Blow-up in Nonlinear Sobolev Type Equations (De Gruyter Series in Nonlinear Analysis and Applications)

De Gruyter Series in Nonlinear Analysis and Applications 15 Editor in Chief Jürgen Appell, Würzburg, Germany Editors Ca...

Author: Alexander B. Al’shin | Maxim O. Korpusov | Alexey G. Sveshnikov

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De Gruyter Series in Nonlinear Analysis and Applications 15 Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Karl-Heinz Hoffmann, Munich, Germany Mikio Kato, Kitakyushu, Japan Umberto Mosco, Worchester, Massachusetts, USA Louis Nirenberg, New York, USA Katrin Wendland, Augsburg, Germany Alfonso Vignoli, Rome, Italy

Alexander B. Al’shin Maxim O. Korpusov Alexey G. Sveshnikov

Blow-up in Nonlinear Sobolev Type Equations

De Gruyter

Mathematics Subject Classification 2010: 35B44, 35D30, 35D35, 35M99, 35Q92, 35Q60, 35J92, 35G20, 35G60.

ISBN 978-3-11-025527-0 e-ISBN 978-3-11-025529-4 ISSN 0941-813X Library of Congress Cataloging-in-Publication Data Al’shin, A. B. Blow-up in nonlinear Sobolev type equations / by A. B. Al’shin, M. O. Korpusov, A. G. Sveshnikov. p. cm. ⫺ (De Gruyter series in nonliniear analysis and applications ; 15) Includes bibliographical references and index. ISBN 978-3-11-025527-0 (alk. paper) 1. Initial value problems ⫺ Numerical solutions. 2. Nonlinear difference equations. 3. Mathematical physics. I. Korpusov, M. O. II. Sveshnikov, A. G. (Aleksei Georgievich), 1924⫺ III. Title. QA378.A47 2011 5151.782⫺dc22 2011003941

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

This monograph is devoted to the study of general problems on the global-in-time solvability and blow-up for a finite time of initial-value and initial-boundary-value problems for nonlinear equations of Sobolev type. Our studies together with an outstanding Russian mathematician S. A. Gabov, who prematurely died in 1989, stimulated further work in this direction. Our study of the blow-up of solutions to pseudoparabolic nonlinear equations was considerably stimulated by the classical work of A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhaylov Blow-Up in Quasilinear Parabolic Equations, which influenced the choice of many themes of our monograph. We express a deep appreciation to V. P. Maslov, S. I. Pokhozhaev and N. N. Kalitkin for useful discussion of certain results presented in the monograph. We thank all participants of the scientific seminar “Nonlinear differential equations” and its supervisor Prof. I. A. Shishmaryov for their valuable comments on various sections of the book. The research being the subject of this monograph were supported by the Russian Foundation for Basic Research (projects No. 02-10-00253, 05-01-00122, 05-01-00144, and 08-01-00376-a) and the Program of the President of the Russian Federation for supporting scientific schools and young candidates and doctors of science (projects No. NH-1918.2003.1, MK-1857.2005.1, MK-1513.2005.9, MD-1006.2007.1 MD99.2009.1). Moscow, January 2011

Alexander B. Al’shin, Maxim O. Korpusov, Alexey G. Sveshnikov

Contents

Preface

v

0 Introduction 0.1 List of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1.1 One-dimensional pseudoparabolic equations . . . . . . . . . 0.1.2 One-dimensional wave dispersive equations . . . . . . . . . . 0.1.3 Singular one-dimensional pseudoparabolic equations . . . . . 0.1.4 Multidimensional pseudoparabolic equations . . . . . . . . . 0.1.5 New nonlinear pseudoparabolic equations with sources . . . . 0.1.6 Model nonlinear equations of even order . . . . . . . . . . . 0.1.7 Multidimensional even-order equations . . . . . . . . . . . . 0.1.8 Results and methods of proving theorems on the nonexistence and blow-up of solutions for pseudoparabolic equations . . . 0.2 Structure of the monograph . . . . . . . . . . . . . . . . . . . . . . . 0.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 3 5 6 7 10 13 14

1

20

Nonlinear model equations of Sobolev type 1.1 Mathematical models of quasi-stationary processes in crystalline semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Model pseudoparabolic equations . . . . . . . . . . . . . . . . . . . 1.2.1 Nonlinear waves of Rossby type or drift modes in plasma and appropriate dissipative equations . . . . . . . . . . . . . 1.2.2 Nonlinear waves of Oskolkov–Benjamin–Bona–Mahony type 1.2.3 Models of anisotropic semiconductors . . . . . . . . . . . . . 1.2.4 Nonlinear singular equations of Sobolev type . . . . . . . . . 1.2.5 Pseudoparabolic equations with a nonlinear operator on time derivative . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Nonlinear nonlocal equations . . . . . . . . . . . . . . . . . 1.2.7 Boundary-value problems for elliptic equations with pseudoparabolic boundary conditions . . . . . . . . . . 1.3 Disruption of semiconductors as the blow-up of solutions . . . . . . . 1.4 Appearance and propagation of electric domains in semiconductors . 1.5 Mathematical models of quasi-stationary processes in crystalline electromagnetic media with spatial dispersion . . . . . . . . . . . . . . .

20 27 27 29 34 37 38 39 46 48 56 60

viii

Contents

1.6 1.7

Model pseudoparabolic equations in electric media with spatial dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model pseudoparabolic equations in magnetic media with spatial dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Blow-up of solutions of nonlinear equations of Sobolev type 2.1 Formulation of problems . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminary definitions, conditions, and auxiliary lemmas . . . . . . . 2.3 Unique solvability of problem (2.1) in the weak generalized sense and blow-up of its solutions . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Unique solvability of problem (2.1) in the strong generalized sense and blow-up of its solutions . . . . . . . . . . . . . . . . . . . . . . . 2.5 Unique solvability of problem (2.2) in the weak generalized sense and estimates of time and rate of the blow-up of its solutions . . . . . . . 2.6 Strong solvability of problem (2.2) in the case where B 0 . . . . . . 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Initial-boundary-value problem for a nonlinear equation with double nonlinearity of type (2.1) . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Local solvability of problem (2.131)–(2.133)in the weak generalized sense . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Blow-up of solutions . . . . . . . . . . . . . . . . . . . . . . 2.9 Problem for a strongly nonlinear equation of type (2.2) with inferior nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Unique weak solvability of problem (2.185) . . . . . . . . . . 2.9.2 Solvability in a finite cylinder and blow-up for a finite time . . 2.9.3 Rate of the blow-up of solutions . . . . . . . . . . . . . . . . 2.10 Problem for a semilinear equation of the form (2.2) . . . . . . . . . . 2.10.1 Blow-up of classical solutions . . . . . . . . . . . . . . . . . 2.11 On sufficient conditions of the blow-up of solutions of the Boussinesq equation with sources and nonlinear dissipation . . . . . . . . . . . . 2.11.1 Local solvability of strong generalized solutions . . . . . . . 2.11.2 Blow-up of solutions . . . . . . . . . . . . . . . . . . . . . . 2.12 Sufficient conditions of the blow-up of solutions of initial-boundaryvalue problems for a strongly nonlinear pseudoparabolic equation of Rosenau type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.1 Local solvability of the problem in the strong generalized sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.2 Blow-up of strong solutions of problem (2.288)–(2.289) and solvability in any finite cylinder . . . . . . . . . . . . . . . . 2.12.3 Physical interpretation . . . . . . . . . . . . . . . . . . . . .

64 66 69 69 70 78 101 111 127 133 141 142 159 164 165 177 183 187 187 196 197 200

203 203 211 215

Contents

3 Blow-up of solutions of strongly nonlinear Sobolev-type wave equations and equations with linear dissipation 3.1 Formulation of problems . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminary definitions and conditions and auxiliary lemma . . . . . . 3.3 Unique solvability of problem (3.1) in the weak generalized sense and blow-up of its solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Unique solvability of problem (3.1) in the strong generalized sense and blow-up of its solutions . . . . . . . . . . . . . . . . . . . . . . . 3.5 Unique solvability of problem (3.2) in the weak generalized sense and blow-up of its solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Unique solvability of problem (3.2) in the strong generalized sense and blow-up of its solutions . . . . . . . . . . . . . . . . . . . . . . . 3.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 On certain initial-boundary-value problems for quasilinear wave equations of the form (3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Local solvability in the strong generalized sense of problems (3.141)–(3.143) . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Blow-up of solutions . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Breakdown of weakened solutions of problem (3.141) . . . . 3.9 On an initial-boundary-value problem for a strongly nonlinear equation of the type (3.1) (generalized Boussinesq equation) . . . . . . . . 3.9.1 Unique solvability of the problem in the weak sense . . . . . 3.9.2 Blow-up of solutions and the global solvability of the problem 3.10 Blow-up of solutions of a class of quasilinear wave dissipative pseudoparabolic equations with sources . . . . . . . . . . . . . . . . . . . 3.10.1 Unique local solvability of the problem in the strong sense and blow-up of its solutions . . . . . . . . . . . . . . . . . . 3.10.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Blow–up of solutions of the Oskolkov–Benjamin–Bona–Mahony– Burgers equation with a cubic source . . . . . . . . . . . . . . . . . . 3.11.1 Unique local solvability of the problem . . . . . . . . . . . . 3.11.2 Global solvability and the blow-up of solutions . . . . . . . . 3.11.3 Physical interpretation of the obtained results . . . . . . . . . 3.12 On generalized Benjamin–Bona–Mahony–Burgers equation with pseudo-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.1 Blow-up of strong generalized solutions . . . . . . . . . . . . 3.12.2 Physical interpretation of the obtained results . . . . . . . . . 3.13 Sufficient, close to necessary, conditions of the blow-up of solutions of one problem with pseudo-Laplacian . . . . . . . . . . . . . . . . . 3.13.1 Blow-up of strong generalized solutions . . . . . . . . . . . . 3.13.2 Physical interpretation of the obtained results . . . . . . . . .

ix

216 216 217 219 244 254 273 278 288 288 295 302 308 309 315 320 320 327 329 330 333 337 337 337 340 341 341 345

x

Contents

3.14 Sufficient, close to necessary, conditions of the blow-up of solutions of strongly nonlinear generalized Boussinesq equation . . . . . . . . 345 4 Blow-up of solutions of strongly nonlinear, dissipative wave Sobolevtype equations with sources 4.1 Introduction. Statement of problem . . . . . . . . . . . . . . . . . . . 4.2 Unique solvability of problem (4.1) in the weak generalized sense and blow-up of its solutions . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Unique solvability of problem (4.1) in the strong generalized sense and blow-up of its solutions . . . . . . . . . . . . . . . . . . . . . . . 4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Blow-up of solutions of a Sobolev-type wave equation with nonlocal sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Unique local solvability of the problem . . . . . . . . . . . . 4.5.2 Blow-up of strong generalized solutions . . . . . . . . . . . . 4.6 Blow-up of solutions of a strongly nonlinear equation of spin waves . 4.6.1 Unique local solvability in the strong generalized sense . . . . 4.6.2 Blow-up of strong generalized solutions and the global solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Physical interpretation of the obtained results . . . . . . . . . 4.7 Blow-up of solutions of an initial-boundary-value problem for a strongly nonlinear, dissipative equation of the form (4.1) . . . . . . . 4.7.1 Local unique solvability in the weak generalized sense . . . . 4.7.2 Unique solvability of the problem and blow-up of its solution for a finite time . . . . . . . . . . . . . . . . . . . . . . . . . 5 Special problems for nonlinear equations of Sobolev type 5.1 Nonlinear nonlocal pseudoparabolic equations . . . . . . . . . . . . . 5.1.1 Global-on-time solvability of the problem . . . . . . . . . . . 5.1.2 Global-on-time solvability of the problem in the strong generalized sense in the case q 1 . . . . . . . . . . . . . . . . 5.1.3 Asymptotic behavior of solutions of problem (5.1), (5.2) as t ! C1 in the case q > 0 . . . . . . . . . . . . . . . . . . . . . 5.2 Blow-up of solutions of nonlinear pseudoparabolic equations with sources of the pseudo-Laplacian type . . . . . . . . . . . . . . . . . . 5.2.1 Blow-up of weakened solutions of problem (5.77) . . . . . . . 5.2.2 Blow-up and the global-on-time solvability of problem (5.78) 5.2.3 Blow-up of solutions of problem (5.79) . . . . . . . . . . . . 5.2.4 Blow-up of weakened solutions of problems (5.80) and (5.81) 5.2.5 Interpretation of the obtained results . . . . . . . . . . . . . .

357 357 358 380 385 391 391 398 402 403 412 417 417 418 435 439 439 439 469 471 475 476 477 479 482 484

xi

Contents

5.3

5.4

5.5

5.6

5.7

5.8

Blow-up of solutions of pseudoparabolic equations with fast increasing nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Local solvability and blow-up for a finite time of solutions of problems (5.112) and (5.113) . . . . . . . . . . . . . . . . 5.3.2 Local solvability and blow-up for a finite time of solutions of problem (5.114) . . . . . . . . . . . . . . . . . . . . . . . . . Blow-up of solutions of nonhomogeneous nonlinear pseudoparabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Unique local solvability of the problem . . . . . . . . . . . . 5.4.2 Blow-up of strong generalized solutions of problem (5.154)– (5.155) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Blow-up of classical solutions of problem (5.154)–(5.155) . . Blow-up of solutions of a nonlinear nonlocal pseudoparabolic equation 5.5.1 Unique local solvability of the problem . . . . . . . . . . . . 5.5.2 Blow-up and global solvability of problem (5.177) . . . . . . 5.5.3 Blow-up rate for problem (5.177) under the condition q D 0 . Existence of solutions of the Laplace equation with nonlinear dynamic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Reduction the problem to the system of the integral equations 5.6.2 Global-on-time solvability and the blow-up of solutions . . . Conditions of the global-on-time solvability of the Cauchy problem for a semilinear pseudoparabolic equation . . . . . . . . . . . . . . . 5.7.1 Reduction of the problem to an integral equation . . . . . . . 5.7.2 Theorems on the existence/nonexistence of global-on-time solutions of the integral equation (5.219) . . . . . . . . . . . . . Sufficient conditions of the blow-up of solutions of the Boussinesq equation with nonlinear Neumann boundary condition . . . . . . . . .

6 Numerical methods of solution of initial-boundary-value problems for Sobolev-type equations 6.1 Numerical solution of problems for linear equations . . . . . . . . . . 6.1.1 Dynamic potentials for one equation . . . . . . . . . . . . . . 6.1.2 Solvability of Dirichlet problem . . . . . . . . . . . . . . . . 6.2 Numerical method of solving initial-boundary-value problems for nonlinear pseudoparabolic equations by the Rosenbrock schemes . . . . . 6.2.1 Stiff method of lines . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Stiff systems of ODE and methods of solving them . . . . . . 6.2.3 Stiff stability . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Schemes of Rosenbrock type . . . . . . . . . . . . . . . . . . 6.2.5 "-embedding method . . . . . . . . . . . . . . . . . . . . . .

484 485 492 496 496 499 502 503 504 506 509 511 511 517 525 525 527 537

543 543 544 548 554 554 555 555 555 557

xii

Contents

6.3

Results of blow-up numerical simulation . . . . . . . . . . . . . . . . 6.3.1 Blow-up of pseudoparabolic equations with a linear operator by the time derivative . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Blow-up of strongly nonlinear pseudoparabolic equations . . 6.3.3 Blow-up of equations with nonlocal terms (coefficients of the equation depend on the norm of the function) . . . . . . . . .

560

Appendix A Some facts of functional analysis s;p A.1 Sobolev spaces W s;p ./, W0 ./, and W s;p ./ . . . . . . . . . . A.2 Weak and -weak convergence . . . . . . . . . . . . . . . . . . . . . A.3 Weak and strong measurability. Bochner integral . . . . . . . . . . . A.4 Spaces of integrable functions and distributions . . . . . . . . . . . . A.5 Nemytskii operator. Krasnoselskii theorem . . . . . . . . . . . . . . A.6 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 Operator calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 Fixed-point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 Weakened solutions of the Poisson equation . . . . . . . . . . . . . . A.10 Intersections and sums of Banach spaces . . . . . . . . . . . . . . . . A.11 Classical, weakened, strong generalized, and weak generalized solutions of evolutionary problems . . . . . . . . . . . . . . . . . . . . . A.12 Two equivalent formulations of weak solutions in L2 .0; TI B/ . . . . . A.13 Gâteaux and Fréchet derivatives of nonlinear operators . . . . . . . . A.14 On the gradient of a functional . . . . . . . . . . . . . . . . . . . . . A.15 Lions compactness lemma . . . . . . . . . . . . . . . . . . . . . . . A.16 Browder–Minty theorem . . . . . . . . . . . . . . . . . . . . . . . . A.17 Sufficient conditions of the independence of the interval, on which a solution of a system of differential equations exists, of the order of this system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.18 On the continuity of some inverse matrices . . . . . . . . . . . . . . .

581 581 583 584 585 586 588 589 589 589 591

561 566 575

592 594 596 604 606 607

608 610

Appendix B To Chapter 6 613 B.1 Convergence of the "-embedding method with the CROS scheme . . . 613 Bibliography

621

Index

647

Chapter 0

Introduction

0.1

List of equations

First, we give a definition of Sobolev-type equations. Under a Sobolev-type equation, we mean an equation which is not resolved with respect to the partial derivative in time of the highest order. In our monograph, we use another term, pseudoparabolic equations, introduced by R. E. Showalter and T. W. Ting in [378]. Pseudoparabolic equations form a subclass of Sobolev-type equations with first-order derivatives in time. As in [378], we study only nonlinear, odd-order, pseudoparabolic equations, but, for completeness, we review results concerning nonlinear even-order equations.

0.1.1 One-dimensional pseudoparabolic equations The Camassa–Holm equation u t uxxt C 3uux D 2ux uxx C uuxxx describes the unidirectional propagation of shallow-water waves over a flat bottom (see [67–72,200–202]). It is completely integrable (see [208]) and admits, in addition to smooth waves, a variety of travelling-wave solutions with singularities: peakons, cuspons, stumpons, and composite waves (see [258, 269, 270, 292]). This equation models wave breaking (see [95–101]). The Degasperis–Procesi equation u t uxxt C 4uux D 3ux uxx C uuxxx models nonlinear shallow-water dynamics. It is completely integrable (see [208]) and has a variety of travelling-wave solutions including solitary-wave solutions, peakon solutions, and shock waves solutions (see [89, 90, 137, 259, 282, 291, 292]). The Fornberg–Whitham equation u t uxxt C ux C uux D uuxxx C 3ux uxx appeared in the study of the qualitative behavior of wave breaking (see [428]). It admits a wave of greatest height, as a peaked limiting form of the travelling-wave solution ˇ ˇ³ ² 4 ˇˇ 1 ˇˇ u.x; t / D A exp ˇx t ˇ 2 3 (see [157]), where A is an arbitrary constant. It is not completely integrable (see [208]).

2

Chapter 0 Introduction

The Benjamin–Bona–Mahony (BBM) equation u t uxxt C ux C uux D 0; which is widely presented in the scientific literature; we note only the most significant works in which initial-value, initial-boundary-value, and periodic initial-boundary value problems were investigated: [2, 5, 24, 37, 39, 42–53, 78, 193, 297]. The modified Benjamin–Bona–Mahony (MBBM) equation u t uxxt C ux C 3u2 ux D 0 has also been investigated by many authors (see, e.g., [38,66,273,310,343,344,438]). The hyperelastic-rod wave equation u t uxxt C 3uux D .2ux uxx C uuxxx / was considered in [208]. The generalized hyperelastic-rod wave equation u t uxxt C @x .~ u C ˛u2 C u3 / D ux uxx C uuxxx is a generalization of the previous equation.

