

ϕ1 γ vt − vtϕ2 ≤ K |ϕ1 − ϕ2 | e 2 |t| for t ∈ lR.

(3.2)

(3.3)


57

Moreover, 0

|Tg (h)ϕ1 − Tg (h)ϕ2 | ≤

−∞

ϕ ϕ ϕ ϕ |U(−τ)P− | g(vτ 1 + h(vτ 1 )) − g(vτ 2 + h(vτ 2 )) dτ

+∞

+ 0

Consequently, 0

|Tg (h)ϕ1 − Tg (h)ϕ2 | ≤ 2

−∞

ϕ ϕ ϕ ϕ |U(−τ)P+ | g(vτ 1 + h(vτ 1 )) − g(vτ 2 + h(vτ 2 )) dτ.

ϕ ϕ Keγτ |P− | ε vτ 1 − vτ 2 dτ + 2

Using inequality (3.3), we obtain

|Tg (h)ϕ1 − Tg (h)ϕ2 | ≤ 2K 2 |P− | ε |ϕ1 − ϕ2 |

0 −∞

+∞ 0

ϕ ϕ Ke−γτ |P+ | ε vτ 1 − vτ 2 dτ. +∞

γ

e 2 τ dτ + 2K 2 |P+ | ε |ϕ1 − ϕ2 |

γ

e− 2 τ dτ.

0

It follows that |Tg (h)ϕ1 − Tg (h)ϕ2 | ≤ If we choose ε such that

4ε 2 K (|P− | + |P+ |) |ϕ1 − ϕ2 | . γ

4ε 2 K (|P− | + |P+ |) < 1, γ

then Tg maps F into itself. The next step is to show that Tg is a strict contraction on F . Let h1 , h2 ∈ F . For ϕ ∈ Y0 and for i = 1, 2, let vti denote the solution of the following equation t

vti = U(t)ϕ + lim

λ→∞ 0

Then, 1 2 vt − vt ≤ εK |P0 |

t

e

γ 3 |t−τ|

0

and

1 vt − vt2 ≤ 2εK |P0 |

t

e 0

0 fλ X0 g(viτ + hi (viτ )) dτ for t ∈ lR. U(t − τ) B 1 2 1 2 2 2 vτ − vτ + h1 (vτ ) − h1 (vτ ) + h1 (vτ ) − h2 (vτ ) dτ ,

γ 3 |t−τ|

1 vτ − v2τ dτ + εK |P0 | |h1 − h2 |

t

e 0

By Gronwall’s lemma, we obtain that

1 γ vt − vt2 ≤ 3εK |P0 | |h1 − h2 | e[ 3 +2K|P0 |ε]|t| for t ∈ lR. γ

By (3.2), we obtain

1 γ vt − vt2 ≤ 3εK |P0 | |h1 − h2 | e 2 |t| for all t ∈ lR. γ

γ 3 |t−τ|

dτ .

58

Mostafa Adimy, Khalil Ezzinbi and Jianhong Wu For i = 1, 2, we have 0

Tg (hi )ϕ = lim

λ→∞ −∞

− fλ X0 g(viτ + hi (viτ )) dτ U(−τ) B

0

+ lim

λ→∞ +∞

It follows that

+ fλ X0 g(viτ + hi (viτ )) dτ. U(−τ) B

K |P− | ε 6ε2 K |P0 | |h1 − h2 | + |h1 − h2 | 2 γ γ K |P+ | ε 6ε2 K |h1 − h2 | . + 2K |P+ | 2 |P0 | |h1 − h2 | + γ γ

|Tg (h1 )ϕ − Tg (h2 )ϕ| ≤ 2K |P− |

Consequently, |Tg (h1 ) − Tg (h2 )| ≤ (|P− | + |P+ |) We choose ε such that

Kε 12εK |P0 | + 1 |h1 − h2 | . γ γ

12εK Kε |P0 | + 1 < 1. (|P− | + |P+ |) γ γ

Then Tg is a strict contraction on F , and consequently it has a unique fixed point hg in F : Tg (hg ) = hg .

Finally, we show that Mg := {ϕ + hg (ϕ) : ϕ ∈ Y0 } is globally invariant under the flow on Y . Let ϕ ∈ Y0 and v be the solution of equation (3.1). ϕ ϕ We claim that t → vt + hg (vt ) is an integral solution of equation (1.1) with initial value ϕ ϕ ϕ + hg (ϕ). In fact, we have Tg (hg )(vt ) = hg (vt ), t ∈ lR. Moreover, for t ∈ lR, one has − 0 ϕ ϕ ϕ fλ X0 g(vt+τ Tg (h)(vt ) = lim U(−τ) B + hg (vt+τ )) dτ λ→∞ −∞

0

+ lim

λ→∞ +∞

which implies that ϕ

+ ϕ ϕ fλ X0 g(vt+τ U(−τ) B + hg (vt+τ )) dτ,

− fλ X0 g(vϕτ + hg (vϕτ )) dτ U(t − τ) B λ→∞ −∞ + t fλ X0 g(vϕτ + hg (vϕτ )) dτ. U(t − τ) B + lim t

hg (vt ) = lim

λ→∞ +∞

Then, for t ∈ lR, we have ϕ

ϕ

0 fλ X0 g(vϕτ + hg (vϕτ )) dτ U(t − τ) B λ→∞ 0 − t fλ X0 g(vϕτ + hg (vϕτ )) dτ + lim U(t − τ) B λ→∞ −∞ + t fλ X0 g(vϕτ + hg (vϕτ )) dτ. U(t − τ) B + lim t

vt + hg (vt ) = U(t)ϕ + lim

λ→∞ +∞

Center Manifold and Stability in Critical Cases for Some Partial Functional ... For any t ≥ a, we have − fλ X0 g(vϕτ + hg (vϕτ )) dτ U(t − τ) B −∞ − a fλ X0 g(vϕτ + hg (vϕτ )) dτ = lim U(t − τ) B λ→∞ −∞ − t fλ X0 g(vϕτ + hg (vϕτ )) dτ, + lim U(t − τ) B t

lim

λ→∞

λ→∞ a

and

− fλ X0 g(vϕτ + hg (vϕτ )) dτ U(t − τ) B −∞ − a ϕ ϕ fλ X0 g(vτ + hg (vτ )) dτ . = U(t − a) lim U(a − τ) B a

lim

λ→∞

λ→∞ −∞

By the same argument as above, we obtain

+ fλ X0 g(vϕτ + hg (vϕτ )) dτ U(t − τ) B +∞ + a fλ X0 g(vϕτ + hg (vϕτ )) dτ = lim U(t − τ) B λ→∞ +∞ + t fλ X0 g(vϕτ + hg (vϕτ )) dτ, + lim U(t − τ) B t

lim

λ→∞

λ→∞ a

and

+ fλ X0 g(vϕτ + hg (vϕτ )) dτ U(t − τ) B +∞ + a ϕ ϕ f = U(t − a) lim U(a − τ) Bλ X0 g(vτ + hg (vτ )) dτ . a

lim

λ→∞

λ→∞ +∞

Note that

+ fλ X0 g(vϕτ + hg (vϕτ )) dτ U(a − τ) B λ→∞ +∞ − a fλ X0 g(vϕτ + hg (vϕτ )) dτ, U(a − τ) B + lim

hg (vϕa ) = lim

a

λ→∞ −∞

and in particular ϕ

vt = U(t − a)vϕa + lim

λ→∞ a

t

0 fλ X0 g(vϕτ + hg (vϕτ )) dτ. U(t − τ) B

Consequently, for any t ≥ a, we obtain ϕ ϕ vt + hg (vt )

0 ϕ ϕ f U(t − τ) B X g(v + h (v )) dτ lim g τ τ λ 0 λ→∞ a + t fλ X0 g(vϕτ + hg (vϕτ )) dτ + lim U(t − τ) B λ→∞ a − t fλ X0 g(vϕτ + hg (vϕτ )) dτ, U(t − τ) B + lim

= U(t − a) (vϕa + hg (vϕa )) +

λ→∞ a

t

59

60


which implies that for any t ≥ a, ϕ

ϕ

vt + hg (vt ) = U(t − a) (vϕa + hg (vϕa )) + lim

λ→∞ a

ϕ

ϕ

t

fλ X0 g(vϕτ + hg (vϕτ )) dτ. U(t − τ) B

Finally, we conclude that vt + hg (vt ) is an integral solution of equation (1.1) on lR with initial value ϕ + hg (ϕ). Theorem 3.2. Let vϕ be the solution of equation (3.1) on lR. Then, for t ∈ lR ϕ γ γ vt ≤ K |ϕ| e 2 |t| and vtϕ + hg (vtϕ ) ≤ 2K |ϕ| e 2 |t| . Conversely, if we choose ε such that

2Kε (|P− | + |P+ | + 3 |P0 |) < 1, γ γ

then for any integral solution u of equation (1.1) on lR with ut = O(e 2 |t| ), we have ut ∈ Mg for all t ∈ lR. Proof. Let vϕ be the solution of equation (3.1). Then, using the estimate (3.3), we obtain that ϕ γ vt ≤ K |ϕ| e 2 |t| for t ∈ lR

and from the fact that Lip(h) ≤ 1 and hg (0) = 0, we obtain ϕ γ vt + hg (vtϕ ) ≤ 2K |ϕ| e 2 |t| for t ∈ lR.

γ

Let u be an integral solution of equation (1.1) such that ut = O(e 2 |t| ). Then there exists γ a positive constant k0 such that |ut | ≤ k0 e 2 |t| for all t ∈ lR. Note that t

ut = U(t − s)us + lim

λ→∞ s

On the other hand, ut+ = U(t − s)u+ s + lim

t

λ→∞ s

fλ X0 g(uτ )dτ for t ≥ s. U(t − τ)B

+ fλ X0 g(uτ ) dτ for s ≥ t. U(t − τ) B

Moreover, for s ≥ t and s ≥ 0, we have γ γ γ(t−s) U(t − s)u+ |P+ us | ≤ k0 K |P+ | eγ(t−s) e 2 |s| = k0 K |P+ | eγt e− 2 s . s ≤ Ke Therefore,

lim U(t − s)u+ s = 0.

s→∞

It follows that ut+ = lim

t

λ→∞ +∞

Similarly, we can prove that ut− = lim

t

λ→∞ −∞

+ fλ X0 g(uτ ) dτ. U(t − τ) B − fλ X0 g(uτ ) dτ. U(t − τ) B


61

We conclude that ut = ut+ + ut− +U(t)u00 + lim

t

λ→∞ 0

0 fλ X0 g(uτ ) dτ for t ∈ lR. U(t − τ) B

Let φ ∈ Y0 such that φ = u0 . By Theorem 3.1, there exists an integral solution w of equation (1.1) on lR with initial value φ + hg (φ) such that wt ∈ Mg for all t ∈ lR and 0 fλ X0 g(wτ ) dτ U(t − τ) B λ→∞ 0 + t fλ X0 g(wτ ) dτ + lim U(t − τ) B λ→∞ −∞ − t fλ X0 g(wτ ) dτ. U(t − τ) B + lim t

wt = U(t)φ + lim

λ→∞ +∞

Then, for all t ∈ lR, we have 0 t f |ut − wt | ≤ lim U(t − τ) Bλ X0 (g(uτ ) − g(wτ )) dτ λ→∞ 0 + t fλ X0 (g(uτ ) − g(wτ )) dτ U(t − τ) B + lim λ→∞ −∞ − t f + lim U(t − τ) Bλ X0 (g(uτ ) − g(wτ )) dτ . λ→∞ +∞

This implies that

|ut − wt | ≤ Kε |P0 | t

+ |P− |

−∞

t 0

γ e 3 |t−τ| |uτ − wτ | dτ

−γ(t−τ)

e

|uτ − wτ | dτ + |P+ |

∞

γ(t−τ)

e t

|uτ − wτ | dτ .

γ

e = sup N(t) < ∞. On the other hand, we Let N(t) = e− 2 |t| |ut − wt | for all t ∈ lR. Then, N t∈lR have t t ∞ γ γ γ |t−τ| − − (t−τ) (t−τ) e |P0 | e 6 e 2 e2 dτ + |P− | dτ + |P+ | dτ . N(t) ≤ KεN −∞

0

t

Finally we arrive at

e ≤ 2Kε (3 |P0 | + |P− | + |P+ |) N. e N γ

4

e = 0 and ut = wt for t ∈ lR. Consequently, N

Attractiveness of the Center Manifold

In this section, we assume that there exists no characteristic value with a positive real part and hence the unstable space Y+ is reduced to zero. We establish the following result on the attractiveness of the center manifold.

62

Mostafa Adimy, Khalil Ezzinbi and Jianhong Wu Theorem 4.1. There exist ε ≥ 0, K1 ≥ 0 and α ∈ 3γ , γ such that if Lip(g) < ε, then any integral solution ut (ϕ) of equation (1.1) on R+ satisfies − ut (ϕ) − hg (ut0 (ϕ)) ≤ K1 e−αt ϕ− − hg (ϕ0 ) for t ≥ 0.

(4.1)

Proof. The proof of this theorem is based on the following technically lemma. Lemma 4.2. There exist ε ≥ 0, K0 ≥ 0 and α ∈ 3γ , γ such that if Lip(g) < ε, then there is a continuous bounded mapping p : lR+ × Y0 × Y− → Y− such that any integral solution ut (ϕ) of equation (1.1) satisfies ut− (ϕ) = p(t, ut0 (ϕ), ϕ− ) for t ≥ 0.

(4.2)

Moreover, |p(t, φ1 , ψ1 ) − p(t, φ2 , ψ2 )| ≤ K0 |φ1 − φ2 | + e−αt |ψ1 − ψ2 |

(4.3)

for all φ1 , φ2 ∈ Y0 , ψ1 , ψ2 ∈ Y− and t ≥ 0.

Idea of the proof of Lemma 4.2. The proof is similar to the one given in [22]. Let ϕ ∈ Y0 and ψ ∈ Y− . For t ≥ 0 and 0 ≤ τ ≤ t, we consider the system t

q(τ,t, ϕ, ψ) = U(τ−t)ϕ− lim

λ→∞ τ

and p(t, ϕ, ψ) = U(t)ψ + lim

t

λ→∞ 0

0 fλ X0 g (q(s,t, ϕ, ψ) + p(s, q(s,t, ϕ, ψ), ψ)) ds, U(τ−s) B

− fλ X0 g(q(s,t, ϕ, ψ) + p(s, q(s,t, ϕ, ψ), ψ)) ds. U(t − s) B

Using the contraction principle, we can prove the existence of q and p. The expression (4.2) and the estimate (4.3) are obtained in a completely similar fashion to that in [22]. Proof of Theorem 4.1. Let Mg be the center manifold of equation (1.1). Then any integral solution lying in Mg must satisfy (4.2). Let ut = ut (ϕ− + ϕ0 ) be an integral solution of equation (1.1) on R+ with initial value ϕ− + ϕ0 . Let τ ≥ 0. Then, u0τ + hg (u0τ ) ∈ Mg and the corresponding integral solution exists on R and lies on Mg . This solution can be considered as an integral solution of equation (1.1) starting from ψ− + ψ0 at 0. Let vt = vt (ψ− + ψ0 ) be the integral solution corresponding to ψ− + ψ0 . Using Lemma 4.2, we conclude that 0 0 − 0 − u− τ − hg (uτ ) = p(τ, uτ , ϕ ) − p(τ, uτ , ψ ),

which implies that

− uτ − hg (u0τ ) ≤ K0 e−ατ ϕ− − ψ− .

Since Lip(h) ≤ 1, we have

− uτ − hg (u0τ ) ≤ K0 e−ατ ϕ− − hg (ϕ0 ) + ϕ0 − ψ0 .

(4.4)


63

The initial values ϕ0 and ψ0 correspond to the solutions of the following equations for 0≤s≤τ 0 s fλ X0 g(vσ + p(σ, vσ , ϕ− )) dσ, vs = U(s − τ)vτ + lim U(s − σ) B λ→∞ τ 0 s ∗ ∗ fλ X0 g(v∗σ + p(σ, v∗σ , ψ− )) dσ. vs = U(s − τ)vτ + lim U(s − σ) B λ→∞ τ

Note that vτ = v∗τ . It follows, for 0 ≤ s ≤ τ, that

|vs − v∗s | ≤ K(1+K0 )ε |P0 |

τ s

τ

γ

e 3 (σ−s) |vσ − v∗σ | dσ+KK0 ε |P0 |

s

γ e 3 (σ−s) e−ασ dσ ϕ− − ψ− .

Then by Gronwall’s lemma, we deduce that there exists a positive constant ν which depends only on constants γ, K, K0 and ε such that, for 0 ≤ s ≤ τ, we have |vs − v∗s | ≤ ν ϕ− − ψ− . If we assume that Lip(g) is small enough such that ν < 1, then 0 ϕ − ψ0 ≤ ν ϕ− − ψ− ≤ ν ϕ− − hg (ϕ0 ) + hg (ϕ0 ) − hg (ψ0 ) ,

which gives that

0 ϕ − ψ0 ≤

We conclude that − ut − hg (ut0 ) ≤

ν − ϕ − hg (ϕ0 ) . 1−ν

1 K0 e−αt ϕ− − hg (ϕ0 ) for t ≥ 0. 1−ν

As an immediate consequence, we obtain the following result on the attractiveness of the center manifold. Corollary 4.3. Assume that Lip(g) is small enough and the unstable space Y+ is reduced to zero. Then the center manifold Mg is exponentially attractive. We also obtain. Proposition 4.4. Assume that Lip(g) is small enough and the unstable space Y+ is reduced to zero. Let w be an integral solution of equation (1.1) that is bounded on R. Then wt ∈ Mg for all t ∈ R. Proof. Let w be a bounded integral solution of equation (1.1). Since, the equation (1.1) is autonomous, then for σ ≤ 0, wt 0 +σ is also an integral solution of equation (1.1) for t 0 ≥ 0 with initial value wσ at 0. It follows by the estimation (4.1) that − 0 0 w 0 (ϕ) − hg (w00 (ϕ)) ≤ K1 e−αt 0 w− σ − hg (wσ ) for t ≥ 0. t +σ t +σ Let t ≥ σ. Then − 0 wt − hg (wt0 ) ≤ K1 e−α(t−σ) w− σ − hg (wσ ) for t ≥ σ.

(4.5)

Since w is bounded on R, letting σ → −∞, we obtain that wt− = hg (wt0 ) for all t ∈ R.

64

5


Flow on the Center Manifold

In this section, we establish that the flow on the center manifold is governed by an ordinary differential equation in a finite dimensional space. In the sequel, we assume that the function g satisfies the conditions of Theorem 4.1. We also assume that the unstable space Y+ is reduced to zero. Let d be the dimension of the center space Y0 and Φ = (φ1 , ...., φd ) be a basis of Y0 . Then there exists d-elements Φ = (φ∗1 , ...., φ∗d ) in Y ∗ , the dual space of Y, such that

∗ φi , φ j := φ∗i (φ j ) = δi j , 1 ≤ i, j ≤ d,

and φ∗i = 0 on Y− . Denote by Ψ the transpose of (φ∗1 , ...., φ∗d ). Then the projection operator P0 is given by P0 φ = Φ hΨ, φi . 0 Since U (t) t≥0 is a strongly continuous group on the finite dimensional space Y0 , by Theorem 2.15, p. 102 in [10], we get that there exists a d × d matrix G such that U 0 (t)Φ = ΦeGt for t ≥ 0. ∗ on E by e and i ∈ {1, ...., d} . We define the function xni Let n ∈ N, n ≥ n0 ≥ ω D E ∗ fn (X0 y) . xni (y) = φ∗i , B

∗ is a bounded linear operator on E. Let x∗ be the transpose of (x∗ , ..., x∗ ), then Then xni n n1 nd D E fn (X0 y) . hxn∗ , yi = Ψ, B

Consequently,

sup |xn∗ | < ∞,

n≥n0

which implies that important result.

(xn∗ )n≥n0

is a bounded sequence in L (E, Rd ). Then, we get the following

Theorem 5.1. There exists x∗ ∈ L (E, Rd ) such that (xn∗ )n≥n0 converges weakly to x∗ in the sense that hxn∗ , yi → hx∗ , yi as n → ∞ for y ∈ E. For the proof, we need the following fundamental theorem [25, pp. 776] ∗ Theorem 5.2. Let X be a separable Banach space ∗and (zn )n≥0 be a bounded sequence∗ in ∗ ∗ X . Then there exists a subsequence znk k≥0 of (zn )n≥0 which converges weakly in X in the sense that there exists z∗ ∈ X ∗ such that

∗ znk , y → hz∗ , yi as k → ∞ for x ∈ X.

Proof of Theorem 5.1. Let Z0 be a closed separable subspace of E. Since (xn∗ )n≥n0 is a bounded sequence, by Theorem 5.2 there is a subsequence xn∗k k∈N which converges weakly to some xZ∗ 0 in Z0 . We claim that all the sequence (xn∗ )n≥n0 converges weakly to xZ∗ 0 in Z0 . This can be done by way of contradiction. Namely, suppose that there exists a

Center Manifold and Stability in Critical Cases for Some Partial Functional ... 65 of (xn∗ )n≥n0 which converges weakly to some xeZ∗ 0 with xeZ∗ 0 6= xZ∗ 0 . Let subsequence xn∗ p p∈N

uet (., σ, ϕ, f ) denote the integral solution of the following equation ( d ue(t) = Ae u(t) + L(e ut ) + f (t), t ≥ σ dt ueσ = ϕ ∈ C,

where f is a continuous function from R to E. Then by using the variation of constants formula and the spectral decomposition of solutions, we obtain t

P0 uet (., σ, 0, f ) = lim

n→+∞ σ

and

0 U (t − ξ) Ben X0 f (ξ) dξ,

D E P0 Ben X0 f (ξ) = Φ Ψ, Ben X0 f (ξ) = Φ hxn∗ , f (ξ)i .

It follows that

t

P0 uet (., σ, 0, f ) = lim Φ

σ t

n→+∞

= lim Φ

σ

n→+∞

D E e(t−ξ)G Ψ, Ben X0 f (ξ) dξ, e(t−ξ)G hxn∗ , f (ξ)i dξ.

For a fixed a ∈ Z0 , set f = a to be a constant function. Then t

lim

k→+∞ σ

e(t−ξ)G xn∗k , a dξ = lim

t

p→+∞ σ

which implies that t σ

e(t−ξ)G xZ∗ 0 , a dξ =

t σ

D E e(t−ξ)G xn∗ p , a dξ for a ∈ Z0 ,

e(t−ξ)G xeZ∗ 0 , a dξ for a ∈ Z0 .

Consequently xZ∗ 0 = xeZ∗ 0 , which yields a contradiction. We now conclude that the whole sequence (xn∗ )n≥n0 converges weakly to xZ∗ 0 in Z0 . Let Z1 be another closed separable subspace of X. By using the same argument as above, we obtain that (xn∗ )n≥n0 converges weakly to xZ∗ 1 in Z1 . Since Z0 ∩ Z1 is a closed separable subspace of E, we get that xZ∗ 1 = xZ∗ 0 in Z0 ∩ Z1 . For any y ∈ E, we define x∗ by hx∗ , yi = hxZ∗ , yi , where Z is an arbitrary given closed separable subspace of E such that y ∈ Z. Then x∗ is well defined on E and x∗ is a bounded linear operator from E to Rd such that |x∗ | ≤ sup |xn∗ | < ∞, n≥n0

and (xn∗ )n≥n0 converges weakly to x∗ in E. As a consequence, we conclude that

66


Corollary 5.3. For any continuous function f : R → E, we have 0 t t lim U (t − ξ) Ben X0 f (ξ) dξ = Φ e(t−ξ)G hx∗ , f (ξ)i dξ for t, σ ∈ R. n→+∞ σ

σ

Let ϕ ∈ Y0 such that ϕ + hg (ϕ) ∈ Mg . From the properties of the center manifold, we ϕ ϕ ϕ know that the integral solution starting from ϕ + h(ϕ) is given by vt + hg (vt ), where vt is the solution of 0 t ϕ ϕ ϕ f U(t − τ) Bλ X0 g(vτ + hg (vτ )) dτ fort ∈ lR. vt = U(t)ϕ + lim λ→∞ 0

ϕ

ϕ

Let z(t) be the component of vt . Then Φz(t) = vt fort ∈ lR. By Theorem 5.1 and Corollary 5.3, we have t

ϕ

Φz(t) = vt = ΦeGt z(0) + Φ

0

We conclude that z satisfies z(t) = eGt z(0) + lim

n→∞ 0

t

ϕ ϕ e(t−τ)G x∗ , g(vτ + hg (vτ )) dτ fort ∈ lR.

ϕ ϕ eG(t−τ) xn∗ , g(vτ + hg (vτ )) dτ fort ∈ lR.

Finally we arrive at the following ordinary differential equation, which determines the flow on the center manifold z0 (t) = Gz(t) + hx∗ , g(Φz(t) + hg (Φz(t)))i for t ∈ lR.

6

(5.1)

Stability in Critical Cases

In this section, we suppose that Lip(g) < ε, where ε is given by Theorem 4.1. Here we study the critical case where the unstable space Y+ is reduced to zero and the exponential stability is not possible, which implies that there exists at least one characteristic value with a real part equals zero. Theorem 6.1. Assume that Lip(g) is small enough. Then there exists a positive constant K2 such that for each ϕ ∈ Y, there exists e z0 ∈ Rd such that 0 ut − Φe z(t) + ut− − hg (Φe z(t)) ≤ K2 e−αt ϕ− − hg (ϕ0 ) for t ≥ 0, (6.1)

where e z is the solution of the reduced system (5.1) with initial value e z0 and u is the integral solution of equation (1.1) with initial value ϕ. Proof. Let ϕ ∈ Y and u be the corresponding integral solution of equation (1.1). Then 0 s 0 fλ X0 g(u− u0s = U(s − t)ut0 + lim U(s − τ) B dτ for 0 ≤ s ≤ t. τ + uτ ) λ→∞ t

Also let s → ws,t be the solution of the following equation ws,t = U(s − t)ut0 + lim

λ→∞ t

s

0 fλ X0 g(wτ,t + hg (wτ,t )) dτ for 0 ≤ s ≤ t. U(s − τ) B

Center Manifold and Stability in Critical Cases for Some Partial Functional ... Then, for 0 ≤ s ≤ t, we have t γ 0 us − ws,t ≤ 2Kε |P0 | e 3 (τ−s) u0τ − wτ,t dτ + Kε |P0 | s

It follows that γ e 3 s u0s − ws,t ≤ 2Kε |P0 |

t s

t s

γ e 3 τ u0τ − wτ,t dτ + Kε |P0 |

67

γ 0 e 3 (τ−s) u− τ − hg (uτ ) dτ. t s

γ 0 e 3 τ u− τ − hg (uτ ) dτ.

Using Gronwall’s lemma, we obtain that t 0 us − ws,t ≤ 2Kε |P0 | eµ(τ−s) u− − hg (u0 ) dτ for 0 ≤ s ≤ t, τ τ s

where µ = 2Kε |P0 | +

γ 3.

By Theorem 4.1, we obtain t 0 0 us − ws,t ≤ 2KK1 ε |P0 | e(µ−α)(τ−s) dτ u− s − hg (us ) for 0 ≤ s ≤ t. s

Let us recall that α ∈ Consequently,

γ 3,γ .

Then we can choose ε small enough such that µ − α < 0.

0 0 us − ws,t ≤ 2KK1 ε |P0 | u− s − hg (us ) for 0 ≤ s ≤ t, α−µ

(6.2)

and for s = 0, we have 0 u0 − w0,t ≤ 2KK1 ε |P0 | u− − hg (u00 ) for t ≥ 0. 0 α−µ

We deduce that {w0,t : t ≥ 0} is bounded in Y0 and then there exists a sequence tn → ∞ as n → ∞ and φ ∈ Y0 such that w0,tn → φ as n → ∞. e φ) be the solution of the following equation Let w(., 0 t f e eτ (., φ) + hg (w eτ (., φ))) dτ for t ≥ 0. wt (., φ) = U(t)φ + lim U(t − τ) Bλ X0 g(w λ→∞ 0

By the continuous dependence on the initial data, we obtain, for all s ≥ 0 es (0, φ) = lim w es (0, w0,tn ) = lim w es (0, w−tn (0, utn )), w n→∞

n→∞

es−tn (0, utn ) = lim ws,tn . = lim w n→∞

n→∞

By (4.4) and (6.2), there exists a positive constant K2 such that 0 0 us − w es (0, φ) ≤ K2 e−αs u− 0 − hg (u0 ) for all s ≥ 0. If we put

then e z(t) = eGt e z(0) +

t 0

et (0, φ) for t ∈ lR, Φe z(t) = w

e(t−τ)G hx∗ , g(Φe z(τ) + hg (Φe z(τ)))i dτ for t ∈ lR.

Finally, the estimation (6.1) follows from (4.4).

68


Now, we can state the following result on the stability by using the reduction to the center manifold. Theorem 6.2. If the zero solution of equation (5.1) is uniformly asymptotically stable (unstable), then the zero solution of equation (1.1) is uniformly asymptotically stable (unstable). Proof. Assume that 0 is uniformly asymptotically stable for equation (5.1). For ς ≥ 0, let Bς = ϕ− + ϕ0 ∈ Y− ⊕Y0 : ϕ− + ϕ0 < ς ,

and Mg ∩ Bρ for some ρ ≥ 0, be the region of attraction of 0 for equation (5.1). First, we prove that 0 is stable for equation (1.1). Let ε ≥ 0. Then there exists δ < ρ such that |z(t)| < ε for t ≥ 0, provided that |z(0)| < δ, where z is a solution of (5.1). As 0 is assumed to be uniformly asymptotically stable for equation (5.1), there exists t0 = t0 (δ) such that |z(t)| < 2δ , for t ≥ t0 . Without loss of generality, we can choose δ and t0 so that max(K1 , K2 )e−αt0 < 21 . By the continuous dependence on the initial value for equation (1.1), there exists δ1 < δ2 such that if 0 δ1 − δ1 − 0 − 0 0 ϕ + ϕ ∈ Vδ1 := ψ + ψ ∈ Y− ⊕Y0 : ψ < , ψ − hg (ψ ) < , 2 2 then the corresponding integral solution ut = ut (ϕ− + ϕ0 ) of equation (1.1) satisfies ut ∈ Bε for t ∈ [0,t0 ] . Moreover,

− ut − hg (ut0 ) < δ1 . 0 0 2

Furthermore, by Theorem 6.1, there exists z0 ∈ Rd such that 0 ut − Φe z(t) ≤ K2 e−αt ϕ− − hg (ϕ0 ) for t ≥ 0,

(6.3)

where e z is a solution of the reduced system (5.1) with initial value e z0 such that |e z0 | < δ. It follows that 0 ut < δ. 0

Consequently, ut0 ∈ Bε and ut must be in Bε for all t ≥ 0. This completes the proof of the stability. Now we deal with the local attractiveness of the zero solution. For a given integral solution u(., ϕ) of equation (1.1) which is assumed to be bounded for t ≥ 0, it is well known that the ω-limit set ω(ϕ) is nonempty, compact, invariant and connected since the map ϕ → ut (·, ϕ) is compact for t ≥ r. For the attractiveness of 0, let Vδ be chosen as above and ϕ ∈ Vδ . Then the integral solution u of equation (1.1) starting from ϕ lies in Bε . The ω-limit set ω(ϕ) of u is nonempty and invariant and must be in Mg ∩ Bε . Since the equilibrium 0 of (1.1) is uniformly asymptotically stable, we deduce by Theorem 11.4, p. 111 [23] and by the LaSalle invariance


69

principle that the only invariant set in Mg ∩ Bε must be zero. Consequently, the ω-limit set ω(ϕ) is zero and ut (., ϕ) → 0 as t → 0. Assume now that the zero solution of the reduced system (5.1) is unstable. Then there exist σ1 ≥ 0, a sequence (tn )n of positive real numbers and a sequence (zn )n converging to 0 such that |z(tn , zn )| ≥ σ1 , where z(., zn ) is a solution of (5.1). On the other hand Φz(., zn ) + hg (Φz(., zn )) is an integral solution of equation (1.1) and |Φz(tn , zn ) + hg (Φz(tn , zn ))| ≥ (1 − Lip(hg )) |Φz(tn , zn )| . Moreover, Lip(hg ) can be chosen such that 1 − Lip(hg ) ≥ 0. It follows that |Φz(tn , zn ) + hg (Φz(tn , zn ))| ≥ (1 − Lip(hg )) σ2 , for some σ2 ≥ 0. Consequently, the zero solution of equation (1.1) is unstable.

7

Local Existence of the Center Manifold

In this section we prove the existence of the local center manifold when g is only defined in a neighborhood of zero. We assume that (H4 ) There exists ρ1 ≥ 0 such that g : B(0, ρ1 ) → E is C1 -function, g(0) = 0 and g0 (0) = 0, where B(0, ρ1 ) = {ϕ ∈ C : |ϕ| < ρ1 } . For ρ < ρ1 , we define the cut-off function gρ : C → E by gρ (ϕ) =

g(ϕ) if |ϕ| ≤ ρ, ρ g( |ϕ| ϕ) if |ϕ| ≥ ρ.

We consider the following partial functional differential equation ( d u(t) = Au(t) + L(ut ) + gρ (ut ) for t ≥ 0 dt u0 = ϕ ∈ C.

(7.1)

Theorem 7.1. Assume that (H1 ), (H3 ) and (H4 ) hold. Then there exist 0 < ρ < ρ1 and Lipschitz continuous mapping hgρ : Y0 → Y− ⊕Y+ such that hgρ (0) = 0 and the local Lipschitz manifold Mgρ = ϕ + hgρ (ϕ) : ϕ ∈ Y0 is globally invariant under the flow associated to equation (7.1).

Proof. Using the same arguments as in [26], Proposition 3.10, p.95, one can show that gρ is Lipschitz continuous with Lip(gρ ) ≤ 2 sup g0 (ϕ) . |ϕ| 0 then the zero solution of equation (1.1) is unstable. Proof. The proof is based on Theorem 6.2 and on the following known stability result.


71

Theorem 8.2. [6] Consider the scalar differential equation z0 (t) = am zm + am+1 zm+1 + ....

(8.2)

If m is odd and am < 0, then the zero solution of equation (8.2) is uniformly asymptotically stable. If am > 0, then the zero solution of equation (8.2) is unstable. Concluding remark. Assumption (8.1) is natural and it is a consequence of the smoothness of the center manifold, which states that if g is a Ck -function, for k ≥ 1, then hgρ is also a Ck -function. Consequently if g is a C∞ -function, then the center manifold hgρ is also a C∞ -function. Assumption (8.1) can be obtained by using the approximation of the center manifold hgρ . The proof of the smoothness result is omitted here and it can be done in similar way as in [11].

Acknowledgements A part of this work has been done when the second author was visiting the Abdus Salam International Centre for Theoretical Physics, ICTP, Trieste-Italy. He would like to acknowledge the centre for the support.

References [1] M. Adimy and K. Ezzinbi, Semi groupes intégrés et e´ quations différentielles a` retard en dimension infinie, C. R. Acad. Sci. Paris, t. 323, série I, 481-486, (1996). [2] M.Adimy and K.Ezzinbi, A class of linear partial neutral functional differential equations with non-dense domain, Journal of Differential Equations, 147, 285-332, (1998). [3] M. Adimy and K. Ezzinbi, Local existence and linearized stability for partial functional differential equations, Dynamic Systems and Applications, Vol. 7, 389-403, (1998). [4] M. Adimy and K. Ezzinbi, Existence and stability of solutions for a class of partial neutral functional differential equations, Hiroshima Mathematical Journal, Vol. 34, No. 3, 251-294, (2004). [5] M. Adimy, K. Ezzinbi and M. Laklach, Spectral decompostion for some partial neutral functional differential equations, Canadian Applied Mathematics Quarterly, Vol. 9, No. 4, 1-34, (2001). [6] J. Carr, Applications of Center Manifold Theory, Applied Mathematical Sciences, Springer-Verlag, Vol. 35, (1981). [7] S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, Journal of Differential equations, 74, 285-317, (1988). [8] G. Da Prato and A. Lunardi, Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach spaces, Archive for Rational Mechanics and Analysis, Vol. 101, 115-141, (1988).

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[9] W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups, in ” Differential equations in Banach spaces”, (A. Favini and E. Obrecht, Eds), Lecture Notes in Mathematics, Springer-Verlag, Vol. 1223, 61-73, (1986). [10] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H-O Walther, Delay Equations, Functional Complex and Nonlinear Analysis, Applied Mathematical Sciences, Springer-Verlag, (1995). [11] T. Faria, W. Huang and J. Wu, Smoothness of center manifolds for maps and formal adjoints for semilinear FDEs in general Banach spaces, SIAM, Journal of Mathematical Analysis, 34, 173-203, (2002). [12] J. K. Hale, Critical cases for neutral functional differential equations, Journal of Differential Equations, 10, 59-82, (1971). [13] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, (1977). [14] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer-Verlag, Vol. 840, (1981). [15] A. Keller, The stable, center-stable, center, center-unstable and unstable manifolds, Journal of Differential Equations, 3, 546-570, (1967). [16] A. Keller, Stability of the center-stable manifold, Journal of Mathematical Analysis and Applications, 18, 336-344, (1967). [17] X. Lin, J. W. H. So and J. Wu, Centre manifolds for partial differential equations with delays, Proceedings of the Royal Society of Edinburgh, 122, 237-254, (1992). [18] N. V. Minh and J. Wu, Invariant manifolds of partial functional differential equations, Journal of Differential Equations, 198, 381-421, (2004). [19] K. Palmer, On the stability of the center manifold, Journal of Applied Mathematics and Physics, (ZAMP), Vol. 38, 273-278, (1987). [20] S. N. Shimanov, On the stability in the critical case of a zero root for systems with time lag, Prikl. Mat. Mekh. 24, 447-457, (1960). [21] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of the American Mathematical Society, 200, 395-418 (1974). [22] J. W. H. So, Y. Yang and J. Wu, Center manifolds for functional partial differential equations: Smoothness and attractivity, Mathematica Japonica, 48, 67-81, (1998). [23] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences, Springer-Verlag, Vol. 14, (1975). [24] A. Vanderbauwhede and S. A. Van Gils, Center manifolds and contractions on a scale of Banach spaces, Journal of Functional Analysis, 72, 209-224, (1987).


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[25] E. Zeidler, Nonlinear Functional Analysis and its Applications, Tome I, Fixed Point theorems, Springer-Verlag, (1993). [26] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, (1985). [27] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, Vol. 119, (1996).


S TABILITY R ADII OF P OSITIVE L INEAR F UNCTIONAL D IFFERENTIAL S YSTEMS IN BANACH S PACES Pham Huu Anh Ngoc 1∗, Nguyen Van Minh 2† and Toshiki Naito 3‡ 1 Department of Mathematics, University of Hue, 32 Le Loi Street Hue City, Vietnam 2 Department of Mathematics, State University of West Georgia, Carrollton, GA 30118 3 Department of Mathematics, The University of Electro-Communications, Chofu, Tokyo 182-8585. Japan

Abstract In this paper we study stability radii of positive linear functional differential systems in Banach spaces under multi-perturbations and multi-affine perturbations. We prove that for the class of positive systems, complex stability radius, real stability radius and positive stability radius of positive systems under multi-perturbations (or multi-affine perturbations) coincide and they are computed via a simple formula. We illustrated the obtained results by an example.

Key Words: linear functional differential equation in Banach space, positive system, stability radius, multi-perturbation, multi-affine perturbation. AMS Subject Classification: 34 K20, 93 D09

1 Introduction Over the last decades there has been an increasing interest in the robust stability of dynamical systems that has many applications in control engineering. In the analysis of robust stability, the notion of stability radius shows to be an appropriate tool. By definition, the stability radius of a given asymptotically stable system x(t) ˙ = Ax(t) is the maximal γ > 0 for which all the systems of the form x(t) ˙ = (A + D∆E)x(t), k∆k < γ are asymptotically stable. Here, ∆ is a unknown disturbance matrix, D and E are given matrices defining the structure of the perturbations. ∗ E-mail

address: [email protected]. Corresponding author address: [email protected] ‡ E-mail address: [email protected] † E-mail

76

Pham Huu Anh Ngoc, Nguyen Van Minh and Toshiki Naito

In general, one may consider the disturbance matrix ∆ as real or complex. Accordingly, the maximal γ is called complex or real stability radius, respectively. The basic problem in the study of robustness of stability of the system is to characterize and compute these radii in terms of given matrices A, D, E. It is worth mentioning that these two stability radii are in general different. The analysis and computation of the complex stability radius for systems under structured perturbations was done first in [8] and extended later in many subsequent papers (see [9] for a survey up till 1990) while the computation of the real stability radius, being a much more difficult problem, was solved only recently, see e.g. [24]. The situation is much simpler for the class of positive systems. It is shown (see [11], [26]) that if A is a Metzler matrix (i.e. all off-diagonal entries of A are nonnegative) and D, E are nonnegative matrices, then the complex and the real stability radii coincide and can be computed directly by a simple formula. These results have been extended recently to infinite dimensional case (see [1], [2]), for positive continuous time-delay systems (see [27]-[30]) and for discrete time-delay systems (see [13], [19]). It is worth noticing that the notion of stability radius can be extended to various perturbation types [9]. Among perturbation types, two of the following perturbation types N

A → A + ∑ Di ∆i Ei

(Multi-perturbation)

(1)

i=1

N

A → A + ∑ δi A i

(Multi-affine perturbation)

(2)

i=1

are most well-known in control theory that include the perturbation types studied in the literature. The problem of computing the stability radii of positive linear systems without delays under multi-perturbations was studied in [20]. Recently, in [21], the authors successfully computed the stability radii of positive linear functional differential equations under multi-perturbations and multi-affine perturbations. Although recently there have been many works dedicated to extending the abovementioned results to the infinite dimensional case (see e.g. [1], [2], [3], [10], [23], [31]), the problem of computing the stability radii of positive linear functional differential equations in Banach spaces under multi-perturbations and multi-affine perturbations as considered in [21] has not been studied yet. And the main purpose of this paper is to extend the results of [21] to general retarded systems described by linear functional differential equations in Banach spaces of the form 0

x(t) ˙ = A0 x(t) +

−h

d[η(θ)]x(t + θ),

where A0 is a generator of a compact C0 -semi-group and η(·) is a function of bounded variation on [−h, 0] with values in a Banach space. Formulae will be derived for the stability radii of these general systems in terms of the operator A0 and the function η which define the initial system. The organization of the paper is as follows. In the next section, we summarize some notations and give preliminary results on C0 - semigroups on Banach spaces and on Banach

Robust Stability of Linear Functional Differential Equations

77

lattice theory which will be used in the sequel. In Section 3, a lower and upper bound for the complex stability radius of a general functional differential equation under multiperturbations are given. Then, for the class of positive functional differential equations it will be shown that the complex, real, positive stability radii under multi-perturbations coincide and a simple formula for their computation can be established. In Section 4, we study stability radii of positive functional differential equations under multi-affine perturbations. We also prove that the positive, real, complex stability radii coincide and can be computed by an explicit formula. An example is given to illustrate the obtained results.

2

Preliminaries

Let (X, k · k) be a Banach space. An X-valued function η(·) : [α, β] → X is said to be of bounded variation if Var(η; α, β) := sup

∑ kη(θk ) − η(θk−1 )k < +∞,

(3)

P[α,β] k

where the supremum is taken over the set of all finite partitions of the interval [α, β]. The set BV([α, β], X) of all X-valued functions η(·) of bounded variation on [α, β] satisfying η(α) = 0 is a Banach space endowed with the norm kηk = Var(η; α, β). Let Y,U be Banach spaces. Throughout this paper, we denote by L(Y,U), (L (Y,U)) the Banach space of all linear operators (bounded linear operators) from Y to U. Given δ(·) ∈ BV([α, β], L (Y,U)) then for any continuous functions γ ∈ C([α, β], K) and φ ∈ C([α, β],Y ), the integrals β α

γ(θ)d[δ(θ)] and

β α

d[δ(θ)]φ(θ)

exist and are defined respectively as the limits of S1 (P) := ∑kp γ(ζk )(δ(θk ) − δ(θk−1 )) and S2 (P) := ∑kp (δ(θk ) − δ(θk−1 ))φ(ζk ) as d(P) := maxk |θk − θk−1 | → 0, where P = {θ1 = α ≤ θ2 ≤ · · · ≤ θn = β} is any finite partition of the interval [α, β] and ζk ∈ [θk−1 , θk ] (see e.g. [5]). It is immediate from the definition that k αβ γ(θ)d[δ(θ)]k ≤ maxθ∈[α,β] |γ(θ)| kδk, k αβ d[δ(θ)]φ(θ)k ≤ maxθ∈[α,β] kφ(θ)k kδk. Then,

β

Lφ =

α

d[δ(θ)]φ(θ), φ ∈ C([−h, 0],Y ),

(4)

(5)

defines a bounded linear operator from C([α, β],Y ) to U. In the subsequent sections the following subspace of BV([−h, 0], X) will be used frequently : NBV([−h, 0], X) := {δ ∈ BV([−h, 0], X); δ(−h) = 0, δ is c.f.l. on (−h, 0)}.

(6)

It is clear that NBV([−h, 0], X) is closed in BV([−h, 0], X) and thus it is a Banach space with the norm kδk = Var(δ; −h, 0). Denote by C([−h, 0], X) the Banach space of all continuous functions on [−h, 0] with values in X normed by the supremum norm.

78


Let (T (t))t≥0 be a strongly continuous semigroup (or shortly, C0 -semigroup) of bounded linear operators on complex Banach space (X, k · k). Denote by A the generator of the semigroup (T (t))t≥0 and D (A) its domain. That is,

T (t)x − x ∈X t→0 t

D (A) := x ∈ X : lim

and

T (t)x − x , x ∈ D (A). t Since A is a closed operator, D (A) is a Banach space with the graph norm Ax = lim t→0

kxkD (A) := kxk + kAxk,

x ∈ D (A).

(7)

The resolvent set ρ(A), by definition, consists of all λ ∈ C for which (λIX − A) has a bounded linear inverse in X. The complement of ρ(A) in C is called the spectrum of A and denoted by σ(A). With the C0 -semigroup (T (t))t≥0 , we associate the following quantities: 1. The spectral bound, s(A) := sup{Re λ λ ∈ σ(A)}, where σ(A) is spectrum of the linear operator A. 2. The abscissa of uniform boundedness of the resolvent of A, s0 (A) := inf ω ∈ R : {Re λ > ω} ⊂ ρ(A) and sup kR(λ, A)k < ∞ . Reλ>ω

3. The growth bound ω1 (A), ω1 (A) := inf{ω ∈ R : there exists M > 0 such that kT (t)xk ≤ Meωt kxkD(A) for all x ∈ D (A) and t ≥ 0}. 4. The uniform growth bound ω0 (A), ω0 (A) := inf{ω ∈ R : there exists M > 0 such that kT (t)k ≤ Meωt for all and t ≥ 0}. It is well-known that −∞ ≤ s(A) ≤ ω1 (A) ≤ s0 (A) ≤ ω0 (A) ≤ +∞, see, e.g [16], [22]. Next, the C0 -semigroup T (t)t≥0 is called : 1. Hurwitz stable if σ(A) ⊂ C− := {λ ∈ C : Re λ < 0}, 2. Strictly Hurwitz stable if s(A) < 0,

(8)


79

3. Exponentially stable if ω1 (A) < 0, 4. Uniformly exponentially stable if ω0 (A) < 0. It is well-known that for an eventually norm continuous semigroup, that is, lim kT (t) − T (t0 )k = 0, for some t0 ≥ 0,

t→t0

s(A) = ω1 (A) = s0 (A) = ω0 (A), see e.g. [16]. So, the notations of strictly Hurwitz stability, exponentially stability, uniformly exponentially stability coincide. To make the presentation self-contained, we give some basic facts on Banach lattices which will be used in the sequel (see, e.g. [15]). Let X 6= {0} be a real vector space endowed with an order relation ≤ . Then X is called an ordered vector space. Denote the positive elements of X by X+ := {x ∈ X : 0 ≤ x}. If furthermore the lattice property holds, that is, if x ∨ y := sup{x, y} ∈ X, for x, y ∈ X, then X is called a vector lattice. It is important to note that X+ is generating, that is, X = X+ − X+ . Then, the modulus of x ∈ X is defined by |x| := x ∨ (−x). If k · k is a norm on the vector lattice X satisfying the lattice norm property, that is, if |x| ≤ |y| ⇒ kxk ≤ kyk,

x, y ∈ X,

(9)

then X is called a normed vector lattice. If, in addition, (X, k · k) is a Banach space then X is called a (real) Banach lattice. We now extend the notion of Banach lattices to the complex case. For this extension all underlying vector lattices X are assumed to be relatively uniformly complete, that is, if for every sequence (λn )n∈N in R satisfying ∑∞ n=1 |λn | < +∞ and for every x ∈ X and every sequence (xn )n∈N in X it holds that 0 ≤ xn ≤ λn x ⇒ sup

n

∑ xi

n∈N i=1

∈ X.

Now let X be a relatively uniformly complete vector lattice. The complexification of X is defined by XC = X + ıX. The modulus of z = x + ıy ∈ XC is defined by |z| = sup |(cos φ)x + (sin φ)y| ∈ X.

(10)

0≤φ≤2π

A complex vector lattice is defined as the complexification of a relatively uniformly complete vector lattice endowed the modulus (10). If X is normed then kxk := k|x|k,

x ∈ XC

(11)

defines a norm on XC satisfying the lattice norm property. If X is a Banach lattice then XC endowed the modulus (10) and the norm (11) is called a complex Banach lattice. Throughout of this paper, for simplicity of presentation, we denote X, XR instead of XC , X, respectively. Let ER , FR be real Banach lattices and T ∈ L(ER , FR ). Then T is called positive and denoted by T ≥ 0 if T (E+ ) ⊂ F+ . By S ≤ T we mean T − S ≥ 0, for T, S ∈ L(ER , FR ).

80


An operator T ∈ L(E, F) is called real if T (ER ) ⊂ FR . Then a operator T ∈ L(E, F) is called positive (T ≥ 0) if T is real and T (E+ ) ⊂ F+ . We introduce the notations LR (E, F) := {T ∈ L(E, F) : T real} ; L R (E, F) := {T ∈ L (E, F) : T real}, L+ := {T ∈ L(E, F) : T ≥ 0} ;

L + (E, F) := {T ∈ L (E, F) : T ≥ 0}.

(12) (13)

For T ∈ L + (E, F), we emphasize the simple but important fact kT k =

sup

kT xk.

(14)

x∈E+ ,kxk=1

Finally, δ ∈ NBV([α, β], L R (E, F)) is called increasing if δ(θ1 ) − δ(θ2 ) ≥ 0 for α ≤ θ1 ≤ θ2 ≤ β. In this paper, we always define inf 0/ = +∞, 0−1 = +∞.

3

Stability Radii of Linear Abstract Functional Differential Equations under Multi-perturbations

3.1

Abstract Functional Differential Equations

We let X be a complex Banach lattices which makes C([−h, 0], X) a Banach lattice as well. We consider a linear retarded system described by the following general functional differential equation x(t) ˙ = A0 x(t) + Lxt , t ≥ 0, x(t) ∈ X (15) x(θ) = φ0 (θ), θ ∈ [−h, 0]. where, for each t ≥ 0, xt ∈ C := C([−h, 0], X) is defined by xt (θ) = x(t + θ), θ ∈ [−h, 0], A0 is the generator of a C0 -semigroup on X and L : C([−h, 0], X) → X is a bounded linear operator defined by 0

Lφ =

−h

d[η(θ)]φ(θ), t ≥ 0, φ ∈ C([−h, 0], X).

(16)

Here η(·) ∈ NBV([−h, 0], L (X)) is a given L (X)-valued function of bounded variation on [−h, 0] such that η vanishes at −h and is c.f.l. on [−h, 0]. We shall extend the definition of η to R by setting η(θ) = η(−h) = 0 for all θ ≤ −h, η(θ) = η(0) for all θ ≥ 0. Definition 3.1. A function x ∈ C([−h, 0], X) is called a solution of the linear retarded system (15)-(16), if a) x is right-size differentiable at 0 and continuously differentiable for t > 0. b) x(t) ∈ D (A0 ) for t ≥ 0. c) (15) is satisfied for t ≥ 0. To (15)-(16), we associate the following operator A on the Banach space C([−h, 0], X). Let A be the differential operator 0

0

A f = f , D (A) := { f ∈ C1 ([−h, 0], X) : f (0) ∈ D (A0 ), f (0) = A0 f (0) + L f }.

(17)


81

Theorem 3.2. [16] The operator A defined in (17) is the generator of a C0 -semigroup (T (t))t≥0 on C([−h, 0], X) satisfying the translation property ( f (t + s) if t + s ≤ 0 (18) T (t) f (s) = T (t + s) f (0) if t + s > 0, f ∈ C([−h, 0], X). Moreover, for φ0 ∈ D (A) define x : [−h, ∞) → X by ( φ0 (t) if − h ≤ t ≤ 0 x(t) = T (t)φ0 (0) if t > 0.

(19)

Then x is the unique solution of the linear retarded system (15)-(16) . The C0 -semigroup (T (t))t≥0 in the above theorem is called solution semigroup of linear retarded system (15)-(16) and its operator A is called the solution operator of the linear retarded system (15)-(16). Definition 3.3. The linear retarded system (15)-(16) is called, respectively, Hurwitz stable, strictly Hurwitz stable, exponentially stable, uniformly exponentially stable if its solution semigroup (T (t))t≥0 is Hurwitz stable, strictly Hurwitz stable, exponentially stable, uniformly exponentially stable. Define

0

H(s) := sIX − A0 −

esθ d[η(θ)].

(20)

−h

H(s) is called the characteristic operator of the linear retarded system (15)-(16). By the retarded resolvent set ρ(A0 , η), we understand the set of all s ∈ C for which H(s) has a bounded linear inverse in X. The complement of ρ(A0 , η) in C is called the retarded spectrum and denoted by σ(A0 , η). Then, the following fact can be found in [16], page 224. σ(A) = σ(A0 , η).

(21)

Assuming that A0 generates a C0 -compact semigroup (U(t))t≥0 of bounded linear operators, it was shown in [32, 33, 17, 18] that the initial value problem t

u(t) = U(t)φ(0) + u0 = φ ∈ C ,

0

U(t − s)L(us )ds,

t ≥ 0,

has a unique continuous solution u(t; φ) for t ≥ −h, and {T (t)}t≥0 , T (t)φ = ut (·; φ), is a C0 -semigroup of linear (and compact for t > h) operators on C , with infinitesimal generator A given as above. Moreover, one has σ(A) = σP (A) = {s ∈ C : H(s)y = 0, for some y ∈ dom(A0 ) \ {0}} where σ p (A) denotes the point spectrum of A. In this case, since (T (t))t≥0 is eventually compact, it is eventually norm continuous. Hence, the above concepts of stability are actually the same if A0 generates a compact semigroup. For more information on the theory and applications of abstract functional differential equations we refer the reader to [6, 7, 17, 18, 33].

82

3.2


Stability Radii

Let X, Y, Ui0 ,Ui1 (i ∈ N := {i ∈ {1, 2, ..., N}) be complex Banach lattices and D0i ∈ L (Ui0 , X), D1i ∈ L (Ui1 , X) (i ∈ N ), E ∈ L (X,Y ). Assume that the retarded system (15)-(16) is Hurwitz stable and A0 , η are subject to multi-perturbations of the type A0 → A0∆ = A0 + ∑Ni=1 D0i ∆i E,

∆i ∈ L (Y,Ui0 ), i ∈ N (22)

η → ηδ =

η + ∑Ni=1 D1i δi E,

δi ∈

NBV([−h, 0], L (Y,Ui1 )),

i∈N

and thus the perturbed system is described by 0 x(t) ˙ = (A0 +∑Ni=1 D0i ∆i E)x(t)+ −h d[η(θ) + ∑Ni=1 D1i δi (θ)E]x(t + θ), t ≥ 0

(23) x(θ) = φ0 (θ), θ ∈ [−h, 0]. Here D0i ∈ L (Ui0 , X), D1i ∈ L (Ui1 , X)(i ∈ N ) and E ∈ L (X,Y ) are fixed and describe the structure of perturbations, while ∆i ∈ L (Y,Ui0 ) and δi (·) ∈ NBV([−h, 0], L (Y,Ui1 )), i ∈ N are unknown disturbances. We shall measure the size of each perturbation ∆˜ := [∆, δ] where ∆ := (∆0 , ..., ∆N ), δ := (δ0 , ..., δN ), ∆i ∈ L (Y,Ui0 ), δi ∈ NBV([−h, 0], L (Y,Ui1 )), i ∈ N , by the norm N

N

i=1

i=1

˜ := ∑ k∆i k + ∑ kδi k, kδi k := Var(δi ; −h, 0), i ∈ N. k∆k

(24)

Set ˜ = [∆, δ] : ∆i ∈ L (Y,Ui0 ), δi ∈ NBV([−h, 0], L (Y,Ui1 )), i ∈ N}, D C := {∆ ˜ = [∆, δ] : ∆i ∈ L R (Y,Ui0 ), δi ∈ NBV([−h, 0], L R (Y,Ui1 )), i ∈ N}, D R := {∆ ˜ = [∆, δ] : ∆i ∈ L + (Y,Ui0 ), δi ∈ NBV([−h, 0], L R (Y,Ui1 )), D + := {∆ δi is increasing for every i ∈ N} Then D C , D R , D + is called respectively the class of complex, real, nonnegative with respect to the perturbation structure (22). Denote by A∆˜ , ((T∆˜ (t))t≥0 the solution operator, solution semigroup of the perturbed system (23), respectively. To study robustness of stability of the retarded system (15)-(16) we introduce the following. Definition 3.4. Let the retarded system (15)-(16) be Hurwitz stable. The complex, real, positive stability radius of the system with respect to perturbations of the form (22), measured by the norm (24), is defined respectively by ˜ : ∆˜ ∈ D C , σ(A ˜ ) 6⊂ C− }, rC = inf{k∆k ∆

(25)

˜ : ∆˜ ∈ D R , σ(A ˜ ) 6⊂ C− }, rR = inf{k∆k ∆ ˜ ˜ r+ = inf{k∆k : ∆ ∈ D + , σ(A∆˜ ) 6⊂ C− }.