0.1.2 One-dimensional wave dispersive equations The Benjamin–Bona–Mahony–Burgers (BBMB) equation u t uxxt C ux C uux uxx D 0 was considered, e.g., in [1, 37, 48–53, 78, 140, 178, 188, 296, 402, 441]. The generalized porous-media equation u t D @x .u˛ ux C uˇ uxt /;

˛; ˇ 0;

which was studied, e.g., in [106]. The Rosenau–Burgers equation uxxxxt C u t ˛uxx C up ux D 0: The breaking result for this equation was obtained in [315, 316, 348–352]. Moreover, in [197, 296, 315, 316], the first terms of asymptotic expansions of solutions for large time were found. In addition, we note the paper [83, 86], where Galerkin approximations were considered.

3

Section 0.1 List of equations

0.1.3 Singular one-dimensional pseudoparabolic equations In these equations, elliptic operators under the highest time derivative are not resolvable. The Coleman–Duffin–Mizel equation u t C uxxt uxx D 0 was considered in [91]. The Hoff equation u t C uxxt D ˛u C ˇu3 was studied, e.g., in [199,395–398]. The semigroup approach to the general theory of Sobolev-type singular equations was developed by G. A. Sviridyuk and V. E. Fyodorov. In an abstract form, degenerate pseudoparabolic equations were considered, e.g., in [148, 431–433]. The Korpusov–Pletner–Sveshnikov equation u t C uxxt C ˛uxx C ˇ.u2 /xx D 0;

˛ > 0; ˇ > 0;

describes nonstationary processes in crystalline semiconductors. The one-dimensional Oskolkov equation u t C uxxt C uux C uxx D 0: The one-dimensional Boussinesq equation u t C uxxt C .jujp2 u/xx D 0 was considered in [130, 395, 398].

0.1.4 Multidimensional pseudoparabolic equations The Barenblatt–Zheltov–Kochina equation @ .u C cu/ C u D 0; @t

c 2 R1 n¹0º;

describes nonstationary filtering processes in fissured-porous media. This equation can be rewritten in a more general form @ A.u/ C B.u/ D 0; @t where A.u/ and B.u/ are nonlinear elliptic operators. In the classical works [365– 371, 374, 378] R. E. Showalter and T. W. Ting considered linear equations of this form.

4


The Showalter equation @ .u C div.jrujp2 ru/ u/ C ˛u C ˛ div.jrujp2 ru/ D 0 @t and initial-value problems for it were considered in [368, 369] and some general results on the unique solvability for abstract pseudoparabolic equations were obtained. The Showalter inclusion @ A.u/ C B.u/ 3 f; @t where A.u/ and B.u/ are maximal monotone elliptic operators, were studied in [113– 120, 365–371, 374, 388–392]. Moreover, we mention the following works devoted to the study of initial-value and initial-boundary-value problems for multidimensional dissipative pseudoparabolic equations: [12, 31–36, 39–42, 57–61, 64, 65, 77–79, 93, 94, 103–106, 113–120, 143–149, 179, 195–197, 217, 251–255, 306–309, 315, 316, 364]. We also note the monograph of H. Gajewski, K. Gröger, and K. Zacharias [168], in which various aspects of the local solvability of pseudoparabolic equations are considered. The central part of this monograph is devoted to the study of operator and operator-differential equations. For pseudoparabolic operator-differential equations, the aspects of C- and L2 -solvability are analyzed. The basis of finite-dimensional approximate methods, especially the Galerkin method, is considered. We note that the method of construction of asymptotical expansions for large time for a wide class of nonlinear evolutionary equations developed in works of N. Hayashi, I. A. Shishmaryov, P. I. Naumkin, and E. I. Kaikina (see [195–197, 306–309]) can be applied to the study of the asymptotical behavior for large time for pseudoparabolic equations. Specifically, in [364] the asymptotic behavior of solutions of the Cauchy problem for the following dissipative pseudoparabolic equation was obtained. The semiconductor equation @ .u u/ C u C ˛u3 D 0; ˛ 2 R: @t was obtained in [236]; it describes nonstationary processes in crystalline semiconductors. The generalized Boussinesq nonlinear equation u t .u/ u t C q.u/ D 0: Based on the comparison principle, A. I. Kozhanov [243] proved the solvability of the first boundary-value problem and the occurrence of the blow-up. The blow-up of positive solutions was proved and existence/nonexistence theorems were obtained. The multidimensional Benjamin–Bona–Mahony–Burgers equation @ .u u/ C .; r/.u C u2 / C u D 0 @t was considered in [217, 429, 441].

5


The linear Rossby wave equation @ @u 2 u C ˇ.y/ D 0; @t @x

2

@2 @2 C @x 2 @y 2

is a linear approximation in the ˇ-plane of the two-dimensional Rossby wave equation (see [164, 165]), where the axes Ox and Oy are directed to the east and to the north, respectively, and ˇ D ˇ.y/ is the Coriolis parameter. The Kadomtsev–Petviashvilli equation @u @u @ .2 u u/ C Cu C J.u; 2 u/ D 0; @t @x @x

J.a; b/ D ax by ay bx ;

is a nonlinear generalization of the two-dimensional equation of Rossby waves (see, e.g., [400]). The three-dimensional Camassa–Holm equation @ .u u/ C .u u/ C u .r .u u// D rp; @t is the viscous version of the three-dimensional Camassa–Holm equations; it was considered in [202].

0.1.5 New nonlinear pseudoparabolic equations with sources Now we list some equations obtained in our works [233–240]. The generalized Benjamin–Bona–Mahony–Burgers equation @ @u @u Cu C u C jujq2 u D 0; .u u jujq1 u/ C @t @x1 @x1 where x D .x1 ; x2 ; x3 / 2 R3 , q1 ; q2 > 0;

@2 @2 @2 C C : @x12 @x22 @x32

The nonlocal pseudoparabolic equation Z q @ .u u/ C u jruj2 dx D 0; @t

q > 1:

The generalized Rosenau–Burgers equation @ .2 u C u C div.jrujp1 2 ru// C u div.jrujp2 2 ru/ D 0; @t

6


where 2 ;

p1 ; p2 2:

The spin-wave equation @ .2 u C u C div.jrujp2 ru// C u div.jruj2 ru/ @t @ @ @ @u @u @u @u @u @u C ˛1 C ˛2 C ˛3 D 0; @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2 where j˛1 j C j˛2 j C j˛3 j > 0, ˛1 C ˛2 C ˛3 D 0; p 2:

0.1.6 Model nonlinear equations of even order For completeness, we list some model, nonlinear, even-order equations although we will not study them in what follows. One-dimensional nonlinear equations The Longern wave equation @2 .uxx ˛u C ˇu2 / C uxx D 0; ˛ > 0; ˇ > 0; @t 2 which describes electric signals in telegraph lines [281]. Moreover, in active nonlinear media, the following equations hold. The Rabinowitz wave equations with nonlinear damping @2 @2 .u u/ C ux .ux /2 C uxx D 0; xx 2 @t @t @x 2 @ @ .uxx u/ C u u2 C uxx D 0; 2 @t @t which describe electric signals in telegraph lines on the basis of the tunnel diode (see [340, 341]). Note that in our book [230], sufficient conditions of the blow-up for the corresponding initial-boundary-value problems were obtained. The nonlinear telegraph equation with nonlinear damping 3 u @4 u @3 @ 2 u @2 u 2 D˛ 2 2 Cˇ 2 u ; ˛ > 0; ˇ > 0; @t 2 @x @x @t @x @t 3 which was considered in [62, 223]. The Pochhammer–Chree equation u t t uxxt t .a1 u C a3 u3 C a5 u5 /xx D 0 was considered in [442–444].

7


The improved Boussinesq equations @2 @2 .u u/ C u C .f .u// D 0; .uxx u/ C uxx C .g.ux //x D 0I xx xx xx @t 2 @t 2 blow up results for these equations were obtained in [102, 107, 125–127, 191]. The higher-order improved Boussinesq equation @2 .2 uxxxx C 1 uxx u/ C uxx C .g.u//xx D 0; @t 2 the global solvability in time was analyzed in [125–127, 424].

1 ; 2 > 0I

Systems of nonlinear equations The system of two improved Boussinesq equations @2 .u1xx u1 / C u1xx C .f1 .u1 ; u2 //xx D 0; @t 2 @2 .u2xx u2 / C u2xx C .f2 .u1 ; u2 //xx D 0: @t 2 The system of two higher-order improved Boussinesq equations @2 .2 u1xxxx C 1 u1xx u1 / C u1xx C .g1 .u1 ; u2 //xx D 0; @t 2 @2 .2 u2xxxx C 1 u2xx u2 / C u2xx C .g2 .u1 ; u2 //xx D 0I @t 2 blow-up results were obtained in [125–127]. The Benney–Luke equation u t t uxx C auxxxx buxxt t C u t uxx C 2ux uxt D 0 was considered in [18].

0.1.7 Multidimensional even-order equations First, we mention researches made in Russia. Of course, we can state that the first who strictly analyzed problems for equations that are not of the Cauchy–Kovalevskaya type was S. L. Sobolev; he invoked a great interest to the study of nonclassical equations, which are now called equations of the Sobolev type. The S. L. Sobolev equation 2 @2 @2 u @2 u @2 u 2@ u C ˛ C C D0 @t 2 @x12 @x22 @x32 @x32 describes small-amplitude oscillation in rotating fluids.

8


Researches of S. L. Sobolev were continued in Russia by R. A. Aleksandryan, A. B. Al’shin, G. I. Barenblatt, V. A. Borovikov, G. V. Demidenko, E. S. Dzektser, G. I. Eskin, V. E. Fyodorov, M. V. Falaleev, S. A. Gabov, S. A. Galperin, B. V. Kapitonov, N. D. Kopachevsky, M. O. Korpusov, A. G. Kostyuchenko, I. N. Kochina, A. I. Kozhanov, S. I. Lyashko, V. N. Maslennikova, V. P. Maslov, A. P. Oskolkov, L. V. Ovsyannikov, Yu. D. Pletner, S. G. Pyatkov, S. V. Popov, S. Ya. SekerzhZenkovich, I. A. Shishmaryov, N. A. Sidorov, A. G. Sveshnikov, G. A. Sviridyuk, M. B. Tverskoy, S. V. Uspensky, T. I. Zelenyak, Yu. P. Zheltov, and others. Linear even-order multidimensional Sobolev-type equations The ion-acoustic wave equation @2 .u u/ C !02 u D 0 @t 2 was derived in and investigated by S. A. Gabov (see [164, 165]). The gravity-gyroscopic wave equation @2 .3 u ˇ 2 u/ C !02 2 u C ˛ 2 Œux3 x3 ˇ 2 u D 0; @t 2 where ˛; ˇ; !0 2 .0; C1/, 3 2 C

@2 ; @x32

2

@2 @2 C ; @x12 @x22

was studied in [165]. Equations of order higher than four The ion-acoustic wave equation in plasma in external magnetic field 2 2 1 @ 2 @2 2 2 @ 2 2 @ u C ! u u C ! u C ! ! D0 3 3 pi pi Bi Bi 2 @t 2 @t 2 @t 2 rD @x32 was derived in [231] and was studied by Yu. D. Pletner. The spin-wave equation in magnetics in an external magnetic field 2 2 2 @ @ 2 2 2 @ u C !7 C !6 C !8 @t 2 @t 2 @x12 2 2 2 @ @ @ u 2 2 2 C C ! C ! C ! 6 7 10 2 2 @t @t @x22 2 2 2 @ @ 2 2 2 @ u C C !7 C !6 C !9 D0 @t 2 @t 2 @x32 was also derived in [231] and was studied by Yu. D. Pletner.

9


Nonlinear multidimensional Sobolev-type equations of even order The Benney–Luke equation u t t bu t t C u C a2 u C u t u C 2 .ru; ru t / D 0 was considered in [339] (see also the references therein). The nonlinear ion-acoustic wave equations with nonlinear damping @2 .2 u C ˛2 u C ux3 x3 C div.jrujp1 2 ru// @t 2 @ C .u div.jrujp2 2 ru// C ux3 x3 D 0; @t 2 @ @ .˛2 u C ux3 x3 jujq1 u/ C .u C jujq2 u/ C ux3 x3 D 0; @t 2 @t

˛ > 0; ˛ > 0;

was derived in [230]. Higher-order nonlinear ion-acoustic wave equations with nonlinear damping 2 2 @ @ @2 2 2 2 p1 2 C ! C ! ru// 2 4 . u C u C div.jruj @t 2 @t 2 @t 2 2 2 2 @2 @ @ 2 2 @ 2 C !12 2 C ! u C ! C ! 2 4 3 2 2 u @t @t 2 @t 2 @t 2 2 2 @ @ 2 2 C C !2 C !4 ux3 x3 @t 2 @t 2 2 @ @ @2 2 2 C C !2 C !4 .u div.jrujp2 2 ru// D 0; @t @t 2 @t 2 2 2 @ @ @2 2 2 C !2 C !4 .u jujq1 u/ @t 2 @t 2 @t 2 2 @2 2 @2 2 @ 2 2 @ 2 C !1 2 C !4 2 u C !3 2 C !2 2 u @t @t 2 @t @t 2 2 2 @ @ 2 2 C C !2 C !4 ux3 x3 @t 2 @t 2 2 @ @ @2 2 2 C C !2 C !4 .u C jujq2 u/ D 0 @t @t 2 @t 2 were also derived in [230]. The dissipative nonlinear ion-acoustic wave equation with nonlinear damping @ @2 div. 1 .x; jruj/ru/ C u div. 2 .x; jruj/ru/ D 0: .2 u C u/ C 2 @t @t

10


The dissipative system of ion-acoustic wave equations q @2 2 . u C u/ div x; jruj2 C jrvj2 ru D 0; 2 @t q @2 2 . v C v/ div x; jruj2 C jrvj2 rv D 0 2 @t was derived in [229]. The Oskolkov system of equations with sources @ .u u/ C u C .u; r/u C juj2 u D rp; @t div u D 0; u D .u1 ; u2 ; u3 /; was analyzed in [240] and a blow up result was obtained. This system generalizes the well-known Navier–Stokes system; it is a Sobolev-type system with sources. The generalized von Karman system u t t u t t C 2 u C D Œu; v; 2 v D Œu; u; Œu; v D

t u t D 0;

@ 2 u @2 v @2 u @2 u @2 v C 2 C @x1 @x2 @x12 @x22 @x22 @x12

was considered in [318]. The improved Boussinesq–Schrödinger system i t C u D 0; u t t au t t u D f .u/ C j j2 was considered by many authors. The nonlocal higher order wave equation Z u t t u t t jruj2 dx u C f .u/ D 0

was considered by J. Fereira.

0.1.8 Results and methods of proving theorems on the nonexistence and blow-up of solutions for pseudoparabolic equations There are three fundamental methods for the study of blow-up. The first is the method of nonlinear capacity of S. I. Pokhozhaev and E. L. Mitidieri, the second is the energy method of H. A. Levine, P. Pucci, and J. Serrin, and, finally, the third is the

11


method of self-simulating modes based on various comparison criteria and developed by A. A. Samarskii, V. A. Galaktionov, S. P. Kurdymov, and A. P. Mikhailov. First, we mention the classical work by Fujita [159] on the nonexistence of positive solutions for semilinear parabolic equations. In this work, in addition to the proof of the blow-up, existence/nonexistence theorems for bounded solutions were proved for the first time, where solutions are treated in the classical sense. In this work, wellknown properties of the fundamental solution of the thermal conduction operator were used for the obtaining an optimal result for the blow-up of positive solutions to the Cauchy problem for the following semilinear parabolic equation: @u D u C u1C˛ : @t In the classical works of Levine [260, 261], the energy approach to the study of the blow-up of strong and weak generalized solutions under sufficiently large initial parameters of the problem. This work is devoted to the study of the global-on-time insolvability for an operator-differential equation of the form A

du C Lu D F.u/; dt

u.0/ D u0 I

the fact that the operators A and L are linear, positive definite, and self-adjoint and F .u/ has a symmetric Fréchet derivative was substantially used. We note that our technique of the proof of the global-in-time nonexistence of solutions to the problems considered is a development of the energy method proposed by Levine, Pucci, and Serrin. We generalize the approach of Levine in the following areas: first, we consider the case of nonlinear operators A and L and obtain two-sided estimates for the blow-up time; second, in the case of linear operator A, we obtain optimal two-sided estimates not only for the blow-up time but for the blow-up speed; third, we consider the case of wave pseudoparabolic equations, for which the technique of Levine is inapplicable. Note that in the work of S. Kaplan [216] the maximum principle was applied to the study of the class of quasilinear parabolic equations. In [15], H. Amann and M. Fila proposed a new problem for which they obtained several Fujita-type optimal results. We also mention the works of M. Chipot, D. Phillips, J. Escer, B. Hu, W. Walter. The wide spectrum of results in the study of solutions was obtained in the classical work of A. A. Samarsky, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov [358]. In this work, the authors study the blow-up of solutions to quasilinear parabolic equations using various techniques. For some problems, by using criteria and upper and lower solutions, existence/nonexistence theorems were proved. For other problems, the method of unlimited Fourier coefficients was applied. Sufficient conditions for the blow-up of solutions of parabolic quasilinear equations were obtained. Note that the technique developed for the proof of the blow-up of solutions of parabolic

12


equations can be applied to the study of questions on the blow-up for pseudoparabolic equations. We also mention the works by V. A. Galaktionov [169–175]. In the monograph [304], S. I. Pokhozhaev and E. Mitidieri deeply developed an original method of the study of questions on the blow-up of solutions of elliptic, hyperbolic, and parabolic differential inequalities. For all these types of equations, the authors proposed a generalization of the concept of capacity, the so-called nonlinear capacity. Note that this method is applicable not only to differential equations in unbounded domains but also to differential equations in bounded domains and it allows one to study many important equation, for example, the Kuramoto–Sivashinsky equation (see, e.g., [172, 305, 334]). The Kuramoto–Sivashinsky equation 1 u t C 2 u C u C jruj2 D 0: 2 Moreover, this method was successfully applied to nonlinear evolutionary third-order equations in bounded and unbounded domains, for example, for the following nonlinear equations: u t C uux C uxxx D 0;

u t D .uux /xx ;

u t D u3 uxxx

(see, e.g., [173]). The method for proving of the nonexistence of solutions for certain classes of boundary-value problems based on the maximum principle was developed in the works of Yu. V. Egorov and V. A. Kondratiev. We note that in the monograph [212], N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, and B. V. Rogov proposed an original method of quasi-uniform grid for numerical simulation of initial-boundary-value problems in unlimited domains without any standard formulation of certain auxiliary boundary conditions. This method allowed numerical analyzing of linear and nonlinear equations of Sobolev type in unlimited domains. In concluding the review of results, we note that, as far as we know, the study of the blow-up of solutions to pseudoparabolic nonlinear equations was fulfilled only in works of H. A. Levine and A. I. Kozhanov and, specifically, theorems on the blow-up of solutions to pseudoparabolic semilinear equations in unbounded domains can be derived by the method of trial functions developed by S. I. Pokhozhaev and E. Mitidieri in [304]. V. K. Kalantarov and O. A. Ladyzhenskaya [210] considered differential inequalities of the form ˆˆ00 .1 C ˛/.ˆ0 /2 C C1 ˆˆ0 C C2 ˆ2 0;

C1 0; C2 0; ˛ > 0:

These inequalities give sufficient conditions for a discontinuity of the second kind of the function ˆ.t / (see [132, 135]). Note that we extensively use this ordinary differential inequality for obtaining sufficient conditions of the blow-up of solutions to initial-boundary-value problems for nonlinear pseudoparabolic equations.

Section 0.2 Structure of the monograph

0.2

13

Structure of the monograph

Now we briefly review the contents of our monograph. The main goal of the monograph is the description of the present state of the studies of the existence/nonexistence of solutions to Cauchy problems and initial-boundary-value problems for linear and nonlinear Sobolev-type equations and also numerical simulation of the blow-up of their solutions. In Chapter 1, mathematical models of quasi-stationary processes in continuous electromagnetic media are considered. It is shown that the Maxwell equations in quasi-stationary approximation and phenomenological equations, which relate the vectors of electric field intensity E, electric induction D, medium polarization P, current density J, magnetic field intensity H, magnetic field induction B, and medium magnetization M are reduced, subject to various nonstationary processes, to a wide spectrum of initial-value and initial-boundary-value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. Therefore, some of the model equations obtained are wave equation and the others are dissipative equations. A qualitative description of certain substantially nonlinear effects observed in experiments is given. In Chapter 2, two different abstract Cauchy problems for pseudoparabolic equations with operator coefficients in Banach spaces are considered. For the first problem, under certain conditions for operator coefficients, necessary and sufficient conditions of the blow-up of solutions are proved, and in the case of the blow-up, lower and upper estimates of the blow-up time are obtained. Finally, for other problems, optimal lower and upper estimates of the speed of blow-up are obtained. For each of the abstract problems, examples of operator coefficients having physical meaning are given. Three model problems are considered in detail. In Chapter 3, abstract Cauchy problems for ordinary first-order differential equations with nonlinear operator coefficients are considered, for example, the Oskolkov– Benjamin–Mahony–Burgers equations with cubic source or pseudo-Laplacian. We also give examples of strongly nonlinear pseudoparabolic equations in bounded domains with smooth boundaries. For problems posed, we obtain sufficient (close to necessary) conditions of the blow-up for a finite time and conditions of the global solvability. Under certain conditions on nonlinear operators, we prove the solvability in an arbitrary finite cylinder, and under certain conditions on the norm of an initial function in a certain Banach space, when the initial function is sufficiently large, we solve the problem on the blow-up of a solution in a finite time. In Chapter 4, abstract Cauchy problems for first-order ordinary differential equations with nonlinear operator coefficients are considered. We also present examples of strongly nonlinear wave dissipative pseudoparabolic equations in bounded domains with smooth boundaries. We obtain sufficient, close to necessary conditions of the blow-up in a finite time and of the global-in-time solvability. Moreover, problems on

14


the blow-up of strongly nonlinear spin-wave pseudoparabolic equation and strongly nonlinear dissipative pseudoparabolic equation are considered. Under certain conditions on nonlinear operators, we prove the solvability in a finite cylinder and, under other conditions on the norm of an initial function, in a certain Banach space, when the initial function is sufficiently large, we prove the blow-up of solutions in a finite time. Problems for strongly nonlinear spin-wave dissipative pseudoparabolic equations are given. In Chapter 5, we discuss methods for solving initial-value and initial-boundaryvalue problems for certain specific pseudoparabolic equations that do not satisfy conditions of Chapters 2 and 3 and consider Cauchy problems in R3 . The results of Chapters 1–5 were obtained in works of M. O. Korpusov and A. G. Sveshnikov. In Chapter 6, numerical methods for equations unsolved with respect to time derivative are considered. First, with a linear pseudoparabolic equation as an example, we consider the possibility of applying the theory of dynamic potentials to numerical simulation of initial-boundary-value problems. Thereby the problem is reduced to an integral equation for which an effective numerical algorithm is constructed, which allows calculating on concentrating grids with accuracy control. Further, certain nonlinear pseudoparabolic equations already considered in the second part of the monograph are taken into consideration. By using the method of lines, the problem is reduced either to the solution of a system of multidimensional implicit ordinary differential equations or to the solution of a system of differential-algebraic equations. For solving such problems, the complex Rosenbrock scheme convenient for large stiffness problems is used. Calculation on concentrating grids allows one to determine the time of blow-up of a solution with the accuracy of grid step degree. The results of this chapter were obtained by A. B. Al’shin.

0.3

Notation

Physical notation E is the vector of electric field; D is the vector of electric displacement field; P is the vector of medium polarization density; J is the vector of current density; H is the vector of auxiliary magnetic field (magnetic field intensity); B is the vector of magnetic induction (magnetic flux density); M is the vector of medium magnetization; n is the density of free electrons;

Section 0.3 Notation

15

Q is the density of current sources – free electrons; ij is the tensor of medium conductivity; ~ij is the tensor of medium electric susceptibility; ij is the tensor of medium magnetic susceptibility; ' is the potential of electric field; is the potential of magnetic field; ~0 is the scalar electric susceptibility; 0 is the scalar conductivity of the medium; n0 is the “quasistationary” free electron distribution; b Te is the temperature of free electrons; T0 is the temperature of phonons; M is the “quasistationary” magnetization; m is the “fast alternating” part of magnetization.

Mathematical notation A ˝ B is the Cartesian product of topological spaces A and B; N is the set of natural numbers; Z is the set of integer numbers; Zn is the set whose elements are ordered n-tuples of integer numbers of the form .z1 ; z2 ; : : : ; zn /, zm 2 Z, m D 1; n; ZnC is the set whose elements are ordered n-tuples of nonnegative integer numbers; RN is the N-dimensional Euclidean space; jj is the norm of the Euclidean space RN ; R1C is the set of nonnegative real numbers; ˛ is the multi-index ˛ D .˛1 ; : : : ; ˛n / ; ˛i 2 R1C ; j˛j denotes ˛1 C ˛2 C C ˛n ; Dx is the gradient operator with respect to the variable x 2 RN ; Dm x , where m D .m1 ; m2 ; : : : ; mN / is a multi-index, is the operator of the form @ m1 @ mN @ m2 m I Dx @x1 @x2 @xN

16


f .m/ .x/, where m D .m1 ; : : : ; mN / is a multi-index and x D .x1 ; : : : ; xN / 2 RN , is mN 1 f .m/ .x/ D Dm x1 DxN f .x/I rx is the gradient with respect to the variable x 2 RN ; @u is the derivative with respect to the direction of the external normal @n n to the “smooth” boundary @ 2 C 1;ı of a bounded domain RN ; @ t is the partial derivative with respect to t ; @kxi is the partial derivative of kth degree with respect to xi 2 R1 ; p u div.jrujp2 ru/ is the pseudo-Laplacian (p-Laplacian); is the Laplace operator; 2 is the biharmonic operator; 2 u @2x1 u C @2x2 u is the two-dimensional Laplace operator; ./1=2 is the square root of the operator , i.e., in the case where RN , Z 1 1=2 d k jkju.k/ O exp.i.k; x//; ./ u .2 /N RN and in the case of a bounded domain with “smooth” boundary @, C1 X 1=2 k .u; wk /wk ; ./1=2 u kD1

where k and wk are the kth eigenvalue and the kth eigenfunction of the first boundary-value problem for the Laplace operator; uO is the Fourier transform of function u; L.X; Y / is the set of linear continuous operators that act from X in Y ; A0u ./ is the Fréchet derivative of an operator A.u/ W X ! X , A0u ./ W X ! L.X; X /; X is the space adjoint to a Banach space X; RN is a domain in RN ; N is the closure of a domain ; @ is the boundary of a domain ; @ 2 C .m;ı/ denotes the fact that the boundary @ of a domain 2 RN can be represented, in a neighborhood of any point x 2 @, by local

17


coordinates i D î .1 ; 2 ; : : : ; N1 ; /;

i D 1; N 1;

where î are m times continuously differentiable functions of their .m/ variables and the functions î , m 2 ZN C , are Hölder functions with index ı 2 .0; 1; k kX is the norm of a Banach space X; C .p/ ./

is the set of all functions on that have p continuous derivatives in , where p 2 N;

.p/

Cb ./ is a Banach space with norm kuk

Cp

sup

p X

x2 mD1

jDm x uj I

C01 ./ is the set of compactly supported, infinitely differentiable functions; supp u is the support of a function u; Lip./ is the Banach space of Lipschitz-continuous functions with norm ju.x/ u.y/j I jx yj x;y2

kukLip sup juj C sup x2

C .0;ı/ ./ is the Banach space of Hölder functions with norm ju.x/ u.y/j ; jx yjı x;y2

kuk0;ı sup juj C sup x2

ı 2 .0; 1I

AC.0; T/ is the space of absolutely continuous functions; BV .0; T/ is the space of functions of finite variation; Lp ./ is the Banach space of measurable functions that are integrable with power p 2 Œ1; C1 in a domain ; the norm of this space is Z kukp dx jujp I

.; / is the scalar product in L2 (the same is sometimes used for the scalar product in RN ); h; i is the duality bracket between a reflexive Banach space X and its adjoint space X ;

18


k k is the norm of the Banach space X if k k is the norm of a Banach space X; H0m ./ is the Hilbert space of the measurable functions that have zero trace on the boundary of a domain and possess m (m 2 N) generalized derivatives from L2 ./; the scalar product has the form X D˛ u; D˛ v I .u; v/Hm D 0

j˛jm

Hm ./ is the Hilbert space dual to H0m ./; any element u of Hm ./ can be represented in the form X uD D˛ g˛ ; g˛ 2 L2 ./I j˛jm

Hs ./ is defined by means of real interpolation: Hs ./ D ŒHm ./; H0 ./ ; .1 /m D s; m 2 Z; 0 < < 1I W k;p ./ is the Banach space of measurable functions that are integrable with power p 2 R1C in the domain and for which k (k 2 N) generalized derivatives exist; the norm is as follows: kukk;p

k X

kDm ukp I

mD1 k;p

W0

./ is the Banach space consisting of elements of the Banach space W k;p ./ that have zero trace on the boundary of a domain ;

0

k;p W k;p ./ is the Banach space dual to the Banach space W0 ./, p 0 D 0 p=.p 1/; elements of W k;p ./ can be represented in the form X 0 uD D˛ g˛ ; g˛ 2 Lp ./I j˛jk

h; is is the duality bracket for the Hilbert spaces H0s ./ and Hs ./; k;p

h; ik;p is the duality bracket for the Banach spaces W0 0 W k;p ./; p 0 D p=.p 1/;

./ and

Lp .0; TI B/ is the Banach space of strongly measurable on an interval .0; T/, B-valued functions for which the Lebesgue integral Z T p dt kukB 0

19


is finite; the norm is as follows: Z kuk D

T

dt 0

kukp B

1=p I

BC.BI L.BI B // is the set of continuous and bounded operators on a Banach space B in the sense of the uniform topology of the space of linear operators L.BI B /; R.; T / D .I T /1 is the resolvent of an operator T ; C m .0; TI B/ is the space of m times continuously differentiable functions with values in a Banach space B; D./ is the space of compactly supported, infinitely differentiable functions; D 0 ./ is the space of generalized functions dual to D./.

Chapter 1

Nonlinear model equations of Sobolev type

In this chapter, mathematical models of quasi-stationary processes in continuous electromagnetic media are considered. It is shown that the initial system of Maxwell equations in the quasi-stationary approximation and phenomenological equations that relate the vectors of electric field E, electric displacement D, medium polarization P, current density J, magnetic field intensity H, magnetic field induction B, and medium magnetization M, can be reduced, for various nonstationary processes, to initial-value and initial-boundary-value problems for strongly nonlinear pseudoparabolic equations. Some of model equations obtained are wave equations and other are dissipative equations. A qualitative description of certain substantially nonlinear effects observed in experiments is given. The results of this chapter are obtained in [233, 234].

1.1

Mathematical models of quasi-stationary processes in crystalline semiconductors

From the mathematical standpoint, the main property of semiconductors is the fact that nonstationary processes observed in them are described by the system of quasistationary field equations, the continuity equation, and constitutive equations. For describing these processes, the explicit form is essential. These equations relate the electric field E and the electric displacement D and also the electric field E and the semiconductor current density J. In the general case, the system of equations in an appropriate Cartesian coordinate system has the following form [54]: div D D 4 e n; @n D div J C Q; @t

rot E D 0; Ji D

3 X

D D E C 4 P;

ij Ej ;

i; j D 1; 3;

(1.1) (1.2)

j D1

where P is the polarization vector; for certain models, the following phenomenological relation holds: Pi D

3 X

~ij Ej ;

i; j D 1; 3:

(1.3)

j D1

Here we, as usual, have divided electrons in the semiconductor lattice into two groups: free and bound charges [399]. Recall that the term “free charges” means charges free

21

Section 1.1 Models of quasi-stationary processes in semiconductors

to move over macroscopic distances. On the other side, the term “bound charges” means charges that cannot move by macroscopic distances; they only initiate the polarization of the semiconductor. Therefore, the value n in Eqs. (1.1) and (1.2) has the sense of the density of free charges and the value div P

(1.4)

means the density of bound charge. The value Q in Eq. (1.2) appropriately defines sources or sinks of the free electron current from or into impurity centers of the semiconductor lattice, depending on the fact whether the impurity centers are donors or acceptors [54]. Now we consider explicit model relations of the form (1.3) for some models and explicit model relations for bound charge density of the form (1.4). For the majority of models, Eq. (1.3) has the form P D ~0 E;

~0 2 R1C :

(1.5)

But in the presence of an external magnetic field B0 , the phenomenological equation (1.3) assumes a substantially anisotropic nature [256]: 1 0 ~11 ~12 ~13 ~ D @ ~21 ~22 ~23 A ; ~31 ~32 ~33 where ~ij depend on the magnetic field B0 . If the intensity of the electric field is comparable with intra-atomic fields, then the polarizability tensor is, generally speaking, a nonlinear function of the components of the intensity of the electric field [256] in the specified three-dimensional Euclidean space: ~ij D ıij ajEjp2 ;

~ij D ıij aj jEj jpj 2 ;

a; aj 2 R1C ;

i; j D 1; 3:

(1.6)

We note that the indices p and pj in (1.6) presume physical sense, being equal to 3 and 4. Consider another class of models for which phenomenological relations of the form (1.3) are not valid, or, more exactly, the polarization vector P should be represented as follows: P D d1 P1 C d 2 P2 ;

d12 C d22 > 0;

d1 ; d2 2 R1C [ ¹0º;

where for the vector P1 , a phenomenological relation of the form (1.3) holds: P1i D

3 X j D1

~1ij Ej ;

i; j D 1; 3;

(1.7)

22

Chapter 1 Nonlinear model equations of Sobolev type

together with any possible constitutive equations (1.5) and (1.6), while, for the vector P2 , we know only the model distribution of the appropriate part of bound charge density of the form (1.4): 2 div P2 :

(1.8)

Under the assumption that Eqs. (1.1) and (1.2) are considered in a cylindric domain G .0; T/, R3 , T > 0, where is a surface-simply-connected domain, the equation rot E D 0 is equivalent (in an appropriate smoothness class) to the existence of an electric field potential ' satisfying the equation E D r'. Now we can propose model distributions of the density of bound charges 2 in a self-consistent electric potential field '. We have the well-known Debye shielding effect, which implies that 2 .'/ has the form e' ; (1.9) 2 D 0 exp kTe where Te is the temperature of bound electrons at main centers of the lattice. On the other hand, “non-Boltzmann” distributions of the form e' (1.10) ; r > 0; q1 > 0; 2 D rj'jq1 exp kTe are known. Thus, under the assumption about strongly heated bound electrons, a good model distribution of charges 2 is 2 D r1 j'jq1 Œ1 C "1 ';

q1 0; r1 > 0;

"1 D

e : kTe

(1.11)

This distribution of bound charges in a self-consistent semiconductor field leads to quasi-elastic link of main centers of the lattice of the semiconductor and bound electrons. However, the following example is valid. As is known, there exist so-called semiconductor ferroelectrics (see, e.g., [152]), i.e., semiconductor crystals with ferroelectric properties. A ferroelectric is a medium whose properties depend on temperature and there exists a critical temperature Tc at which a so-called second-order phase transition occurs. There is an substantial difference between medium properties at T < Tc (pyroelectric phase) and at Tc < T (nonpyroelectric phase). In the book [256, p. 128], a graph of Pz against Ez at T < Tc is given; this graph shows that the derivative is negative on a certain interval: d Pz < 0; d Ez which corresponds to metastable modes; these modes are very interesting for analysis.


23

Further, we will consider cubic ferroelectric semiconductor. We define the electric displacement of the ferroelectric semiconductor as D D E C 4 P;

P D P.E/;

and use the fact that div P 2 has the sense of bound charges. If in the standard case d Pi > 0; Ei D .ei ; E/; Pi D .ei ; P/; i D 1; 3; d Ei and an appropriate model distribution of charges has the form (1.9)–(1.11), then the appropriate model distribution of charges in our “nonclassical” case is d Pi < 0; Ei D .ei ; E/; Pi D .ei ; P/; d Ei and in the linear approximation we can write 2 D n0 Œ1 r2 ';

r2 > 0:

i D 1; 3;

(1.12)

The conditions of negativity of the derivatives of the components of the polarization vector with respect to appropriate components of the electric field can be described in the linear approximation as a negative addition to the electric polarizability. As is known (see [256]), the situation where the electric polarizability is negative is excluded from physical consideration. However, in quasi-stationary cases, disbalance metastable modes are possible, as in the case of a ferroelectric semiconductor where the electric polarizability ~ is negative locally in time, and hence the dielectric permittivity 1 C 4 ~ is less than one. However, we note that just a negative addition to the electric polarizability appears, i.e., it can be proved that the total electric polarizability is always positive. The study of initial-value and initial-boundary-value problems for evolutionary nonlinear and linear partial differential equations appearing in such physical situations is very difficult. In our research, such cases are reduced to so-called singular Sobolev-type equations, i.e., to equations with noninvertible elliptic operators under time derivative. Such problems appear in the theory of viscoelastic liquids. Specifically, they are considered in works of Sviridyuk (see, e.g., [395]). Now we consider model relations between the electric field density J and intensity E according to formula (1.2). First, we note that, under an external magnetic field B0 , the conductivity tensor in the three-dimensional Euclidean space in the approximation of a weak magnetic field has the form 1 0 a2 B0 0 a1 A; 0 (1.13) D @ a2 B0 a1 2 0 0 a1 C a3 B0 where al 0, l D 1; 3, are certain constants.

24


Second, in a strongly “overheated” semiconductor, the conductivity tensor is a nonlinear function of the intensity of the electric field [29]: ij D ıij 0 .1 C ˇjEj2 /;

0 > 0;

(1.14)

and ˇ can be positive or negative. In the case where ˇ < 0, the breakdown of a semiconductor is observed experimentally [54]; this can be proved theoretically in a certain generalized smoothness class by using the method of energy estimates developed by Korpusov (see, e.g., [226–228]). Moreover, the conductivity tensor can have the form ij D ıij 0 jEjp ;

0 > 0; p > 0

(1.15)

(see [54]). Of course, taking into account an external magnetic field, we see that Eqs. (1.14) and (1.15) become more complicated because of anisotropic character of the conductivity tensor (1.13). As in the case of the tensor of electric polarizability, the phenomenological relation (1.2) requires some modification if a constant external electric field E0 presents. In this case, the phenomenological equation has the form J D J1 C bJ2 ; J1 D e n0 .'/E0 ;

b 2 ¹0º [ ¹1º; J2i D

3 X

2ij Ej ;

(1.16) (1.17)

j D1

where is the mobility of free electrons, n0 .'/ is a certain (generally speaking, nonlinear) function of the self-consistent field potential ' that has the sense of quasistationary distribution of free electrons [272] and can be described by the same equations as the density of bound electron, i.e., (1.9)–(1.12), but, in the general case, with other parameters and, specifically, with the other index q2 0. In the case of strongly overheated free electrons, we have from (1.10) 1 n0 .'/ D r3 j'jq2 1 C "2 ' C "22 ' 2 ; q2 0; r3 > 0; "2 D e=.k TO e /; (1.18) 2 where TO e is the temperature of free electrons and "2 is a small parameter. Finally, the function Q D Q.'/ in (1.2) depends on the density of sources or sinks of free electrons and has the form similar to the distributions of free and bound electrons of the lattice main centers of the semiconductor. We use the following model distribution (see, specifically, [160]): Q.'/ D j'jq3 ';

q3 0;

(1.19)

where < 0 for donor impurity centers and > 0 for acceptor impurity centers, respectively. Obviously, D 0 holds in the absence of impurity centers.