(26) (27)


83

We define the associated transfer functions of the perturbed system (23) by setting Gki (s) = EH(s)−1 Dki

i ∈ N , k ∈ {0, 1},

(28)

−1 0 θs where H(s)−1 := sIX − A0 − −h e d[η(θ)] , Re s > s(A). We need the following technical lemmas. Lemma 3.5. Let X,Y be Banach spaces and S ∈ L (Y, X), T ∈ L (X,Y ). Then, IX − ST is invertible if and only if so is IY − T S. Proof. Assume that IX − ST is invertible. Setting R := (IX − ST )−1 , we can check that (IY − T S)(IY + T RS) = IY = (IY + T RS)(IY − T S). By the same way, we can prove that if IY − T S is invertible then so is IX − ST. Lemma 3.6. Assume that the sups∈C,Res≥0 kH(s)−1 k < +∞. Then

system

(15)-(16)

is

Hurwitz

stable

and

(i) If sups∈C,Res≥0 kG0i0 (s)k 6= 0 for some i0 ∈ N then, for every ε > 0, there exists a complex perturbation ∆˜ := [∆, δ] ∈ D C , such that ˜ < k∆k

1 sups∈C,Res≥0 kG0i0 (s)k

+ ε,

(29)

and σ(A∆˜ ) 6⊂ C− .

(30)

(ii) If sups∈C,Res≥0 kG1i0 (s)k 6= 0 for some i0 ∈ N then, for every ε > 0, there exists a complex perturbation ∆˜ := [∆, δ] ∈ D C , such that ˜ < k∆k

1 + ε, sups∈C,Res≥0 kG0i0 (s)k|e−hs |

(31)

and σ(A∆˜ ) 6⊂ C− .

(32)

(iii) In particular, if G0i (0) ∈ L + (Ui0 ,Y ), G1i (0) ∈ L + (Ui1 ,Y ) for every i ∈ N and max{maxi∈N kG0i (0)k, maxi∈N kG1i (0)k} 6= 0 then, for every ε > 0, there exists a nonnegative perturbation ∆˜ ∈ D + satisfying ˜ < k∆k

1 max{maxi∈N kG0i (0)k,

maxi∈N kG1i (0)k}

+ ε.

(33)

and σ(A∆˜ ) 6⊂ C− .

(34)

84


Proof. (i) It follows from sups∈C,Res≥0 kH(s)−1 k < +∞ that sups∈C,Re s≥ 0 kG0i0 (s)k < +∞. For ε > 0, there exists s0 ∈ C, Re s0 ≥ 0 such that 1 ε 1 < + kG0i0 (s0 )k sups∈C,Re s≥ 0 kG0i0 (s)k 2 1 < kG0 1(s )k kG0i (s0 )u0 k i0 0 0 (Y )∗ , ky∗0 k = 1 satisfying

Let u0 ∈ Ui00 ku0 k = 1 satisfy Theorem, there exists y∗0 ∈

+ 2ε . Then, by the Hahn-Banach

y∗0 (G0i0 (s0 )u0 ) = kG0i0 (s0 )u0 k. Define ∆0i0 := It is easy to see that k∆0i0 k = ∆0i0 Ex0 = u0 . Therefore,

1 u0 y∗ (·) ∈ 0 kGi0 (s0 )u0 k 0 1 . kG0i (s0 )u0 k 0

L (Y,Ui00 ).

Setting x0 := H(s0 )−1 D0i0 u0 ∈ D (A0 ), we have

0 6= x0 = H(s0 )−1 D0i0 ∆0i0 Ex0 .

This implies ((A0 + D0i0 ∆0i0 E) +

0 −h

es0 θ d[η(θ)])x0 = s0 x0 ,

x0 ∈ D (A0 ), x0 6= 0.

We define ∆˜ := [∆, δ] = ((∆0 , ..., ∆N ), (δ0 , ..., δN )) , where ∆i = ∆0i0 if i = i0 otherwise ∆i = 0, i ∈ N and δi = 0 for all i ∈ N . Then, we have ˜ = k∆0i k < k∆k 0

1 sups∈C,Re s≥ 0 kG0i0 (s)k

+ε

and

s(A∆˜ ) ≥ Re s0 ≥ 0.

(ii) The proof is similar that of (i). For ε > 0, there exists s1 ∈ C, Re s1 ≥ 0 such that 1 1 ε < + kG1i0 (s1 )k|e−s1 h | sups∈C,Re s≥ 0 kG1i0 (s)k|e−sh | 2 1 < kG1 1(s )k kG1i (s1 )u1 k i0 1 0 (Y )∗ , ky∗1 k = 1 satisfying

Let u1 ∈ Ui10 , ku1 k = 1 satisfy Theorem, there exists y∗1 ∈

+ 2ε |e−hs1 |. By the Hahn-Banach

y∗1 (G1i0 (s1 )u1 ) = kG1i0 (s1 )u1 k. Define ∆1i0 := It is easy to see that k∆1i0 k =

1 u1 y∗ (·) ∈ 1 kGi0 (s1 )u1 k 1 1 . kG1i (s1 )u1 k

L (Y,Ui10 ).

We now consider the following step function

0

( 0 if θ = −h, δi0 (θ) = ∆1i0 ehs1 if θ ∈ (−h, 0].


85

It is clear that δi0 ∈ NBV([−h, 0], L (Y,Ui10 )), kδ1 k = Var(δ1 ; −h, 0) = k∆1i0 k|ehs1 |, Setting x1 := H(s1 )−1 D1i0 u1 ∈ D (A0 ), we have Therefore, 0

0 6= x1 = H(s1 )−1 D10

−h

0 −h

es1 θ d[δ1 (θ)] = ∆1i0 .

−h es1 θ d[δi0 (θ)]Ex1 0

= ∆1i0 Ex1 = u1 .

es1 θ d[δi0 (θ)]Ex1 .

This implies 0

(A0 +

−h

es1 θ d[η(θ) + D1i0 δi0 E])x1 = s1 x1 ,

x1 ∈ D (A0 ), x1 6= 0.

We define ∆˜ := [∆, δ] = ((∆0 , ..., ∆N ), (δ0 , ..., δN )) , where δi = δi0 if i = i0 otherwise δi = 0, i ∈ N and ∆i = 0 for every i ∈ N . Then, we have ˜ = k∆1i k|es1 h | < k∆k 0

1 + ε and sups∈C,Re s≥ 0 kG1i0 (s)k|e−sh |

s(A∆˜ ) ≥ Re s1 ≥ 0.

(iii) If G0i (0) ∈ L + (Ui0 ,Y ) and G1i (0) ∈ L + (Ui1 ,Y ) for every i ∈ N then, by (14), kG0i (0)k = supu∈(Ui0 )+ ,kuk=1 kG0i (0)uk, kG1i (0)k = supu∈(Ui1 )+ ,kuk=1 kG1i (0)uk. Thus we can choose u0 ∈ (Ui0 )+ , u1 ∈ (Ui1 )+ ku0 k = ku1 k = 1 such that kG0i (0)u0 k = kG0i (0)k , kG1i (0)u1 k = kG1i (0)k. Since G0i (0)u0 ≥ 0, G1i (0)u1 ≥ 0, by the Hahn-Banach theorem for positive operators (see e.g. [34] page 249), there exist positive linear forms y∗0 ∈ (Y )∗ , y∗1 ∈ (Y )∗ of dual norms ky∗0 k = ky∗1 k = 1 such that y∗0 Gi (0)u0 = kGi (0)u0 k, y∗1 Gi (0)u1 = kGi (0)u1 k. Hence the perturbation ∆˜ constructed as in (i), (ii) (where s0 = s1 = 0) is nonnegative. The proof is complete. Using Lemma 3.6 we obtain the following estimates for the complex radius rC . Theorem 3.7. Let the system (15)-(16) be Hurwitz stable and sups∈C,Res≥0 kH(s)−1 k < +∞. Assume that A0 , η are subjected to multi-perturbations of the form (22). Then, we have

max ≤

max

1 maxi∈N { sups∈C,Res≥0 kG0i (s)k},

≤ rC maxi∈N { sups∈C,Res≥0 kG1i (s)k}

1 . (35) maxi∈N { sups∈C,Res≥0 kG1i (s)k|e−hs |}

maxi∈N { sups∈C,Res≥0 kG0i (s)k},

Proof. Assume that rC < +∞. Let ∆˜ := [∆, δ],

∆ := (∆0 , ..., ∆N ), δ := (δ0 , ..., δN ), ∆i ∈ L (Y,Ui0 ), δi ∈ NBV([−h, 0], L (Y,Ui1 )), i ∈ N

86


satisfy σ(A∆˜ ) 6⊂ C− . By (21), there exists s0 ∈ C, Res0 ≥ 0 such that s0 ∈ σ (A0 + 0 ∑Ni=1 D0i ∆i E)+ −h es0 θ d[η(θ)+ ∑Ni=1 D1i δi (θ)E] . Since the system (15)-(16) is Hurwitz sta0 s0 θ e d[η(θ)]) has a bounded linear inverse. Since ble, H(s0 ) = (s0 IX − A0 − −h ! N

=

∑

IX −

N

N

0

i=1

−h

s0 IX − (A0 + ∑ D0i ∆i E) − N

D0i ∆i −

i=1

∑

es0 θ d[η(θ) + ∑ D1i δi (θ)E] i=1

0

D1i

s0 θ

e −h

i=1

! −1 d[δi (θ)] EH(s0 ) H(s0 ),

and since this operator is not invertible, the operator N

IX −

∑

D0i ∆i −

i=1

N

∑

D1i

i=1

0

s0 θ

e −h

! −1 d[δi (θ)] EH(s0 ) ,

is not invertible. Then, it follows from Lemma 3.5 that the operator N

N

0

i=1

i=1

−h

IY − EH(s0 )−1 ( ∑ D0i ∆i + ∑ D1i

! es0 θ d[δi (θ)]

is not invertible. Hence −1

kEH(s0 )

N

∑

D0i ∆i +

i=1

N

∑

D1i

i=1

0

s0 θ

e −h

d[δi (θ)] k ≥ 1.

Using the inequalities in (4), we get max{max kG0i (s0 )k, max kG1i (s0 )k} i∈N

i∈N

N

N

i=1

i=1

∑ k∆i k + ∑ kδi k

Hence

max max { i∈N

sup s∈C,Res≥0

kG0i (s)k},

max { i∈N

sup s∈C,Res≥0

≥ 1.

˜ ≥ 1, k∆k

kG1i (s)k}

(36)

or equivalently, ˜ ≥ k∆k

max

1 maxi∈N { sups∈C,Res≥0 kG0i (s)k},

. maxi∈N { sups∈C,Res≥0 kG1i (s)k}

By the definition of the complex stability radius rC , we have rC ≥

1 max maxi∈N { sups∈C,Res≥0 kG0i (s)k}, maxi∈N { sups∈C,Res≥0 kG1i (s)k}

It remains to prove that rC ≤

max

1 . maxi∈N { sups∈C,Res≥0 kG1i (s)k|e−hs |}

maxi∈N { sups∈C,Res≥0 kG0i (s)k},

Robust Stability of Linear Functional Differential Equations Fix ε > 0. It follows from (36) that max max { sup kG0i (s)k}, max { i∈N

i∈N

s∈C,Res≥0

sup s∈C,Res≥0

87

kG1i (s)k} 6= 0.

Taking the Lemma 3.6 (i), (ii) into account, there exists a complex perturbation ∆˜ := [∆, δ] ∈ D C such that ˜ < k∆k and

1 + ε, max maxi∈N { sups∈C,Res≥0 kG0i (s)k}, maxi∈N { sups∈C,Res≥0 kG1i (s)k|e−hs |} σ(A∆˜ ) 6⊂ C− .

By the definition of the complex stability radius rC , rC
λ ⇐⇒ s(A) > λ.

In the rest of this paper, we suppose that Assumption 3.11. A0 generates a positive C0 -semigroup (U(t))t≥0 of bounded compact / linear operators on X, L ∈ L (C[−h, 0], X , X is a positive operator and σ(A) 6= 0.

Remark 3.12. As noted in the beginning of this section, if A0 generates a C0 -compact semigroup then the solution semigroup (T (t))t≥0 of the system (15)-(16) is eventually compact. Hence, the concepts of stability in Definition 3.3 are actually the same. Furthermore, if Assumption 3.11 is fulfilled then it is easy to see that all conditions of Lemma 3.10(ii) is satisfied. 0 sθ Lemma 3.13. Let H(s) = sIX − A0 − −h e d[η(θ)]. Then i) H(t1 )−1 ≥ H(t2 )−1 ≥ 0 f or t2 > t1 > s(A).

(37)

(ii) Let s = α + ıβ ∈ C, α > s(A). Then |H(s)−1 x| ≤ H(α)−1 |x|, x ∈ X. (38) Proof. (i) It follows from L ∈ L (C[−h, 0], X), X being a positive operator and the equality 0

(

−h

etθ d[η(θ)])x =

0

−h

(etθ x)d[η(θ)], x ∈ X,

0 tθ e d[η(θ)] ∈ L + (X) for every t ∈ R. Since A0 is a generator of a positive C0 that −h 0 tθ 0 tθ semigroup on X and −h e d[η(θ)] ∈ L + (X), A0 + −h e d[η(θ)] also generates a positive 0 tθ e d[η(θ)]). From C0 -semigroup on X. By Lemma 3.10 (ii), if t > s(A) then t > s(A0 + −h Theorem 1.1 (C-III), page 292 [16], we have 0

(tIX − A0 −

−h

etθ d[η(θ)])−1 = H(t)−1 ≥ 0

for every t > s(A).

(39)


89

It is easy to verify that the following resolvent equation holds 0

H(t1 )−1 − H(t2 )−1 = (t2 − t1 )H(t1 )−1 H(t2 )−1 + H(t1 )−1

−h

(et1 θ − et2 θ )d[η(θ)]H(t2 )−1 .

It follows from (39) that H(t1 )−1 ≥ H(t2 )−1 ≥ 0 f or t2 > t1 > s(A). (ii) Let A0 be the generator of the positive semigroup (U(t))t≥0 on X. Set B(s) := 0 −h esθ d[η(θ)], s ∈ C. It is well-known that A0 + B(s) with domain D (A0 + B(s)) = D (A0 ) is the generator of a C0 -semigroup (Vs (t))t≥0 satisfying n t t Vs (t)x = lim U( )e n B(s) x, n→∞ n

for t ≥ 0, x ∈ X,

(40)

see e.g [16], page 44. It is easy to see that 0

|(

−h

esθ d[η(θ)])x| ≤ (

0

es(A)θ d[η(θ)])|x|,

−h

x ∈ X,

where s ∈ C, Res > s(A). Using the above inequality, we could check that ω0 (A0 + 0 0 −h esθ d[η(θ)]) ≤ ω0 (A0 + −h es(A)θ d[η(θ)]), s ∈ C, Res > s(A), where ω0 (·) is growth bound of a C0 -semigroup with the generator (·). On the other hand, since A0 is a generator 0 sθ 0 s(A)θ of a compact C0 -semigroup, so do (A0 + −h e d[η(θ)]), (A0 + −h e d[η(θ)]), see, e.g. [22], page 79. It follows from the spectral mapping theorem that 0

ω0 (A0 +

ω0 (A0 +

−h 0

−h

esθ d[η(θ)]) = s(A0 +

es(A)θ d[η(θ)]) = s(A0 +

0

esθ d[η(θ)]),

(41)

es(A)θ d[η(θ)]).

(42)

−h 0 −h

0 sθ From (21), s = (α + ıβ) ∈ ρ(A0 + −h e d[η(θ)]), for every s ∈ C, Re s > s(A). By 0 s(A)θ 0 s(A)θ e d[η(θ)]) = ω0 (A0 + −h e d[η(θ)]). Lemma 3.10 (ii) and (42), s(A) = s(A0 + −h 0 s(A)θ Then, we derive that if Re s = α > s(A) then Re s > ω0 (A0 + −h e d[η(θ)]) ≥ ω0 (A0 + 0 sθ −h e d[η(θ)]). So, we can represent the following

H(s)−1 x = (sIX − A0 −

0 −h

∞

esθ d[η(θ)])−1 x = 0

e−st Vs (t)xdt.

From the formulas (40), (43), it is easy to see that ∞

|H(s)−1 x| ≤ 0

e−αt |Vs (t)||x|dt ≤

∞ 0

e−αt Vα (t)|x|dt = H(α)−1 |x|.

(43)

90


Remark 3.14. It follows from the above lemma that if the system (15)-(16) is Hurwitz stable and Assumption 3.11 is fulfilled then sups∈C,Res≥0 kH(s)k < +∞. In fact, by (37) and (38), we have |H(s)−1 x| ≤ H(Re s)−1 |x| ≤ H(0)−1 |x|, x ∈ X, for every s ∈ C, Re s ≥ 0. From the lattice norm property (9) and the definition of the operator norm, we get kH(s)−1 k ≤ kH(0)−1 k, for every s ∈ C, Res ≥ 0. We are now in the position to prove the main result of this section. Theorem 3.15. Assume that the linear retarded system (15)-(16) is Hurwitz stable and A0 , η are subjected to multi-perturbations of the form (22) where D0i ∈ L + (Ui0 , X), D1i ∈ L + (Ui1 , X) (i ∈ N ) and E ∈ L + (X,Y ). Then, we have rC = rR = r+ =


maxi∈N kG1i (0)k}

.

(44)

Proof. Suppose that rC < +∞, as otherwise there is nothing to show. We prove that rC =


maxi∈N kG1i (0)k}

(45)

.

It follows from (37), (38) and D0i ∈ L + (Ui0 , X), D1i ∈ L + (Ui1 , X) (i ∈ N ) that |G0i (λ)x| = |(EH(λ)−1 D0i )x| ≤ (EH(Reλ)−1 D0i )|x| ≤ (EH(0)−1 D0i )|x|,

x ∈ Ui0 , i ∈ N ,

|G1i (λ)y| = |(EH(λ)−1 D1i )y| ≤ (EH(Reλ)−1 D0i )|y| ≤ (EH(0)−1 D1i )|y|,

y ∈ Ui1 , i ∈ N ,

for every λ ∈ C, Re λ ≥ 0. From the lattice norm property (9) and the definition of the operator norm, we have kG1i (λ)k ≤ kG1i (0)k

and kG1i (λ)k ≤ kG1i (0)k, i ∈ N ,

(46)

for every λ ∈ C, Re λ ≥ 0. Then (45) now follows from (35) and (46). On the other hand, by D0i ∈ L + (Ui0 , X), D1i ∈ L + (Ui1 , X) (i ∈ N ) and (37), we have G0i (0) ∈ L + (Ui0 ,Y ), G1i (0) ∈ L + (Ui1 ,Y ), i ∈ N . It follows from Lemma 3.6 and the definition of r+ that r+ ≤


maxi∈N kG1i (0)k}

.

(47)

Finally, (44) follows from (45), (47) and the inequalities rC ≤ rR ≤ r+ . This completes our proof. We illustrate the above result by an example.


91

Example 3.16. Consider the equation ∂u(t,x) ∂t

∂2 u(t,x) ∂2 x

=

− d(x)u(t, x) + b(x)u(t − 1, x)

(t ≥ 0, x ∈ [0, 1])

(48)

with boundary condition ∂u(t,x) ∂x |x=0

=0=

∂u(t,x) ∂x |x=1

(49)

(t ≥ 0)

and initial condition u(s, x) = ψ(s, x),

s ∈ [−1, 0], x ∈ [0, 1].

Let X be the function space C[0, 1] equipped with sup-norm, and let B be defined by 00

Bh = h , 0

0

with domain D(B) := {h ∈ C2 [0, 1] : h (0) = h (1) = 0}. Denote by Mb , Md the respective multiplication operators for b, d ∈ X, b, d ≥ 0. Then (48)-(49) takes the abstract time-delay system of the form u(t) ˙ = Bu(t) − Md u(t) + Mb u(t − 1), u0 = ψ ∈ C([−1, 0] × [0, 1]).

t ≥ 0,

(50)

It is well-known that B generates a positive compact semigroup, see [16], page 230. The same is true for the operator B − Md , see [22], page 79. Moreover, if maxx∈[0,1] (b(x) − d(x)) < 0 then the delay system (50) is uniformly exponentially stable, see [16], page 230. This is equivalent to the uniform exponential stability of the solutions of (48)-(49). We now assume that the functions b, d are subjected to perturbations of the form b(x) −→ b(x) + δ0 (x),

d(x) −→ d(x) + δ1 (x),

δ0 , δ1 ∈ X.

From Theorem 3.15, the solutions of the following perturbed equation ∂u(t,x) ∂t

∂2 u(t,x) ∂2 x

− (d(x) + δ1 (x))u(t, x) + (b(x) + δ0 (x))u(t − 1, x) (t ≥ 0, x ∈ [0, 1]) (51) with boundary condition =

∂u(t,x) ∂x |x=0

=0=

∂u(t,x) ∂x |x=1

(t ≥ 0)

and initial condition u(s, x) = ψ(s, x),

s ∈ [−1, 0], x ∈ [0, 1],

are uniformly exponentially stable if max |δ0 (x)| + max |δ1 (x)|
sin kr

0 0 where < with r> = max(r, r ) and r< = min(r, r ). The asymptotic form of Ψ− (k, r) for large r is given by Eq. (12) with S+ replaced by S− (k) = [S+ (k)]∗ where ∗ represents complex conjugation. The Fredholm determinants associated with the kernels of Eq. (13) are denoted by D±(k). The integral equations for the Jost solutions are

f ± (k, r) = e±ikr −

Z ∞Z ∞ r

G(k, r, r 0)V (r 0 , s) f ±(k, s) ds dr 0,

(14)

0

The Fredholm determinant associated with the kernel of Eq. (14) is denoted by ∆(k). The Fredholm determinants have the following properties: ℜ D± (k), D(k), and ∆(k) are even functions of k, while ℑ D± (k) is an odd function of k. Both D(k) and ∆(k) are real for real k, and D(k) = ∆(k) for symmetric nonlocal potentials [46].

104

S.B. Qadri, B. Mulligan, M.F. Mahmood et al.

For a local potential, the reason that one can start either from a Jost function point of view or from a physical integral equation — Fredholm determinant point of view lies in the fact [45] that for a local potential the Jost functions L ± (k) and the Fredholm determinants D± (k) are identically equal,

L ± (k) = D± (k)

[ local potential ].

(15)

Also, with the exception of Eq. (13), for a local potential the integral equations become Volterra equations. Thus, the Fredholm determinants D(k) and ∆(k) become unity in the case of a local potential and can play no explicit role in the description of the scattering process. The Fredholm determinants D± (k) are not unity in the case of a local potential, consistent with the fact that they are the only Fredholm determinants needed to establish all of the scattering information. For a nonlocal potential, the integral equations with which D(k) and ∆(k) are associated are Fredholm integral equations rather than Volterra integral equations. Thus, the Fredholm determinants D(k) and ∆(k) are not unity, and it might be anticipated that a complete description of the scattering process would require these determinants in addition to the determinants D± (k). Nevertheless, zeros of the Fredholm determinants D± (k) and D(k) are not independent. Whenever there is a zero of D± (k) for real k there is also associated a zero of D(k) at that same value of k [40]. A zero of D± (k) for real k 6= 0 is called a continuum bound state [6,8], often referred to by the abbreviation CBS. The existence of a continuum bound state results from the fact that when D± (k) = 0 for real k 6= 0 solutions of the homogeneous integral equations associated with Eq.(l3), namely Ψ± h (k, r) =

Z∞Z∞

0 G± (k, r, r0)V (r0 , s) Ψ± h (k, s) ds dr ,

(13)

0 0

are normalizable. Trivial solutions Ψ± h (k, r) = 0 are the only solutions allowed when D± (k) 6= 0. A spurious state [40,43] occurs when D(k) is zero and D± (k) is nonzero for a real value of k. It is demonstrated in Ref. 40 that at a spurious state the regular solution ϕ(k, r) does not exist, although it is possible to obtain a real solution of Eq. (9) which is regular at the origin. In the case of a local potential, it is customary [47] to define the Jost functions L ± (k) in terms of the Jost solutions at r = 0: that is

L ± (k) = f ± (k, r)|r=0 .

(17)

No ambiguity exists in showing that this definition leads to Eq. (15). Thus Eq. (15) could equally well be used as a definition of L ±(k) for a local potential. For a nonlocal potential, difficulties can arise in using Eq. (17) as the definition of ± L (k). The Jost solutions f ± (k, r) do not exist at a spurious state. At a continuum bound state, f ± (k, r) may or may not exist. Under circumstances in which there are no difficulties with the existence of f ± (k, r), that is, at values of k for which D(k) 6= 0, for a symmetric nonlocal potential it has been shown [48-51] that D±(k) and L ±(k) are related by the

Effects of Nonlocality and Phase Shift Definitions...

105

expression

L ±(k) = D±(k)/D(k).

[ symmetric nonlocal potential ]

(18)

However, the derivation of Eq. (18) breaks down when f ± (k, r) do not exist. Correspondingly, the definition of the Jost functions as given in Eq. (17) will no longer hold if f ± (k, r) do not exist. This is especially a problem in the case of redundant states, since it would mean a failure of the existence of the Jost functions at any energy. However, it has been shown [52], using the Green’s function cancellation method of Krause and Mulligan [53] to construct a potential which produces a redundant state, that under these circumstances Eq. (18) can nevertheless be used as a definition of L + (k). In addition, the results in Ref. [52] demonstrate that the Jost functions L ±(k) can be directly related to the kernels of integral equations by means of Fredholm determinants without referring to the solutions of the integral equations involved. Consequently, in defining the Jost functions, the question of the existence of Jost solutions at a CBS or in the presence of a redundant state can be avoided. As we shall see, by taking the definition of the Jost functions as the ratio of Fredholm determinants D±(k) to D(k) and proceeding with the derivation of Levinson ’s theorem, we run into no problems. There are, nevertheless, other difficulties associated with the derivation of Levinson’s theorem in the case of a nonlocal potential. Consider Eqs. (l5) and (18). The expression for L ±(k) for a local potential is a special case of the expression for L ± (k) for a nonlocal potential. Because of the presence of D(k) in Eq.(18), when considering the definition of the phase shift for a nonlocal potential an ambiguity arises. This ambiguity has been discussed in detail in Ref. [41]. The presence of the Fredholm determinant D(k) results in two possible definitions of the phase shift: δD = -phase of D+(k) and δL =-phase of L +(k). The phase shift δL gives a continuous phase at a CBS and discontinuity of −π in the phase shift as the energy increases through each spurious state. The phase shift δD is continuous at a spurious state, but gives a discontinuity of π with increasing energy to the phase shift at a CBS. Since for a local potential the phase of L ±(k) is the same as the phase of D±(k), the phase shifts δL and δD are identical in that case. For a nonlocal potential, a derivation of Levinson’s theorem can proceed in much the same way as that described in Sec. 2 in the case of a local potential. However, because of the two possible definitions of the phase shift, there will be two possible results for Levinson’s theorem. In one case, the function f (z) to be employed in Eq. (4) will be the Jost function L +(k) defined in Eq. (l8); in the other case one would use D+ (k).

4

Properties of D+ (k), D(k), and L + (k) for a Nonlocal Potential

For a local potential, it is known that L + (k) and therefore D+ (k) have the property L +(k) ≡ D+ (k) 6= 0 for real k 6= 0. As we have already noted, in the case of a local potential the Fredholm determinant D+ (k) cannot have zeros on the real k-axis. Also, we will see that for a nonlocal potential zeros of D(k) [and thus poles of the Jost function L +(k)] are found on and above the real axis in some cases. For these reasons, in the case of a nonlocal potential the techniques from complex analysis [54] used to obtain the character of the Fredholm determinant D+ (k) and the Jost function L + (k) for a local potential are not sufficient for a general discussion of D+(k), D(k), and L +(k). However, we have demonstrated [55] that

106


techniques from functional analysis can be used for a wide class of nonlocal potentials to obtain the analytic properties of D+(k), D(k), and L +(k). These techniques were developed by Iwasaki and Mulligan [56], who focused on the fact that the kernels of integral Eqs. (11), (13), and (14) are not square integrable. Noting that the usual derivations of the Fredholm alternative [57] do not apply in cases where kernels are not square integrable, they make use of Riesz-Schauder theory [58], which demonstrates that the Fredholm alternative easily can be extended to integral equations for which the operators are compact. The conditions on the nonlocal potential V (r, r0) necessary for compactness of the kernels of (11) and (13), are given in Ref. [55] or Ref. [56]. Complete continuity of the kernels of the integral Eqs. (11) and (13) is crucial to establishing the existence of the Fredholm determinants D+ (k) and D(k) for a nonlocal potential. It is shown in Ref. [55] that these kernels are compact for k real, from which it follows that the Fredholm determinants D+ (k) and D(k) exist for k real. Fredholm determinants can then be defined for complex k by extending these definitions, established on the real axis, to the complex plane. The properties of these determinants, established in Ref. [55], are summarized below. The character of D+ (k) and D(k) in the upper half plane are such that D+ (k) when so extended will be analytic in that portion of the k plane, while D(k) may not. Thus with nonlocal potentials, as with local potentials, poles of D+ (k) are confined to the bottom half of the complex plane. Zeros of D+ (k) in the upper half of the complex k plane must lie either on the real axis or on the positive imaginary axis. In the case of a local potential, D+ (k) cannot have a zero for k real [40]; since for a local potential L +(k) 6= 0 for real k 6= 0 [47], it follows that for a local potential D+ (k) must not be zero for real k 6= 0. On the other hand, the modified definition of L +(k) in the case of a nonlocal potential, given by Eq. (l8), allows for a zero of D+ (k) on the real k axis as long as such a zero is accompanied by a corresponding zero of D(k), resulting in a bound state in the continuum. Such zeros of D+ (k) and D(k) on the real k axis will occur in pairs, symmetric about the imaginary axis. That such a zero of D(k) is possible on the real k axis has been established by many examples [40]. However, because the boundary conditions associated with the equation for the regular solution are not the same as those associated with the physical solution, in general it would not be expected that D(k) would be zero for the same values of k as is D+ (k). Indeed, for a local potential D(k) is unity at all zeros of D+ (k). Thus, as mentioned earlier, a zero of D(k) at the same value of k as that for which D+(k) is zero is a special circumstance; usually zeros of D(k) can be expected to occur at values of k for which D+ (k) is not zero. However, zeros of D(k) which occur on the real k axis always occur in pairs, symmetric with respect to the origin. Zeros of D(k) off of the real axis occur in sets of four, symmetric with respect to both the real and imaginary k-axes. Since D(k) is a ratio of two polynomials of the same order, the number of zeros and number of poles in the entire complex plane are equal. We already have pointed out that D(k) will not have poles on the real axis. Its poles will be symmetrically distributed in the upper and lower half planes. If a pair of zeros of D happen to occur on the real axis, then the number of poles in either of the half planes, excluding the real axis, exceeds the number of zeros by 1. If 2m zeros of D(k) lie on the real axis (symmetrically located about the imaginary axis) then in either half plane the number of poles exceeds the number of zeros by m, excluding the real axis.