25

Now we consider possible boundary conditions for Eqs. (1.1) and (1.2). Introduce the normal unit vector n.x/ to the interface @ of two media, i.e., the vector directed from the domain into the domain R3 n . We denote by @˙ the two sides of the oriented surface @; then the boundary conditions for the electric displacement and the electric field take the form (see, e.g., [399]) ŒE1 ; nj@C ŒE2 ; nj@ D 0; .D1 ; n/j@C .D2 ; n/j@ D 4 j@ ;

(1.20)

where is the surface density of free charges (the boundary @ is sufficiently smooth). If we substitute expressions (1.3) in Eqs. (1.20) and take into account the fact that the field is potential, i.e., E D r', then the boundary conditions (1.20) take the form

.Iıij C 4 ~1ij /

@'1 ˇˇ ni @C @xj

.r'1 ; /j@C D .r'2 ; /j@ ; (1.21) @'2 ˇˇ .Iıij C 4 ~2ij / ni @ D 4 e .x; t /j@ ; (1.22) @xj

where .x/ is an arbitrary vector in the tangent plane at the point x 2 @, e .x; t / D .x; t / C 0 .x; t / is the total surface density of free and bound charges, 0 .x; t / D D1i0 n0i j@C D2i 0 n0i j@ ; and ~ij is the tensor of electric polarizability. Using the potential of the electric field introduced above, we rewrite the boundary conditions (1.21)–(1.22) on the interface between the semiconductor and the conductor in the form

1 4

'.r; t /jx2@ D C.t /;

(1.23)

Lx '.x; t /jx2@ d.@/ D Q.t /I

(1.24)

Z @

here Q.t / is the total charge on the semiconductor surface, the operator Lx '.x; t /jx2@

3;3 X @' @' C 4 ~ik cos.nx ; ei /; @nx @xk i;kD1;1

where the right-hand side of (1.23) is either an explicitly defined function or an unknown function of time to be defined. In the latter case, for the problem being well stated, it suffices to define the total charge on semiconductor surface, i.e., the nonlocal boundary condition (1.24). The boundary condition in the interface semiconductorconductor of the form .J; n/j@ D 0

26


is interesting. Under the relation (1.2) between the current density J and the electric field intensity E will be of the form Lx '.x; t /jx2@

3;3 X i;kD1;1

ˇ @' ˇˇ ik cos.nx ; ei / D 0: @xk ˇx2@

(1.25)

Now we consider physical conditions in physics of semiconductors that lead to Cauchy problems for evolutionary equations. Note that Cauchy problems appear for describing of experiments with fast running laser raying of semiconductors. The laser raying time is incomparable less than the relaxation time of nonstationary free electron density distribution that appears owing to the raying and the appropriate selfconsistent electric field. In addition to Cauchy problems and arbitrary boundary-value problems, periodic boundary problem are often used due to the periodicity of the structure of crystalline semiconductors. Now we consider model nonlinear boundary conditions. For simplicity, we assume that in a Cartesian coordinate system ¹Ox1 ; Ox2 ; Ox3 º, the plane x3 D 0 is the interface between an ideal conductor (that fills the half-space x3 < 0) and a semiconductor (x3 > 0). We choose the direction of the external normal n to the plane x3 D 0 relative to the domain x3 < 0. The words “ideal conductor” means a medium where the intensity of an electric field vanishes for any external conditions. As is known, in the stationary case, conductors satisfy this condition. On the other hand, in the quasi-stationary case, which we will consider, conductors approximately satisfy the condition of the absence of an electric field inside the conductor [399]. We consider the case where the field E is potential in the domain x3 > 0 and has the potential '. Assume that on the plane x3 D 0, we have a quasi-stationary free charge distribution with the evolution governed by the following dynamic system of boundary conditions [399]: .D; n/ D 4 e!;

x3 D 0;

@! D .J; n/ C j'jq3 '; @t

(1.26) x3 D 0;

(1.27)

where ! is the areal density of free electrons. For boundary values of the vectors of electric displacement D and current density J, we assume that they satisfy all phenomenological relations mentioned in this paragraph in the limit sense near the plane x3 D 0. In the next section, we consider model initial-boundary value problems for elliptic equations that describe some class of physical phenomena in crystalline semiconductors, i.e., for equations with pseudoparabolic boundary conditions that are consequences of dynamic boundary conditions (1.26) and (1.27).

27

Section 1.2 Model pseudoparabolic equations

1.2

Model pseudoparabolic equations

In this section, we consider a wide spectrum of mathematical models that can be reduced to evolutionary pseudoparabolic equations. Therefore, certain models are related to analyzing nonstationary processes in semiconductors in external electric and magnetic fields. In the case of external electric fields, electric potentials are solutions of pseudoparabolic wave equations that coincide with the Oskolkov–Benjamin– Bona–Mahony-type wave equations or are similar to them. On the other hand, model equations in the case of external magnetic fields are anisotropic, but in the absence of external fields they are isotropic. For the model equations considered, the blow-up of solutions is of great theoretical and practical interest because it means – from the physical standpoint – an electric breakdown in a semiconductor. In general, the equations considered here play a more significant role than the model equations in physics of semiconductor since they also appear in other branches of mathematical physics. On the other hand, the results obtained can be physically interpreted by problems of physics of semiconductors. First, we consider the case of linear (elliptic) operators with time derivative. We study system (1.1), (1.2) with the phenomenological equation (1.5) or (1.7), (1.8) and Eqs. (1.11) or (1.12) with q1 D 0 and the additional phenomenological equations (1.13)–(1.19). We consider the simplest class of pseudoparabolic equations.

1.2.1 Nonlinear waves of Rossby type or drift modes in plasma and appropriate dissipative equations There are many works devoted to analyzing planetary waves or Rossby waves. Such waves have small frequency and long-wave character. One can observe them in the atmosphere and great oceans. Similar equations describe so-called drift modes in plasma that caused by various factors, for example, an inhomogeneity of electron distribution in plasma or toroidal magnetic trap. For these equations, we consider diffraction problems, excitation problems, and questions on the existence of solitons (see [25, 164, 165, 222, 346, 401, 437]). In the linear approximation of ˇ-plane, the two-dimensional Rossby waves equation has the form (see [164]) @ 2 @t

C ˇ.y/

@ D 0; @x

2

@2 @2 C ; @x 2 @y 2

where the axes Ox and Oy are directed to the east and to the north, respectively, and ˇ D ˇ.y/ is the Coriolis parameter. One of the existing nonlinear generalizations of the two-dimensional equation of Rossby waves is the two-dimensional Petviashvili equation (see, e.g., [401]) @ .2 @t

/C

@ C @x

@ C J. ; 2 / D 0; @x

J.a; b/ D ax by ay bx :

28


The equation given is adjoint to the one-dimensional Camassa–Holm equation [67] @u @3 u @u @2 u @ @2 u u 3u C u D 0: C 2 @t @x 2 @x @x @x 2 @x 3 We assume that the temperature of bound electrons in a crystalline semiconductor is sufficiently high; we can also assume that their density is constant. In this case, we use the phenomenological equation (1.5). On the other hand, assume that the temperature of free electrons is incomparable with the temperature of bound electrons; for quasistationary free electron distribution we take (1.18) with q2 D 0. For the current density equation (1.16), we assume that J2 D 0 E; moreover, let the direction of the axis Ox be the same as the direction of the constant external field E0 . Then, taking into account system (1.1), (1.2), we obtain a nonlinear Rossby-type equation

@' @' @' C a1 C a2 ' C a3 ' D 0; @t @x @x

@2 @2 @2 C C ; @x 2 @y 2 @z 2

(1.28)

where a1 D 4 e 1 1 "2 jE0 j, a2 D 4 e 1 21 "22 jE0 j, a3 D 4 e 1 b0 , and D 1 C 4 ~0 . Note that the inequality a3 0 is implied by (1.16). If we take into account the presence of sources (sinks) (1.19) and the nonlinear dependence of the conductibility tensor on electric field (1.14), we obtain the following equation:

@' @' @' C a1 C a2 ' C a3 ' C a4 div.jr'j2 r'/ C j'jq3 ' D 0; @t @x @x

(1.29)

where q3 0, a4 D 1 b0 ˇ, D 1 C 4 ~0 , and D 4 e 1 . We note that, owing to (1.14) and (1.19), we have a4 ; 2 R1 . We can mark an important property of the problems considered. In the absence of external electric field (that is, quite naturally in a real physical situation), Eq. (1.29) implies the dissipative equation @' (1.30) C a3 ' C a4 div.jr'j2 r'/ C j'jq3 ' D 0; q3 0; @t where the constants a3 , a4 , and are defined as for Eqs. (1.28), (1.29), and (1.19) and a3 0, a4 2 R1 , and 2 R1 . Finally, consider a model that leads to dissipative equations similar to (1.30). Assume that the following relation between the current density of free charges J and the electric field E holds: 1 2 2 J D en0 .'/E; n0 .'/ D r3 1 C "2 ' C "2 ' ; n0 .'/ D r3 j'jq2 : (1.31) 2

Then from system (1.1) and (1.2) and phenomenological relations (1.5) and (1.19), taking into account (1.31), we obtain 1 1 @' C a5 ' C "2 ' 2 C "22 ' 3 C j'jq3 ' D 0; q3 0; (1.32) @t 2 6

29


or

@' 1 C a5 .j'jq2 '/ C j'jq3 ' D 0; @t q2 C 1

q2 ; q3 0;

(1.33)

where a5 D 4 e 1 r3 e and D 1 C 4 ~0 . Note that entirely incomparable by their scales wave processes in atmosphere, plasma and semiconductors are described by the same pseudoparabolic type equations (1.28), (1.29), (1.30), (1.32), (1.33). Note that rather different wave processes in atmosphere, plasma, and semiconductors that, at first glance, are incomparable by their scales, are described by the same pseudoparabolic equations (1.28), (1.29), (1.30), (1.32), and (1.33).

1.2.2 Nonlinear waves of Oskolkov–Benjamin–Bona–Mahony type In [37, 312, 313], the following model, nonlinear, one-dimensional, pseudoparabolic equation was obtained: @u @u @u @3 u Cu 2 D 0: @t @x @x @x @t It describes nonlinear surface waves that spread along the axis Ox. If the viscosity presents, the waves are described by the Oskolkov–Benjamin–Bona–Mahony– Burgers equation @u @2 u @3 u @u @u Cu 2 2 D 0: @t @x @x @x @x @t These equations together with the Korteweg–de Vries and Burgers equations have become classical test equations for group-theoretic and inverse-scattering methods for divergent nonlinear evolutionary equations [139, 180, 437]. Multidimensional generalizations of Oskolkov–Benjamin–Bona–Mahony-type equations are considered, for example, in [37, 39, 78, 188, 193, 209, 217, 271, 294, 296, 297, 317, 441]. However, the question on the applicability of these results to some multidimensional Oskolkov–Benjamin–Bona–Mahony-type model equations remains open. In our models with finite temperature of bound electrons we can use distribution (1.11) with q2 D 0, which corresponds to the first two summands in the decomposition of the Boltzmann distribution (1.9) by powers of e'=.kTe /. We assume that the inequality je'=.kTe /j < 1 holds; this is valid under the condition of the boundedness of the electric potential ' for sufficiently high temperature Te of bound electrons. In this case, we have an additional term in (1.28)–(1.30), (1.32), and (1.33): r1 "1 1

@' : @t

30


Then the equations become @ @' @' .' a'/ C a1 C a2 ' C a3 ' D 0; @t @x @x @ @' @' .' a'/ C a1 C a2 ' C a3 ' @t @x @x Ca4 div.jr'j2 r'/ C j'jq3 ' D 0;

(1.34)

(1.35)

@ (1.36) .' a'/ C a3 ' C a4 div.jr'j2 r'/ C j'jq3 ' D 0; @t @ 1 1 .' a'/ C a5 ' C "2 ' 2 C "22 ' 3 C j'jq3 ' D 0; (1.37) @t 2 6 1 @ (1.38) .' a'/ C a5 .j'jq2 '/ C j'jq3 ' D 0; @t q2 C 1 q2 ; q3 0; where @2 @2 @2 C C ; @x 2 @y 2 @z 2 e e a D r1 1 "1 ; "1 D ; "2 D ; kTe k TOe

a1 D 4 e1 1 "2 jE0 j; a3 D 4 e 1 b0 ;

a2 D 8 e1 1 "22 jE0 j;

a4 D 4 e 1 b0 ˇ;

D 1 C 4 ~0 ;

a5 D 4 e 2 1 r3 ;

D 4 e 1 I

here q2 ; q3 0, a1 ; a2 ; a3 0, a; a5 > 0, a4 ; 2 R1 . We note that, subject to an external constant electric field, Eqs. (1.36)–(1.38) contain additional terms with convective nonlinearities of the form '

@' : @x1

In [20, 160], a theoretical description of an interesting phenomenon that occurs in transitional processes in certain crystalline semiconductors is proposed. This effect consists of the appearing of layers of volumetric charge with alternating sign in a semiconductor activated by a laser radiation and then placed in the electric field of a constant capacitor. In the course of time, the thickness of layers decreases and their number and the absolute value of the density of volumetric charge in each layer increases (see [20]). This phenomenon is called the stratification effect of volumetric charges in semiconductors.

31


The initial system of equations that describe transitional processes in semiconductor has the form (see [160]) 4 @ n0 n ; D e.n0 n/; De ; (1.39) " @t where is the volumetric charge density in the crystal, is the density of the volumetric charge bound on semiconductor impurity centers, n0 is the balanced concentration of electrons, n is the concentration of free electron related to the electric potential '.x; t / by the Boltzmann law: e' n D n0 exp ; (1.40) kTe 3 ' D

is the characteristic lifetime of free electrons, " is the dielectric permittivity of the semiconductor, Te the temperature of free electrons, and x 2 is a bounded domain in R3 with a piecewise smooth boundary @ 2 A.1;/ . Using general methods developed in [164], we can reduce system (1.39), (1.40) with the initial condition .x; t /j tD0 D 0 in the function class '.x; t / 2 C.Œ0; C1/I C .2/ .// to the nonlinear integro-differential equation Z 4 e n0 1 t 3 '.x; t / C ds f .'/.x; s/ D 0; (1.41) f .'/.x; t / C " 0 where f .'/ D 1 exp.e'=kTe /. In the case where the electric potential is given on the boundary of the domain , we arrive at the following boundary condition: 'jx2@ D a.x; t /;

x 2 @;

t 0;

(1.42)

where a.x; t / 2 C .0/ .@ Œ0; C1// is a given function. Under the additional condition jea.x; t /j < 1; kTe Eq. (1.42) can be simplified. Consider the first two summands in the expansion of the exponent exp.e'=kTe / by powers of e'=.kTe /; we obtain the linear equation Z 1 1 t 3 '.x; t / 2 '.x; t / C ds '.x; s/ D 0; (1.43) 0 rd p where rd D "kTe =.4 e 2 n0 / is the Debye radius. Since the initial and boundary conditions of Eq. (1.43) in the considered function class '.x; t / must be consistent, we see that the initial distribution of electric field potential in semiconductor '.x; 0/ D '0 .x/ satisfies the following boundary-value problem: 3 '0 .x/ rd2 '0 .x/ D 0;

x 2 ;

'0 jx2@ D a.x; 0/; x 2 @:

(1.44)

32


Moreover, if we assume that '.x; t / 2 C .1/ ..0; C1/I C .2/ .//, then the integrodifferential equation (1.43) is equivalent in this expanded smoothness class to a partial differential equation of order 3 with an initial condition of composite type, which satisfies the boundary-value problem (1.44): 1 1 @ 3 '.x; t / 2 '.x; t / 2 '.x; t / D 0: @t rd rd

(1.45)

Thus, we arrive at the following initial-boundary-value problem. Problem D. Find a continuous function '.x; t / in Œ0; C1/ that belongs to the class C.Œ0; C1/I C .2/ .// \ C .1/ ..0; C1/I C .2/ .// and satisfies, in the classical sense, Eq. (1.41), the boundary condition (1.42), and the initial condition (1.44). Note that the Oskolkov–Benjamin–Bona–Mahony–Burgers wave equation (1.34) and the dissipative equation (1.45) contain the common linear operator under the time derivative. The existence and uniqueness of solutions of this initial-boundary-value problem can be proved by methods developed [165]. Now we consider quasi-stationary processes in two-component liquid semiconductors. The standard model for this case is the ambipolar diffusion model [54, 55, 256, 272]: eNe @Ne r' ; D De div rNe @t Te eNi @Ni r' ; D Di div rNi C @t Ti

(1.46)

' D 4 e.Ni Ne /: In the approximation of quasi-neutral plasma, from (1.46) we obtain the following simplified model system: Ne D N0e exp .eˇu/; @Ni D eDi divŒNe ru; @t 4 e .Ni Ne /; u D Ti where ' D Ti u and ˇ D Ti =Te .

(1.47)

33


Now we assume that ˇjeuj < 1. We consider the first two summands in the expansion of the function exp .eˇu/ in the Taylor series; from (1.47) we have Ne D N0e C N0e eˇu; @Ni D eDi N0e eu C Di N0e e 2 ˇ divŒuru; @t 4 e .Ni N0e N0e eˇu/: u D Ti

(1.48)

Integrating the second equation of system (1.48) by t and substituting the resulting expression of Ni in the third equation, we obtain the integro-differential equation Z t Z t d u.r; / C ˛3 d divŒuru.r; / D 0; (1.49) u ˛1 u C ˛2 0

0

2 , ˛ D D =r 2 , ˛ D D eˇ=r 2 , and r 2 D 4 e 2 N =T . where ˛1 D ˇ=rD 2 i D 3 i 0e i D D We note that the natural boundary condition defines the electric potential on the domain boundary :

u.x; t /jx2@ D a.x; t /;

x 2 @:

(1.50)

Since the initial and boundary conditions for Eq. (1.49) in the function class C .1/ .Œ0; C1/I C .2/ ./ \ C.// must be consistent, we see that the initial distribution u.x; 0/ D u0 .x/ of the electric potential in the semiconductor satisfies the following boundary-value problem: u0 .x/ ˛1 u0 .x/ D 0;

x 2 ;

u0 jx2@ D a.x; 0/; x 2 @:

(1.51)

In this case, the integro-differential equation (1.49) is equivalent to the partial differential equation of third order of composite type with the initial condition that satisfies the boundary-value problem (1.51): @ Œu ˛1 u C ˛2 u.r; t / C ˛3 divŒuru.r; t / D 0: @t

(1.52)

Thus, we arrive at the following formulation of the initial-boundary-value problem. Problem A. Find a continuous function '.x; t /, .x; t / 2 Œ0; C1/, of the class C.Œ0; C1/I C .2/ .// \ C .1/ ..0; C1/I C .2/ .// and satisfies in the classical sense Eq. (1.52), the boundary condition (1.50), and the initial condition (1.51). Equations (1.52) and (1.34) are linked by the common linear operator on time derivative.

34


1.2.3 Models of anisotropic semiconductors Anisotropy of a conductive medium can be caused by various factors. As in [399], we assume that electrodynamics charges of a continuous medium are conditionally divided into free and bound charges. Free charges are macroscopic current carriers in the medium and bound charges determine medium polarization. Then the anisotropy of the conductive medium can be caused either by anisotropy of the current of external charges in a magnetic field (Hall’s effect) or by anisotropic distribution of bound charges – anisotropy of medium polarizability (for example, ferroelectrics). As we will see below, mathematical models of quasi-stationary processes in anisotropic conductive media in both cases lead to problems for nonclassical pseudoparabolic differential equations of third order. The presence of an external magnetic field leads to peculiar quasi-stationary processes even in electrically isotropic media. In particular, we consider a model problem on dissipative processes in a crystalline semiconductor in an external magnetic field. Let us choose a right-hand Cartesian coordinate system ¹Ox1 ; Ox2 ; Ox3 º with the unit vectors ¹ei º3iD1 , where the axis Ox3 is directed along the magnetic field B0 . In the quasi-stationary approximation, the system of equations that describe dissipative processes in a unipolar semiconductor in a constant magnetic field has the form [399] div D D 4 e n;

rot E D 0;

D D E C 4 P;

@n D div J C j'jq '; @t Ji D

3 X

ik .B0 /Ek ;

kD1

Pi D

3 X

(1.53) B0 D B0 e3 ;

~ik .B0 /Ek ;

i D 1; 3;

kD1

where r D .x1 ; x2 ; x3 /, D.r; t / and E.r; t / are the vectors of electric displacement and electric field, respectively, P.r; t / is the vector of medium polarization, J.r; t / is the current density of free charges, 2 R1 , 1 ~11 ~12 ~13 ~ D @ ~21 ~22 ~23 A ; ~31 ~32 ~33 0

1 11 12 13 D @ 21 22 23 A 31 32 33 0

are the coordinates of the electric polarizability and medium conductivity tensors in the basis considered; they depend on the magnetic field B0 D B0 e3 . Under the assumption that the domain R3 is surface-simply-connected, we introduce the Coulomb potential '.r; t /: E.r; t / D r'.r; t /:

35


From system (1.53) we obtain the following nonlinear equation of anisotropic dissipation in crystalline semiconductors: @ @2 ' @2 '.r; t / C 4 eik ' C 4 ~ik 4 e j'jq ' D 0; (1.54) @t @xi @xk @xi @xk where the summation with respect to repeating indices is assumed. The boundary conditions on the interface “semiconductor–conductor” have the form (1.21)–(1.22) with the operator of directional derivative 3;3 X @ @ C ~ik cos.n; ek / : Lx @nx @xi i;kD1;1

Note that for many isotropic semiconductor crystals, the electric polarizability tensor has the coordinates ~ik D ~ıik , where ~ > 0 and ıik is the Kronecker delta and is independent of the magnetic field and the conductivity tensor in the approximation of weak field has the coordinates (see [54]) 1 0 0 a1 a2 B0 A; 0 D @ a2 B0 a1 2 0 0 a1 C a 3 B 0 where al 0 (l D 1; 3) are certain constants. In this case, (1.54) takes the form @ @2 '.r; t / C j'jq ' D 0; 3 ' C ˇ1 2 '.r; t / C ˇ2 @t @x32

(1.55)

and the boundary condition .J; n/ D 0 on the interface “semiconductor–conductor” takes the form (1.25) with the operator of directional derivative Lx Nx C Tx ; @ @ C a3 B02 cos.nx ; e3 / ; @nx @x3 @ @ Tx a2 B0 cos.nx ; e2 / cos.nx ; e1 / ; @x1 @x2

N tx a1

where ˇ1 D a1 =, ˇ2 D .a1 C a3 B02 /=, D 1 C 4 ~, and D 4 e=. Thus, the model problem on quasi-stationary processes in semiconductors in the presence of external magnetic fields and the Hall effect is reduced to Eq. (1.55). Now we consider quasi-stationary processes in monoaxial ferroelectric semiconductors. On one hand, these media have a marked direction of spontaneous polarization and the energy needed for changing this direction is comparatively small (see [256]); on the other hand, it is characterized by all properties of crystalline

36


semiconductors, for example, the Debye screening effect with characteristic radius p rd D T=4 e 2 n0 . We choose a Cartesian coordinate system ¹Ox1 ; Ox2 ; Ox3 º with the axis Ox3 directed along the ferroelectric polar axis. In the quasi-stationary approximation, small ferroelectric-semiconductor polarization fluctuations near the equilibrium state are described by the system (see [152]) div D D 4 e n;

rot E D 0;

D D E C 4 Pe3 ;

@P C ˛1 P D ˛2 E3 ; @t

(1.56)

@n D div J C j'jq '; J D 0 E; @t where ˛1 0, ˛2 > 0, and E3 D .E; e3 /. From the fourth equation of system (1.56) we obtain Z t @'.r; / P D P0 .x/ exp .˛1 t / ˛2 d exp .˛1 .t // : @x3 0 From the latter equality and the first three equations of system (1.56) we obtain the integro-differential equation of the anisotropic relaxation of a ferroelectric semiconductor: @2 '.r; / @ 2 ' C ŒI C 4 ˛2 K C ˛3 ' C j'jq ' @t @x32 D 4 where ˛3 D 4 e0 , D 4 e, 2

@2 @2 C ; @x12 @x22

K u

Z 0

t

@P0 .x/ exp .˛1 t /; (1.57) @x3

d exp .˛1 .t //u. /:

(1.58)

Typical boundary conditions on the interface “conductor–semiconductor” have the form (1.24) with the boundary operator Lx

@ @ I C 4 ˛2 cos.nx ; e3 / K; @nx @x3

where the operator K is defined by formula (2.31). Note that Eq. (1.57) differs from all equations obtained in this monograph because it takes into account the time dispersion of the dielectric permittivity tensor; the paper [228] is devoted to such equations. We also note that in works of Yu. D. Pletner, the operator method for analyzing linear Sobolev-type equations of arbitrary time order was proposed. By using this method, Pletner constructed solutions of initial- and initial-boundary-value problems for Sobolev-type equations and stated the relationship between Sobolev-type

37


equations and certain elliptic equations. He proved that the Sobolev-type equations can be represented as elliptic equations regularly disturbed by Volterra convolution operators. Then Pletner analyzed the question on properties of solutions of initialboundary-value problems for Sobolev-type equations on space variables with zero initial conditions and defined boundary conditions. In [8, 227], this method was applied to initial-boundary-value problems for certain Sobolev-type equations.

1.2.4 Nonlinear singular equations of Sobolev type As is known, for first- and second-order linear evolutionary equations solved relative to the highest time derivative, the semigroup method developed in works [198] is effective. On the other side, singular Sobolev-type equations are partial differential equations unsolved (and unsolvable) relative to the highest time derivative, i.e., contain an irreversible operator (in applications – elliptic operator) at the highest time derivative. In this case, classical semigroup methods are inapplicable. However, G. A. Sviridyuk and V. E. Fyodorov [398] considered operator semigroups with nonzero kernels in appropriate Banach spaces. For this, they introduced the notions of relatively limited, relatively sectorial, and relatively radial operators, that generalize definitions of the classical operator theory. With aid of the developed method of semigroups with kernels, the authors proved theorems on the solvability of linear, singular, Sobolev-type equations and applied them to analyzing nonlinear Sobolev equations; we also mention the works [144–149, 388–392]. Model singular Sobolev-type equations can be formally derived from (1.28)–(1.30) and (1.32) and (1.33) by adding the summand r2 n0 1

@' I @t

then they take the form @' @ @' .' C a'/ C a1 C a2 ' C a3 ' D 0; @t @x @x @ @' @' .' C a'/ C a1 C a2 ' C a3 ' @t @x @x Ca4 div.jr'j2 r'/ C j'jq3 ' D 0;

(1.59)

(1.60)

@ .' C a'/ C a3 ' C a4 div.jr'j2 r'/ C j'jq3 ' D 0; (1.61) @t 1 1 @ (1.62) .' C a'/ C a5 ' C "2 ' 2 C "22 ' 3 C j'jq3 ' D 0; @t 2 6 1 @ (1.63) .' C a'/ C a5 .j'jq2 '/ C j'jq3 ' D 0; @t q2 C 1 q2 ; q3 0;

38


where @2 @2 @2 C C ; @x 2 @y 2 @z 2 e e a D r2 n0 1 ; "1 D ; "2 D ; kTe k TOe

a1 D 4 e1 1 "2 jE0 j; a3 D 4 e 1 b0 ;

a2 D 8 e1 1 "22 jE0 j;

a4 D 4 e 1 b0 ˇ;

D 1 C 4 ~0 ;

a5 D 4 e 2 1 r3 ;

D 4 e 1 ;

q2 ; q3 0, a1 ; a2 ; a3 0, a; a5 > 0, and a4 ; 2 R1 . Thus, Eqs. (1.59)–(1.63) are basic model Sobolev-type equations that describe metastable states in semiconductors in the presence of negative differential polarizability.

1.2.5 Pseudoparabolic equations with a nonlinear operator on time derivative As we have already noted above, in the case where the value of the electric field is comparable with intra-atomic fields, we must take into account the dependence of the electric polarizability on the electric field of the form (1.6). Here we assume that some part of bound charges has a power distribution in a self-consistent field and the distribution vanishes as the electric potential ' vanishes: 2 D r4 j'jq1 ': In this case, the summand @ .a div.jr'jp2 r'/ r4 j'jq1 '/ @t is added to (1.28)–(1.30), (1.32), and (1.33) and the equations take the following form: @ .' C a div.jr'jp2 r'/ r4 j'jq1 '/ @t @' @' Ca1 C a2 ' C a3 ' D 0; @x @x @' @' @ .' C a div.jr'jp2 r'/ r4 j'jq1 '/ C a1 C a2 ' @t @x @x 2 q3 Ca3 ' C a4 div.jr'j r'/ C j'j ' D 0; @ .' C a div.jr'jp2 r'/ r4 j'jq1 '/ @t Ca3 ' C a4 div.jr'j2 r'/ C j'jq3 ' D 0;

(1.64)

(1.65)

(1.66)

39


@ .' C a div.jr'jp2 r'/ r4 j'jq1 '/ @t 1 2 21 3 Ca5 ' C "2 ' C "2 ' C j'jq3 ' D 0; 2 6 @ .' C a div.jr'jp2 r'/ r4 j'jq1 '/ @t 1 .j'jq2 '/ C j'jq3 ' D 0; Ca5 q2 C 1

(1.67)

(1.68)

where @2 @2 @2 C C ; @x 2 @y 2 @z 2 e e q1 ; q2 ; q3 0; "1 D ; "2 D ; kTe k TOe

a1 D 4 e1 "2 jE0 j; a3 D 4 eb0 ;

a4 D 4 eb0 ˇ;

a2 D 8 e1 "22 jE0 j; a5 D 4 e 2 r3 ;

D 4 e;

with r4 ; a1 ; a2 ; a3 ; a 0, a5 > 0, and a4 ; 2 R1 . Thus, the model equations (1.64)–(1.68) describe either the nonlinear dependence of the electric polarizability on the intensity of the electric field or the nonlinear dependence of div P on the potential of the electric field.

1.2.6 Nonlinear nonlocal equations In [54, 55], the authors consider the electron mobility as a nonlocal function of the space variables depending on the intensity E of the electric field. On the other side, the first nonlinear correction to the electron mobility is quadratic with respect to the field intensity. Therefore, it is reasonable to assume that the first nonlinear correction to the electron mobility in the case where R3 has the following form: Z Z .r; t / D 0 C c1 d r1 d r2 K.r; r1 ; r2 ; t /E.r1 r2 ; t /E.r2 ; t /; (1.69) R3

R3

R1

and 0 0. An important special case occurs when the initial distriwhere c1 2 bution of the potential of the electric field is quasi-homogeneous, i.e., the mobility is independent of r 2 R3 . In this case, from (1.69) we obtain that K K.r1 r2 ; t /. Moreover, in sufficiently strong fields E, the kernel K can be considered equal to zero if jr1 r2 j > T=.eE0 / and it has a sharp peak r1 D r2 (see [55]). Therefore, in the first approximation, we can assume that K.r1 r2 / D ı.r1 r2 / in the sense of D 0 .R3 /. The following relation succeeds these assumptions and Eq. (1.69): Z d r jE.r; t /j2 : (1.70) .t / D 0 C c1 R3

40


Because of the known relationship between the mobility of free electrons and the electric conductivity D e n0 , from (1.70) we obtain Z d r jE.r; t /j2 ; 0 D e n0 0 ; c2 D c1 e n0 : (1.71) .t / D 0 C c2 R3

Now, using system (1.1)–(1.2) and phenomenological equations (1.6)–(1.8), subject to (1.71) we obtain @ ' C a div.jr'jp2 r'/ r4 j'jq1 ' C a3 ' C a4 kr'k22 ' D 0; (1.72) @t where a; r4 ; a3 ; q1 0, a4 2 R1 , and Z 2 kr'k2

R3

d r jr'.r; t /j2 :

Note that in the case of a bounded domain R3 , we can formally consider the equation (1.72), where Z 2 d r jr'.r; t /j2 : kr'k2

In the case of bounded semiconductors, there are other reasons that lead to nonlinear, nonlocal, pseudoparabolic equations. Let us consider the so-called two-temperature superheating mechanism when the free-electron temperature Te is higher than the phonon temperature T0 (see [29]). In this case, there is the following dependence of the electric conductivity on the temperature of free electrons averaged over the domain (see [29, 418]): q Z Te 1 .Te / D 0 ; Te dx Te ; q 2 R1 : (1.73) T0 meas./ On the other hand, the free-electron temperature in the extreme case Te T0 is related to the intensity of the electric field by the relation Te D w0 jEj2 ;

w0 > 0;

(1.74)

or by the relation Te D w1 C w0 jEj2 ;

w0 > 0;

w1 > 0:

(1.75)

From (1.73)–(1.75), we obtain the following relationship between the electric conductivity and the intensity of the electric field: Z q w2 2 .jEj/ D dx ; jEj Œmeas./q (1.76) q Z 1 2 dx jEj ; .jEj/ D w2 w0 C w1 meas./

41


where w2 D 0 =Tq0 . Now, using system (1.1)–(1.2) and the phenomenological equations (1.6)–(1.8) subject to (1.76), we obtain @ .' C a div.jr'jp2 r'/ r4 j'jq1 '/ @t q Z 1 2 dx jr'j ' D 0; Cw2 w3 C w1 meas./

(1.77)

where a; r4 ; w3 ; q1 0, w2 ; w1 > 0, and q 2 R1 . In mathematics, an equation with the same nonlinear nonlocality is known already since 1876; it is the wave Kirchhoff-type wave equation. In [221], Kirchhoff proposed the integro-differential equation Z 1 l 2 2 ds ss D 0 t t " C 2l 0 s for description of small transversal oscillations of an elastic rod of length l when longitudinal oscillations are small in comparison with transversal oscillations. A detailed statement of the Kirchhoff model can be found in [385]. For the Kirchhoff equation, different initial-value and initial-boundary-value problems [329] and stability problems [27] were considered. Now we describe an interesting effect of relaxation for a finite time, which is related to the fact that the index q in Eq. (1.77) can be negative. Assume that q 2 .1; 0/ and consider the following model problem for Eq. (1.77), where a D q1 D w3 D 0 and r4 D w2 D w1 D 1: q Z @ .' '/ C c1 ./ dx jr'j2 ' D 0; (1.78) @t '.x; 0/ D '0 .x/;

x 2 ;

'.x; t /j@ D 0;

(1.79) (1.80)

where q 2 .1; 0/, c1 ./ D Œmeas./q , and the bounded, surface simple-connected domain R3 has smooth boundary @ 2 C 2;ı , ı 2 .0; 1. We consider problem (1.78)–(1.80) in the general sense L2 .0; T0 I H1 .//. Assume that this problem has a solution (generally speaking, nonunique), which is global-on-time and belongs to the class ' 2 L1 .0; 1I H01 .//;

' t0 2 L2 .0; 1I H01 .//:

Then, multiplying (in the sense of the inner product) both parts of Eq. (1.78) by ' and ' t with respect to the duality bracket Z T dt h; i 0

42


between the Hilbert spaces L2 .0; TI H01 .// and L2 .0; TI H1 .//, T > 0, where h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./, and integrating by parts we obtain Z T 2 2 ds kr'k2.1Cq/ .s/ D kr'0 k22 C k'0 k22 ; (1.81) kr'k2 C k'k2 C 2c1 ./ 2 0

Z

T 0

ds Œkr's0 k22 .s/ C k's0 k22 .s/ C

c1 ./ c1 ./ 2.1Cq/ 2.1Cq/ D ; kr'k2 kr'0 k2 2.1 C q/ 2.1 C q/ (1.82)

which is possible in the smoothness class considered. In this smoothness class, owing to the absolute continuity of the expressions Z t Z t 2.1Cq/ kr'k2 .s/ ds; Œkr's0 k22 .s/ C k's0 k22 .s/ ds 0

0

with respect to t 2 .0; T/, from (1.81) and (1.82) respectively, we obtain the following differential equations: d 2.1Cq/ .t / D 0; Œkr'k22 C k'k22 C 2c1 ./kr'k2 dt c1 ./ d 2.1Cq/ kr' t0 k22 .t / C k' t0 k22 .t / C D 0: kr'k2 2.1 C q/ dt

(1.83) (1.84)

Due to the conditions on the domain , the inclusion H01 ./ L2 ./ is valid: c2 ./k'k22 kr'k22 with a certain constant c2 ./. From this inclusion and Eq. (1.83), we obtain the following inequality: k'k22 1d 0: Œkr'k22 C k'k22 C c1 ./c2 ./ 2 dt kr'k2q

(1.85)

2

Dividing both parts of (1.85) by c2 ./ and adding the inequality obtained with Eq. (1.83), we have kr'k22 C k'k22 d 1 0: Œ1 C c21 ./ Œkr'k22 C k'k22 C c1 ./ 2 dt kr'k2q

(1.86)

2

From (1.86), since q is negative, we have the following inequality: kr'k22 C k'k22 d 1 Œ1 C c21 ./ Œkr'k22 C k'k22 C c1 ./ 0: 2 dt .kr'k22 C k'k22 /q

(1.87)

43


Introduce the notation F .t / kr'k22 C k'k22 ;

c4 ./ D 2c1 ./c2 ./=.1 C c2 .//:

Then from (1.87) we obtain the following first-order ordinary differential inequality: dF C c4 ./F .t /1Cq 0; dt

F .0/ D kr'0 k22 C k'0 k22 > 0;

(1.88)

where q 2 .1; 0/. From (1.88) we have the inequality 0 F .t / .F0jqj jqjc4 ./t /1=jqj : Analyzing this inequality, we arrive at the following conclusion: if F0 D kr'0 k22 C jqj k'0 k22 > 0, then there exists a moment of time 0 < t0 F0 jqj1 c41 ./ such that kr'k22 .t0 / D k'0 k22 .t0 / D 0: Thus, for the finite time t0 , all solutions of problem (1.78)–(1.80) of the smoothness class ' 2 L1 .0; T I H01 .//; ' t0 2 L2 .0; T I H01 .//; T > 0; vanish for almost all x 2 . This is the essence of the relaxation effect for a finite time. Here every solution of problem (1.78)–(1.80) vanish for almost all x 2 for a certain time t0 (in general, these time instants are different for different solutions). But if for a given solution ', the relaxation time is equal to t0 , then, considering the new problem (1.78)–(1.80) with the initial condition vanishing for t D t0 , we obtain that in the generalized smoothness class, owing to (1.81), the unique solution is trivial. It is easy to verify that the solution glued under t D t0 will belong the above mentioned smoothness class because of (1.81)–(1.82). Really, it suffices to verify that the solution obtained ´ '.x; t /; t 2 Œ0; t0 ; '.x; t / D 0; t t0 ; belongs to the class ' 2 L1 .t0 ; t0 C I H01 .//;

' 0t 2 L2 .t0 ; t0 C I H01 .//

in certain neighborhood t 2 .t0 I t0 C /, 2 .0; t0 =2/. From (1.81) and (1.82) it follows that Z

kr'k22 .t / C k'k22 .t / kr'0 k22 C k'0 k22 ; t0 C t0

ds Œkr' s k22 .s/ C k' s k22 .s/

t 2 .t0 ; t0 C /;

c1 ./ 2.1Cq/ kr'0 k2 : 2.1 C q/

44


Now we obtain optimal upper and lower estimates of the relaxation time and rate for finite time. For this, introduce the notation M.t / .kr'k22 .t / C k'k22 .t //1=2 ; u.s; x/

1 '.s C t ; x/; M.t /

M2q .t /; t 2 .0; t0 /; q E.u/ kruk22 .s/ C kuk22 .s/; x 2 :