107

A zero of D+ (k) on the real axis can be of any order. That is, a potential can be constructed which at a given real value k0 of k produces any specified number of continuum bound states. It has been demonstrated [59] that if the order of the zero of D+ (k) at k = k0 is m, then the order of the zero of D(k) at k = k0 is at most m + 1 and at least m. Furthermore, a zero of D(k) at k = k0 which occurs for real k0 6= 0 and for which D+ (k0 ) 6= 0 must be a simple zero [59]. That is, a zero of D(k) at k = k0 of order m where m > 1 must be accompanied by a zero of D+ (k0 ) of at least order m − 1. As discussed earlier, the Jost function L + (k)is given by the ratio of D+(k)to D(k). In taking this ratio, poles of D+(k)can be cancelled by poles of D(k) and zeros of D+ (k)can be cancelled by zeros of D(k). Thus, as already has been pointed out [52], the Jost function for a nonlocal potential does not usually contain as complete a set of information about the system as the determinants D+ (k)and D(k). The zeros of D(k) which do not cancel become poles of L +(k)and the poles of D(k) which do not cancel become zeros of L +(k). Thus, when m continuum bound states occur, there will be 2 m zeros of D+(k)and 2m zeros of D(k) located on the real k axis. These zeros will cancel, and as a result of the m continuum bound states the Jost function L +(k)will, for ℑk ≥ 0 have m zeros more than the number of its poles. Since bound states result in zeros of D+ (k)on the positive imaginary axis and since these zeros are not cancelled by zeros of D(k), these zeros of D+ (k)also will be zeros of L + (k)in the upper half plane. Thus if a potential is such that it has m continuum bound states and n bound states, the Jost function L +(k)for that potential will have (m + n) more zeros in the upper half plane than its number of poles in the upper half plane. For a spurious state, D(k) = 0 at the value of k at which the state occurs and D(k) = 0 at that same k. As discussed above, when D+ (k) 6= 0 only a first order zero of D(k) can occur. Thus at any given value of k there can be only one spurious state. If there are 2s such isolated zeros of D(k) on the real axis, then in the upper half plane (and, for that matter, in the lower half plane as well) D(k) will have s more poles than the number of zeros, excluding the real axis. Since D+(k)is analytic in the upper half-plane, if there are no bound states it follows that L + (k)will have s more zeros than its number of poles (excluding the poles on the real axis) in the upper half of the complex k-plane. Thus, if in addition to s spurious states there are n bound states, in the upper half plane L +(k)will have (s + n) more zeros than the number of poles. We are now in a position to discuss Levinson’s theorem in terms of L +(k)or in terms of D+ (k). Before beginning this discussion, however, we investigate the properties of (k) and D (k) at k=0, and associate these properties with solutions of Eq.(9).

5

Solutions at k = 0

Newton [60] has pointed out that for a nonsingular, central, local potential L + (k)can be zero at k = 0. Such a zero must be simple, and does not correspond to a normalizable solution of the radial equation. Newton refers to the solution of the radial equation under these circumstances as a half-bound state [61]. In the case of a nonlocal potential, such a state is also possible. It will be characterized by a zero of D+ (k)but not a zero of D(k); L +(k)thus will be zero under this circumstance. It is also possible in the case of a nonlocal potential to have normalizable states at k = 0. The Saito potential is an example of a nonlocal potential with a zero-energy bound state, although this case is complicated by the fact that the Saito

108


potential is not symmetric [62]. A nonlocal potential can have more than one of these states, which are characterized by zeros of both D+ (k)and D(k), and thus no zero of L +(k). Furthermore, if there are q of these states at k = 0, then the order of the zeros of D+ (k)and D(k) is 2q. This can easily be seen by realizing that such a state is a continuum bound state which happens to occur at k = 0 rather than some positive value of k. A CBS at k0 , where k0 > 0, is characterized by a pair of zeros of D+(k), one at k0 and the other at −k0 , and a pair of zeros of D(k) at those same positions; zeros of D+ (k)and D(k) at k = 0 clearly will occur as double zeros. The Levinson’s theorem contour for a half-bound state, since it implies a zero of

L +(k)and D+ (k), is handled for both δL and δD in exactly the same manner as for a local potential. Integration of either L + (k)0/L +(k) or D+ (k)0/D+ (k) clockwise around a semicircle located at the origin yields −πi, as discussed in Sec. 2. The treatment of the contour for Levinson’s theorem in the case of a zero energy bound state differs depending upon whether one is considering δL or δD . For a continuum bound state at zero energy, L +(k)will not have a zero at k = 0, and thus no adjustment of the contour is necessary to accommodate the presence of the state. As discussed above, however, D+ (k)will have a double zero at k = 0. In order to evaluate the integral of D+ (k)0/D+ (k) around an unambiguous contour in terms of the zero energy phase shift we can slightly displace the double zeros about the origin by η and let η → 0 as shown in Fig. 2. The contribution from each of the contours γ in Fig. 2 will be −πi, resulting in a total contribution as η → 0 of −2πi. It is also necessary to consider how the presence of the pair of zeros of D+ (k)affects the phase shift δD at k = 0. This question, however, is deferred until the next section.

Figure 2. Contour C for integration of D(k)+/D(k) in the case double zeros of D+(k) at the origin. These zeros, displaced by η in order to define an unambiguous contour, are indicated by the symbol o.


6 6.1

109

Derivation of the Generalized Levinson’s Theorem for the Phase Shift Case of n Bound States and One Half-Bound State

For a nonlocal potential with n bound states and one half-bound state, the Jost function L +(k)will have n zeros on the positive real axis and one zero at the origin, as shown in Fig. 3. There will be no poles on or above the real axis. Thus, making use of Eq. (4) for the contour C of Fig. 3 gives Z L +(k)0 dk = 2πin (19) + C L (k) As discussed in the case of a local potential in Sec. 2, the explicit line integral yields Z

L +(k)0 dk = 2i[δL (0) − δL (∞)] − πi. + C L (k)

(20)

Thus we obtain the same result as in the case of a local potential, namely 1 δL (0) − δL (∞) = (n + )π. 2

(21)

Figure 3. Contour C for the derivation of the Generalized Levinson’s Theorem for δL when there are n bound states and a half-bound state at k = 0. Zeros of L + (k)are indicated by the symbol o.

6.2

Case of n Bound States and m Continuum Bound States

As discussed in Sec. 4, the Jost function L +(k)for a potential with n bound states and m continuum bound states has n + m more zeros than poles in the upper half plane, and has no zeros or poles on the real axis. This set of conditions is indicated in Fig. 4, and applies whether the continuum bound states are at k = 0 or k > 0.

110

S.B. Qadri, B. Mulligan, M.F. Mahmood et al. Calculating the logarithmic derivative of L + (k)by the two different methods yields Z

and

L +(k)0 dk = 2πi(n + m) + C L (k)

(22)

L +(k)0 dk = 2i[δL (0) − δL (∞)]. + C L (k)

(23)

Z

Figure 4. Contour C for the derivation of the Generalized Levinson’s Theorem for δL when there are n bound states and m continuum bound states. Zeros and poles of L +(k)are indicated by o and χ, respectively. At least n zeros of L +(k)must be on the positive imaginary axis. Additional zeros of L +(k)and the poles of L +(k)in the upper half plane may be either on the positive imaginary axis or off the axis in pairs symmetrically located with respect to the imaginary axis. The assertion that there be n + m more zeros than poles in the upper half plane is a condition which takes into account all of the zeros and poles. Thus, in this case δL (0) − δL (∞) = (n + m)π.

(24)

If in addition there is a zero of L +(k)at k = 0 we get 1 δL (0) − δL (∞) = (n + m )π. 2

6.3

(25)

Case of n Bound States and p Spurious States

As discussed in Sec. 4, the Jost function L + (k)for a potential with n bound states and s spurious states has n + s more zeros than poles above the real axis. In addition, L +(k)is characterized by 2s poles located in pairs on the real axis. This situation is shown in Fig. 5. Using Eq. (4) for the contour C of Fig. 5 we get Z

L +(k)0 dk = 2πi(n + s). + C L (k)

(26)


111

Figure 5. Contour C for the derivation of the Generalized Levinson’s Theorem for δL when there aren bound states and s spurious states. Zeros and poles of L + (k)are indicated by o and χ respectively. At least n zeros of L +(k)must be on the positive imaginary axis and 2s poles of L +(k)must be on the real axis in pairs symmetric with respect to the imaginary axis. The additional zeros and poles of L +(k)in the upper half plane must either be on the positive imaginary axis, or in pairs located symmetrically with respect to that axis. The assertion that there be n + s more zeros than poles in the upper half plane is a condition which takes into account all of the zeros and poles of L + (k)above the real axis. Direct integration of the logarithmic derivative of L + (k)around C yields Z s L +(k)0 dk = 2i δL (0) − δL (∞) + ∑ [δL (ki + 0) − δL (ki − 0)] + 2πis, + C L (k) i=1

(27)

where the term 2πis arises from integrating around the 2s semicircles on the real axis, as shown in Fig. 5. When a spurious state occurs at k = ki there is a phase change of −π in going from k = ki − ε to k = ki + ε in the limit as ε → 0. This is δL (ki + 0) − δL (ki − 0) = −π.

(28)

L + (k)0 dk = 2i[δL (0) − δL (∞) − sπ] + 2πis. + C L (k)

(29)

Thus Eq. (27) becomes Z

Taken together, Eqs. (26) and (29) imply δL (0) − δL (∞) = (n + s)π.

(30)

If, in addition, there is a zero of L + (k)at k = 0 then we get 1 δL (0) − δL (∞) = (n + s )π. 2

(31)

112

6.4


Summary

We have seen that if we use δL as the definition of the phase shift, then Levinson’s theorem becomes (32) δL (0) − δL (∞) = (n + p)π, where n is the number of bound states and p is the number of continuum bound states (including those at k = 0) and/or spurious states. If, also, we have a zero of L + (k)at k = 0 then Eq. (32) becomes 1 (33) δL (0) − δL (∞) = (n + p + )π, 2

7 7.1

Derivation of the Generalized Levinson ’s Theorem for the Phase Shift Case of n Bound States and One Half-Bound State

With respect to bound states and half-bound states, the structure of the poles and zeros of D+ (k)is the same as that of L +(k). Thus, as shown in Fig. 6, there are no poles of D+(k)on or above the real axis, n zeros of D+ (k)on the positive imaginary axis corresponding to n bound states, and a simple zero of D+ (k)at k = 0 corresponding to a half-bound state. Applying Eq. (4) to the logarithmic derivative of D+ (k)for the contour C shown in Fig. 6, we get Z D+(k)0 dk = 2πin. (34) + C D (k) Direct integration yields Z C

Therefore

7.2

D+ (k)0 dk = 2i[δD(0) − δD (∞)] − πi. D+ (k) 1 δD (0) − δD (∞) = (n + )π. 2

(35)

(36)

Case of n Bound States and m Continuum Bound States

As discussed in Sec. 4, D+ (k) for a nonlocal potential has no poles on or above the real axis. If the potential has n bound states, then there will be n zeros of D+(k)on the positive imaginary axis, as shown in Fig. 7. If, in addition, the potential has m continuum bound states, then there will be 2m zeros of D+ (k)on the real axis, in pairs symmetric about k = 0. This also is shown in Fig. 7. Using Eq. 4 and integrating around the contour C shown in Fig. 7 yields Z D+(k)0 dk = 2πin. (37) + C D (k) From the line integral around the contour we get Z m D+(k)0 dk = 2i δD (0) − δD (∞) + ∑ [δD (ki + 0) − δD (ki − 0)] − 2mπi, + C D (k) i=1

(38)


113

Figure 6. Contour C for the derivation of the Generalized Levinson’s Theorem for δD when there are n bound states for k 6= 0 and a half-bound state at k = 0. Zeros of D+ (k) are indicated by o and χ respectively.

Figure 7. Contour C for the derivation of the Generalized Levinson’s Theorem for δD when there are n bound states and m continuum bound states. Zeros of D+ (k) are indicated by o. where the term −2mπi results from the integrals around the 2m semicircles along the real axis. Since δD (ki + 0) − δD (ki − 0) = π. (39) Eq. (39) becomes Z

D+ (k)0 dk = 2i[δD(0) − δD (∞) + m]π − 2mπi, + C D (k)

(40)

Combining Eqs. (37) and (40) gives δD (0) − δD (∞) = nπ.

(41)

If, in addition, there is a zero of D+ (k)of order q at k = 0, it is necessary to discuss this derivation further. A continuum bound state at k = 0 requires a double zero of D+(k). Thus, if q is odd, there will be a half-bound state at k = 0. Since only one such state is

114


possible according to the result in part A of this section, an odd value of q will add 12 π to the right hand side of Eq. (41). There will be no contribution to the right hand side of Eq. (41), however, in the case of a zero-energy continuum bound state. As discussed in Sec. 3 one can move the pair of zeros to give a zero at k = −ε and one at k = ε; integrating around these two semicircles will yield −2πi. But, as indicated above, the phase shift is now discontinuous at k = −ε and one at k = ε, and this discontinuity yields 2 πi. This cancels the contribution of the integrals around the semicircles. Thus for n bound states, m continuum bound states (either at k = 0 or k > 0), and a half-bound state, Levinson’s theorem for δD (k) is 1 (42) δD (0) − δD (∞) = (n + )π. 2 Actually, there is an ambiguity in the zero energy phase shift δD . As the parameters of a nonlocal potential are adjusted, a zero of D+ (k)can move from a position on the positive imaginary axis down to k = 0. At that point, the bound state becomes a continuum bound state, at zero energy. Whereas the bound state contributed π to the zero energy phase shift δD the continuum bound state will contribute nothing. Thus a very small change in the potential parameters can result in a discontinuity of the phase shift δD by π at zero energy. Such a discontinuity did not occur in the case of the phase shift δL under these circumstances.

7.3

Case of n Bound States and p Spurious States

Corresponding to n bound states there will be n zeros of D+ (k)on the positive imaginary axis, as shown in Fig. 8. Since p spurious states correspond to p zeros of D(k), without corresponding zeros of D+ (k)on or above the real axis, the logarithmic derivative of D+ (k)for the contour C of Fig. 8 will not be affected by the p spurious states. Applying the argument principle then will give Z D+(k)0 dk = 2πin. (43) + C D (k)

Figure 8. Contour C for the derivation of the Generalized Levinson’s Theorem for when there are n bound states and p spurious states. Zeros of D+ (k) are indicated by the symbol o.


115

Direct integration yields Z C

D+(k)0 dk = 2i[δD (0) − δD (∞)]. D+ (k)

(44)

Therefore δD (0) − δD (∞) = nπ.

7.4

(45)

Summary

We have seen that if we use δD as the definition for the phase shift, then Levinson’s theorem becomes (46) δD (0) − δD (∞) = nπ. where n is the number of bound states. The phase shift does not depend upon the number of continuum bound states and/or spurious states. If, also, we have a first-order zero of D+ (k)at k = 0, which would be the case if and only if a half-bound state is present, then Eq. (46) becomes 1 (47) δD (0) − δD (∞) = (n + )π. 2

8

Examples

In this section we consider two examples of nonlocal potentials and calculate the phase shift δ(0)−δ(∞) using both of the definitions, D± (k) = |D± (k)|e∓iδD and L ±(k) = |L ±(k)|e∓iδL . Both examples will make use of nonlocal potentials of the form V (r, r0) = λg(r)g(r 0).

(48)

These two examples have been chosen because the behavior in the complex plane of D+ (k)and D(k) for these potentials has been analyzed in a previous paper [55].

8.1

Yamaguchi Potential

In 1954 Yamaguchi [63] introduced a one-term separable nonlocal potential to describe nucleon-nucleon scattering. The Yamaguchi potential is of the form shown in Eq. (48) with g(r) = e−αr .

(49)

The expressions for D+ (k)and D(k) for this potential are given in Refs. [40] and [55]. They can be written in the form (50) D+(k) = N + (k)/M +(k), where N + (k) = 2αk2 + i4α2 k − (λ + 2α3 )

(51)

M +(k) = 2α(k + iα)2

(52)

and

116


and D(k) = N(k)/M(k),

(53)

N(k) = 2αk2 − (λ − 2α3 )

(54)

M(k) = 2α(k − iα)(k + iα).

(55)

where and From these expressions we see that the roots of N + (k) are r λ . k1,2 = −iα ± 2α

(56)

No values of λ,α will make D+(k)zero for real values of k; therefore no continuum bound state can be associated with the Yamaguchi form factor. The roots of N(k) are r

k1,2 = ±

λ − α2 . 2α

(57)

Thus D(k) can be zero for a wide range of values of λ and α. Although the values of λ and α used by Yamaguchi do not generate a spurious state at any energy, if λ > 2α3 a spurious state will occur. The zeros of M +(k) are k1,2 = −iα.

(58)

k1,2 = ±iα.

(59)

The zeros of M(k) are

8.1.1

Phase Shift Calculation Using δL

We now use the definition L +(k) = |L +(k)|e−iδL (k) to obtain the phase shift in two cases. + Case 1. When λ < 2α3 , there is no p spurious state. L (k)has one zero in the upper half plane, at k = iα, and a pole at k = i (2α3 − λ)/2α. Using Eq. (4) for L + (k)to obtain the integral around the contour C shown in Fig. 9 yields Z L + (k)0 dk = 0. (60) + C L (k)

The line integral around the contour C gives

Z

K L +(k)0 + dk = ln L (k) + + C L (k) 0

Z K

0

−

0 L +(Keiθ )0 iθ + Kiθe dθ + ln L (k) . L + (Keiθ ) −K

(61)

or Z

L +(k)0 L + (K) L + (0−) dk = ln + ln + + L +(−K) L + (0+) C L (k)

Z π + L (Keiθ )0 0

L +(Keiθ )

Kiθeiθ dθ.

(62)


117

Figure 9. Contour C for integration for Levinson’s Theorem for δL , showing the poles and zeros of L + (k)in the upper half of the k plane for the Yamaguchi potential with parameters for which there is no spurious state. [There are no zeros or poles of L +(k)on the real axis.] Zeros and poles of L + (k)for this potential are indicated by o and χ, respectively. The integral on the right hand side of Eq. (62) vanishes as K → ∞. Using the fact that δ(−k) = −δ(k), Eq. (62) implies lim

K→∞

Z

L +(k)0 dk = 2i[δL (0) − δL (∞)]. + C L (k)

(63)

Therefore we get δL (0) − δL (∞) = 0.

(64)

there is a spurious state. The Jost function has one zero at k = iα Case 2. When λ > 2α3, p and two poles, at k = ± (λ − 2α3)/2α = ±k1 , on the real axis. Therefore, integrating around the contour C shown in Fig. 10 yields Z

L +(k)0 dk = 2πi. + C L (k)

(65)

In the limits ε → 0 and K → ∞, direct integration around the contour shown in Fig. 10 gives + + + + Z L +(k)0 L (−k1 −0) L (−∞) L (−0) L (−k1 +0) dk = ln −ln + +ln + −ln +2πi. + L +(k1 +0) L (+∞) L (+0) L +(k1 −0) C L (k) (66) or Z L + (k)0 (67) dk = 2i [δL (0) − δL (∞) + δL (k1 +0) − δL (k1 −0)] + 2πi. + C L (k) Because of the spurious state at k = k1, there is a phase change of −π in going from k = k1 − ε to k = k1 + ε That is, in the limit as ε → 0, δL (k1 +0) − δL (k1 −0) = −π.

(68)

118


Figure 10. Contour C for integration for Levinson’s Theorem for δL , showing the poles and zeros of L + (k)in the upper half of the k plane and on the real axis for the Yamaguehi potential with parameters for which there is a spvrious state at k = k1 . Zeros and poles of L +(k)for this potential are indicated by o and χ, respectively. Combining Eqs. (65), (67), and (68) yields 2πi = 2i [δL (0) − δL (∞) − π] + 2πi,

(69)

δL (0) − δL (∞) = π.

(70)

which implies

8.1.2

Phase Shift Calculation Using δD

Next we use the definition D+(k) = |D+(k)| e−iδD (k) to get the zero-energy phase shift. In both the case λ > 2α3 and λ < 2α3 the result will be the same, since for all values of λ and α the zeros of D+ (k) lie in the lower half plane. Using Eq. (4) for D+(k)and integrating around the contour C shown in Fig. 11, we get Z C

D+(k)0 dk = 0. D+ (k)

(71)

The line integral around the contour yields, in the limit as K → ∞ and ε → 0, Z

D+ (k)0 dk = ln D+ (−0) − ln D+(−∞) + ln D+(∞) − ln D+ (0). + (k) D C

(72)

Combining Eqs. (7l) and (72) gives δD (0) − δD (∞) = 0.

8.2

(73)

Beregi Potential

The second example to be considered here is the Beregi potential. Beregi [33] suggested a one-term separable nonlocal potential of the form given by Eq. (48) with g(r) = e−α1 r − ae−α2 r .

(74)


119

Figure 11. Contour C for integration for Levinson’s Theorem for δD the Yamaguchi potential. [Note that there are no poles or zeros of D+ (k) in the upper half-plane.] This potential yields a continuum bound state at 259 .3 MeV and a bound state at −2.225 MeV [41]. The parameters for the Beregi potential are λ = −302.73 f m−3 , α1 = 2.67 f m−1, α2 = 5.34 f m−1, a = 3.0854. The Fredholm determinants for this potential are given in Refs. [41] and [55]. They can be written as N + (k) (75) D+ (k) = + M (k) where N + (k) = k4[2α1α2(α1 + α2 )] + ik3 (α1 + α2 )[4α21α2 + 4α1α22 ] + k2 [(α1 + α2 )(−2α31α2 − 2α1 α32 − λα2 − λa2 α1 − 8α21 α22 + 4λaα1α2 ] + ik[(α1 + α2 )(−4λaα1α2 − 2λa2α21 − 4α21 α32 − 4α31 α22 − 2λα22) + 8λaα21 α2 + 8λaα1 α22] + [(α1 + α2 )(2α31α32 + λα32 + λa2 α31 ) − 4λaα21 α22]

(76)

and M +(k) = 2α1 α2 (α1 + α2 )(k + iα1)2(k + iα2 )2

(77)

and D(k) =

N(k) M(k)

(78)

120

S.B. Qadri, B. Mulligan, M.F. Mahmood et al. Table 1. Zeros of N + (k) and M + (k) for the Beregi Potential ( f m−1 ) Zeros of N(k)+ (1.77,0) (1.77,0) (0,0.23) (0,-16.25)

Zeros of M +(k) (0,-2.67) (0,-2.67) (0,-5.34) (0,-5.34)

Table 2. Zeros of N(k) and M(k) for the Beregi Potential ( f m−1 ) Zeros of N(k) (1.77,0) (-1.77,0) (0,11.49) (0,-11.49)

Zeroso f M(k) (0,2.67) (0,-2.67) (0,5.34) (0,-5.34)

where N(k) = k4 [2α1α2(α1 + α2 )] + k2 [(α1 + α2 )(2α1α32 + 2α31 α2 − λα2 − λa2 α1) + 4α1 α2 λa] + [(α1 + α2 )(2α31α32 − λα32 + 2α1 α2 λa(α1 + α2 ) − λa2 α31) − 4α21 α22 λa]

(80)

and M(k) = 2α1α2(α1 + α2 )(α21 + k2 )(α22 + k2 ) N + (k)

(80a)

M +(k)

and are tabulated in Table I, whi1e the zeros of N(k) and The zeros of M(k) are tabulated in Table II. The information from these tables is combined in Table III to give the zeros and poles of L + (k)for this potential. 8.2.1

Phase Shift Calculation Using δL (k)

The zero-energy phase shift δc l associated with the Beregi potential can be calculated as discussed in Sec. 6. Using Eq. (4) for L +(k)and integrating around the contour C of Fig. 12, we have Z L +(k)0 dk = 4πi. (81) + C L (k) Explicit calculation of the line integral around the contour C yields Z

L +(k)0 dk = 2i[δL (0) − δL (∞)]. + C L (k)

Table 3. Zeros of L +(k) for the Beregi Potential Zeros (0,0.23) (0,2.67) (0,5.34) (0,-16.25)

Poles (0,11.49) (0,-11.49) (0,-2.67) (0,-5.34)

(82)


121

Figure 12. Contour C for integration for Levinson’s Theorem for δL , showing the poles of L +(k)in the upper half-plane for the Beregi potential [There are no zeros or poles of L +(k)on the real axis.] Zeros and poles are indicated by o and χ, respectively. This gives δL (0) − δL (∞) = 2π 8.2.2

(83)

Phase Shift Calculation Using δL

The information necessary for the calculation of δL for the Beregi potential is given in Table I. Using the argument principle for the contour C given in Fig. 13, we obtain Z

D+ (k)0 dk = 2πi. + C D (k)

(84)

Figure 13. Contour C for integration for Levinson’s Theorem for δD showing the zeros of D+ (k) on the real axis for the Beregi potential. These zeros are indicated by o. Taking k1 = 1.77 f m−1, the explicit line integral yields Z

D+ (k)0 dk = 2i[δD (0) − δD (∞) + δD (k1 +0) − δD (k1 −0)] − 2πi. + C D (k)

(85)

122

S.B. Qadri, B. Mulligan, M.F. Mahmood et al. The discontinuity in the phase shift at k = k1 is given by δD (k1 +0) − δD (k1 −0) = π.

(86)

From Eqs. (84), (85), and (86) we get 2i[δD (0) − δD (∞)] = 2πi

(87)

δD (0) − δD (∞) = π.

(88)

or

9

Summary and Conclusion

In this section we compare the proofs of Levinson’s theorem given in Secs. 6 and 7 with other proofs in the literature. In so doing, it is important to note that in derivations of Levinson’s theorem in the literature some authors have used L + (k)or D+ (k) and others L − (k) or D− (k). The use of L −(k) or D− (k) instead of L +(k)or D+(k) leads to a corresponding change in the contour integral considered from the upper half-plane to the lower half-plane. As mentioned in the introduction, one of the earliest derivations of the generalized Levinson’s theorem for a nonlocal potential was that due to P. Swan [2]. The problem he was considering was that of nonlocality due to antisynimetrization of the incident particle with respect to particles in the target. He found that the natural definition to use was the phase of L −(k), that is the phase shift δL . [In his paper the notation for L −(k) is f (k).] The contour used for his calculation and the pertinent zeros and poles are displayed in Fig. 5 of Ref. [2] and the phase shift calculation is carried out on p. 23 of Ref. [2]. It is interesting to note that although he showed that L − (k) could have poles on the real axis, he obtained a zero energy phase shift of zero. Using the approach presented here, for his case we would obtain δL (0) = π instead of δL (0) = 0. That is, although he recognized the presence of poles on the real axis, Swan did not take into consideration a discontinuity of −π in the phase shift as the energy increases through a spurious state. Also, in his derivation of the generalized Levinson’s theorem redundant states occur in his Eq. (54), which reads as follows: Z C1

R

f 0 (k) − φ0 0∞ φ(r0) f 0 (k, r 0) dr0 R dk = f (k) − φ0 0∞ φ(r0) f (k, r0) dr 0

Z C1

f 0 (k) dk. f (k)

(87a)

He assumes the existence of the Jost solution f (k, r) under these conditions. As pointed out in Ref. [52], a Jost solution may or may not exist when a continuum bound state or a redundant state occurs. Thus his subsequent derivation is applicable only to those cases in which Jost solutions exist. Jauch’s [5] generalized Levinson’s theorem was undertaken with the assumption that a nonlocal potential does not have bound states in the continuum. In his generalization of Levinson’s theorem, Jauch used the definition of the phase shift as the phase of the S-matrix, that is, (88a) S(k) = e2iδ(k) . As pointed out in Ref. 41, this definition is not sensitive to the presence of bound states in the continuum or to discontinuities in the phase shift due to poles of L +(k)on the real


123

axis. Thus Jauch left out the important consideration of the existence of continuum bound states and spurious states in the case of a nonlocal potential. Shortly after Jauchs paper, Gourdin and Martin [6] noted that a continuum bound state could be associated with a nonlocal potential and that such a state required a modification of Levinson’s theorem. Martin [8] showed that for a nonlocal potential with continuum bound states the index in Eq. (2) should be replaced by ν + ν0 ), where ν is the number of bound states with energy less than zero and ν0 is the number of continuum bound states. He used the definition δL in his derivation, and his result can be expressed in the following form, δ(0) − δ(∞) = (ν + ν0 )π.