Here we consider ' as a solution of problem (1.78)–(1.80) with a fixed “relaxation” jqj point 0 < t0 F0 jqj1 c41 ./. It is not difficult to verify that the function u.s; x/ satisfies the following initialboundary-value problem in the sense of L2 .t =; T = t = I H1 .//: @ 2q .u u/ C c1 ./kruk2 u D 0; @s

u.t =; x/ D

'0 ; M.t / (1.89)

u.s; x/ 2 L1 .t =; T = t = I H01 .//; u0s .s; x/ 2 L2 .t =; T = t = I H01 .//;

T > 0;

q 2 .1; 0/:

We prove that the given solution of problem (1.89) corresponding to the fixed solution '.x; t / with the certain relaxation time t0 is such that c6

ˇ d E.u/ˇsD0 c5 ; ds

c5 > 0;

c6 > 0:

(1.90)

Note that in the given smoothness class, the given solution of problem (1.89) satisfies (1.81) and (1.82) in which '.x; t / is replaced by u.x; s/ and T by s in the upper limits. We see that u.x; s/ also satisfies (1.83) and (1.84). In this case, u.x; s/ satisfies also estimate (1.88) which implies ˇ d E.u/ˇsD0 c5 ; ds

c5 ./

c1 ./c2 ./ ; 1 C c2 ./

(1.91)

where c2 ./ is the maximum inclusion constant for H01 ./ L2 ./: c2 ./kvk22 krvk22 : On the other hand, the following inequality holds: ˇ2 ˇZ ˇ ˇ 0 0 ˇ dxŒ.ru ; ru/ C .u ; u/ˇ Œkruk2 C kuk2 Œkru0 k2 C ku0 k2 ; s s 2 2 s 2 s 2 ˇ ˇ

(1.92)

where ˇ ˇ2 ˇ2 ˇZ ˇ ˇ ˇ ˇ 1 0 0 ˇ dx Œ.ru ; ru/ C .u ; u/ˇ D ˇ d F ˇ ; s s ˇ ˇ 4 ˇ ds ˇ

(1.93)

45


where F.s/ kruk22 .s/ C kuk22 .s/. Taking into account (1.83) and (1.84) in which '.x; t / is replaced by u.x; s/, we obtain from (1.92) and (1.93) ˇ ˇ2 ˇdFˇ d 2F F 2 .q C 1/ ˇˇ ˇˇ 0: (1.94) ds ds Dividing both parts of Eq. (1.94) by F2Cq .s/ > 0, where s 2 Œt =; t0 = t =/; after a simple transformation we obtain d 1 dF 0; ds FqC1 ds

q 2 .1; 0/:

(1.95)

Integrating the differential inequality (1.95) by s 2 Œt =; t , where t 2 .t =; t0 = t =/, we obtain ˇ ˇ 1 d F ˇˇ 1 d F ˇˇ qC1 ds ˇsDt = FqC1 ds ˇsDt F D 2c1 ./

kr'0 k2.qC1/ M.t /2.qC1/ 2 M.t /2.qC1/ Œkr'0 k22 C k'0 k22 qC1

D 2c1 ./

2.qC1/ kr'0 k2 2c6 : Œkr'0 k22 C k'0 k22 qC1

Since F0 D 2EE0 , from (1.96) we have ˇ d E ˇˇ c6 : ds ˇsD0

(1.96)

(1.97)

From (1.91) and (1.97) we obtain that (1.90) is valid. It is not difficult to see that, subject to the definition of E.s/, relation (1.90) is equivalent to the following differential inequalities: c6

1 M.t /1C2q

d M.t / c5 ; dt

t 2 .0; t0 /;

q 2 .1; 0/:

(1.98)

Integrating relation (1.98) by t 2 .t; t0 /, M.t0 / D 0, we obtain the inequality c6 2jqj.t0 t / M2jqj .t / c5 2jqj.t0 t /: Hence we obtain the needed estimate: .c5 2jqj/1=jqj .t0 t /1=jqj kr'k22 .t / C k'k22 .t / .c6 2jqj/1=jqj .t0 t /1=jqj ; (1.99)

46


where 2.qC1/

c6 c1 ./ c5 ./

kr'0 k2 ; Œkr'0 k22 C k'0 k22 qC1

c1 ./c2 ./ ; 1 C c2 ./

c1 ./ D Œmeas./q ;

c2 ./ is the maximum inclusion constant of H01 ./ L2 ./: c2 ./kvk22 krvk22 : From (1.99) we easily obtain the upper and lower estimates of relaxation time for all solutions of problem (1.78)–(1.80).

1.2.7 Boundary-value problems for elliptic equations with pseudoparabolic boundary conditions Consider the following model problem. Let the half-space x3 > 0 of the space R3 be filled with a semiconductor without free charges satisfying the following system of quasi-stationary field equations: div D D 0;

rot E D 0;

D D E C 4 P;

x3 > 0:

(1.100)

Obviously, the electric field E is potential in the domain x3 > 0: E D r'. Assume that, according to the phenomenological equations (1.3)–(1.12), Eqs. (1.100) can be written in the following general form: ˇ ˇ 3 X @ ˇˇ @' ˇˇp2 @' ' C a C r4 j'jq1 ' D 0; @xi ˇ @xi ˇ @xi

x3 > 0;

(1.101)

i D1

where p > 2, q1 0, a 0, and r4 2 R1 . Moreover, let the boundary conditions of the form (1.26) and (1.27) with the limit phenomenological relations (1.3)–(1.12) hold on the boundary x3 D 0. In addition, we impose the limit phenomenological relation similar to (1.14): ij D ıij .0 C ˇjEj j2 /: As a result, from system (1.26), (1.27), we obtain the following nonlinear, dynamic, pseudoparabolic boundary condition:

ˇ ˇp2 ˇ @' ˇ @ @' @' q1 ˇ ˇ C aˇ ˇ C r4 j'j ' @t @n @n @n ˇ ˇ2 ˇ ˇ @' ˇ @' ˇ @' q3 ˇ ˇ C a3 C a4 ˇ ˇ C j'j ' ˇˇ D 0; (1.102) @n @n @n x3 D0

47


where a3 0, a4 ; 2 R1 , q3 0, and the constants a and r4 are the same as in (1.101). To the boundary-value problem (1.101) with the boundary condition (1.102), we add the initial conditions on the boundary x3 D 0: ˇ '.x 0 ; x3 ; t /ˇx3 D0 D '0 .x 0 /; x 0 D .x1 ; x2 / 2 R2 tD0

or the condition

ˇ @' ˇˇ D En0 .x 0 /; 3 D0 @n ˇxtD0

x 0 D .x1 ; x2 / 2 R2 :

(1.103)

Analysis of problems for the Laplace equation with dynamic pseudoparabolic boundary conditions can be found in [15, 232]. For example, let us consider an initial-boundary-value problem for the Laplace equation with a nonlinear boundary condition whose solution can be obtained in the explicit form. For simplicity, we consider the quite “good” case En0 .x 0 / 2 C01 .R2 /. We analyze problem (1.101)–(1.103) with a D r4 D D 0 and a3 D a4 D 1:

' D 0; x3 > 0; ˇ ˇ ˇ @2 ' @' ˇˇ @' ˇˇ2 @' ˇˇ D 0; C C @n@t @n ˇ @n ˇ @n ˇx3 D0

ˇ @' ˇˇ D f0 .x 0 / 2 C01 .R2 /; 3 D0 @n ˇxtD0

(1.104) (1.105)

x 0 D .x1 ; x2 /:

(1.106)

Introduce the notation

ˇ @' ˇˇ ; f .x ; t / @n ˇx3 D0 0

f .x 0 ; 0/ D f0 .x 0 /:

(1.107)

Then, because of (1.105), we obtain the initial-value problem @f C f C f 3 D 0; @t which can be easily integrated:

f .x 0 ; 0/ D f0 .x 0 /;

f0 e t : f .x 0 ; t / D q 1 C f02 Œ1 e 2t

(1.108)

Subject to (1.107) and (1.108), problem (1.104)–(1.106) is reduced to the following Neumann problem for the Laplace equation with the parameter t > 0: ' D 0; x3 > 0; ˇ f0 .x 0 /e t @' ˇˇ ; f0 .x 0 / 2 C01 .R2 /; D q @n ˇx3 D0 2 0 2t 1 C f0 .x /Œ1 e

t 0:

48


The solution of the given problem in the class of functions '.x1 ; x2 ; x3 ; t / that decrease as jxj ! C1 is unique because of the maximum principle and has the form Z 1 1 f0 .y1 ; y2 /e t 'D dy1 dy2 q q : 4 R2 x32 C .x1 y1 /2 C .x2 y2 /2 1 C f02 .y1 ; y2 /Œ1 e 2t

1.3

Disruption of semiconductors as the blow-up of solutions

The obvious initial step in the study of initial-boundary-value problems listed above is the proof of the local-on-time solvability in one or another functional class. For some equations and appropriate initial-boundary-value problems, this study can be very difficult because of absence of a priori estimates and, hence, the method of a priori estimates is inapplicable. Moreover, in the case of the presence of local-on-time solvability of the considered initial-boundary-value problems, other difficulties can appear, for example, the proof of the global-on-time solvability or insolvability. The most interesting result from physical and mathematical points of view is the result on the blow-up of solutions, i.e., the simultaneous local-on-time solvability and globalon-time insolvability in a certain class. These difficulties become especially hard in the case of nonlinear operators on time derivatives. We note that we have proposed an approach to the study of just this class of initial-boundary-value problems for pseudoparabolic equations (see [226–240]). Then we consider the problem on the blow-up of solutions for two different initial-boundary-value pseudoparabolic problems without the proof of the local solvability in the required class. The phenomenon of disruption of semiconductors consists of the avalanche growth of the concentration of free charges. This pathological phenomenon leads to the failure of semiconductor devices. One of the main problem of semiconductor physics consists of the search for the cause of disruption and the creation semiconductor devices with parameters that would provide the needed stability in their functioning. On the other hand, a controlled growth of the concentration of free charges is the necessary condition for the creation of electromagnetic power generators. Therefore, it is necessary to produce such devices on the basis of semiconductors that, on one hand, would guarantee unlimited growth of the concentration of free charges and, on the other hand, would prevent disruption. From the mathematical point of view, the disruption of a semiconductor is described as the blow-up of a solution for one or another initial-value or initial-boundary-value problem for a finite time. In other words, there exists a time moment T0 depending on the parameters of the problem, till which the solution of the problem considered belongs to a certain function class, but for t T0 , does not already belong to it. For the problems considered in this work, the appearance of a disruption is caused either by the presence of sources of free charges from donor impurities (1.19) or the

Section 1.3 Disruption of semiconductors as the blow-up of solutions

49

negativity condition ˇ < 0 for the differential conductivity in Eq. (1.14). In this section, we consider two model disruption problems. Here we assume the local, unique solvability of these problems in a required smoothness class, although, certainly, the proof of the local-on-time existence and the uniqueness of a solution with required smoothness is, as a rule, an exceedingly complicated problem. However, here we demonstrate only one method of the proof of the blow-up of solutions, which was successfully applied in [229, 230], where many theorems on the local-on-time unique solvability were proved. First, we consider the first initial-boundary-value problem for Eq. (1.66) with r4 ; a3 ; a4 D 0: ˇ ˇ 3 X @ ˇˇ @u ˇˇp2 @u @ u C C jujq u D 0; @t @xi ˇ @xi ˇ @xi

(1.109)

i D1

u.x; t /j@ D 0; u.x; 0/ D u0 .x/;

t 0;

(1.110)

x 2 ;

(1.111)

where p > 2, q > 0, and .x; t / 2 .0; T. Equation (1.109) is considered in the 0 sense of L2 .0; T0 I W 1;p .//; p 0 D p=.p 1/. We assume that there exists T0 > 0 such that a unique solution of problem (1.109)–(1.111) of the class u 2 L1 .Œ0; T0 /I W0

1;p

u0t 2 L2 .Œ0; T0 /I W01;p .//;

.//;

exists, where R3 is a bounded domain and @ 2 C 2;ı , ı 2 .0; 1. We have 1;p u0 .x/ 2 W0 ./. For convenience, we introduce the notation @v ; @t p 3 p 1 X @u 1 2 E.t / kruk2 C @x ; 2 p v 0t

qC2 B2 B1

i D1

p p1

i p

.qC2/=p ;

˛

E0 E.0/; qC2 ; p

1;p where B1 is the best-inclusion constant of the spaces W0 ./ LqC2 ./ under 5p 6 if 2 < p < 3; thus guarantees the the conditions q > 0 if p 3 or 0 < q 3p inclusion in the three-dimensional case (see, e.g., [26]). Multiplying Eq. (1.109) by u.x; t / and u0t .x; t / in the sense of the duality bracket Z T dt h; i 0

50

Chapter 1 Nonlinear model equations of Sobolev type 0

between the Banach spaces L2 .0; T0 I W01;p .// and L2 .0; T0 I W 1;p .//, where 0 h; i is the duality bracket between the Banach spaces W01;p ./ and W 1;p ./ and integrating by parts (this can be done in the considered smoothness class), we obtain p 3 p 1 X 1 @u kruk22 C 2 p @xi p i D1

p Z t 3 @u 1 p 1 X 0 qC2 2 C D kru0 k2 C kukqC2 .s/ds; 2 p @xi p 0 i D1 # " ˇ ˇ ˇ ˇ Z Z t 3 ˇ @u ˇp2 ˇ @u0s ˇ2 4.p 1/ X 0 2 ˇ ˇ ˇ ˇ ds krus k2 .s/ C dx ˇ ˇ @x ˇ p2 @xi ˇ i 0 i D1

C

1 1 qC2 qC2 ku0 kqC2 D kukqC2 .t /: qC2 qC2

In the smoothness class considered, we obtain that, for t 2 .0; T0 /, the following differential relations hold: p 3 p 1 X @u d 1 2 D kukqC2 .s/; (1.112) kruk2 C qC2 dt 2 p @xi p kru0t k22 C

4.p 1/ p2

3 Z X i D1

i D1

ˇ ˇ ˇ ˇ ˇ @u ˇp2 ˇ @u0t ˇ2 1 d qC2 ˇ ˇ ˇ ˇ dx ˇ ˇ ˇ @x ˇ D q C 2 dt kukqC2 : @x i i

The following inequalities also hold: ˇZ ˇ ˇ ˇ2 ˇ dx ru0 ; ru ˇ kru0 k2 kruk2 ; t t 2 2 ˇ ˇ ˇ Z ˇZ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ @u ˇp2 @u @u0t ˇˇ ˇ @u ˇp1 ˇ @u0t ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ dx ˇ ˇ ˇ dx ˇ ˇ @x ˇ ˇ @xi ˇ @xi @xi ˇ @xi ˇ i ˇ ˇ ˇ ˇ Z ˇ @u ˇp2 ˇ @u0t ˇ2 1=2 @u p=2 ˇ ˇ ˇ dx ˇˇ : ˇ @x ˇ @x @xi ˇ i p i

(1.113)

(1.114)

(1.115)

By the Schwarz inequality (see, e.g., [293]) we have ˇ Z ˇ ˇ ˇ 3 X ˇ @u ˇp2 ˇ @u0t ˇ2 1=2 @u p=2 ˇ ˇ ˇ ˇ dx ˇ @x ˇ @x ˇ @x ˇ i p i i iD1

ˇ 3 Z ˇ ˇ ˇ X 3 ˇ @u ˇp2 ˇ @u0t ˇ2 1=2 @u p 1=2 X ˇ ˇ ˇ dx ˇˇ : (1.116) @x ˇ @x ˇ @xi ˇ i p i i D1

i D1


51

Because of (1.112)–(1.116), the following chain of inequalities holds: jhu; Œu p u0t ij2

jhru0t ; ruij2

ˇ ˇX ˇ ˇ Z ˇ @u ˇp2 @u @u0t ˇ2 ˇ 3 ˇ ˇ ˇ C .p 1/ ˇ dx ˇ @xi ˇ @xi @xi ˇ 2ˇ

i D1

ˇ 3 Z ˇ ˇ ˇ ˇX ˇ @u ˇp2 @u @u0t ˇ 0 ˇ ˇ ˇ ˇ C 2.p 1/j.ru t ; ru/jˇ dx ˇ @xi ˇ @xi @xi ˇ i D1

kru0t k22 kruk22 C .p 1/2

3 X @u p @x i D1

i p

ˇ ˇ ˇ ˇ 3 Z X ˇ @u ˇp2 ˇ @u0t ˇ2 0 ˇ ˇ ˇ ˇ dx ˇ ˇ @x ˇ C 2.p 1/kru t k2 @xi ˇ i i D1

ˇ 3 Z ˇ ˇ ˇ X 3 ˇ @u ˇp2 ˇ @u0t ˇ2 1=2 @u p 1=2 X ˇ ˇ ˇ ˇ kruk2 dx ˇ @x ˇ @x ˇ @xi ˇ i p i i D1

i D1

3 X @u p 2 kuk2 C .p 1/ @x i p i D1

ˇ ˇ ˇ ˇ 3 Z X ˇ @u ˇp2 ˇ @u0t ˇ2 0 2 ˇ ˇ ˇ ˇ k C .p 1/ dx kru t 2 ˇ @x ˇ ˇ @x ˇ i i i D1

ˇ ˇ ˇ ˇ 3 Z X ˇ @u ˇp2 ˇ @u0t ˇ2 ˇ ˇ ˇ p kru0t k22 C .p 1/ dx ˇˇ ˇ @x ˇ @xi ˇ i

i D1

p 3 @u 1 p 1 X 2 kruk2 C @x : 2 p i p i D1

From (1.112), (1.113), and the latter inequality we obtain d 2E dE 2 qC2 E ˛ 0; ˛ D : 2 dt dt p Now we separately consider three cases: 0 < ˛ < 1, ˛ D 1, and 1 < ˛. In the case where 0 < ˛ < 1, inequality (1.117) yields 0 0 qC2 ku0 kqC2 Et E t0 0; ; E˛ E˛ E0˛ qC2 1 ku0 kqC2 1˛ 1˛ t : E.t / E0 C .1 ˛/ E0˛

(1.117)

(1.118)

52


In the case where ˛ D 1, from (1.117) we have E t00t E

2 E t0

E t0 0 ln 0; E t

0;

² E.t / E0 exp

qC2

ku0 kqC2 E0

³ t :

(1.119)

In the case where ˛ > 1, from (1.117) we have

qC2 ku0 kqC2 E t0 E t0 0 0; ; E˛ t E˛ E0˛ 1=.˛1/ ku0 kqC2 qC2 E E0 1 .˛ 1/ t : E0

(1.120)

From (1.112) we obtain p 3 1 p 1 X @u kruk22 C 2 p @xi p i D1

p Z t 3 1 p 1 X @u0 2 D kru0 k2 C kukqC2 qC2 .s/ ds: (1.121) @x C 2 p i p 0 i D1

1;p

Because of the conditions p > 2 and q > 0, the inclusion W0 holds. Now owing to (1.121) we have kukqC2 B1

3 X @u p @x

./ LqC2 ./

!1=p ;

i p

i D1

p 3 3 X @u0 p p 1 X 1 @u 1 kru0 k2 C p 1 kruk22 C 2 @x @x 2 p 2 p i p i p i D1

C B1qC2

i D1

Z

t

0

X 3 @u p .qC2/=p ds .s/: @x i D1

i p

Thus, we have Z E.t / E0 C B2

t

E ˛ .s/ ds;

(1.122)

0

where qC2 ˛D ; p

B2

qC2 B1

p p1

.qC2/=p :

53


Using the Gronwall–Bellman and Bihari lemmas [112], from (1.122) we obtain E.t / ŒE01˛ C .1 ˛/B2 t 1=.1˛/

for 0 < ˛ < 1;

(1.123)

E.t / E0 exp¹B2 t º

for ˛ D 1;

(1.124)

for ˛ > 1:

(1.125)