(89)

His method does not take into consideration possible zeros of D(k) and thus the possible presence of spurious states. The derivation of Levinsons theorem presented by Bertero, Talenti, and Viano [29] is based on consideration of the Fredholm determinant D+ (k). Their derivation, which begins on p.635 of Ref. [29], considers the phase shift associated with D+ (k) [denoted in the paper by F(−k)] and the contour is closed in the upper half-plane. They consider N zeros in the upper half-plane and 2N 0 zeros on the real axis symmetrically situated about the imaginary axis. There are also q zeros at the origin. In the nomenclature of Bertero et al. a continuum bound state is called a spurious state and states labeled spurious states in our nomenclature are not considered. Figure 1 on p. 637 of Ref. [29] displays the zeros mentioned above. The contour integration in Ref.[ 29] yields the following equation N0 F(k)0 dk = 2i η(0) − eta(∞) + ∑ [η(ki +0) − η(ki −0] − 2iπN 0 − iπq. C F(−k) i=1

Z

(90)

However, although F(−k) [and thus η(k)] is not continuous at a continuum bound state, Bertero et al. took the phase shift to be continuous at the continuum bound states (which are in their terminology spurious states), and obtained the following relation q δ(0) − δ(∞) = π(N + N 0 + ). 2

(91)

Had they considered the phase shift to be discontinuous at the continuum bound states they would have obtained the relation (36), but with an ambiguity at zero energy as to whether one should consider zeros of D+ (k) as zero-energy continuum bound states or zeroenergy bound states. Also, in addition to not taking into consideration the phase change at continuum bound states, their proof does not refer to the possibility of a spurious state occurring. Glöckle and LeTourneux [36] studied the generalized Levinson’s theorem for composite particles in the framework of the Saito model. They discussed the zeros of D+ (k) [which they designated by D(k)] and used the phase of D+ (k) for getting the zero energy phase shift. In their proof, which is given on p. 21 of Ref. [36], they assume m bound states and n continuum bound states. The zeros of D+ (k) corresponding to n continuum bound states lie on the real axis symmetrically about the imaginary axis. They used the integral in Eq. (4)

124


and obtained the contribution −2πin from the 2n semicircles around the zeros on the real axis. However in arriving at their Eq. (38), which reads as follows, Z ∞ −∞

dk

d ln D(k) = 2i[δ(0) − δ(∞)], dk

(92)

they did not take into account the phase change at a continuum bound state. Thus they obtained the relation δ(0) − δ(∞) = (m + n)π. (93) Since there is discontinuity of π in the phase of D(k) at each continuum bound state, their result should have been δ(0) − δ(∞) = mπ. (94) For a large class of nonlocal potentials giving rise to continuum bound states, Buslaev obtained Levinson’s theorem in its original form. Buslaev’s work is consistent with Bolsterli’s results. It is also consistent with our derivation of Levinson’s theorem using the definition δD when continuum bound states are present. Dreyfus has also studied Levinson’s theorem for nonlocal interactions. He pointed out that owing to the two defintions of the scattering phase shift we get two different versions of Levinson’s theorem; one of them takes the continuum bound states into account while the other does not. However, neither Buslaev nor Dreyfus take into consideration the possibility of the existence of spurious states. Finally, Newton has considered the point k = 0 and proved a generalized Levinson’s theorem which includes the possiblity of a bound state or a half-bound state at k = 0. His derivation is based on the Fredholm determinant D+ (k). By using the phase of D+ (k) to define a phase shift continuous at a continuum bound state, he obtained a statement of Levinson’s theorem which is the same as that which we find here for δL . He also presents strong arguments as to why this is the proper statement of the theorem. He does not, however, consider the possibility of spurious states in his derivation. In studying Levinson’s theorem for a nonlocal potential we have taken into account the fact that redundant states, continuum bound states, and spurious states are all possible in the case of a nonlocal potential. This has been done for both of the possible definitions, δD and δL , of the phase shift for scattering by a nonlocal potential outlined in Ref.[ 41]. Thus the present work discusses a variety of aspects of Levinson’s theorem not previously treated in the case of a nonlocal potential. An important aspect of our ability to extend Levinsons theorem to nonlocal potentials rests upon the results of Ref. [55], in which we introduced functional analytic techniques for determining the character of the Fredholm determinants D+ (k) and D(k) associated with integral equations for scattering solutions of the radial equation. In discussing the derivation of the generalized Levinson’s theorem for both the definitions of the phase shifts we have seen that the generalized Levinson’s theorem associated with definition δD does not take into account the presence of continuum bound states or spurious states. Also there is ambiguity in δD when there are zeros of D(k) at the origin. The definition δL takes into account the presence of continuum bound states as well as spurious states and half-bound states or zero-energy bound states. The definition δD fails completely in the presence of redundant states, while the definition δL gives consistent results. All things considered, we find that the phase shift δL seems to be more useful in all


125

the cases considered for the generalization of Levinson’s theorem. In this respect we agree with the conclusion of Ref. [41] which prefers the definition δL on the basis of different considerations.

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A SYMPTOTIC B EHAVIOR OF THE F ITZ H UGH -N AGUMO S YSTEM Weishi Liu1∗and Bixiang Wang 2† 1 Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA 2 Department of Mathematics, New Mexico Institute of Mining and Technology Socorro, NM 87801, USA

Abstract For the FitzHugh-Nagumo system defined on IR, we prove the existence of a compact global attractor in a weighted Sobolev space which contains bounded solutions, in particular, traveling wave and spatially periodic solutions. We also study the behavior of the global attractors as a parameter goes to zero. Although the limiting system for = 0 does not possess a bounded attracting set, we show that there exists a constant 0 such that global attractors for 0 < ≤ 0 are all contained in a compact subset of the phase space. Furthermore, we construct a compact local attractor for the limiting system and establish the upper semicontinuity of the global attractors of perturbed system and the local attractor of the limiting system.

AMS Subject Classification: Primary 35B40. Secondary 35B41, 35K57, 92C55 Key Words: FitzHugh-Nagumo system, asymptotic behavior, global attractor

1.

Introduction

We are interested in the long-time behavior of the FitzHugh-Nagumo system on IR: ∂v ∂t ∂w ∂t

∗

=

∂ 2v + h(v) − w, ∂x2

= (v − γw),

(1.1) (1.2)

E-mail address: [email protected], supported in part by NSF Grant DMS-0406998 and a General Research Fund of University of Kansas. † E-mail address: [email protected], supported in part by the Start-up Fund of New Mexico Institute of Mining and Technology.

130

Weishi Liu and Bixiang Wang

where, and γ are positive constants, h is a given nonlinear function (typically, a cubic like function). The FitzHugh-Nagumo system was introduced as a simplification of the Hodgkin-Huxley equation that was derived as a model for the propagation of action potentials in the giant nerve axon of the squid (see, for example, [6, 10, 21]). The nonlinearity in the Hodgkin-Huxley is obtained by assuming a certain form and then fitting the nonlinearity to experimental data by nonlinear least squares or by solving an inverse problem [30], and the cubic like function h in the FitzHugh-Nagumo equation is a simplification that captures some main qualitative behavior of the Hodgkin-Huxley equation. System (1.1)-(1.2) has been studied extensively along two major directions: the existence and stability of traveling waves, and the long-time behavior of the solutions. The existence of traveling waves for this system was studied in detail through bifurcation analysis in [8], a combination of center manifold reduction and the Lyapunov-Schmidt reduction in [25], and the geometric singular perturbation method in [14, 16]. The stability of the traveling waves was first studied in [13], where the Evan’s function method was successfully developed. The method has been extended in [1, 25, 26] and becomes predominant in the investigation of stability problems. The asymptotic behavior of solutions for system (1.1)(1.2) has been studied in [18, 19, 27]) when the system is defined in a bounded domain. In the case of an unbounded domain, the existence of a global attractor for this system was proved in [23] in the standard L2(IRn ) × L2 (IRn) space, which is compact in the phase space but does not contain traveling wave solutions. The existence of a global attractor in a locally uniform Sobolev space was showed in [7], which contains traveling wave solutions but is not compact with respect to the norm topology of the phase space. As a consequence, the global attractor in [7] attracts solutions only in a weak topology, but not in the norm topology, of the phase space. In the present paper, as a necessary step toward the understanding of the global asymptotic behavior, we consider existence of global attractors and study the behavior of these attractors (at the level of invariant sets) as the parameter → 0. Note that, if is set to zero in (1.1)-(1.2), then the component w is conserved, and hence, there cannot be any bounded set that attracts all solutions of the limiting system. However, we will establish, in a suitably weighted space, the existence of a compact global attractor that contains all bounded solutions for all positive but small and show that the union of all global attractors of the perturbed system (1.1)-(1.2) for positive but small is actually contained in a compact subset in the phase space. Furthermore, we will construct a compact local attractor for the limiting system and establish the upper semicontinuity of global attractors for the perturbed system at the compact local attractor for the limiting system. Our method is based on the idea of “tail ends” estimates on solutions, by showing that the solutions of system (1.1)(1.2) are uniformly small when space and time variables are sufficiently large. Actually, along with other things, the uniform smallness of “tail ends” of solutions for large time is not only sufficient, but also necessary for the asymptotic compactness of solution operators of dissipative evolution equations. The idea of “tail ends” estimates on solutions was used in [31] for the Reaction-Diffusion equation and in [23] for the FitzHugh-Nagumo system in the standard L2(IRn ) space, respectively. This paper demonstrates that the method of “tail ends” estimates is effective not only for the standard Sobolev spaces, but also for weighted Sobolev spaces. The existence of global attractors for dissipative equations on unbounded domains can be found in [2, 4, 7, 9, 11, 17, 20, 22, 24] and the references therein.

Asymptotic Behavior of the FitzHugh-Nagumo System

131

In order to include bounded solutions, the weight we choose decays with a polynomial order at infinity. A clear drawback of any non-trivial weight is that the weighted norm destroys the transition (in space) invariance and some traveling waves (those connects zero steady-state to other steady-states) approaches zero in a decaying weighted norm as t → ∞. Nonetheless, some dynamics of traveling waves are kept in the weighted norm space and, most importantly, the compactness and upper semicontinuity of perturbed global attractors provide a good platform for further study of asymptotic behavior of the dynamics of the FitzHugh-Nagumo system. For example, an important question is how one defines the semi-flow of the slow dynamics–the limiting system resulting from a time re-scaling by – on the compact local attractor to capture the essence of asymptotic behavior of the system for > 0 small. An answer to this question and other related ones will be our further investigation. Another interesting question is what happens if the parameter γ equals zero. Note that γ > 0 is a dissipative condition for w. While for γ = 0 the system still defines a continuous semi-group in the weighted space (Theorem 2.1), our approach does require γ > 0 for existence of compact global attractors. We believe that a certain dissipative condition for w is needed and would also like to examine the situation more carefully in the future. The paper is organized as follows. In the next section, we describe our main results. In Section 3, the existence and uniqueness of solutions for system (1.1)-(1.2) are proved for initial data in H 1 (IR)×H 1(IR). We then extend the solution operator to define a continuous dynamical system in a weighted Sobolev space. Section 4 is devoted to uniform estimates of solutions as t → ∞, which are necessary to establish the point dissipativeness and asymptotic compactness of solutions. Particularly, we show that the solutions are uniformly small when space and time variables are large enough. In Section 5, we prove the existence of a global attractor for system (1.1)-(1.2) in a weighted Sobolev space. We first establish the asymptotic compactness of the solutions, and then conclude the existence of a global attractor by a standard result. The behavior of the global attractors as → 0 is discussed in the last three sections. We will show that not only global attractors are bounded uniformly in (in Section 6) but also their union for small > 0 is contained in a compact subset of the phase space (in Section 7). Finally, in Section 8, we construct a compact local attractor for the limiting system and prove the upper semicontinuity of global attractors as → 0.

2.

Main results

Consider the following FitzHugh-Nagumo equations: ∂v ∂t ∂w ∂t

=

∂ 2v + h(v) − w, ∂x2

(2.1)

= (v − γw),

(2.2)

for (t, x) ∈ (0, ∞) × IR with the initial data v(0, x) = v0 (x),

w(0, x) = w0(x),

x ∈ IR

(2.3)

132


where and γ are positive constants, h is a smooth function that satisfies, for some positive constants a, α and β and for all v ∈ IR, h(v)v ≤ −av 2 + β,

h(0) = 0,

h0 (v) ≤ α.

(2.4)

As a particular example, the cubic function h = v(1 − v)(v − a) for a > 0 satisfies (2.4) with 27 a2 − a + 1 (a + 1)4 and α = . β= 256 3 To present our results, we introduce the notation used in the paper. Fix σ > 1/2 and define the weighted Sobolev space L2σ (IR) as Z 2 2 −σ 2 Lσ (IR) = u : u is measurable, (1 + |x| ) |u(x)| dx < ∞ with the norm kukσ =

Z

2 −σ

(1 + |x| )

2

1/2

|u(x)| dx

.

We also denote Hσ1(IR) ⊂ L2σ (IR) the weighted space with the norm defined by kuk21,σ = kuk2σ + k

∂u 2 k . ∂x σ

Due to the weight (1+|x|2)−σ , a function in L2σ (IR) can have a certain growth as |x| → ∞; in particular, the space L2σ (IR) contains all bounded measurable functions such as traveling wave and spatially periodic solutions. For later purpose, we define a function φδ : φδ (x) = (1 + |δx|2)−σ , where δ=

x ∈ IR,

1 min{1, a} where a is the number in (2.4). 2σ

(2.5)

(2.6)

Then φδ satisfies the following inequalities: |

∂ φδ (x)| ≤ δσφδ (x), ∀x ∈ IR, ∂x

and kuk2σ

≤

Z

φδ (x)|u(x)|2dx ≤ δ −2σ kuk2σ .

(2.7)

(2.8)

R 1/2 φδ (x)|u(x)|2dx Inequality (2.8) shows that the norm k · kσ is equivalent to the norm for u ∈ L2σ (IR). As usual, we denote the norm of L2(IR) by k · k. We now describe the main results of the paper and delay their proofs to later sections. We emphasize that we will always assume that h satisfies (2.4) and σ > 1/2. Our first result shows that one can define a continuous dynamical system for problem (2.1)-(2.3) in the space L2σ (IR) × L2σ (IR). More precisely, we have


133

Theorem 2..1. For > 0, one can associate problem (2.1)-(2.3) with a continuous semigroup S(t)t≥0 in L2σ (IR) × L2σ (IR) such that, when the initial datum (v0, w0) ∈ H 1(IR) × H 1(IR), S(t)(v0, w0) is the unique solution of problem (2.1)-(2.3) in the sense of (3.19)-(3.21). Our next result is on the existence of a global attractor of the dynamical system S(t)t≥0 . Theorem 2..2. For > 0, the dynamical system S(t)t≥0 has a global attractor in L2σ (IR) × L2σ (IR), which is a compact invariant set and attracts every bounded set with respect to the norm topology of L2σ (IR) × L2σ (IR). In the sequel, we denote by A the global attractor corresponding to a given . An interesting question is: what happens to the set A as → 0 ? Naturally, one would like to compare the dynamics for > 0 with that for = 0. When = 0, problem (2.1)-(2.3) reduces to ∂v ∂t ∂w ∂t

=

∂ 2v + h(v) − w, ∂x2

(2.9)

= 0,

(2.10)

for (t, x) ∈ (0, ∞) × IR with the initial data v(0, x) = v0 (x),

w(0, x) = w0(x).

(2.11)

Given initial data (v0, w0), by (2.10), we see w(t) = w0 for all t ≥ 0. Therefore, problem (2.9)-(2.11) cannot have a bounded set that attracts all solutions in the space L2σ (IR) × L2σ (IR). Interestingly enough, we will show that all global attractors A for small > 0 are uniformly bounded in L2σ (IR) × L2σ (IR) (in fact, in Hσ1(IR) × Hσ1(IR)). Theorem 2..3. The global attractors A are uniformly bounded in for 0 < < a a min{1, 2γ } in the space Hσ1(IR) × Hσ1(IR). More precisely, for all 0 < < min{1, 2γ } and (v, w) ∈ A , (2.12) k(v, w)kσ ≤ M1 , and k(v, w)k1,σ ≤

q

M12 + M22 ,

(2.13)

where M1 and M2 are given by s M1 =

β(a2 + 2aγ + 2) a2 γ

Z

φδ dx,

(2.14)

and 1 M2 = γ

s

(γ 2 +

2 1)eσ +σ+2α

β(σ + 1) 4β(a + 1)(aγ + 1) + 2 2 γ(σ + σ + 2α) 3a2 γ

Z

φδ dx, (2.15)

where a and β are the constants in (2.4) and δ is the constant given by (2.6).

134


The next result shows that the union of all global attractors A is not only bounded, but also compact in L2σ (IR) × L2σ (IR). More precisely, we have the following conclusion. a Theorem 2..4. Let 0 = min{1, 2γ }. Then the union

S

A of all global attractors is

0 0, there is a constant C depending only on , α, T and M (but independent of L), such that, the solution (v, w) of problem (3.1)-(3.4) satisfies k (v(t), w(t)) kH 1(ΩL )×H 1 (ΩL ) ≤ C, and

Z

k

Z

T

kv(t)k2H 2(ΩL ) dt ≤ C,

0

∂w k 1 ≤ C, ∂t H (ΩL ) T

k

0

t ∈ [0, T ],

∂v 2 k 2 dt ≤ C. ∂t L (ΩL )

(3.13)

(3.14)

Proof. Multiplying (3.1) and (3.2) by ∂xx v and ∂xx w, respectively, and integrating with respect to x over ΩL , one has Z Z Z Z 1d 2 2 |∂x v| dx + |∂xxv| dx = − h(v)∂xxvdx − ∂x v ∂x wdx, 2 dt ΩL ΩL ΩL ΩL and

1d 2 dt

Z

2

|∂x w| dx =

ΩL

Z

∂x v ∂x wdx − γ ΩL

Z

|∂xw|2dx.

ΩL

Therefore, 1 d 2 dt

Z

2

2

Z

|∂x w| dx + |∂xxv|2dx ΩL ΩL Z Z h(v)∂xxvdx + ( − 1) ∂x v ∂x wdx − γ

|∂xv| dx +

ZΩL

Z

|∂x w|2dx.

(3.15)

Using (2.4) to estimate the first term on the right-hand side of (3.15), we obtain Z Z − h(v)∂xxvdx ≤ α |∂x v|2dx.

(3.16)

=−

ΩL

ΩL

ΩL

ΩL

ΩL

Then, it follows from (3.15) and (3.16) that, for all t ≥ 0, Z Z Z d |∂x v|2dx + |∂xw|2dx + 2 |∂xxv|2dx dt ΩL ΩL Ω Z ZL 2 2 ≤ C1 |∂xv| dx + |∂x w| dx , ΩL

ΩL

(3.17)


137

where C1 is a constant depending on and α. By Gronwall’s Lemma, we find that, for t ≥ 0, Z Z Z Z 2 2 C1 t 2 2 |∂xv(t)| dx + |∂x w(t)| dx ≤ e |∂x v0 | dx + |∂x w0| dx ΩL ΩL ΩL ΩL Z Z C1 t 2 2 ≤ e |∂x v0| dx + |∂xw0 | dx , IR

which shows that, for t ∈ [0, T ], Z Z |∂x v(t)|2dx + ΩL

IR

|∂xw(t)|2dx ≤ eC1 T M 2.

(3.18)

ΩL

Integrating (3.17) with respect to t and using (3.18), we obtain Z

T

Z

0

|∂xxv|2dxdt ≤ C,

ΩL

which, along with (3.1)-(3.2), (3.18) and Lemma 3..1, implies (3.13) and (3.14). We are now in a position to establish the existence of solutions for problem (2.1)-(2.3). Suppose (v, w) ∈ C([0, ∞), L2(IR) × L2(IR)). Then we say (v, w) is a solution of problem (2.1)-(2.3) if, (v(0), w(0)) = (v0, w0) and, for every T > 0, ∂v ∂w ∈ L2((0, T ), H −1(IR)), ∈ L2 ((0, T ), L2(IR)), (3.19) ∂t ∂t

v ∈ L2 ((0, T ), H 1(IR)),

and (v, w) satisfies the following equalities, for any κ ∈ C0∞ ([0, T ] × IR): Z

T 0

∂v h , κi(H −1,H 1) dt + ∂t Z 0

T

Z IR

Z 0

T

Z

∂x v ∂x κ dxdt =

Z

T 0

IR

∂w κ dxdt = ∂t

Z

T 0

Z

Z

(h(v) − w)κ dxdt,

(3.20)

IR

(v − γw)κ dxdt,

(3.21)

IR

where (H −1 , H 1) denotes the duality pairing between H −1 (IR) and H 1(IR). Lemma 3..3. Suppose (v0, w0) ∈ H 1(IR) × H 1 (IR). Then problem (2.1)-(2.3) has a solution (v, w) in the sense of (3.19)-(3.21). In addition, v ∈ L∞ ((0, T ), H 1(IR) ∩ L∞ (IR))

\

L2((0, T ), H 2(IR)),

w ∈ L∞ ((0, T ), H 1(IR) ∩ L∞ (IR)),

∂v ∈ L2((0, T ), L2(IR)), ∂t

∂w ∈ L∞ ((0, T ), H 1(IR)). ∂t

Proof. Let θ be a smooth function satisfying 0 ≤ θ(s) ≤ 1 for s ≥ 0, and θ(s) = 1 for 0 ≤ s ≤

1 ; 2

θ(s) = 0 for s ≥ 1.

138


Then, given (v0, w0) ∈ H 1(IR) × H 1(IR), we denote by |x|2 |x|2 (v0,L(x), w0,L(x)) = θ( 2 )v0(x), θ( 2 )w0(x) , L L

x ∈ IR.

It is evident that (v0,L, w0,L) −→ (v0, w0) in H 1 (IR) × H 1(IR) as L → ∞. Let (vL, wL) be the solution of problem (3.1)-(3.4) defined on ΩL with the initial datum (v0,L, w0,L). Then, we extend (vL, wL) to IR by defining (vL (t, x), wL(t, x)) = (0, 0) for |x| > L. From Lemma 3..2, for any given T > 0, there is a constant C depending only on , α, T and k(v0 , w0)kH 1×H 1 (but independent of L) such that k (vL (t), wL(t)) kH 1 (IR)×H 1 (IR) ≤ C, and

Z 0 1

T

kvL (t)k2H 2(IR) dt

≤ C,

k

∂wL k 1 ≤ C, ∂t H (IR) Z 0

T

k

t ∈ [0, T ],

∂vL 2 k 2 dt ≤ C. ∂t L (IR)

(3.22)

(3.23)

By embedding H (IR) ⊂ L∞ (IR) and (3.22), we have kvL (t)kL∞ (IR) + kwL(t)kL∞ (IR) ≤ C,

t ∈ [0, T ].

(3.24)

It follows from (3.22)-(3.24) that there exists a pair of functions (v, w) satisfying v ∈ L∞ ((0, T ), H 1(IR) ∩ L∞ (IR))

\

L2((0, T ), H 2(IR)),

∂v ∈ L2((0, T ), L2(IR)), ∂t (3.25)

∂w ∈ L∞ ((0, T ), H 1(IR)), ∂t and a subsequence of (vL , wL) (still denoted by (vL , wL)) such that, as L → ∞, w ∈ L∞ ((0, T ), H 1(IR) ∩ L∞ (IR)),

vL −→v, wL −→w,

(3.26)

∂vL ∂v −→ weakly in L2 ((0, T ), L2(IR)), ∂t ∂t ∂wL ∂w −→ star-weakly in L∞ ((0, T ), H 1(IR)). ∂t ∂t

Next, we prove (v, w) is a solution of problem (2.1)-(2.3). For that purpose, we take a test function κ ∈ C0∞ ([0, T ] × IR). Then, by (3.1)-(3.2) we have Z TZ Z TZ Z TZ ∂vL ∂x vL ∂x κ dxdt = (h(vL ) − wL ) κ dxdt, (3.27) κ dxdt + 0 0 0 IR ∂t IR IR Z TZ Z TZ ∂wL (vL − γwL) κ dxdt. (3.28) κ dxdt = 0 0 IR ∂t IR We now deal with the nonlinear term on the right-hand side of (3.27). Choose a constant K large enough such that supp(κ) ⊆ [0, T ] × ΩK . Then, by a standard compactness argument (see, e.g., [29]), from (3.25), we infer that vL −→ v in L2((0, T ), L2(ΩK )).

(3.29)


139

Therefore, by (3.24)-(3.25) the following holds for the nonlinear term h: |

Z

T

Z

h(vL ) κ dxdt

Z

−

IR

0

Z

T

0

Z

≤

Z

T

0

≤

C

h(v) κ dxdt| ≤ IR

0

≤

C

Z

0

−→ 0,

T

|(h(vL) − h(v)) κ| dxdt

IR

|h0 (ξ)(vL − v)| |κ| dxdt |vL − v| |κ| dxdt

ΩK

T

Z

0

ΩK T Z

Z

Z

Z

|vL − v|2dxdt

Z

T 0

ΩK

Z

|κ|2 dxdt

ΩK

as L → ∞.

(3.30)

Taking the limits of (3.27)-(3.28) as L → ∞, we find that (v, w) satisfies Z

T 0

Z IR

Z

∂v κ dxdt + ∂t Z 0

T

T 0

Z IR

Z

∂x v ∂x κ dxdt =

Z 0

IR

∂w κ dxdt = ∂t

Z

T 0

Z

T

Z

(h(v) − w)κ dxdt,

(3.31)

IR

(v − γw)κ dxdt,

(3.32)

IR

By a standard continuity argument (see, e.g., [28]), from (3.25)-(3.26) we can prove (v, w) ∈ C([0, T ], H 1(IR) × H 1(IR)) and (v(0), w(0)) = (v0 , w0).

(3.33)

Then, by (3.24)-(3.26) and (3.31)-(3.33), Lemma 3..3 follows. Note that for any σ > 1/2 (in fact for any σ ≥ 0), the standard space L2 (IR) is contained in the weighted space L2σ (IR). Thus the solutions obtained in Lemma 3..3 are actually functions in L2σ (IR) × L2σ (IR). Next, we show that the solutions are Lipschitz continuous in the space L2σ (IR) × L2σ (IR). As an immediate consequence, the solutions in Lemma 3..3 are unique. Lemma 3..4. Suppose (v0 , w0) ∈ H 1(IR) × H 1(IR). Then problem (2.1)-(2.3) possesses a unique solution satisfying (3.19)-(3.21). Furthermore, for every T > 0, there exists a constant C depending only on , α, σ and T such that any two solutions (v1, w1) and (v2, w2) satisfy the inequality, for t ∈ [0, T ], kv1 (t) − v2 (t)kσ + kw1(t) − w2 (t)kσ ≤ C (kv1 (0) − v2 (0)kσ + kw1(0) − w2(0)kσ ) . (3.34) Proof. Let V = v1 − v2 and W = w1 − w2. By Lemma 3..3, both (v1 , w1) and (v2 , w2) satisfy equations (2.1) and (2.2) in the distribution sense. Therefore, V and W satisfy ∂V ∂t ∂W ∂t

=

∂ 2V + (h(v1) − h(v2 )) − W, ∂x2

= (V − γW ).

(3.35) (3.36)

140


Multiplying (3.35) by φ1 V where φ1 is the function given by (2.5) with δ = 1 and integrating the resulting identity with respect to x, we obtain Z Z Z 1d 2 2 φ1 |V | dx + φ1 |∂x V | = (h(v1 ) − h(v2 )) φ1 V dx 2 dt Z Z (3.37) − ∂x φ1 (∂xV ) V dx − φ1 W V dx. The first term on the right-hand side of (3.37) can be estimated from (2.4) as Z Z Z 0 2 (h(v1 ) − h(v2 )) φ1V dx = h (ξ)φ1|V | dx ≤ α φ1 |V |2dx.

(3.38)

Using (2.7), the second term on the right-hand side of (3.37) satisfies Z Z Z Z 1 1 2 2 φ1|∂x V | dx + σ φ1 |V |2dx. | ∂x φ1 (∂x V ) V dx| ≤ σ φ1 |∂x V | |V |dx ≤ 2 2 (3.39) For the last term on the right-hand side of (3.37), we have Z Z Z 1 1 φ1 |W |2dx + φ1 |V |2 dx. | φ1 W V dx| ≤ (3.40) 2 2 It follows from (3.37)-(3.40) that there is a constant C1 depending on α and σ such that Z Z Z d 2 2 2 (3.41) φ1 |V | dx ≤ C1 φ1 |V | dx + φ1 |W | dx . dt Similarly, multiplying (3.36) by φ1 W and integrating, we find there is a constant C2 depending on such that Z Z Z d 2 2 2 φ1 |W | dx ≤ C2 (3.42) φ1|V | dx + φ1 |W | dx . dt Therefore, we have Z Z Z Z d 2 2 2 2 φ1 |V | dx + φ1 |W | ≤ C φ1 |V | dx + φ1 |W | dx . dt Thus, (3.34) follows from Gronwall’s inequality. The proof is complete. Next, we prove Theorem 2..1 and define a dynamical system for problem (2.1)-(2.3). Proof of Theorem 2..1. Given T > 0, by Lemma 3..4, there is a mapping G from H 1(IR) × H 1 (IR) into C([0, T ], L2σ(IR) × L2σ (IR)) such that for each (v0, w0) ∈ H 1(IR) × H 1(IR), G(v0, w0) is the unique solution of problem (2.1)-(2.3) in the sense of (3.19)(3.21). It follows from (3.34) that the mapping G is continuous from H 1(IR) × H 1(IR) ⊆ L2σ (IR)×L2σ (IR) into C([0, T ], L2σ(IR)×L2σ (IR)). Since H 1(IR)×H 1 (IR) is a dense subset of L2σ (IR) × L2σ (IR), G can be extended uniquely to a mapping G˜ from L2σ (IR) × L2σ (IR) into C([0, T ], L2σ(IR)×L2σ (IR)). We then define a semigroup S(t)t≥0 : L2σ (IR)×L2σ (IR) → L2σ (IR) × L2σ (IR) such that, for every t ≥ 0 and for (v0 , w0) ∈ L2σ (IR) × L2σ (IR), ˜ 0 , w0)(t). It follows from (3.34) that S(t)t≥0 is a continuous semiS(t)(v0, w0) = G(v group.


4.

141

Uniform estimates in time

In this section, we derive a priori estimates on solutions of problem (2.1)-(2.3) as t → ∞. We first establish uniform estimates in time in the space L2σ (IR) × L2σ (IR) and then improve the estimates in Hσ1(IR) × Hσ1(IR). Finally, we show that all solutions are uniformly small when space and time variables are sufficiently large. Lemma 4..1. Suppose k(v0 , w0)kσ ≤ R and let (v(t), w(t)) = S(t)(v0, w0). Then, k(v(t), w(t))kσ ≤ M, and

Z

for all t ≥ T,

(4.1)

t+d

k∂x v(t)k2σ dt ≤ C(1 + d),

for any t ≥ T and d > 0,

(4.2)

t

where M is a constant depending only on the data (, γ, a, β, σ), T depending on the data (, γ, a, β, σ) and R, while C depending on the data (, γ, a, β, σ), but not on d. Proof. Multiplying (2.1) and (2.2) by φδ v and φδ w, respectively, and integrating with respect to x, we find Z Z Z Z Z 1d φδ |v|2dx+ v∂x v ∂x φδ dx+ φδ |∂x v|2dx = φδ h(v)vdx− φδ wvdx, 2 dt IR IR IR IR IR and 1d 2 dt

Z

2

φδ |w| dx + γ

Z

IR

2

φδ |w| dx = IR

Z

φδ vwdx. IR

Therefore, Z Z Z Z 1d φδ |v|2dx + φδ |w|2dx + v∂x v ∂x φδ dx + φδ |∂x v|2dx 2 dt IR IR IR Z Z IR + γ φδ |w|2dx = φδ h(v)vdx. (4.3) IR

IR

Since h satisfies (2.4), we have the following estimates for the right-hand side of (4.3): Z Z Z φδ h(v)vdx ≤ −a φδ |v|2dx + β φδ dx. (4.4) IR

IR

IR

By (4.3) and (4.4), we get 1 d 2 dt

Z

2

φδ |v| dx + IR

Z IR

2

φδ |w| dx

+

Z

φδ |∂x v|2dx

IR Z

Z φδ |v|2dx + γ φδ |w|2dx IR Z ZIR φδ dx − v∂x v ∂x φδ dx. (4.5) ≤ β + a

IR

IR

142


Next, we estimate the last term on the right-hand side of (4.5). By (2.6) and (2.7) we have Z Z Z Z 1 2 2 | v∂xv ∂x φδ dx| ≤ δσ |v||∂xv|φδ dx ≤ δσ φδ v dx + φδ (∂x v) dx 2 IR IR IR IR Z Z 1 1 φδ v 2dx + φδ (∂x v)2dx. (4.6) a ≤ 4 4 IR IR Note that the last inequality is obtained by (2.6). From (4.5)-(4.6), it follows that Z Z Z 1 d 1 2 2 φδ |v| dx + φδ |w| dx + φδ |∂x v|2dx 2 dt 2 IR IR IR Z Z Z 1 2 2 a + φδ |v| dx + γ φδ |w| dx ≤ β φδ dx. (4.7) 2 IR IR IR R Let C0 = min{a, 2γ} and C1 = 2β IR φδ dx. Then from (4.7) we get Z Z Z Z d 2 2 2 2 φδ |v| dx + φδ |w| dx + C0 φδ |v| dx + φδ |w| dx ≤ C1 , (4.8) dt IR IR IR IR Applying Gronwall’s inequality, by k(v0 , w0)kσ ≤ R and (2.8), we obtain Z Z Z Z C1 2 2 −C0 t 2 2 φδ |v(t)| dx + φδ |w(t)| dx ≤ e φδ |v0| dx + φδ |w0| dx + C0 IR IR IR IR C 1 ≤ δ −2σ max{, 1}e−C0 t k(v0, w0)k2σ + (4.9) C0 C 2C1 ≤ δ −2σ R2 max{, 1}e−C0 t + 1 ≤ , (4.10) C0 C0 for t ≥ T = C10 (ln(C0 δ −2σ R2 max{, 1}) − ln C1 ). Note that (4.10) implies (4.1). Integrating (4.7) with respect to t, (4.2) follows from (4.10). The proof is complete. a Notice that if ≤ min{1, 2γ }, then the above constant C0 = min{a, 2γ} = 2γ. In this case, for all t ≥ 0, it follows from (4.9) that Z Z Z β φδ |v|2dx + φδ |w|2dx ≤ δ −2σ e−2γt kv0 k2σ + kw0k2σ + φδ dx, (4.11) γ IR IR IR

which proves useful to establish uniform bounds of global attractors, independent of small , in Section 6. In the sequel, we denote by B the set: B = (v, w) ∈ L2σ (IR) × L2σ (IR) : k(v, w)kσ ≤ M , (4.12) where M is the constant in (4.1). Then Lemma 4..1 implies the set B is an absorbing set for the dynamical system S(t)t≥0 . Since B itself is bounded, it follows from Lemma 4..1 that there exists a constant TB depending on the data (, γ, a, β, σ) such that S(t)B ⊆ B,

for all t ≥ TB ,

(4.13)

which shows that B is positively invariant for large time, and is useful when we investigate the large-time behavior of solutions. Next, we estimate the component v of the solution in Hσ1(IR).


143

Lemma 4..2. Suppose k(v0 , w0)kσ ≤ R and let (v(t), w(t)) = S(t)(v0, w0). Then we have (4.14) k∂x v(t)kσ ≤ M, for all t ≥ T, where M is a constant depending only on the data (, γ, a, β, α, σ), T depending on the data (, γ, a, β, σ) and R. Proof. Multiplying (2.1) by φ1 ∂xx v and integrating, we get Z Z Z Z ∂v ∂xx vdx = φ1 |∂xx v|2dx + φ1 h(v)∂xxvdx − φ1 w∂xxvdx. φ1 ∂t

(4.15)

Note that Z

∂v 1d ∂xx vdx = − φ1 ∂t 2 dt

Z

2

φ1 |∂x v| dx −

Z

∂v ∂x φ1 ∂x vdx, ∂t

(4.16)

and Z

φ1h(v)∂xx vdx = −

Z

h(v)∂x φ1 ∂x vdx −

Z

φ1 ∂x (h(v)) ∂x vdx.

Then, it follows from (4.15)-(4.17) that Z Z Z ∂v 1d 2 2 φ1 |∂xv| dx+ φ1 |∂xxv| dx = − − h(v) ∂x φ1 ∂x vdx 2 dt ∂t Z Z + h0 φ1|∂x v|2dx + φ1 w ∂xx vdx.

(4.17)

(4.18)

By (2.1) we have ∂v − h(v) = ∂xx v − w. ∂t Using (2.4) and (4.19), from (4.18) we find Z Z Z 1 d 2 2 φ1 |∂x v| dx+ φ1 |∂xx v| dx ≤ − ∂xx v ∂x φ1 ∂x vdx 2 dt Z Z Z 2 + w∂x φ1 ∂x vdx + α φ1 |∂xv| dx + φ1w ∂xx vdx. By (2.7) we have Z Z Z 1 φ1 |∂xx v|2dx | ∂xx v ∂x φ1 ∂x vdx| ≤ σ |∂xx v| |φ1 | |∂xv|dx ≤ 2 Z 1 2 φ1|∂x v|2dx. + σ 2

(4.19)

(4.20)

(4.21)

The second and the last term on the right-hand side of (4.20) satisfy Z Z Z Z 1 1 2 | w∂x φ1 ∂x vdx| ≤ σ |φ1 ||w||∂xv|dx ≤ σ φ1 w dx + σ φ1|∂x v|2dx, (4.22) 2 2

144


and |

Z

φ1 w ∂xx vdx| ≤

1 2

Z

φ1|w|2dx +

1 2

Z

φ1|∂xx v|2dx.

(4.23)

By (4.20)-(4.23) we get Z Z Z 1d 1 2 1 1 2 φ1 |∂x v| dx ≤ σ + σ+α φ1 |∂x v|2dx + (σ + 1) φ1 |w|2dx, 2 dt 2 2 2 that is,

d (4.24) k∂xvk2σ ≤ σ 2 + σ + 2α k∂x vk2σ + (σ + 1) kwk2σ . dt Applying the uniform Gronwall’s Lemma (see, e.g., [28]), by (4.1)-(4.2) we obtain k∂x vk2σ ≤ C, for all t ≥ T + 1, where T is the constant in (4.1).

Next, we improve the estimates given by (4.1) for the component w of the solution. Since equation (2.2) has no smoothing effect on w, we cannot obtain an estimate for w in the space Hσ1(IR) if the initial datum w0 ∈ L2σ (IR). Nevertheless, we can prove w is a sum of two functions: one is regular in the sense it belongs to Hσ1(IR) and the other converges to zero uniformly as t → ∞. This decomposition technique has been used by several authors for the FitzHugh-Nagumo equations (see, for example [18]). We split w as w = w1 + w2 where w1 is the solution of the initial value problem ∂w1 = −γw1, ∂t

w1(0) = w0,

(4.25)

and w2 is the solution of ∂w2 = −γw2 + v, ∂t

w2 (0) = 0.

(4.26)

It is evident the following energy equation holds for w1: kw1kσ = e−γt kw0kσ ,

for all t ≥ 0,

(4.27)

which implies that w1 converges to zero uniformly in bounded initial data when t → ∞. Next, we show that w2 is bounded in the space Hσ1(IR) for large time. Lemma 4..3. w2 satisfies k∂xw2 (t)kσ ≤ M,

for all t ≥ T,

(4.28)

where M is a constant depending only on the data (, γ, a, β, α, σ), T depending on the data (, γ, a, β, σ) and R when k(v0, w0)kσ ≤ R. Proof. Multiplying (4.26) by φ1 ∂xx w2 and then integrating, we get Z Z Z ∂w2 ∂xx w2 dx = −γ w2 φ1∂xx w2 dx + vφ1∂xx w2 dx. φ1 ∂t Note that the left-hand side of (4.29) can be rewritten as Z Z Z 1 d ∂w2 ∂w2 2 ∂xx w2dx = − φ1 |∂xw2 | dx − ∂x φ1 ∂x w2dx. φ1 ∂t 2 dt ∂t

(4.29)

(4.30)


145

We now deal with the right-hand side of (4.29). For the first term on the right-hand side, we have Z Z Z 2 (4.31) −γ w2 φ1∂xx w2 dx = γ φ1 |∂xw2 | dx + γ w2 ∂x φ1 ∂x w2dx. Integrating by parts, we get the identity for the last term on the right-hand side: Z Z Z vφ1 ∂xx w2dx = − φ1 ∂x v ∂x w2dx − v∂x φ1 ∂x w2dx. By (4.29)-(4.32) we find Z Z Z 1d 2 2 φ1 |∂x w2| dx + γ φ1 |∂x w2| dx = φ1 ∂x v∂xw2 dx 2 dt Z ∂w2 + γw2 − v ∂x φ1∂x w2dx. − ∂t

(4.32)

(4.33)

By (4.26), the last term on the right-hand side of (4.33) is zero. While, the first term on the right can be estimated as follows. Z Z Z 1 2 (4.34) φ1 ∂x v ∂x w2dx ≤ γ φ1 |∂x w2| dx + φ1 |∂x v|2dx. 2 2γ From (4.33) and (4.34) we obtain Z Z d 2 φ1 |∂x w2| + γ φ1 |∂xw2 |2dx ≤ dt

γ

Z

φ1|∂x v|2dx ≤ C,

for t ≥ T,(4.35)

where the last inequality is obtained by (4.14). Then, (4.28) follows from (4.35) and Gronwall’s Lemma. In what follows, we derive estimates on solutions for large time and large space variables. Actually, we show that the solutions are uniformly small for large time in the complement of a bounded set, which is crucial for the asymptotic compactness of the dynamical system S(t)t≥0 associated with problem (2.1)-2.3). The uniform estimates outside a bounded set are stated as follows. Lemma 4..4. For every η > 0, there exists T (η) depending only on η, (, γ, a, β, σ) and R when k(v0, w0)kσ ≤ R, such that, for all t ≥ T (η), the following inequality holds: Z φ1|v|2 + φ1|w|2 dx ≤ η, (4.36) |x|≥K(η)

where K(η) depends only on η and (, γ, a, β, α, σ). Proof. Let θ be a smooth function satisfying 0 ≤ θ(s) ≤ 1 for s ≥ 0, and θ(s) = 0 for 0 ≤ s ≤ 1;

θ(s) = 1 for s ≥ 2.

146


Then, there is a positive constant C such that |θ0 (s)| ≤ C for all s ≥ 0. Multiplying (2.1) 2 )φδ v and then integrating, we find by θ( |x| k2 1d 2 dt

Z

θ(

|x|2 )φδ |v|2dx = k2 +

Z Z

∂xx v θ(

|x|2 )φδ vdx k2

|x|2 θ( 2 )φδ h(v)vdx − k

Z

θ(

|x|2 )φδ wvdx.(4.37) k2

2

)φδ w and integrating, we get Multiplying (2.2) by θ( |x| k2 1d 2 dt

Z

θ(

|x|2 )φδ |w|2dx = k2

Z

θ(

|x|2 )φδ vwdx − γ k2

Z

θ(

|x|2 )φδ |w|2dx k2

(4.38)

By integration by parts, we have the following identity for the first term on the right-hand side of (4.37):

Z

∂xx v θ(

Z Z |x|2 |x|2 |x|2 2 )φ vdx = − θ( )φ |∂ v| dx − θ( )v ∂x v ∂x φδ dx δ δ x k2 k2 k2 Z |x|2 (4.39) − φδ v ∂x v ∂x θ( 2 ) dx. k

By (4.37)-(4.39) we obtain Z

Z |x|2 |x|2 2 2 θ( 2 )φδ |v| dx + θ( 2 )φδ |w| dx k k Z Z 2 |x|2 |x| + θ( 2 )φδ |∂xv|2dx + γ θ( 2 )φδ |w|2dx k k Z Z 2 |x|2 |x| = θ( 2 )φδ h(v)vdx − θ( 2 )v ∂x v ∂x φδ dx k k Z 2 |x| − φδ v ∂x v ∂x θ( 2 ) dx. k

1d 2 dt

(4.40)

Using (2.4) we have Z |x|2 |x|2 2 2 )φ |v| dx + θ( )φ |w| dx δ δ k2 k2 Z Z Z |x|2 |x|2 |x|2 2 2 + θ( 2 )φδ |∂x v| dx + a θ( 2 )φδ |v| dx + γ θ( 2 )φδ |w|2dx k k k Z Z Z 2 2 |x| |x| |x|2 ≤ β θ( 2 )φδ dx − θ( 2 )v ∂x v ∂x φδ dx − φδ v ∂x v ∂x θ( 2 ) dx. k k k (4.41)

1d 2 dt

Z

θ(

Next, we estimate terms on the right-hand side of (4.41). By (2.7) and (2.6), we have the

Asymptotic Behavior of the FitzHugh-Nagumo System following bounds for the second term on the right-hand side. Z Z |x|2 |x|2 | θ( 2 )v ∂x v ∂x φδ dx| ≤ δσ θ( 2 )φδ |v||∂xv|dx k k Z 1 |x|2 ≤ θ( 2 )φδ |∂x v|2dx 4 k Z |x|2 1 a θ( 2 )φδ |v|2dx. + 4 k

147

(4.42)

By properties of θ, the last term on the right-hand side of (4.41) is bounded by Z Z |x|2 2x |x|2 | φδ v ∂x v ∂x θ( 2 ) dx| = | φδ v ∂x v θ0 ( 2 ) 2 dx| k k k √ √ Z C2 2C1 2C1 ≤ k∂x vkσ kvkσ ≤ , (4.43) √ |φδ ||v||∂xv|dx ≤ k k k k≤|x|≤ 2k for all t ≥ T , where the last inequality is obtained by Lemmas 4..1 and 4..2, C2 and T are constants independent of k. By (4.42) and (4.43), it follows from (4.41) that Z Z |x|2 |x|2 1d 2 2 θ( 2 )φδ |v| dx + θ( 2 )φδ |w| dx 2 dt k k Z Z Z 2 |x| |x|2 1 1 |x|2 + θ( 2 )φδ |∂x v|2dx + a θ( 2 )φδ |v|2dx + γ θ( 2 )φδ |w|2dx 2 k 2 k k Z 2 C2 |x| , for all t ≥ T. (4.44) ≤ β θ( 2 )φδ dx + k k Let C3 = min{a, 2γ}. Then we get from (4.44) that, for all t ≥ T , Z Z |x|2 d |x|2 θ( 2 )φδ |v|2dx + θ( 2 )φδ |w|2dx dt k k Z Z 2 |x|2 |x| 2 2 + C3 θ( 2 )φδ |v| dx + θ( 2 )φδ |w| dx k k Z 2 |x| 2C2 . ≤ 2β θ( 2 )φδ dx + k k

(4.45)

Next, we manipulate the first term on the right-hand side of (4.45), which is bounded by Z Z Z |x|2 |x|2 |2β θ( 2 )φδ dx| ≤ 2β θ( 2 )φδ dx ≤ 2β φδ dx. (4.46) k k |x|≥k |x|≥k Therefore, by (4.45) we find there exists a constant K(η) such that for k ≥ K(η) and t ≥ T, Z Z d |x|2 |x|2 2 2 θ( 2 )φδ |v| dx + θ( 2 )φδ |w| dx dt k k Z Z 2 |x|2 |x| (4.47) θ( 2 )φδ |v|2dx + θ( 2 )φδ |w|2dx ≤ 2η. +C3 k k

148


By Gronwall’s Lemma, for all t ≥ T and k ≥ K(η), we get Z

Z |x|2 |x|2 2 )φ |v(t)| dx + θ( )φδ |w(t)|2dx δ 2 2 k k IR IR Z Z 2 |x| |x|2 2η −C3 (t−T ) 2 2 ≤e θ( 2 )φδ |v(T )| dx + θ( 2 )φδ |w(T )| dx + k k C3 IR IR 2η ≤ C4 e−C3 (t−T ) + . (4.48) C3 θ(

Note the last inequality is obtained by Lemma 4..1. Let T1(η) = T + 1/C3 ln C3ηC4 . Then, by (4.48) we obtain, for t ≥ T1(η) and k ≥ K(η), Z

|x|2 θ( 2 )φδ |v(t)|2dx + k IR

Z

θ(

IR

|x|2 3η )φδ |w(t)|2dx ≤ , 2 k C3

which implies (4.36) since θ(s) = 1 for s ≥ 2. The proof is complete. As an immediate consequence of Lemma 4..4 and (4.27), we have the following estimates for (v, w2), which are useful when we establish the asymptotic compactness of the solution in the next section. Corollary 4..5. For every η > 0, there exists T (η) depending only on η, (, γ, a, β, σ) and R when k(v0, w0)kσ ≤ R, such that, for all t ≥ T (η), the following inequality holds: Z φ1 |v|2 + φ1 |w2|2 dx ≤ η, |x|≥K(η)

where K(η) depends only on η and (, γ, a, β, α, σ).

5.

Existence of global attractors

In this section, we show that the dynamical system S(t)t≥0 has a global attractor in the space L2σ (IR) × L2σ (IR). It is known that the global attractor exists if S(t)t≥0 is asymptotically compact and has a bounded absorbing set. We have already proved that S(t)t≥0 has a bounded absorbing set, which is given by (4.12). The next lemma establishes the asymptotic compactness of S(t)t≥0. Lemma 5..1. The dynamical system S(t)t≥0 is asymptotically compact, that is, if (vn , wn) is bounded in L2σ (IR)×L2σ (IR) and tn → ∞, then S(tn )(vn , wn) is precompact in L2σ (IR)× L2σ (IR). Proof. Assume (vn , wn) is bounded in L2σ (IR) × L2σ (IR) and tn → ∞. We want to prove the sequence {S(tn )(vn , wn)} has a convergent subsequence in L2σ (IR) × L2σ (IR). To this end, we decompose {S(tn )(vn , wn)} as a sum of two sequences: one is precompact and the other converges to zero as n → ∞. Given (v0 , w0) ∈ L2σ (IR) × L2σ (IR), denote by w1 and w2 the solutions of problem (4.25) and (4.26), respectively. Then, for t ≥ 0, we define two maps S1 (t) and S2(t) from L2σ (IR)×L2σ (IR) into itself such that S1 (t)(v0, w0) = (0, w1(t))


149

and S2 (t)(v0, w0) = (v(t), w2(t)) for (v0, w0) ∈ L2σ (IR) × L2σ (IR). Thus, S(t)(v0, w0) = S1 (t)(v0, w0) + S2 (t)(v0, w0). Particularly, for the sequence {S(tn )(vn, wn )}, we have S(tn )(vn , wn) = S1(tn )(vn , wn) + S2 (tn )(vn , wn).

(5.1)

Since (vn , wn) is bounded in L2σ (IR) × L2σ (IR), there exists R > 0 such that k(vn , wn)kσ ≤ R,

for n ≥ 1.

(5.2)

It then follows from (4.27) that kS1(tn )(vn , wn)kσ ≤ e−γtn kwn kσ ≤ Re−γtn → 0,

as n → ∞.

(5.3)

Thus, if S2 (tn )(vn , wn) has a convergent subsequence, then so does S(tn )(vn , wn). We now prove the sequence S2 (tn )(vn , wn) is precompact in L2σ (IR) × L2σ (IR), that is, for any η > 0, the set {S2 (tn )(vn , wn)} has a finite covering of balls of radii less than η. For a positive constant K, let’s denote ¯ K = {x : |x| ≤ K} and ΩcK = {x : |x| > K}. Ω Then by Corollary 4..5, given η > 0, there exist K(η) and T (η) such that kS2(t)(vn , wn)kL2σ (Ωc

K(η)

)×L2σ (ΩcK(η))

≤

η , 4

for t ≥ T (η).

(5.4)

Since tn → ∞, there exists N1(η) such that tn ≥ T (η) for n ≥ N1. Therefore, kS2(tn )(vn , wn)kL2σ (Ωc

K(η)

)×L2σ (ΩcK(η) )

≤

η , 4

for n ≥ N1 .

(5.5)

On the other hand, by Lemmas 4..1-4..3, there exist M1 and T1 such that kS(t)(vn , wn)kHσ1 (IR)×Hσ1 (IR) ≤ M1

for t ≥ T1.

(5.6)

Take N2 large enough such that tn ≥ T1 for n ≥ N2. Then it follows from (5.6) that kS(tn )(vn , wn)kHσ1 (IR)×Hσ1 (IR) ≤ M1

for n ≥ N2.

(5.7)

Taking (5.3) into account, by (5.7) we find that there exists M2 such that kS2(tn )(vn , wn)kHσ1 (IR)×Hσ1 (IR) ≤ M2

for n ≥ N2 .

(5.8)

Note (5.8) implies kS2(tn )(vn , wn)kHσ1 (Ω¯ K(η) )×Hσ1 (Ω¯ K(η)) ≤ M2

for n ≥ N2,

(5.9)

¯ K(η) is bounded, the norm k · k 1 ¯ where K(η) is the constant in (5.4). Since Ω Hσ (ΩK(η)) is equivalent to the norm k · kH 1 (Ω¯ K(η)) . Therefore, by (5.9) we see that there exists M3 such that kS2(tn )(vn , wn )kH 1(Ω¯ K(η))×Hσ1 (Ω¯ K(η) ) ≤ M3 for n ≥ N2. (5.10)

150


¯ K(η)) ⊂ L2 (Ω ¯ K(η)), the sequence S2 (tn )(vn , wn) By the compactness of embedding H 1 (Ω 2 2 ¯ ¯ ¯ K(η)) × L2σ (Ω ¯ K(η)). is precompact in L (ΩK(η)) × L (ΩK(η)) and hence in L2σ (Ω This implies that, for the given η > 0, {S2(tn )(vn , wn)} has a finite covering in ¯ K(η)) × L2σ (Ω ¯ K(η)) of balls of radii less than η/4, which along with (5.4) shows that L2σ (Ω {S2(tn )(vn , wn)} has a finite covering in L2σ (IR) × L2σ (IR) of balls of radii less than η. In other words, the sequence {S2 (tn )(vn , wn)} is precompact in the space L2σ (IR) × L2σ (IR). Therefore, by (5.1) and (5.3), we know {S(tn )(vn , wn)} is precompact in L2σ (IR)×L2σ (IR). The proof is complete. We are now ready to complete Theorem 2..2 which is concerned with the existence of a global attractor. Proof of Theorem 2..2. By a standard result (see, for example, [3, 5, 12, 15, 28]), we know the dynamical system S(t)t≥ possesses a global attractor if it is asymptotically compact and has a bounded absorbing set. In our case, the asymptotic compactness of S(t)t≥0 was established by Lemma 5..1 and the bounded absorbing set was given by (4.12).

6.

Uniform bounds of global attractors in

In this section, we show that the global attractors A are uniformly bounded as → 0. We first derive the uniform bounds in L2σ (IR) × L2σ (IR). a Lemma 6..1. For any < min{1, 2γ } the solution (v, w) of (2.1)-(2.3) satisfies, for t ≥ 0, Z β 2 −2σ −2γt 2 2 kv0 kσ + kw0kσ + e (6.1) kwkσ ≤ δ φδ dx, γ

and kvk2σ

2β(1 + aγ) a(a − 2γ) + 2 −2σ −2γt ≤ e δ kv0k2σ + kw0k2σ + a(a − 2γ) a2 γ

Z

φδ dx,

(6.2)

where a and β are the constants in (2.4), and δ is the constant given by (2.6). Furthermore, for t ≥ 0, Z t+1 2(a + 2)(a − 2γ) + 4 −2σ −2γt kv0k2σ + kw0k2σ k∂x vk2σ dt ≤ e δ 3a(a − 2γ) t Z 4β(a + 1)(aγ + 1) + φδ dx. (6.3) 3a2 γ Proof. Multiplying (2.1) by φδ v and integrating with respect to x, we get Z Z Z Z Z 1d 2 2 φδ |v| dx+ φδ |∂x v| dx = − v∂x v ∂x φδ dx+ φδ h(v)v− φδ vwdx. (6.4) 2 dt By inequality (2.4) for h, we obtain from (6.4) that Z Z Z 1 d 2 2 φδ |v| dx + φδ |∂x v| dx ≤ − v∂x v ∂x φδ dx 2 dt Z Z Z 2 − a φδ |v| dx + β φδ dx − φδ vwdx.

(6.5)


151

We now estimate the terms on the right-hand side of (6.5) as follows. By (2.6) and (2.7) we see the first term on the right-hand side is bounded by Z Z Z Z 1 φδ |v|2dx + φδ |∂x v|2dx | v∂x v ∂x φδ dx| ≤ δσ φδ |v||∂xv|dx ≤ δσ 2 Z Z 1 1 (6.6) a φδ |v|2dx + φδ |∂x v|2dx. ≤ 4 4 For the last term on the right-hand side of (6.5), we have the bounds: Z Z Z a 1 | φδ vwdx| ≤ φδ |v|2dx + φδ |w|2dx. 4 a

(6.7)

It follows from (6.5)-(6.7) that Z Z Z Z Z 3 a 1 1 d 2 2 2 φδ |v| dx+ φδ |∂x v| dx+ φδ |v| dx ≤ β φδ dx+ φδ |w|2dx. (6.8) 2 dt 4 2 a a }, (4.11) implies Since < min{1, 2γ Z Z β 2 −2σ −2γt 2 2 φδ |w| dx ≤ δ e φδ dx, kv0kσ + kw0kσ + γ IR

for t ≥ 0.