.˛1/ 1=.˛1/

E.t / E0 Œ1 .˛ 1/B2 E0

t

The following statement for (1.118)–(1.120) and (1.123)–(1.125) holds. Theorem 1.3.1. Let there exist a unique maximum solution of problem (1.109)–(1.111) of the class u 2 L1 .Œ0; T0 /I W0

1;p

.//;

u0t 2 L2 .Œ0; T0 /I W01;p .//:

Then (1) if p > q C 2, then T0 D C1: qC2 1 ku0 kqC2 1˛ 1 1˛ t E.t / ŒE01˛ C .1 ˛/B2 t 1˛ I E0 C .1 ˛/ ˛ E0 (2) if p D q C 2, then T0 D C1: E0 exp¹ku0 kqC2 E01 t º E.t / E0 exp¹B2 t ºI qC2

(3) if p < q C 2 (i.e., ˛ > 1), then T0 D T0 .u0 / is such that lim sup t"T0

N X @u p @x D C1; i D1

i p

q2 T1 .˛ 1/1 B21 E0˛C1 T0 .˛ 1/1 ku0 kqC2 E0 T2 ;

˛

qC2 : p

Now we interpret the result of this theorem. The term jujq u in Eq. (1.109) corresponds to the presence of the source of free-electron current from impurity donor centers, which, from the standpoint of the band theory, transferred from the valence band of the semiconductor crystal into the conduction band. Meanwhile, they have spent certain energy for overcoming barrier. The dependence of the dielectric permittivity tensor on the field increases the barrier energy and thus prevents the transfer from the valence band into the conduction band. The combination of these two factors leads either to disruption of the semiconductor or to gradual growth of energy of electrons, which, with account of heat loss, leads to heating of the semiconductor (see [29]). The increasing of the energy of the electric field means that, starting

54


from some time instant, the value of electric field turns out to be comparable with intra-atomic electric fields. Now we consider the model of semiconductor disruption caused by the negativity of the differential conductivity ˇ < 0 in Eq. (1.14). We consider the simplest possible model, which is the first initial-boundary-value for Eq. (1.30) with D 0, a3 D 1, and a4 D 1:

@u C u div.jruj2 ru/ D 0; @t u.x; t /j@ D 0;

(1.126) t 0; x 2 :

u.x; 0/ D u0 .x/;

(1.127) (1.128)

Equation (1.126) is considered in the sense of L2 .0; T0 I W 1;4=3 .//. Assume that there exists T0 > 0 for which problem (1.126)–(1.128) has a unique maximal solution of the class u 2 L1 .Œ0; T0 /I W01;4 .//;

u0t 2 L2 .Œ0; T0 /I W01;4 .//;

where R3 is a bounded domain and @ 2 C 2;ı , ı 2 .0; 1. We have u0 .x/ 2 W01;4 ./. Note that, subject to the linearity of the operator on time derivative in Eq. (1.126), the approach described in [260] is applicable. However, in this case, the necessary result can be received directly. Let us substitute u D e t v in Eq. (1.126); then we obtain @ v e 2t div .jrvj2 rv/ D 0: @t

(1.129)

Multiplying both parts of Eq. (1.129) by v and v 0t in the sense of the duality bracket Z 0

T

dt h; i

between the Banach spaces L2 .0; T0 I W01;4 .// and L2 .0; T0 I W 1;4=3 .// (see [347]), where h;i is the duality bracket between the spaces W01;4./ and W 1;4=3./, and integrating by parts, we obtain Z krvk22 Z

T 0

D

kru0 k22

C2

1 1 dt e 2t krv 0t k22 C kru0 k44 D krvk44 .t / 4 4

0

T

dt e 2t krvk44 .t /;


55

because of the mentioned smoothness of solution of problem (1.126)–(1.128). We directly derive that in the considered smoothness class, the following relations hold for t 2 .0; T0 /: 1d krvk22 D e 2t krvk44 ; 2 dt e 2t d krvk44 : krv 0t k22 D 4 dt

(1.130) (1.131)

Because of the Cauchy–Bunyakovskii inequality we have jhv 0t ; vij2

ˇZ ˇ2 ˇ ˇ 0 D ˇˇ .rv t ; rv/ dx ˇˇ krv 0t k22 krvk22 ;

(1.132)

where .; / is the inner scalar product in R3 . From (1.130)–(1.132) we come to the following second-order differential inequality: Ee

2t

d dt

dE 2 2t dE 0; e 2 dt dt

E krvk22 :

(1.133)

Dividing both parts of (1.133) by E 3 , we obtain the inequality E0 .E 0 /2 E 00 C 2 2 0 E2 E2 E3 and hence e 2t

E0 E2

0

0:

(1.134)

Integrating (1.134) subject to (1.130), we have 1 kru0 k44 2t E E01 .1 e / ; kru0 k42

E0 D E.0/ D kru0 k22 :

(1.135)

Now we assume that the initial distribution of the potential of the electric field satisfies the condition kru0 k44 > kru0 k22 :

(1.136)

By (1.136), it follows from (1.135) that there exists T0 such that kru0 k22 1 T0 ln 1 ; 2 kru0 k44

(1.137)

56


and, since kruk22 .t / is monotone, the expression lim kruk22 D C1

t"T0

(1.138)

follows from (1.130). Now we analyze the obtained results (1.136)–(1.138). From the physical point of view, (1.136) means that the potential energy of system (1.126)–(1.128) exceeds its kinetic energy. In this case, the negative conductivity leads to disruption (see (1.138)). However, it is quite possible that the contrary condition kru0 k44 kru0 k22 can lead to certain stabilization. From the mathematical point of view, it is necessary for this to prove the global-on-time solvability of problem (1.126)–(1.128) under the mentioned condition, which is rather complicated, in our opinion, because we do not have any necessary a priori estimate. Also, at first, we must prove the local-on-time solvability in one or another sense. Thus, we have shown that sources of free charge current (1.19) or the negativity of the differential conductivity (ˇ < 0) in Eq. (1.14) lead to the disruption of a semiconductor. However, these factors also lead to certain stabilization.

1.4

Appearance and propagation of electric domains in semiconductors

Under large deviations from equilibrium in homogeneous semiconductor, areas of strong (or weak) electric field – electric domains – appear [55]. These domains can be motionless or movable; movable domains move in the direction of the motion of principal charge carriers; for example, in the case of electronic conductivity, the direction of this motion is opposite to the external electric field. As a rule, the domain creation time and the collapse time in semiconductors are much less than the time of domain motion in semiconductor. Note that under one or another mode of observing domain motion, one can clearly see that domain do not change their shape. The basic cause of the domain creation is assumed to be the presence of a voltagecurrent characteristic descending branch or, in other words, the negativity of differential conductivity [54]. Indeed, as we have seen in Section 1.3, under a sufficiently large disturbance of the distribution of the potential of the electric field in a semiconductor with a negative differential conductivity, the appearance of an avalanche growth mode of the intensity of the electric field is possible. Unfortunately, our considerations cannot afford to make a conclusion about the way how the intensity of the electric field will behave pointwise in this case. But we can consider a mechanism of domain appearance itself. Let a certain initial distribution of the potential of the electric field such that the semiconductor occupying a surface-simply-connected, bounded

57

Section 1.4 Electric domains in semiconductors

domain R3 can be divided into two domains: D1 and D2 whose closures D 1 and D 2 have a common nonempty boundary 2 C .2;ı/ , ı 2 .0; 1, which is unknown beforehand and is to be defined, so the domains depend on . Moreover, in the bounded domain D 1 , let condition (1.136) be valid, and in the domain D 2 , let the contrary condition hold. We assume that the condition contrary to (1.136) guarantees the global-on-time solvability of problem (1.126)–(1.128). Hypothesis. In this case, in the domain D1 , we will have the growth of the intensity of the electric field in time, while in the domain D2 , the intensity of the electric field will remain bounded at following time moments. Of course, in order to solve the problem on the appearance of electric domains, one must consider the problem for Eq. (1.126) in the domain with conditions of conjugation on alternating boundary . From mathematical point of view, this is a very complicated problem. We consider the simplest one-dimensional model problem on the appearance of electric domains, which shows that our hypothesis is valid. Consider Eq. (1.126) on the segment x 2 .0; 1/ with the additional boundary Neumann conditions ux .0; t / D ux .1; t / D 0 and the initial condition u.x; 0/ D u0 .x/: uxxt C uxx .jux j2 ux /x D 0;

(1.139)

ux .0; t / D ux .1; t / D 0;

(1.140)

u.x; 0/ D u0 .x/:

(1.141)

Assume that u0 .x/ 2 C .2/ Œ0; 1 \ ¹u0x .0/ D u0x .1/ D 0º; we search for a local-ontime solution of problem (1.139)–(1.141) in the class u.x; t / 2 C .1/ .Œ0; T/I C .2/ Œ0; 1 \ ¹ux .0; t / D ux .1; t / D 0º/;

T > 0:

Integrating Eq. (1.139) by x 2 Œ0; y in the considered smoothness class, we obtain uyt C uy .uy /3 D 0;

y 2 .0; 1/:

(1.142)

Since the potential u.x; t / of the electric field is related with the intensity E of the field, E D ux , Eq. (1.142) can be rewritten in the form E t C E E3 D 0:

(1.143)

We also impose condition (1.141), E.0; x/ D E0 .x/ u0x .x/:

(1.144)

58


Thus, we have reduced problem (1.139)–(1.141) to the Cauchy problem (1.143), (1.144) with a parameter x 2 Œ0; 1. The ordinary differential equation (1.143) with the initial condition (1.144) can be easily integrated and we have ED q

E0 1

E20 Œ1

e t :

(1.145)

e 2t

Formula (1.145) implies that at points of the interval .0; 1/ where jE0 .x/j > 1, a time instant t0 D t0 .x/ exists, 1 (1.146) t0 ln.1 E2 0 .x//; 2 such that the avalanche growth in time of the intensity of the electric field occurs: lim jE.x; t /j D C1:

t"t0

(1.147)

On the other hand, at points of the interval .0; 1/ where the initial intensity of the electric field is equal to one by absolute value, we have E.x; t / D E0 .x/:

(1.148)

Finally, at points of the segment Œ0; 1 where jE0 .x/j < 1, the intensity of the electric field exponentially tends to zero: jE.x; t /j q

jE0 .x/j 1

e t :

(1.149)

E20 .x/

The cases (1.147)–(1.149) describe the appearance of electric domains. Really, let us consider the following initial distribution of the intensity of the electric field on the segment Œ0; 1: 8 ˆ x 2 I1 ; E0 > 1; ˆ ˆ ˆ <E < 1; x 2 I ; 0 2 E0 .x/ D ˆ D ˙1; x 2 I E 0 3; ˆ ˆ ˆ :jE j < 1; x 2 I ; 0 4 I1 [ I2 [ I3 [ I4 D Œ0; 1;

E0 .x/ 2 C .1/ Œ0; 1;

where I1 , I2 , and I4 are simple-connected open sets. Because of (1.146) and (1.147), for finite times we have ³ ² 1 2 t C inf t0 W t0 .x/ D ln.1 E0 .x//; x 2 I1 ; 2 ² ³ 1 t inf t0 W t0 .x/ D ln.1 E2 .x//; x 2 I 2 I 0 2

Section 1.4 Electric domains in semiconductors

59

in the domain I1 , a sharpening mode with a positive intensity of the electric field will appear and in the domain I2 , a mode with negative intensity of the electric field appear. On the sets I3 and I4 , the intensity of the electric field will remain bounded. Thus, in a semiconductor for a finite time t D min¹t C ; t º, electric domains will appear; this time decreases if the initial distribution of the intensity of the electric field in the domains I1 and I2 increases. Note that stationary electric domains are nontrivial stationary solutions of wave equations, for example, of (1.126). Now we consider the question on the propagation of electric domains. An obvious cause of the motion of domains is an external electric field. If we take into account the listed experimental facts, it becomes clear that we deal with solitary waves, which, as is known, propagate without changing their shape. Only the question on the fact, which wave equations describe solitary waves observed in experiment, remains open. We recall some facts on wave equations with dissipation derived in this chapter, that take into account the negative differential conductivity, i.e., Eqs. (1.29), (1.35), (1.60), and (1.65) under the condition D 0: @' @' @' C a1 C a2 ' C a3 ' a4 div.jr'j2 r'/ D 0; @t @x @x @ @' @' .' a'/ C a1 C a2 ' C a3 ' a4 div.jr'j2 r'/ D 0; @t @x @x @' @' @ .' C a'/ C a1 C a2 ' C a3 ' a4 div.jr'j2 r'/ D 0; @t @x @x @ .' C a div.jr'jp2 r'/ r4 j'jq1 '/ @t @' @' C a2 ' C a3 ' a4 div.jr'j2 r'/ D 0: Ca1 @x @x

One can assume that the propagation of domains observed in certain experiments is described in the framework of solitary waves by the classical Oskolkov–Benjamin– Bona–Mahony-type equations (1.34) or by nonlinear Rossby waves (1.28) under the condition a3 D 0: @' @ @' .' a'/ C a1 C a2 ' D 0; @t @x @x @' @' @' C a1 C a2 ' D 0: @t @x @x It is natural to formulate two various problems: on the appearance of electric domains in semiconductors and on their motion. These problems will be the subject of our further research.

60

1.5


Mathematical models of quasi-stationary processes in crystalline electromagnetic media with spatial dispersion

In the case of conducting media, for an adequate description it is necessary sometimes to take into account a spatial dispersion of nonstationary processes considered. We mean by a spatial dispersion a nonlocal relation of some characteristics of electromagnetic media, for example, the electric displacement D and the intensity of the electric field E: Z Di .r/ D Ei .r/ C ~ij .r; r0 /Ej .r0 / d r0 ;

the magnetic induction B and the magnetic field intensity H: Z Bi .r/ D Hi .r/ C ij .r; r0 /Hj .r0 / d r0 ;

the current density J in a medium and the intensity E of the electric field" Z ij .r; r0 /Ej .r0 / d r0 ; Ji .r/ D

where summation by repeating indices is assumed. Note that in this monograph, we assume that the kernels ~ij .r; r0 /, ij .r; r0 /, and ij .r; r0 / of the integral operators depend on the differences of the arguments jr r0 j: ~ij .jr r0 j/;

ij .jr r0 j/;

ij .jr r0 j/:

The domain R3 is, generally speaking, an arbitrary surface-simply-connected, bounded or unbounded (or coinciding with the whole space R3 ). Note that in what follows, we assume more general relations between parameters of electromagnetic media. The fundamental property of media such as semiconductors consists of the fact that nonstationary processes observed in them are described by a system of quasistationary field equations, a continuity equation, and constitutive equations. In their description, the explicit form of constitutive equations that link the intensity E of the electric field and the electric displacement D, on one hand, and the intensity E of the electric field and the current density J in the semiconductor, on the other hand, is essential. The system of equation in a certain Cartesian coordinates in the general case has the form [54, 55] div D D 4 e n;

rot E D 0;

@n D div J C Q; @t

D D E C 4 P;

(1.150) (1.151)

Section 1.5 Models of quasi-stationary processes in media with spatial dispersion

61

where P is the polarization vector and J is the electric current density in the medium. In the surface-simply-connected domain , we can introduce the potential ' of the electric field E: E D r'. In (1.151), Q are sources or sinks of free charges (electrons). On the other hand, spin waves in magnets in absence of an external magnetic field are described in a quasi-stationary approximation by the following system of equations: div B D 0;

rot H D 0;

B D H C 4 M;

@M D ŒH; M C R; @t

(1.152) (1.153)

where M is the magnetization vector and R is the vector of sources (or sinks) of magnetic domains. In the surface-simply-connected domain , we can introduce the potential of the magnetic field H: H D r . Note that the systems of equations (1.150)–(1.151) and (1.152)–(1.153) have a similar form. System (1.150)–(1.151) can be reduced to a system with a greater number of equations. System (1.152)–(1.153) is very complicated; therefore, we further obtain a simpler systems. To this end, we impose certain physical assumptions. Let us represent the magnetization vector as the sum of the quasi-stationary summand M and the “fast alternating” summand m (see [192, 247] M D M C m;

(1.154)

and the following relations are valid: jmj jMj ;

ˇ ˇ ˇ ˇ ˇ ˇ ˇ @m ˇ ˇ ˇ @M ˇ : ˇ ˇ @t ˇ ˇ @t ˇ

Subject to these relations, (1.154) can be represented in the form @m D ŒH; M C R: @t

(1.155)

Relative to the value M, we assume that it is related to the intensity H of the magnetic field in a medium by the certain phenomenological (generally speaking, nonlinear), nonlocal relation ij .H/Hj ; Mi D b

(1.156)

where b ij is a certain nonlinear, nonlocal integral operators. In what follows, instead of system (1.152)–(1.153), we consider system (1.152), (1.154)–(1.156). Now we consider possible phenomenological relations of characteristics of electromagnetic media: the electric displacement D, the density J of the electric current,

62


the intensity E of the electric field, the magnetization M, and the intensity H of the magnetic field. All nonlocal expressions will be written for the case D R3 , and then we rewrite relations required to us in the case of an arbitrary domain . We start from the vector P of the medium polarization. In the absence of an external magnetic field in an isotropic medium, a “good” phenomenological relation that links the polarization vector P with the intensity of the electric field and takes into account a spatial dispersion in the medium with strong fields is as follows: P D P 1 C P2 C P3 ;

(1.157)

where Z P1 D jEjp2 E;

P2 .x; t / D

R3

dy S1 .jx yj/E.y; t /;

div P3 D 1 .'/;

p D 3; 4. The Fourier transform of the kernel S1 .jxj/ will be denoted by b S1 .k/. The explicit form of the first summands in the decomposition of the Fourier transform b S1 .k/ of the kernel S1 .jxj/ by powers of k in the case of crystals whose symmetry does not allow natural optical activity, has a quadratic form in the simplest case [256]: b S1 .k/ D ~0 jkj2 : Thus, in a certain function class E.x; t / for the kernel of the mentioned form the following relation is valid: P2 D ~0 E:

(1.158)

Now we consider the explicit expression for the density 1 .'/ of bound charges. A sufficiently good model distribution of the density of bound charges is the power function 1 .'/ D a1 j'jq1 ';

q1 > 0;

a1 > 0:

(1.159)

Consider phenomenological relations for the current density J in the medium where an external electric field E0 possibly exists: Z J1 D 0 jEjp2 E;

J2 D

J D J1 C J2 C J3 ; R3

dy K2 .jx yj/E.y; t /;

(1.160) J3 D 2 .'/E0 ;

(1.161)

where p D 3; 4. We denote by b S2 .k/ the Fourier transform of the kernel K2 .jxj/. We consider the well-known model expression for b S2 .k/ D 1 jkj1 . Then for the value

Section 1.5 Models of quasi-stationary processes in media with spatial dispersion

63

div J2 in a certain smoothness class of functions '.x; t / that decrease as jrj ! C1, we obtain the following relation: div J2 D 1 ./1=2 ';

1 > 0;

(1.162)

where we have used the fact that E D r' in the considered surface-simply-connected domain. Consider the case where for 2 the following explicit expressions hold: 2 .'/ D a2 j'jq2 C1 ;

2 .'/ D a2 j'jq2 ';

a2 > 0;

q2 > 0:

(1.163)

Consider the model distribution of sources or sinks of the current of free charges in a medium. The function Q D Q.'/ in Eq. (1.151) depends on the density of sources of donor centers of the lattice or on the density of sinks of acceptor centers and has the form similar to the distributions of free and bound electrons of main centers of the semiconductor. We will use the following model distribution: Q.'/ D j'jq3 ';

q3 0;

(1.164)

where < 0 for donor and > 0 for acceptor impurity centers, respectively. It is obvious that D 0 in the absence of impurity centers. Consider the explicit form of quasi-stationary magnetization M:

M1i D ij Hj ;

(1.165) M D M1 C M2 C M3 ; Z M2 D 0 jHj2 H; M3 D dy K3 .jx yj/H.y; t /: (1.166) R3

S3 .k/. Similarly to the case We denote the Fourier transform of the kernel K3 .jxj/ by b of the kernel K1 .jxj/, the first summand in the decomposition b S3 .k/ by powers of jkj in certain crystalline media has the form b S3 .k/ D 1 jkj2 :

(1.167)

Thus, in a certain function class E.x; t /, for the kernel of the mentioned form the following relation holds: M3 D ~0 H:

(1.168)

Like in the case of an electric medium, the dissipation in a magnetic medium can be taken into account on the phenomenological level. It seems to us very natural to consider the case where, instead of the dissipation in a magnetic medium, there are magnetic domains “sources.” But, unlike electric media, the factors considered have a vector character. In many works, the following relation cubic in H is usually used: R D 1 jHj2 H; where 1 < 0 for “sources” and 1 > 0 for “sinks.”