By (6.8) and (6.9) we get, for t ≥ 0, Z Z Z 3 d 2 2 φδ |v| dx + φδ |∂x v| dx + a φδ |v|2dx dt 2 Z 2 1 φδ dx + δ −2σ e−2γt kv0k2σ + kw0k2σ . ≤ 2β 1 + aγ a Dropping the second term on the left side , by Gronwall’s inequality we obtain Z Z Z 2β 1 2 −at φδ |v| dx ≤ 1+ φδ dx + e φδ |v0|2 dx a aγ 2δ −2σ e−2γt kv0k2σ + kw0k2σ . + a(a − 2γ) a Since < 2γ by assumption, we have Z Z e−at φδ |v0 |2dx ≤ e−2γt φδ |v0 |2dx ≤ δ −2σ e−2γt kv0 k2σ + kw0k2σ .

(6.9)

(6.10)

(6.11)

(6.12)

It follows from (6.11) and (6.12) that Z

2β φδ |v| dx ≤ a 2

1 1+ aγ

Z

2 φδ dx + 1 + a(a − 2γ)

δ −2σ e−2γt kv0 k2σ + kw0k2σ . (6.13)

By (6.9) and (6.13) we get (6.1) and (6.2), respectively. Next, we prove (6.3). Integrating (6.10) from t to t + 1, we obtain Z Z Z 1 3 t+1 2 2 k∂xvkσ dt ≤ φδ |v(t)| dx + 2β 1 + φδ dx 2 t aγ Z t+1 −2γt 2 −2σ 2 2 kv0kσ + kw0kσ e dt. (6.14) + δ a t

152


We now deal with the last term on the right-hand side of (6.14). Notice that Z

t+1

e−2γt dt ≤ e−2γt

for all > 0 and t ≥ 0.

(6.15)

t

Substituting (6.13) and (6.15) into (6.14), then (6.3) follows. The proof is complete. Notice that by adding (6.1) and (6.2) together we get the inequality: kS(t)(v0, w0)k2σ ≤ K1e−2γt kv0k2σ + kw0k2σ + K2,

for all t ≥ 0, (6.16)

where K1 and K2 are given by 2 (1 + a(a − 2γ)) −2σ β(a2 + 2aγ + 2) , K2 = δ K1 = a(a − 2γ) a2 γ

Z

φδ dx.

(6.17)

The uniform boundedness of A in L2σ (IR) × L2σ (IR) can now be easily obtained. Proof of Theorem 2..3. Suppose (v, w) ∈ A and tn → ∞. Then by invariance of A , there exists a sequence of (vn , wn ) ∈ A such that (v, w) = S(tn )(vn, wn ) for all n ≥ 1.

(6.18)

Since A ⊆ B, where B is the bounded absorbing set given by (4.12), we have k(vn , wn )kσ ≤ M,

for all n ≥ 1,

(6.19)

where M depends only on the data (, γ, a, β, σ). On the other hand, by applying (6.16) to S(tn )(vn , wn) we get kS(tn )(vn , wn)k2σ ≤ K1e−2γtn k(vn , wn)k2σ + K2 ,

for all n ≥ 1,

(6.20)

where K1 and K2 are constants given by (6.17). It follows from (6.19) and (6.20) that kS(tn )(vn , wn)k2σ ≤ K1 M 2e−2γtn + K2,

for all n ≥ 1,

which along with (6.18) implies that for any (v, w) ∈ A , the following inequality holds k(v, w)k2σ ≤ K1 M 2e−2γtn + K2,

for all n ≥ 1.

(6.21)

Taking the limit as n → ∞, we find k(v, w)k2σ ≤ K2 from which (2.12) follows. To complete the proof, we also need to establish the estimates claimed in (2.13). In what follows, we first derive various uniform bounds in for the solution in Hσ1(IR) × Hσ1(IR) via a sequence of lemmas, and then complete the proof of Theorem 2..3 in the end of this section. a Lemma 6..2. For any < min{1, 2γ }, the solution (v, w) of (2.1)-(2.3) satisfies

k∂x v(t)k2σ ≤ K3 e−2γt kv0k2σ + kw0k2σ + K4 ,

for t ≥ 1,

(6.22)


153

where K3 and K4 are given by σ+1 −2σ 2γ+σ 2 +σ+2α 2(a + 2)(a − 2γ) + 4 e + , (6.23) K3 = δ 3a(a − 2γ) (2γ + σ 2 + σ + 2α)2 and σ 2 +σ+2α

K4 = e

β(σ + 1) 4β(a + 1)(aγ + 1) + 2 2 γ(σ + σ + 2α) 3a2 γ

Z

φδ dx,

(6.24)

where a and β are the constants in (2.4), and δ is the constant given by (2.6). Proof. We start with inequality (4.24): d k∂x vk2σ ≤ σ 2 + σ + 2α k∂xvk2σ + (σ + 1) kwk2σ , dt

for t ≥ 0.

(6.25)

Substituting (6.1) into (6.25) we get, for all t ≥ 0, β(σ + 1) d k∂x vk2σ − kk∂x vk2σ ≤ (σ + 1) δ −2σ e−2γt kv0k2σ + kw0k2σ + dt γ

Z

φδ dx.

(6.26) Here and after, we denote by κ = σ 2 + σ + 2α. Let t and s be positive numbers satisfying 0 ≤ t ≤ s ≤ t + 1. Then multiplying (6.26) by e−κt and integrating with respect to t between s and t + 1, we obtain k∂x v(t + 1)k2σ e−κ(t+1) ≤e

−κs

k∂x v(s)k2σ

−2σ

+ (σ + 1) δ Z t+1 Z β(σ + 1) e−κt dt. φδ dx + γ s

kv0 k2σ

+

kw0k2σ

Z

t+1

e−(κ+2γ)t dt s

The two integrals on the right-hand side of (6.27) have the estimates Z t+1 Z t+1 1 1 −(κ+2γ)t −(κ+2γ)s e e dt ≤ and e−κt dt ≤ e−κs . κ + 2γ κ s s

(6.27)

(6.28)

Substituting (6.28) into (6.27), and then multiplying the resulting inequality by eκ(t+1) we find Z β(σ + 1) κ(t+1−s) 2 κ(t+1−s) 2 k∂xv(t + 1)kσ ≤ e e φδ dx k∂x v(s)kσ + γκ σ + 1 −2σ (6.29) δ kv0k2σ + kw0k2σ eκ(t+1−s)−2γs . + κ + 2γ Since 0 ≤ t ≤ s ≤ t + 1, integrating (6.29) with respect to s between t and t + 1 we have Z Z t+1 Z β(σ + 1) t+1 κ(t+1−s) 2 κ(t+1−s) 2 k∂x v(t + 1)kσ ≤ e k∂xv(s)kσ ds + e ds φδ dx γκ t t Z t+1 κ(t+1−s)−2γs (σ + 1)δ −2σ 2 2 e ds. (6.30) kv0 kσ + kw0kσ + κ + 2γ t

154


We now deal with the right-hand side of (6.30). By (6.3), the first term on the right-hand side of (6.30) satisfies Z

t+1

e

κ(t+1−s)

k∂xv(s)k2σ ds

≤ e

t

κ

Z

t+1

k∂x v(s)k2σ ds t

≤

2(a + 2)(a − 2γ) + 4 −2σ −2γt+κ δ kv0 k2σ + kw0k2σ e 3a(a − 2γ) Z 4β(a + 1)(aγ + 1) κ (6.31) + e φδ dx. 3a2 γ

The integrals in the last two terms of the right-hand side of (6.30) can be estimated Z

t+1

e

κ(t+1−s)−2γs

t

Z

1 ds ≤ e−2γt+κ and κ + 2γ

t+1

eκ(t+1−s) ds ≤ t

1 κ e . κ

(6.32)

It follows from (6.30)-(6.32) that, for all t ≥ 0, k∂x v(t + 1)k2σ

≤

σ+1 2(a + 2)(a − 2γ) + 4 δ −2σ e−2γt+κ kv0 k2σ + kw0 k2σ + 2 3a(a − 2γ) (κ + 2γ) Z β(σ + 1) 4β(a + 1)(aγ + 1) κ + + e φδ dx, γκ2 3a2 γ

which along with κ = σ 2 + σ + 2α implies (6.22). As an immediate consequence of Lemma 6..2, we have the following estimates for global attractors. a }. If (v, w) ∈ A , then Lemma 6..3. Let 0 < < min{1, 2γ

k∂x vk2σ ≤ eσ

2 +σ+2α

4β(a + 1)(aγ + 1) β(σ + 1) + γ(σ 2 + σ + 2α)2 3a2 γ

Z

φδ dx,

(6.33)

where a, β and α are the constants in (2.4) and δ is the constant given by (2.6). Next, we establish the uniform bounds in for the solution w2 of problem (4.26) in Hσ1(IR). a }. If (v0 , w0) ∈ A , then the solution w2 of problem Lemma 6..4. Let 0 < < min{1, 2γ (4.26) satisfies:

k∂x w2k2σ

≤

eσ

2 +σ+2α

γ2

β(σ + 1) 4β(a + 1)(aγ + 1) + 2 2 γ(σ + σ + 2α) 3a2γ

Z

φδ dx,

(6.34)

where a, β and α are the constants in (2.4) and δ is the constant given by (2.6). Proof. We begin with inequality (4.35): Z Z d 2 φ1 |∂xw2 | + γ φ1|∂x w2|2 dx ≤ dt

γ

Z

φ1 |∂x v|2dx,

for t ≥ 0. (6.35)


155

Since (v0, w0) ∈ A , we know for all t ≥ 0, (v(t), w(t)) = S(t)(v0, w0) ∈ A . Then it follows from Lemma 6..3 that Z (6.36) φ1|∂x v(t)|2dx ≤ C, for all t ≥ 0, where C is given by σ2 +σ+2α

C=e

β(σ + 1) 4β(a + 1)(aγ + 1) + 2 2 γ(σ + σ + 2α) 3a2γ

By (6.35) and (6.36) we get Z Z d 2 φ1|∂x w2| + γ φ1 |∂x w2|2dx ≤ dt Applying Gronwall’s Lemma to (6.38), we obtain Z C φ1 |∂xw2 (t)|2dx ≤ 2 , γ

Z

C , γ

φδ dx,

for t ≥ 0.

(6.37)

(6.38)

for all t ≥ 0.

(6.39)

Then (6.34) follows from (6.39) and (6.37). As an immediate consequence of Lemmas 6..3 and 6..4, we have the following bounds in the space Hσ1(IR) × Hσ1(IR) for the dynamical system S(t)t≥0. a Corollary 6..5. If (v0, w0) ∈ A for 0 < < min{1, 2γ }, then the solution (v, w) of problem (2.1)-(2.3) and the solution w2 of problem (4.26) satisfy:

k∂x v(t)k2σ + k∂x w2(t)k2σ ≤ K5,

for all t ≥ 0,

(6.40)

where K5 is given by (γ 2 + 1)eσ K5 = γ2

2 +σ+2α

4β(a + 1)(aγ + 1) β(σ + 1) + 2 2 γ(σ + σ + 2α) 3a2γ

Z

φδ dx, (6.41)

where a, β and α are the constants in (2.4) and δ is the constant given by (2.6). We are now ready to complete the proof of Theorem 2..3. Proof of Theorem 2..3 (Continued). By invariance of A , there exist tn → ∞ and (vn , wn) ∈ A such that (v, w) = S(tn )(vn , wn ),

for all n ≥ 1.

(6.42)

Decompose S(t) as in (5.1): S(t)(v0, w0) = S1 (t)(v0, w0)+S2 (t)(v0, w0),

for (v0 , w0) ∈ L2σ (IR)×L2σ (IR), (6.43)

where S1 (t)(v0, w0) = (0, w1(t)), S2 (t)(v0, w0) = (v(t), w2(t)), w1 and w2 are solutions of problem (4.25) and problem (4.26), respectively. It follows from (6.42) and (6.43) that (v, w) = S1 (tn )(vn , wn) + S2(tn )(vn , wn),

for all n ≥ 1.

(6.44)

156


Since (vn , wn) ∈ A , we find k(vn , wn)kσ ≤ C,

for all n ≥ 1,

(6.45)

where C is the constant on the right-hand side of (2.12). By (4.27) and (6.45) we get kS1 (tn )(vn , wn)kσ ≤ e−γtn kwn kσ ≤ Ce−γtn → 0, On the other hand, kS2(tn )(vn, wn )kHσ1

as n → ∞.

q ≤ K62 + K72 ,

(6.46)

(6.47)

where K6 and K7 are given by s K6 =

β(a2 + 2aγ + 2) a2 γ

Z

φδ dx,

(6.48)

and 1 K7 = γ

s

(γ 2 + 1)e

σ2 +σ+2α

β(σ + 1) 4β(a + 1)(aγ + 1) + 2 2 γ(σ + σ + 2α) 3a2γ

Z

φδ dx.

(6.49) By (6.47) we see the sequence {S2 (tn )(vn , wn)} is bounded in Hσ1(IR) × Hσ1 (IR). Therefore there exists (˜ v, w) ˜ ∈ Hσ1(IR) × Hσ1 (IR) such that v , w) ˜ weakly in Hσ1 (IR) × Hσ1 (IR). S2(tn )(vn , wn) → (˜

(6.50)

Furthermore, by (6.47) and (6.50) we have k(˜ v, w)k ˜ Hσ1 ×Hσ1 ≤ lim inf kS2(tn )(vn , wn)kHσ1 ×Hσ1 ≤ n→∞

q

K62 + K72.

(6.51)

For any (f, g) ∈ L2σ (IR) × L2σ (IR), it follows from (6.44) that h(v, w), (f, g)i = hS1 (tn )(vn , wn), (f, g)i + hS2(tn )(vn , wn), (f, g)i,

(6.52)

where h·, ·i denotes the inner product of L2σ (IR) × L2σ (IR). Taking the weak limit of (6.52) in L2σ (IR) × L2σ (IR) as n → ∞, by (6.46) and (6.50) we get h(v, w), (f, g)i = h(˜ v, w), ˜ (f, g)i,

for all (f, g) ∈ L2σ (IR) × L2σ (IR).

(6.53)

By (6.53) we find (v, w) = (˜ v, w) ˜ ∈ Hσ1(IR) × Hσ1(IR). (6.54) p It follows from (6.51) and (6.54) that k(v, w)kHσ1×Hσ1 ≤ K62 + K72 , which, together with (6.48) and (6.49), concludes the proof.


7.

157

Compactness of the union of global attractors

In this section, we show that all global attractors are contained in a compact subset of L2σ (IR) × L2σ (IR). The idea is to prove that the functions in global attractors are uniformly small in when the space variable is sufficiently large. a }. Then for every η > 0, there exist a constant Lemma 7..1. Suppose 0 < < min{1, 2γ K(η) depending only on η and the data (γ, a, β, α, σ), but independent of , such that for all (v, w) ∈ A , the following inequality holds: Z φ1|v|2dx + φ1 |w|2dx ≤ η. (7.1) |x|≥K(η)

Proof. Let tn → ∞. Since (v, w) ∈ A , there exists a sequence of (vn , wn) ∈ A such that (v, w) = S(tn )(vn , wn),

for n ≥ 1.

(7.2)

Set (vn (t), wn(t)) = S(t)(vn, wn ). Then Theorem 2..3 implies that there exists a constant C independent of such that for all n ≥ 1 and t ≥ 0: k(vn (t), wn(t))kHσ1 ×Hσ1 ≤ C.

(7.3)

Proceeding as the proof of Lemma 4..4 but using (7.3) instead of Lemmas 4..1 and 4..2, we find that, given η > 0, there exists a constant K1(η) depending only on η and the data (γ, a, β, α, σ) such that for k ≥ K1(η) and t ≥ 0: Z Z |x|2 |x|2 d 2 2 θ( 2 )φδ |vn (t)| dx + θ( 2 )φδ |wn(t)| dx dt k k Z Z 2 |x| |x|2 +2γ θ( 2 )φδ |vn (t)|2 dx + θ( 2 )φδ |wn(t)|2dx ≤ 2η, (7.4) k k which is the analogue of (4.47) with C3 = min{a, 2γ} = 2γ since < √ Lemma, we can verify from (7.4) that, for t ≥ 0 and k ≥ 2K1(η): Z η φδ |wn (t)|2dx ≤ C1 e−2γt + . γ |x|≥k

a 2γ .

By Gronwall’s

(7.5)

Next, we derive the estimates similar to (7.5) for vn (t). For that purpose, multiplying (2.1) 2 )φδ vn (t) we get by θ( |x| k2 Z Z 1d |x|2 |x|2 θ( 2 )φδ |vn (t)|2dx = ∂xx vn (t) θ( 2 )φδ vn (t)dx 2 dt k k Z Z 2 |x|2 |x| (7.6) + θ( 2 )φδ h(vn (t))vn (t)dx − θ( 2 )φδ wn (t)vn (t)dx. k k By Young’s inequality and (7.5), we obtain the following bound for the last term on the right-hand side of (7.6). Z Z Z |x|2 |x|2 a 1 |x|2 2 θ( 2 )φδ |vn (t)| dx + θ( 2 )φδ |wn (t)|2dx | θ( 2 )φδ wn (t)vn (t)dx| ≤ k 4 k a k Z 2 |x| a C1 −2γt η θ( 2 )φδ |vn (t)|2 dx + e , (7.7) ≤ + 4 k a aγ

158

Weishi Liu and Bixiang Wang √ for all t ≥ 0 and k ≥ 2K1 (η), where C1 is independent of and the last inequality is obtained by (7.5). Dealing with the first two terms on the right-hand side of (7.6) as the proof of Lemma 4..4 but using (7.3) instead of Lemmas 4..1 and 4..2, after detailed computations, by (7.7) we find that there exists K2 (η) such that for t ≥ 0 and k ≥ K2(η): Z Z |x|2 2C1 −2γt d |x|2 2 2 2 θ( 2 )φδ |vn (t)| dx + a θ( 2 )φδ |vn (t)| dx ≤ 1 + η+ e . dt k k aγ a By Gronwall’s Lemma, for all t ≥ 0 and k ≥ K2(η), we get: Z 2C1 |x|2 2 η 2 + e−2γt , θ( 2 )φδ |vn (t)| dx ≤ 1 + k aγ a a(a − 2γ) IR √ which implies for t ≥ 0 and k ≥ 2K2 (η): Z 2 η 2C1 2 φδ |vn (t)| dx ≤ 1 + (7.8) + e−2γt . aγ a a(a − 2γ) |x|≥k √ Let K3 (η) = 2 max{K1(η), K2(η)}. Then it follows from (7.2), (7.5) and (7.8) that for all n ≥ 1 and k ≥ K3 (η): Z Z η 2 φδ |w| dx = φδ |wn(tn )|2dx ≤ C1 e−2γtn + , (7.9) γ |x|≥k |x|≥k and Z

φδ |v|2dx = |x|≥k

Z

φδ |vn (tn )|2dx ≤

1+

|x|≥k

2 aγ

η 2C1 + e−2γtn . (7.10) a a(a − 2γ)

Taking the limits of (7.9) and (7.10) as n → ∞, then (7.1) follows. We now establish the compactness of the union of all global attractors. S A has a finite Proof of Theorem 2..4. Given η > 0, we want to show that the set 0 0 such that Z t0 (M ) kΦ(Stx + T0(t)f )kdt ≤ q1 kxk + c1 kf k 0

for all fx ∈ D(A). Then, the matrix operator A generates a strongly continuous semigroup (T (t))t≥0 on E1 satisfying the variation of constants formula T

(t) ( xf )

=

T0 (t) ( xf ) +

Z

t

T0(t − s)BT (s) ( xf ) ds

(3.7)

0

for all ( fx ) ∈ D(A) and t ≥ 0. Proof. Let c > c1, and the new norm k fx kc = kxk + ckf kL1 on E1, which is equivalent to the original norm k · kE1 . Hence, by (M ) Z t0 Z t0 x kBT0(t) ( f )kc = kΦ(Stx + T0(t)f )kdt 0

0

≤ q1 kxk + c1 kf kL1 c 1 k ( fx ) kc . ≤ max q1 , c

c 1 Setting q = max q1 , , we have q < 1 and c Z t0 kBT0 (t) ( fx ) kc ≤ qk ( xf ) kc .

(3.8)

0

Since A0 is a generator of a strongly continuous semigroup on (E1, k · kE1 ), and being k · kE1 equivalent to k · kc , one has that A0 is a generator on (E1, k · kc ). By assumptions, B ∈ L(D(A0), (E1, k · kc )) and, by (3.8), B is a small Miyadera-Voigt perturbation of the semigroup T0 . Hence, the Miyadera-Voigt perturbation Theorem implies that also A = A0 + B generates a strongly continuous semigroup on E1 and satisfies (3.7). Observe that, for r = +∞, C([−∞, 0], X) denotes the space C0((−∞, 0], X)( i.e. the space of continuous functions vanishing at −∞). Remark 3..1. In the references [1, 2, 15, 16, 22], the above result was shown under a different Miyadera-Voigt condition (M )0 : there is t0 > 0, q ∈]0, 1[ such that Z t0 kΦ(Stx + T0(t)f )kdt ≤ qk ( xf ) kE1 . 0

The Miyadera-Voigt condition (M ) has appeared for the first time in [25] in the autonomous past delay case.

A Partial Differential Equation with Nonautonomous Past Delay

171

In the following lemma we give a class of operators Φ which satisfies the MiyaderaVoigt assumption (M ). Lemma 3..2. Let r ≤ ∞ and η : [−r, 0] −→ L(X) be of bounded variation such that |η|([−r, 0]) < ∞, where |η| is the positive Borel measure of [−r, 0] defined by the total variation on η. Let Φ : C([−r, 0], X) ∩ L1([−r, 0], X) −→ X the operator given by Z 0 dη(θ)f (θ). (3.9) Φ(f ) := −r

Then, Φ fulfills the condition (M ). Proof. Let 0 < t0 < r. We have Z t0 Z t0 Z kΦ(Stx + T0(t)f )kdt = k 0

≤

Z

t0

k 0

≤

Z

t0

Z

Z

0

0

dη(θ)(Stx + T0(t)f )(θ)kdt

−r

0

dη(θ)U (θ, 0)S(θ + t)xkdt +

Z

−t 0

t0

k 0

d|η|(θ)kU (θ, 0)kkS(θ + t)xkdt +

−t

dη(θ)U (θ, θ + t)f (θ + t)kdt

−r

Z

−t

0

Z

t0 0

Z

−t

d|η|(θ)kU (θ, θ + t)kkf (θ + t)kdt. −r

By Fubini’s theorem and a change of variables, we obtain Z t0 Z t0 Z 0 kΦ(Stx + T0 (t)f )kdt ≤M e|ω|t0 d|η|(θ)kS(θ + t)xkdt 0

0

+ M e|ω|t0

Z

−t 0 Z 0 −r

d|η|(θ)kf (t)k dt

θ

≤ Mω ( sup kS(t)kt0|η|([−r, 0])kxk + |η|([−r, 0])kf kL1 ) 0≤t≤t0

≤ Mw |η|([−r, 0])(t0 sup kS(t)kkxk + kf kL1 ) 0≤t≤t0

≤ q1 kxk + c1kf kL1 , with Mw := M e|w|t0 , q1 := sup kS(t)kMw |η|([−r, 0])t0 and c1 := Mw |η|([−r, 0]). 0≤t≤t0

Choosing t0 small so that q1 < 1, the condition (M ) is then satisfied. Corollary 3..1. Let Φ given as in Lemma 3..2. Then, (A, D(A)) generates a strongly continuous semigroup (T (t))t≥0 on E1 satisfying the variation of constants formula Z t T0(t − s)BT (s) ( xf ) ds (3.10) T (t) ( xf ) = T0 (t) ( xf ) + 0 x

for all ( f ) ∈ D(A) and t ≥ 0. Moreover, there is a unique mild solution u of the equation (3.6) and it is given by x π1T (t) ( f ) , t ≥ 0, u(t) = f (t), −r ≤ t ≤ 0.

172

4.

S. Boulite, G. Fragnelli, M. Halloumi and L. Maniar

Asymptotic behaviour of partial differential equations with nonautonomous past delay

We study the persistence of some asymptotic properties of the semigroup (S(t))t≥0 under the modified delay term. More precisely, we give conditions under which the trajectories of the semigroups S(·) and T (·) have the same asymptotic behavior. First, we show that this holds for S(·) and T0 (·), and by perturbation techniques the claim will be obtained. In the following theorem, we show first that S(·) and (T0(t))t≥0 have some common asymptotic properties. The finite delay case can be shown similarly as in [7]; for the asymptotic almost automorphy see [6]. We give here the proof for the infinite delay case. Theorem 4..1. Assume that the evolution family (U (t, s))t≤s≤0 is exponentially stable, and let x ∈ X. If the map t 7−→ S(t)x is (i) in C0 (R+ , X), the space of functions vanishing at infinity, or (ii) asymptotically almost periodic, or (iii) asymptotically almost automorphic, or (iv) uniformly ergodic, or (v) totally uniformly ergodic. then t 7→ T0(t) ( xf ) has the same property for every f ∈ L1(IR− , X). Proof. Let f ∈ L1(IR− , X). Since x

T0(t) ( f ) =

S(t)x St x+T0 (t)f

,

we have only to verify that the map t 7→ St x + T0(t)f has the same property as S(·)x. To show (i), we should show that t 7→ St x + T0(t)f ∈ C0(R+ , L1(IR− , X)), i.e., Z

0

kSt x(θ) + T0 (t)f (θ)kdθ −→ 0. t→∞

−∞

We have Z 0

kStx(θ) + T0(t)f (θ)kdθ ≤

−∞

Z

0

kStx(θ)kdθ + −∞

≤

Z

≤

−∞ Z 0

0

1t+θ≥0 kU (θ, 0)S(t + θ)xkdθ +

−∞

1θ≥−t kU (θ, 0)S(t + θ)xkdθ +

Z

Z

Z

0

kT0(t)f (θ)kdθ −∞

0

1t+θ≤0 kU (θ, t + θ)f (t + θ)kdθ −∞ −t

kU (θ, t + θ)f (t + θ)kdθ.

−∞

Since S(·)x is vanishing at +∞, there exists M 0 > 0 such that supkS(t)xk ≤ M 0, and t≥0

then 1θ≥−t kU (θ, 0)S(t + θ)xk ≤ M 0M ewθ := g(θ).


173

The function g is integrable on R− and a.e. θ ∈ R− .

1θ≥−t kU (θ, 0)S(t + θ)xk −→ 0

t−→+∞

Hence, the Lebesgue dominated convergence Theorem implies that Z 0 kU (θ, 0)S(t + θ)xkdθ −→ 0 . t−→+∞

−t

We have also kT0(t)f kL1 =

Z

−t

kU (θ, t + θ)f (t + θ)kdθ =

Z

−∞

0

kU (θ − t, θ)f (θ)kdθ −∞ −tw

≤ Me

kf kL1 −→ 0 . t−→+∞

Assume now (ii). It is clear that T0(·)f is asymptotically almost periodic. Now it is sufficient to show that R+ 3 t 7−→ St x is asymptotically almost periodic. From definition there exist a unique g ∈ AP (R, X) and a unique h ∈ C0 (R+ , X) such that S(t)x = h(t) + g(t),

t ≥ 0.

(4.11)

Hence, St x(θ) = =

U (θ, 0)S(t + θ)x, θ + t > 0 0, θ+t≤0 U (θ, 0)g(t + θ) + U (θ, 0)h(t + θ), θ + t > 0 0, θ + t ≤ 0.

Since h ∈ C0 (R+ , X) and U (θ, 0) is exponentially stable, then, as in (i), h1 (t)(θ) := 1θ≥−t U (θ, 0)h(t + θ) ∈ C0(R+ , L1(R− , X)). Set g1 (t)(θ) := U (θ, 0)g(t + θ) and h2 (t)(θ) := h1 (t)(θ) − 1t+θ≤0 U (θ, 0)g(t + θ) for all t ∈ R and θ ∈ R− . Then, ( g1(t)(θ) + h1 (t)(θ) for θ + t > 0, (g1(t) + h2(t))(θ) = g1(t)(θ) − g1(t)(θ) = 0 for θ + t ≤ 0 =St x(θ).

(4.12)

Since h1 ∈ C0(R+ , L1(R− , X)) and Z 0 Z 1t+θ≤0 kU (θ, 0)g(t + θ)kdθ ≤ kgk∞ −∞

−t

kU (θ, 0)kdθ −→ 0 , −∞

t−→+∞

it follows that h2 ∈ C0(R+ , L1(R− , X)). Since g is almost periodic, then for all > 0, there exists l() > 0 such that for all a ∈ R, there exists τ ∈ [a, a + l()] such that kg(t + θ + τ ) − g(t + θ)k ≤

for all

t ∈ R, θ ∈ IR− ,

174


and moreover kg1(t + τ ) − g1 (t)kL1 = ≤

Z

0

kU (θ, 0)g(t + θ + τ ) − U (θ, 0)g(t + θ)kdθ

−∞ Z 0

≤

kU (θ, 0)kkg(t + θ + τ ) − g(t + θ)kdθ

−∞ Z 0

kU (θ, 0)kdθ = kU (·, 0)kL1 .

−∞

This means that g1 is almost periodic, and finally t 7−→ St x is asymptotically almost periodic. For the asymptotic almost automorphy, assume that S(t)x = h(t) + g(t), t ≥ 0, where g ∈ AA(R, X) and h ∈ C0 (R+ , X). From (4.12), Stx = g1 (t) + h2 (t), t ≥ 0, with h2 ∈ C0(R+ , X). To conclude we have to show that g1 is almost automorphic on IR. For this, consider a sequence (σn )n∈N in IR. Since g ∈ AA(IR, X), there is a subsequence (sn )n∈N such that lim g(t + θ + sn − sm ) = g(t + θ) n,m→+∞

for each t ∈ IR, θ ∈ IR− . Therefore, as for the almost periodicity, one can see that g1 ∈ AA(IR, L1(IR−, X)). As (T0(t))t≥0 is exponentially stable then it is asymptotically almost automorphic. Z +∞

For (iv), we have that the limit F (·)

=

lim α

α→0

e−αt S(· + t)xdt exists in

0 BU C(R+ , X).

BU C(R+ , X) for each x ∈ X, and then F (·) ∈ U (θ, 0)F (t + θ) Ft (θ) = U (θ, 0)F (0)

Put

for θ + t > 0 for θ + t ≤ 0.

Show first that t 7→ Ft ∈ BU C(R+ , L1(R− , X)). For h > 0, we have Z 0 kFs+h (θ) − Fs (θ)kdθ −∞

≤ + ≤ +

Z

0

−∞ Z 0

k1s+h+θ≥0 U (θ, 0)F (s + h + θ) − 1s+θ≥0 U (θ, 0)F (s + θ)kdθ k1s+h+θ≤0 U (θ, 0)F (0) − 1s+θ≤0 U (θ, 0)F (0)kdθ

−∞ Z −s −s−h Z −s

kU (θ, 0)F (s + h + θ)kdθ +

Z

0

kU (θ, 0)kkF (s + h + θ) − F (s + θ)kdθ −s

kU (θ, 0)F (0)kdθ

−s−h

≤ 2hkU (·, 0)k∞kF k∞ +

Z

0

kU (θ, 0)kkF (s + h + θ) − F (s + θ)kdθ. −s

Since F (·) ∈ BU C(R+ , X) then, for > 0 there exists h > 0 such that kF (s + h + θ) − F (s + θ)k ≤ for all s + θ ≥ 0 and 0 < h < h . Hence, kFs+h (·) − Fs (·)k∞ ≤ hkU (·, 0)k∞kF k∞ + kU (·, 0)kL1 .

A Partial Differential Equation with Nonautonomous Past Delay 175 Z +∞ The same for h < 0. To achieve (iv) we show that the limit lim α e−αt S·+t xdt = F· α→0

exists in BU C(R+ , L1(R− , X)). Let s ≥ 0. Z +∞ Z 0 kα e−αt Ss+t xdt(θ) − Fs (θ)kdθ −∞

= =

Z

0 0

kα

−∞ Z 0

kα

Z Z

−∞

0

+∞

e−αt Ss+t x(θ)dt − Fs (θ)kdθ 0 +∞

1s+t+θ≥0 e−αt U (θ, 0)S(s + t + θ)xdt − 1s+θ>0 U (θ, 0)F (s + θ) 0

− 1s+θ≤0 U (θ, 0)F (0)kdθ Z +∞ Z 0 kU (θ, 0)kkα e−αt S(s + t + θ)xdt − F (s + θ)kdθ ≤ −s 0 Z +∞ Z −s kU (θ, 0)kkα e−αt S(s + t + θ)xdt − F (0)kdθ + −∞ −s−θ Z +∞ e−αt S(s + t)xdt − F (s)kkU (·, 0)kL1 ≤ sup kα s≥0 −s

+ +

Z

0

e

−∞ Z −s

α(s+θ)

kU (θ, 0)kkα

Z

+∞

e−αt S(t)xdt − F (0)kdθ 0

kU (θ, 0)kkF (0) − eα(s+θ) F (0)kdθ.

−∞

Hence Z

0

kα −∞

Z

+∞

e−αt Ss+t xdt(θ) − Fs (θ)kdθ 0

≤ 2 sup kα

Z

s≥0

+ M kF (0)k

+∞

e−αt S(s + t)xdt − F (s)kkU (·, 0)kL1 0

Z

0

eωσ (1 − eασ )dσ,

−∞

and this ends the proof of (iv). The same proof yields the assertion (v). Using [3, Lemma 7.2] and the variation of constants formula (3.7), we obtain the asymptotic properties of the perturbed semigroup T (·) and then the ones of the trajectories of the solutions to equation (3.6). Theorem 4..2. Assume that the evolution family (U (t, s))t≤s≤0 is exponentially stable, and that Φ is bounded linear from C([−r, 0], X) ∩ L1([−r, 0], X) to X. Assume also that S(·)x satisfies one of the asymptotic properties (i)-(v) for all x ∈ X, and there are 0 ≤ q1 < 1 and c1 > 0 such that Z ∞ kΦ(Stx + T0 (t)f )kdt ≤ q1 kxk + c1kf k (4.13) 0

for all

( fx )

∈ D(A0). Then, T (·) ( xf ) has the same asymptotic properties for all ( xf ) ∈ E1.

176


Proof. As in the proof of Theorem 3..1, one can find 0 ≤ q < 1 and c > 0 such that Z ∞ kBT0(t) ( xf )kc ≤ qk ( xf ) kc (4.14) 0

( fx )

∈ D(A), where k · kc is a norm equivalent to k · kE1 . Fix now ( xf ) ∈ D(A). for all x From Theorem 4..1, the trajectory T0(·) ( f ) has the same asymptotic properties of S(·)x with respect to the new norm k · kc . From Theorem 3..1, we have Z t T (t) ( xf ) = T0 (t) ( xf ) + T0(t − s)BT (s) ( xf ) ds, t ≥ 0. (4.15) 0

Hence, it is sufficient to show that the function Z IR+ 3 t 7−→ f (t) :=

t

T0(t − s)g(s)ds,

−∞

satisfies the same properties, where g is defined by ( x BT (s) ( f ) , s ≥ 0, g(s) := 0, s ≤ 0, For assertions (i)-(ii)-(iv)-(v), by [3, Lemma 7.2] it suffices to show g ∈ L1(IR, (E1, k·kC )), and this can be deduced from (4.14) and (3.7). As the semigroup S(·) is bounded, the assertion (iii) follows from [6, Proposition 2.3] and (ii). Finally, all the assertions follow for all ( fx ) ∈ E1 by density of D(A). The asymptotic behavior of the solutions of (3.6) can be now obtained. Corollary 4..1. Assume that S(·)x satisfies one of the asymptotic properties (i)-(v) for all x ∈ X and that Φ is a bounded linear from C([−r, 0], X) ∩ L1 ([−r, 0], X) to X satisfying (4.13). Then, t 7−→ u ˜t and t 7−→ u(t) have the same asymptotic properties as S(·)x.

5.

Application

In this section, we use the abstract results of the previous ones to study the wellposedness and asymptotic behavior of the dynamical population equation  Rr 0  (t, x) = ∆ u(t, x) − du(t, x) + u  N 0 b(a)v(t, a, x)da − b1 v(t, r, x), t ≥ 0, x ∈ Ω,   v 0 (t, a, x) = − ∂ v(t, a, x) + ∆ v(t, a, x) − dv(t, a, x) − b(a)v(t, a, x), t ≥ 0, D ∂a  x ∈ Ω, 0 ≤ a ≤ r,    v(t, 0, x) = f (x)u(t, x), t ≥ 0, x ∈ Ω, (5.16) where ∆D and ∆N denote the Laplace operators whit Dirichlet and Neumann conditions, respectively. Here t, r are positive, a ∈ [0, r], d, b1 > 0 and b ∈ L1 ([0, r], IR+). The space variable x is supposed to vary in Ω ⊂ Rn where Ω is open, connected and bounded with smooth boundary. The condition ∂ u(t, x) = 0 in ∂Ω, ∂n

(5.17)


177

where n denotes, as usual, the outward normal, states that the population cannot cross the boundary. The condition v(t, a, x) = 0 if x ∈ ∂Ω, t ≥ 0, a ∈ [0, r] (5.18) says that no pregnant individual reaches the borderline. Moreover the condition f (x) = 0 for x ∈ ∂Ω is required. The initial conditions u(0, x) = u0 (x), u(τ, x) = g(τ, x), τ ∈ (−r, 0),

(5.19)

v(0, a, x) = v0 (a, x), v0 (a, x) = f (x)g(−a, x), a ∈ [0, r],

(5.20)

where v0 , g and u0 are given functions.

5.1.

The population equation as an abstract equation with nonautonomous past delay

In this subsection, we show in details how to transform the population equation (5.16)(5.20) in a partial differential equation with nonautonomous past delay, and then study its wellposedness and asymptotic behavior. To this purpose, we start by solving the second equation along the characteristic lines in the plane (t, a), namely in the strip [0, +∞)×[0, r]. Set V (s, x) = v(t0 + s, a0 + s, x), where a0 ∈ [0, r], t0 ≥ 0 are fixed, while x varies in Ω and s in [0, +∞). Rewriting the second equation in (5.16) with V (s, x), one obtains ∂ ∂ ∂ V (s, x) = v(t0 + s, a0 + s, x) + v(t0 + s, a0 + s, x) ∂s ∂t ∂a = ∆D v(t0 + s, a0 + s, x) − dv(t0 + s, a0 + s, x) − b(a0 + s)v(t0 + s, a0 + s, x) = ∆D V (s, x) − dV (s, x) − b(a0 + s)V (s, x). To solve the above equation we follow the abstract approach choosing X = L1(Ω) as a Banach space and denoting D(∆D ) the domain of ∆D on X. For t < a, putting t0 = 0, we obtain ( V 0(s, x) = ∆D V (s, x) − dV (s, x) − b(a0 + s)V (s, x) V (0, x) = v(0, a0, x) = v0 (a0, x), which has the unique solution V (s, ·) = e−

Rs 0

b(a0 +σ)dσ s(∆D −d)

e

V (0, ·),

where d denotes the multiplication operator and et(∆D −d) the strongly continuous semigroup generated by the linear operator ∆D − d on X with the appropriate domain. Hence Rs

b(a0 +σ)dσ s(∆D −d)

Ra

b(σ)dσ s(∆D −d)

v(s, a0 + s, ·) = e =e

0

0 +s a0

e

e

v(0, a0, ·)

v0 (a0, ·).

178


Setting t = s and a0 + s = a, this becomes Ra

v(t, a, ·) = e

a−t


e

v0 (a − t, ·).

For t > a, putting a0 = 0, we obtain ( V 0 (s, x) = ∆D V (s, x) − dV (s, x) − b(s)V (s, x) V (0, x) = v(t0 , 0, x) = f (x)u(t0, x). Solving this equation we obtain Rs 0


Rs

V (0, ·) v(t0, 0, ·) f (·)u(t0, ·).

v(t0 + s, s, ·) = V (s, ·) = e

e

=e

0


Rs

=e

0


e e

Setting t0 + s = t, this becomes v(t, a, ·) = e

Ra 0

b(σ)dσ a(∆D −d)

e

f (·)u(t − a, ·).

Substituting v(t, a, x) in the first equation of (5.16), one has, for t > r, Z r u0 (t, x) = (∆N − d)u(t, x) + b(a)v(t, a, x)da − b1v(t, r, x) 0 Z r Ra b(a)[e 0 b(σ)dσ ea(∆D −d) f (x)u(t − a, x)]da = (∆N − d)u(t, x) + 0

Rr 0

b(σ)dσ r(∆D −d)

Rr

b(σ)dσ r(∆D −d)

− b1 e

f (x)u(t − r, x) Z 0 R −a = (∆N − d)u(t, x) + b(−a)[e− 0 b(σ)dσ e−a(∆D −d) f (x)u(t + a, x)]da

− b1 e

0

e

−r

e

f (x)u(t − r, x).

For t < r, we have Z r u0 (t, x) = (∆N − d)u(t, x) + b(a)v(t, a, x)da − b1 v(t, r, x) 0 Z t Z r = (∆N − d)u(t, x) + b(a)v(t, a, x)da + b(a)v(t, a, x)da − b1v(t, r, x) = (∆N − d)u(t, x) + +

Z

Z

0

t 0

b(−a)[e−

R −a 0

b(σ)dσ −a(∆D −d)

e

f(x)u(t + a, x)]da

−t −t

b(−a)[e−

R −a

a−t

b(σ)dσ t(∆D −d)

e

v0 (−a − t, x)]da − b1e−

Rr

r−t

b(σ)dσ t(∆D −d)

e

v0 (r − t, x).

−r

We put U (t, s) = e− and Φ(ϕ) =

Z

R −t −s

b(σ)dσ (s−t)(∆D −d)

e

,

t ≤ s ≤ 0,

0

−r

b(−s)f (·)ϕ(s)ds − b1f (·)ϕ(−r),

ϕ ∈ C([−r, 0], X).


179

It is easy to verify that (U (t, s))t≤s≤0 is an evolution family. The modified history function u ˜t is U (s, 0)u(t + s, ·) if t + s > 0 u ˜t (s, ·) := U (s, s + t)g(t + s, ·) if t + s ≤ 0. Hence, Φ(˜ ut(·)) =

Z

0

b(−s)f (·)˜ ut(s, ·)ds − b1f (·)˜ ut(−r, ·).

−r

Since for t > r, u ˜t (−r, ·) = U (−r, 0)u(t − r, ·) and s ≥ −r, t + s ≥ t − r ≥ 0 =⇒ u ˜t (s, ·) = U (s, 0)u(t + s, ·), through Z 0 Φ(˜ ut ) = b(−s)f (·)U (s, 0)u(t + s, ·)ds − b1f (·)U (−r, 0)u(t − r, ·). −r

If we set B = ∆N − d, then Bu(t, ·) + Φ(˜ ut) = (∆N − d)u(t, ·) +

Z

0

b(−a)f (·)U (a, 0)u(a + s, ·)da

−r

− b1f (·)U (−r, 0)u(t − r, ·) = u0(t, ·). Thus, by setting u(t) := u(t, ·), we obtain ut ). u0 (t) = Bu(t) + Φ(˜ For t ≤ r, u ˜t (−r, ·) = U (−r, t − r)g(t + s, ·) and Z 0 Φ(˜ ut) = b(−s)f (·)˜ ut(s, ·)ds − b1f (·)U (−r, t − r)g(t − r, ·) = +

Z

−r −t

r Z 0

b(−s)f (·)U (s, s + t)g(t + s, ·)ds b(−s)f (·)U (s, 0)u(t + s, ·)ds − b1 f (r)U (−r, t − r)g(t − r, ·).

t

Summarizing, the population equation is written as the abstract equation with nonautonomous past ( ut), t ≥ 0, u0 (t) = Bu(t) + Φ(˜ (5.21) u(0) = x, u0 = g, where the operator B = ∆N − d generates an exponentially stable C0 -semigroup S := (S(t))t≥0 on X = L1(Ω).

5.2.

Wellposedness and asymptotic behavior of the population equation

We have transformed the population equation (5.16)- (5.20) into the abstract equation with nonautonomous past ( ut), t ≥ 0, u0 (t) = Bu(t) + Φ(˜ (5.22) u(0) = x, u0 = g,

180


where the operator B = ∆N − d generates an exponentially stable C0 -semigroup S R:= (S(t))t≥0 on X = L1 (Ω) and the backward evolution family U (t, s) = −t

e− −s b(σ)dσ e(s−t)(∆D −d) , t ≤ s ≤ 0, is exponentially stable. Moreover, the operator Φ, given by Z 0 b(−s)f (·)ϕ(s)ds − b1f (·)ϕ(−r), Φ(ϕ) = −r

can be written as a Stieltjes integral Z

Φ(ϕ) =

0

dη(s)ϕ(s),

−r

where η : [−r, 0] −→ L(X) is the function with a bounded variation on [−r, 0] given by Z s b(−τ )f (·)dτ. η(s) = −b1f (·)1[−r,0] + 0

Here, we have considered a delay operator Φ with one discrete delay, but one can consider several discrete delays and the delay operator Z 0 n X Φ(ϕ) = b(−s)f (·)ϕ(s)ds − bi f (·)ϕ(−ri), −r

i=1

with 0 ≤ r1 < r2 < ... < rn = r. In this case η is the bounded variation function Z s n X bi f (·)1[−ri,0] + b(−τ )f (·)dτ. η(s) = − 0

i=1

Hence, by Lemma 3..2, these operators Φ satisfy the Miyadera-Voigt condition (M ). Therefore, by Corollary 3..1, there is a unique mild solution of equation (5.22). Moreover, we have the following result. Theorem 5..1. There is a unique mild solution u : IR+ −→ L1(Ω) of the population equation (5.16)-(5.20). Moreover, if |η|([−r, 0]) < d then ku(t)kL1(Ω) −→ 0, t → +∞. Proof. One has Z Z +∞ kΦ(Stx + T0(t)f )kdt = 0

≤

Z

+∞

k 0

≤

Z

0

≤

+∞

Z

Z

+∞

k

0

Z

0

dη(θ)(Stx + T0 (t)f )(θ)kdt

−r

0

dη(θ)U (θ, 0)S(θ + t)xkdt +

−t 0

Z

r

k 0

d|η|(θ)kU (θ, 0)kkS(θ + t)xkdt + −t

Z

−t

dη(θ)U (θ, θ + t)f (θ + t)kdt −r

r Z −t

Z 0

d|η|(θ)kU (θ, θ + t)kkf (θ + t)kdt

−r

1 |η|([−r, 0])kxk + |η|([−r, 0])kf k. d

1 Hence, by assumption q1 = |η|([−r, 0]) < 1, and since S is exponentially stable, by d Theorem 4..2, the result follows.


181

References [1] A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl. 264 (2001), 1-20. [2] A. Bátkai and S. Piazzera, Semigroups for Delay Equations on LP -phase Spaces, book manuscript, Research Notes in Mathematics 10, A. K. Peters, Ltd., Wellesley, MA, (2005). [3] C. J. K. Batty and R. Chill, Bounded convolutions and solutions of inhomogeneous Cauchy problems, Forum Math. 11 (1999), 253–277. [4] S. Boulite, L. Maniar and G. M. N’Guérékata, Almost automorphic solutions for hyperbolic semilinear evolution equations, Semigroup Forum 71 (2005), 231–240. [5] S. Brendle and R. Nagel, Partial functional differential equations with nonautonomous past, Disc. cont. Dyn. Syst. 8 (2002), 1-24. [6] V. Casarino, Almost automorphic groups and semigroups. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 24 (2000), 219–235. [7] V. Casarino and S. Piazzera, On the stability of asymptotic properties of perturbed C0 -semigroups, Forum Math. 13 (2001), 91-107. [8] V. Casarino, L. Maniar and S. Piazzera, The asymptotic behaviour of perturbed evolution families, Differential Integral Equations 15 (2002), 567–586 [9] C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs 70, American Mathematical Society, Rhode Island, (1999). [10] C. Corduneanu, Almost Periodic Functions , 2nd Edition, Chelsea-New York, 1989. [11] K.J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations , Graduate Texts in Mathematics 194, Springer-Verlag (2000). [12] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics 377, Springer-Verlag, New York, 1974. [13] G. Fragnelli, Classical solutions for PDEs with nonautonomous past in Lp − spaces, Bulletin of the Belgian Mathematical Society 11 (2004), 133–148. [14] G. Fragnelli and G. Nickel, Partial functional differential equations with nonautonomous past in Lp −phase spaces, Diff. Int. Equ. 16 (2003), 327–348. [15] G. Fragnelli and L. Tonetto, A population equation with diffusion, J. Math. Anal. Appl. 289 (2004), 90–99. [16] G. Fragnelli, An age dependent population equation with diffusion and delayed birth process, Int. J. Math. Math. Sci. 20 (2005), 3273–3289.

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[17] I. Miyadera, On perturbation theory for semi-groups of operators, Tôhoku Math. J. 18 (1966), 299–310. [18] R. Nagel and G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems, Progr. Nonlinear Differential Equations Appl. 50, Birkhäuser, (2002), 279–293. [19] G. M. N’Guérékata, Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, Kluwer Academic / Plenum Publishers, New York, 2001. [20] G. M. N’Guérékata, Topics in Almost Automorphy , Springer-Verlag, New York, 2005. [21] G. Nickel, Evolution semigroups for nonautonomous Cauchy problems, Abstr. Appl. Anal. 2 (1997), 73–95. [22] G. Nickel and A. Rhandi, Positivity and Stability of Delay Equations with Nonautonomous Past, Math. Nachr. 278 (2005), 1–13. [23] F. Räbiger, A. Rhandi, R. Schnaubelt and J. Voigt, Non-autonomous Miyadera perturbations, Diff. Int. Eq. 13 (2000), 341–368. [24] J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann. 229 (1977), 163–171. [25] M. Stein, H. Vogt, J. Voigt, The modulus semigroup for linear delay equations III. J. Funct. Anal. 220 (2005), 388–400.


S OLVING THE H YPERBOLIC P ROBLEM O BTAINED BY T RANSMUTATION O PERATOR Hikmet Koyunbakan ∗ Firat University, Department of Mathematics, 23119 Elazig, TURKEY

Abstract In this paper, we solve the problem of constructing the kernels of transmutation operator and give the generalized transmutation operator for a singular problem.

Key Words: Inverse Problem, Hyperbolic Equation AMS Subject Classification: 31A25, 35L20.

1.

Introduction

The standard problem on spectral theory is to solve (1.1) for y with appropriate boundary conditions, for a given potential q(x). In the unusual application all that can be observed is the asymptotic behaviour of y for large |x| , which is taken for the whole spectrum comprising at the scattering data for q(x). Inverse eigenvalue problems, usually for equations of the form −y 00 + q(x)y = λy

(1.1)

with a variety of boundary conditions, have an extensive literature [3], [4], [5], [6], [14] . Unless q(x) is constrained, one needs the spectra from two sets of boundary conditions, or one spectrum and a sequence of norming constants to uniquely determine q(x). In the past, important works on these problems were done by Gelfand and Levitan [4], Levinson [8] and Levitan [9]. These authors show that the inverse problem is solved by q(x) = 2

∗

E-mail address: [email protected]

d H(x, x), dx

184

Hikmet Koyunbakan

where H(x, t) is nucleus function of transmutation operator. One topic of inverse problem for spectral analysis is transmutation operator [1], [2], [7], [15]. In [5], [11] Gilbert examined some problems for boundary value problems by using this method, in [7], Hryniv and Mykytyuk, constructed transformation operators for Srurm Liouville operators with singular potentials, in [1], Boumenir and Tuan proved the existence of a transmutation operator between two weighted Sturm-Liouville operator and in [15] Volk solved Hyperbolic equation having singularity type operator.

m2 − 14 x2

at zero by using transmutation

m2 − 1 In this paper, we obtain the problem which is singularity type kx + x2 4 and solve it by using the Riemann method. Before giving the main results, we will mention some well-known trues. We consider the confluent hypergeometric equation d2 W dW − aW = 0 (1.2) + [c − x] 2 dx dx in which a and c are parameters. In this equation, by considering transmutation w = 1 exp(− x2 )xm+ 2 W, we get # " 1 k m2 − 14 d2 w 0 w=0 (1.2 ) − 2 + − + dx 4 x x2 x

where k =

c 2

− a, m =

c 2

0

− 12 . Standard solution of (1.2 ) is

1 x 1 M (k, m, x) = exp(− )xm+ 2 F (m − k + , 2m + 1, x), (1.3) 2 2 where F is solution of (1.2) and sometimes known as Kummer 0s function. [13, p:255-261] Let E be a topological linear space and L1 and L2 be two linear, but not necessarily continuous, operators from E to E, where E1 and E2 are closed subspaces in E.

Definition 1.1 A linear invertible operator, X, defined in the whole space E and acting from E1 to E2 is called a transmutation operator for a pair of operators L1 and L2 if it satisfies the following two conditions [9, p:4] i) The operator X and its inverse X −1 are continuous in space E. ii) The operator equation is of the form L1X = XL2 or L1 = XL2X −1. We consider the problem # " 1 k m2 − 14 d2 w w = 0, 0<x 0, we consider characteristic z = s + ε. (Figure 3.1) Hence, the problem (2.7)-(2.9) transforms to ∂ 2U 1 ∂U 1 ∂U 1 − + − √ qU + R (z, s) U = 0, ∂z∂s 4 (z − s) ∂z 4 (z − s) ∂s 4 zs

(3.3)

q U dU + 3 = 1 , dz 2z 4 4z 2

(3.4)

U (z, z − ε) = 0.

(3.5)

In this section, we shall describe an approach which has considerable affiliation with the Green’s function and which is restricted in its application to hyperbolic equation. Now,

188

Hikmet Koyunbakan

Figure 3.1: Singularity of (3.3) on plane (z,s).

we shall find out solution of problem (3.3)-(3.5) by Riemann method. We consider two operators 1 ∂U 1 ∂U ∂ 2U − + + R (z, s) U, ∂z∂s 4 (z − s) ∂z 4 (z − s) ∂s

L [U ] =

L∗ [V ] =

∂ 2V 1 ∂V 1 ∂V 1 V + R (z, s) V, + − − ∂z∂s 4 (z − s) ∂z 4 (z − s) ∂s 2 (z − s)2

where L∗ is the adjoint operator of L and V is a solution of conjugate equation. If the first equation multiply by V and the second equation multiply by U , after subtracting and integrating on Ω, it yields ZZ

V L [U ] − U L∗ [V ] dzds

Ω

=

ZZ

1 ∂ 2 ∂z

∂U ∂V 1 V −U − UV ∂s ∂s 2 (z − s)

dzds

Ω

ZZ

+

1 ∂ 2 ∂s

∂U ∂V 1 V −U + UV ∂z ∂z 2 (z − s)

Ω

Using the Green theorem, we yield

dzds.

Solving the Hyperbolic Problem Obtained by Transmutation Operator ZZ Ω

189

∂U ∂V 1 1 −U − U V dz V L [U ] − U L [V ] dzds = − V 2 ∂z ∂z 2 (z − s) AM BA 1 ∂U ∂V 1 + V −U − U V ds. (3.6) 2 ∂s ∂s 2 (z − s) Z

∗

As shown in the Figure 3.1, since s and z are fixed on AM and M B respectively, it follows that Z 1 ∂U ∂V 1 V −U − U V ds = 0 2 ∂s ∂s 2 (z − s) AM

and Z

∂U ∂V 1 1 −U + U V dz = 0. − V 2 ∂z ∂z 2 (z − s)

MB

Furthermore, we have Z AM

−

1 ∂U ∂V 1 1 1 V −U + U V dz = − U V (M ) + U V (A) 2 ∂z ∂z 2 (z − s) 2 2 Z V ∂V − dz U + ∂z 4 (z − s) AM

and Z

1 ∂U ∂V 1 V −U + U V ds = 2 ∂s ∂s 2 (z − s)

MB

1 1 U V (B) − U V (M ) 2 2 Z V ∂V + ds. U − ∂s 4 (z − s) MB

Inserting these values in (3.6), it is obtained that ZZ Ω

Z

∗

{V L [U ] − U L [V ]} dzds =

1 ∂U ∂V UV − V −U − dz 2 ∂z ∂z 2 (z − s)

BA

1 ∂U ∂V UV 1 V −U − ds − U V (M ) 2 ∂s ∂s 2 (z − s) 2 Z V 1 ∂V 1 − dz + U V (B) U + U V (A) + 2 ∂z 4 (z − s0 ) 2 AM Z V ∂V 1 + ds. U − U V (M ) − 2 ∂s 4 (z0 − s) +

MB

(3.7)

190

Hikmet Koyunbakan

On the right hand, the values of integrals on AM and M B are not known and on these lines, M is not vanished, then we have to investigate solution of V the following problem. L∗ [V ] = 0,

(3.8)

∂V V − = 0 on line AM , ∂z 4 (z − s0 )

(3.9)

V ∂V + = 0 on line M B, ∂s 4 (z0 − s)

(3.10)

V = 1 at M point.

(3.11)

Solution of this problem is called Riemann-Green function and it is in the form [12] 1

(s − z) 2

V (z, s, z0, s0) =

1

F

[(s0 − z) (s − z0 )] 4

1 1 , , 1, σ , 4 4

where σ=

(s − s0 ) (z − z0 ) (s − z0 ) (z − s0 )

and F is a Hypergeometric function such that F

1 1 , , 1, σ 4 4

= 1+

1 4

· 14 σ + . . .. 1! · 1

Hence, we have ZZ U V (A) + U V (B) U (M ) = − V L [U ] dzds + 2 Ω Z ∂U ∂V UV 1 V −U − dz + 2 ∂z ∂z 2 (z − s) BA 1 ∂U ∂V UV − V −U − ds. 2 ∂s ∂s 2 (z − s)

(3.12)

Because of being z = s + ε on BA line, 1 ∂U ∂V UV 1 ∂U ∂V UV dz = ds and V −U − = V −U − . 2 ∂z ∂z 2 (z − s) 2 ∂s ∂s 2 (z − s) On the other hand L [U ] =

√1 qU. 4 zs

From this point of view and (3.12) we obtain

U V (A) + U V (B) U (M ) = − + 2

z−ε Z Zz0 s0 s0 +ε

1 √ qU V dzds. 4 zs

(3.13)

Solving the Hyperbolic Problem Obtained by Transmutation Operator

191

Last equation is a Volterra- type integral equation. Therefore, its solution is unique and the solution can be proved by successive approximations . Now, we generalize transmutation operator for the problem (1.6),(1.7). Taking q1 (x), q2(x), . . . , qn (x) as real-valued integrable functions on (0, 1), we show that the solution of the following problem can be established by using transmutation operator " # n d2 w 1 k m2 − 14 X − + − + qk (x) w = 0, 0<x

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