(1.169)

64


We have discussed all model phenomenological relations between medium parameters. Now we consider possible boundary conditions for systems (1.150)–(1.151) and (1.152), (1.154)–(1.156). Introduce the unit normal vector n.x/ to the boundary media interface @, which is directed from the domain to the domain R3 n. We denote by @˙ the two sides of the oriented surface @. Then the boundary conditions for the electric displacement and the intensity have the form (see, e.g., [399]) ŒE1 ; nj@C ŒE2 ; nj@ D 0; .D1 ; n/j@C .D2 ; n/j@ D 4 j@ ;

(1.170)

where is the surface density of free charges on the boundary @. For system (1.152), (1.154)–(1.156), similar boundary conditions are valid: 4 I; c D 0;

ŒH1 ; nj@C ŒH2 ; nj@ D .B1 ; n/j@C .B2 ; n/j@

(1.171)

where I is the surface current density. We note that on the interface “semiconductor– ideal conductor,” in the case of a “grounded” conductor, the boundary conditions have the form ˇ @' ˇˇ D0 (1.172) 'j@ D @n ˇ @

and a similar boundary condition holds for the interface “magnet–ideal conductor”: ˇ @ ˇˇ D 0: (1.173) j@ D @n ˇ@

1.6

Model pseudoparabolic equations in electric media with spatial dispersion

In this section, we consider certain model problems on quasi-stationary processes in electric media with a spatial dispersion. Equations that describe such processes have wave ot dissipative character. In this section, we derive certain pseudoparabolic equations of fifth degree with time derivative of first degree that have the form of model three-dimensional equations of Rosenau and Rosenau–Buergers type: @u @ D 0; .2 u C u/ C u @t @x1 @u @ .2 u C u/ C u C u D 0: @t @x1

Section 1.6 Pseudoparabolic equations in electric media with dispersion

65

These equations are model wave equations in media with strong spatial dispersion. For results of the one- or multi-dimensional Cauchy problems and initial-boundaryvalue problems for Rosenau and Rosenau–Buergers equations, see [81–87, 277–279, 295–301, 315, 316]. First, consider a model problem on waves in a conductive medium without spatial dispersion of the dielectric permittivity but with spatial dispersion of the medium conductivity of the form (1.161) in the presence of “sources” or “sinks” of the current of free charges (1.164) and a constant external electric field E0 D E0 e1 , where e1 is the unit vector along the axis Ox 1 . Moreover, we assume that the distribution of bound charges of the form (1.159) with q1 D 0 is linear. Then from system (1.150), (1.151), (1.157), (1.159)–(1.164) we obtain receive the following equation: @j'jq2 C1 @ .' d1 '/ d2 ./1=2 ' C d3 C d4 j'jq3 ' D 0; @t @x1

(1.174)

where d1 D 4 a1 , d2 D 4 e1 , d3 D 4 ea2 jE0 j, d4 D 4 e1 , d1 ; d2 ; d3 > 0, q2 ; q3 > 0, d4 2 RC . An important particular case of Eq. (1.174) is the following equation in the dimensionless variables: @' @ .' '/ ./1=2 ' C ' C d' 3 D 0; @t @x1

d 2 R1C ;

(1.175)

which is called the nonlinear, nonlocal Oskolkov–Benjamin–Bona–Mahony-type equation with a cubic source (d > 0) or a cubic sink (d < 0). Now if we take into account the spatial dispersion of the electric permittivity of the form (1.158), we obtain the following equation of fifth order: @j'jq2 C1 @ C d4 j'jq3 ' D 0; .d5 2 ' C ' d1 '/ d2 ./1=2 ' C d3 @t @x1 (1.176) where d1 D 4 a1 , d2 D 4 e1 , d3 D 4 ea2 jE0 j, d4 D 4 e1 , d5 D 4 ~0 , d1 ; d2 ; d3 ; d5 ; q2 ; q3 > 0, d4 2 RC . The case where a1 D 0 and 1 D 0 also has a physical sense; then in the dimensionless variables from (1.176) we obtain the following equation: @j'jq2 C1 @ C d4 j'jq3 ' D 0: .2 ' C '/ C @t @x1

(1.177)

Now we write the general equation that follows from (1.150), (1.151), (1.157)–(1.164). For convenience, we write it in the dimensionless variables: @ .a1 2 ' C ' C a2 div.jr'jp1 2 r'/ a3 j'jq1 '/ a4 ./1=2 ' @t @ .j'jq2 C1 / C a5 ' C a6 div.jr'jp2 2 r'/ C a7 j'jq3 ' D 0; (1.178) C @x1

66


where a1 ; a2 ; a3 ; a4 ; a5 0, a6 ; a7 2 R1 , q1 ; q2 0, p1 ; p2 2. From this general equation, in addition to Eqs. (1.174)–(1.177), we obtain the following partial differential equations: @ @u div.jruj2 ru/ D 0; .2 u C u/ C u @t @x1 @ .2 u u/ C div.jrujp2 ru/ D 0; @t @ .2 u C u C div.jrujp1 2 ru// div.jrujp2 2 ru/ D 0; @t @ .2 u C u C div.jrujp1 2 ru// C u div.jrujp2 2 ru/ D 0; @t @ .u C div.jrujp2 ru// ./1=2 u C jujq u D 0; @t

(1.179) (1.180) (1.181) (1.182) (1.183)

where p; p1 ; p2 > 2 and q > 0. Note that to Eqs. (1.174)–(1.183), we must add the boundary conditions (1.170)– (1.173).

1.7

Model pseudoparabolic equations in magnetic media with spatial dispersion

In this section, we obtain model equations of the theory of spin waves; we take into account the presence of a spatial dispersion. First, consider the simplest case where the relation between the quasi-stationary magnetization M and the intensity of the magnetic field is linear: Mi D ai Hi :

(1.184)

Then from system (1.152), (1.154)–(1.156) in the absence of a spatial dispersion and sources or sinks of magnetic domains (R D 0), owing to (1.184) we obtain the following: div

@m D div ŒH; M D .H; rot M/ (1.185) @t @ @ @ @ @ @ @ @ @ ˇ2 ˇ3 ; D ˇ1 @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2

where ˇ1 D a2 a3 , ˇ2 D a3 a1 , and ˇ3 D a1 a2 . Note that ˇ1 C ˇ2 C ˇ3 D 0 and assume that jˇ1 j C jˇ2 j C jˇ3 j > 0. Then from system (1.152), (1.154)–(1.156),

67

Section 1.7 Pseudoparabolic equations in magnetic media with dispersion

subject to (1.185) we obtain receive the following three-dimensional equation of spin waves: 3 @ X @2 .1 C ˛1 / 2 @t @xi iD1

@ C 4 ˇ1 @x1

@ @ @x2 @x3

@ C ˇ2 @x2

@ @ @x3 @x1

@ C ˇ3 @x3

@ @ @x1 @x2

D 0;

where ˛i D 4 ai . This equation can be reduced to the form @ @ C ˇ1 @t @x1

@ @ @x2 @x3

@ C ˇ2 @x2

@ @ @x3 @x1

@ C ˇ3 @x3

@ @ @x1 @x2

D 0; (1.186)

where ˇ1 C ˇ2 C ˇ3 D 0 and jˇ1 j C jˇ2 j C jˇ3 j > 0. Equation (1.186) is new. We emphasize that it possesses essentially three-dimensional wave character. It is very interesting to apply the group approach for the study of this equation and find nontrivial self-similar solutions of it. Moreover, because of the quadratic character of the nonlinearity of Eq. (1.186), it is interesting to find whether “turn-over” of given solutions take place. Note that Eq. (1.186) has the form similar to that of the essentially two-dimensional equation of flat waves in an ideal incompressible liquid [163]: @ @ @ @ @ C D 0: @t @y @x @x @y For this equation, in [164], questions on the existence of solitary-wave-type solutions were studied by reducing the problem to the analysis of the nonlinear Hammerstein equation. Now we take into account the spatial dispersion of the form (1.166) and the presence of “sources” or “sinks” of the form (1.169). Then from system (1.152), (1.154)– (1.156) we obtain the following model, three-dimensional equation of spin waves in a media with a spatial dispersion, which in the dimensionless variables has the form 3 @2 @ X 2 .1 C ˛i / 2 0 C div.jr j2 r / @t @xi

(1.187)

iD1

@ C ˇ1 @x1

@ @ @x2 @x3

@ C ˇ2 @x2

@ @ @x3 @x1

@ C ˇ3 @x3

@ @ @x1 @x2

D 0;

68


2 R1 , ˇ1 C ˇ2 C ˇ3 D 0, jˇ1 j C jˇ2 j C jˇ3 j > 0. Finally, we write the general nonlinear wave equation that follows from system (1.152), (1.154)–(1.156), (1.165)– (1.169) and the equation 3 @ X @2 .1 C ˛i / 2 0 2 C div.jr j2 r / C div.jr j2 r / (1.188) @t @x i iD1 @ @ @ @ @ @ @ @ @ C ˇ2 C ˇ3 D 0; C C ˇ1 @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2 which generalizes (1.187); here 2 R1 , ˇ1 Cˇ2 Cˇ3 D 0, and jˇ1 jCjˇ2 jCjˇ3 j > 0. Initial-boundary-value problems for Eq. (1.188) will be considered in Chapter 6. Thus, we have considered certain wave equations in various electromagnetic media, and Eqs. (1.174)–(1.183) in the absence of an external electric field become purely dissipative.

Chapter 2

Blow-up of solutions of nonlinear equations of Sobolev type

In Chapter 2, we consider two different abstract Cauchy problems for pseudoparabolic equations with operator coefficients in Banach spaces. For the first problem, under certain conditions on its operator coefficients, necessary and sufficient conditions of the blow-up of solution for a finite time and of the global-on-time solvability are obtained. In the case of the blow-up, upper and lower estimates for the blow-up time are obtained. For one problem, optimal upper and lower estimates for the blow-up rate of solution are obtained. For each of these abstract problems, examples of operator coefficients with physical sense are given. Certain model problems are discussed in detail. The results of this chapter were obtained in [226–230, 235–240] we also mention [267].

2.1

Formulation of problems

In this chapter, we obtain necessary and sufficient conditions of the blow-up of solutions of strongly nonlinear Sobolev-type equations in an abstract formulation, as Cauchy problems for equations with operator coefficients in Banach spaces: N X d Aj .u/ D F.u/; A0 u C dt

u.0/ D u0 ;

(2.1)

j D1

and estimate upper and lower estimates for the blow-up time of solutions for problem (1.1). Moreover, we also obtain optimal upper and lower estimates for the blow-up rate of solutions of the abstract Cauchy problem d A0 u C B.u/ D F.u/; dt

u.0/ D u0 :

(2.2)

By optimal results, we mean the following: for certain values of input parameters of the problem, the blow-up of solutions occurs, and for other conditions, the global solvability holds. Therefore, we intentionally restrict the class of considered Sobolevtype equations to a class for which we can obtain optimal results. Moreover, we consider only problems for which the uniqueness of solution (meant in one or another sense) can be proved.

70

Chapter 2 Blow-up of solutions of nonlinear equations of Sobolev type

Let us present some examples of certain model, three-dimensional strongly nonlinear Sobolev-type equations derived in Chapter 1: n X @ u C div.jrujpj 2 ru/ C jujq u D 0; @t

(2.3)

@ .u jujq1 u/ C jujq u D 0; @t

(2.4)

j D1

@ .u C div.jrujp2 ru/ jujq1 u/ C jujq u D 0; @t @ .u u/ C div.jrujp2 ru/ C jujp2 u D 0; @t @ 2 . u u/ C div.jrujp2 ru/ D 0; @t @ .2 u C u C div.jrujp1 2 ru// div.jrujp2 2 ru/ D 0; @t

(2.5) (2.6) (2.7) (2.8)

where pj > 2, q1 ; q 0, p1 ; p2 ; p > 2, pj > 2, > 0, and j D 1; n.

2.2

Preliminary definitions, conditions, and auxiliary lemmas

Below for an arbitrary Banach space X, we denote by X its dual space. For h 2 X and u 2 X, we denote by hh; ui the duality bracket between the spaces X and X (see [347, 436]). The strong convergence and the weak convergence are denoted by ! and *, respectively. We denote the norms in X and X by k k and k k , respectively. For two Banach spaces X and Y , we denote by X Y the topological inclusion and by X ,! Y we denote the compact and continuous inclusion of X in Y for which X Y holds. Moreover, by ds

XY we denote the dense and continuous inclusion of X in Y . Definition 2.2.1. Let X be a Banach space, D be a subset of X, and A be a mapping of D into X . Then A is called monotonic mapping if for arbitrary u; v 2 D, the following inequality holds: hA.u/ A.v/; u vi 0:

71

Section 2.2 Preliminary definitions, conditions, and auxiliary lemmas

Definition 2.2.2. Let X be a Banach space, D be a subset of X, and F be a mapping of D into X . Then an operator F.u/ is said to be boundedly Lipschitz continuous if kF .u/ F .v/k .R/ku vk;

R D max.kuk; kvk/;

8u; v 2 D;

where .R/ is an increasing and continuous function and k k and k k are the norms in X and X , respectively. Definition 2.2.3. A functional J W D 2 X ! R1 is said to be weakly lower semicontinuous if for any sequence um 2 D that weakly converges to u0 2 D, the following inequality holds: J .u0 / lim inf J .um /: m!C1

Definition 2.2.4. An operator A W D X ! X is said to be semicontinuous on D if for any u; v; w 2 D, the function ! hA.u C v/; wi is continuous as a function from R1 into R1 . Definition 2.2.5. An operator F.u/ W D X ! X is said to be completely continuous if for any sequence um 2 D that weakly converges to u0 2 D, we have lim kF.um / F.u/k D 0:

m!C1

Definition 2.2.6. If at the point u 2 D X, where D is a convex set, the relation F.u C h/ F.u/ D .u; h/ C !.u; h/ holds, where .u; h/ is a linear operator of h 2 X and k!.u; h/k D0 khk khk!C0 lim

for fixed u 2 D X, then .u; h/ is called the Fréchet differential of F and is denoted by dF .u; h/. Definition 2.2.7. We say that an operator F.u/ has a locally uniform Fréchet differential dF .u; h/ on the set w if for each " > 0 and arbitrary u0 2 w, we have appropriate positive numbers ."; u0 / and ı."; u0 / such that for each u0 from a ball U .u0 ; /, the following inequality holds: k!.u; h/k < "khk; if khk < ı.

72


Definition 2.2.8. A linear operator A acting from a reflexive Banach space X into the dual space X is said to be symmetric if for any u; v 2 X the following relation holds: hAu; vi D hAv; ui: Let us give definitions, assumptions, and auxiliary results necessary for further analysis (see, e.g., [168]). Let Banach spaces Vi , V0 , W0 , and W1 be infinite-dimensional, separable, and reflexive; their dual Banach spaces be Vi , V0 , W0 , and W1 ; the corresponding duality brackets be h; ii , h; i0 , .; /0 , and .; /1 , i D 1; N . Moreover, let the Banach spaces Vi and V0 be uniformly convex. We denote by k kj , j D 0; N , the norms of the Banach spaces Vj , and by k kj the norms of the Banach spaces Vj . The norms of the Banach spaces Wi , i D 0; 1, we denote by j ji and the norms of the Banach spaces Wi by j ji . Let H be a Hilbert space identified with its dual space. Further, we assume that the following conditions hold. Conditions (V1). (1) Let Vi , i D 0; N , be continuously embedded in H. (2) The following chains of continuous and dense embeddings hold: ds

ds

ds

ds

V Vi H Vi V ; ds

ds

ds

i D 0; N ;

ds

V Wj H Wj V ;

j D 0; 1;

i.e., V is dense in Vi , Vi is dense in H, V is dense in Wj , and Wj is dense in H. (3) The space V is infinite-dimensional and separable. Here by V we denote the dual space of V . Remark 2.2.9. Conditions (V1) imply the fact that the intersection V

N \

Vi

i D0

with the norm kkD

N X i D0

can be transformed into a Banach space.

k ki ;

73


Denote by h; i the duality bracket between the Banach spaces V and V . Let operators Ai , A0 , F, and B be defined on the spaces Vi , V0 , W0 , W1 and take their values in the corresponding dual spaces: Ai W Vi ! Vi ;

A0 W V0 ! V0 ;

F W W0 ! W0 ;

B W W1 ! W1 ;

i D 1; N :

Here the sets of values of the operators Aj coincide with the corresponding Banach spaces Vj , j D 0; N . Introduce certain conditions on operator coefficients of Eqs. (1.1) and (1.2). Conditions (A). (1) Let the operators Aj W Vj ! Vj , j D 1; N , be monotonic semilinear operators. (2) The operators Aj have symmetric, bounded Fréchet derivatives for any fixed u 2 Vj : 0 .u/ W Vj ! L.Vj ; Vj /; j D 1; N ; Aj;u which are nonnegative definite and, moreover, the Fréchet derivative is semilinear in each Vj in the sense that for any u; v; h1 ; h2 2 D Vj , j D 1; N , and 0 .u C v/h ; h i is continuous as a function from R1 2 R1 , the function hAj;u 1 2 j into R1 with respect to . (3) The following lower and upper inequalities hold: p 1

kAj .u/kj Mj kukj j

;

p

hAj .u/; uij mj kukj j ;

pj > 2, j D 1; N , Mj > 0, and mj > 0. 1=p

(4) hAj .u/; uij j is a norm in the Banach space Vj , which is equivalent to the initial norm of this Banach space Vj . (5) The operators Aj are positively homogeneous of degree pj 1: Aj .sv/ D s pj 1 Aj .v/; (6) The functionals differentiable.

j .u/

s > 0:

hAj .u/; uij W Vj ! R1 are continuously Fréchet

Additional properties (A). 0 .u/ are boundedly Lipschitz-continuous: (1) The operators Aj;u 0 0 .u1 / Ajf .u2 /kVj !Vj j .R/ku1 u2 kj kAjf

8u1 ; u2 2 Vj ;

where kkVj !Vj is the norm of the Banach space L.Vj ; Vj /, and j D j .R/ is bounded on a compact and nondecreasing function and R D max¹ku1 kj ; ku2 kj º. 0 .u/ are strongly continuous in the Banach space L.V ; V /. (2) The operators Aj;u j j Property (1) will be used in theorems on the strong generalized solvability and property (2) will be used in theorems on the weak generalized solvability.

74


Conditions (A0 ). (1) hA0 u A0 v; u vi0 m0 ku vk20 for any u; v 2 V0 , where m0 > 0 is a constant. (2) The operator A0 is symmetric. (3) The operator A0 W V0 ! V0 satisfies the condition kAuk0 M0 kuk0 ; where M0 > 0 is a constant. 1=2

(4) hA0 u; ui0 is a norm on V0 , which is equivalent to the initial norm of this Banach space. (5) The functional entiable.

0 .u/

hA0 u; ui0 W V0 ! R1 are continuously Fréchet differ-

Conditions (F). (1) The operator F.u/ W W0 ! W0 is boundedly Lipschitz-continuous: jF.u/ F.v/j0 .R/ju vj0 ;

R D max.juj0 ; jvj0 /;

where .R/ is an increasing and continuous function. (2) The operator F.u/ is Fréchet differentiable and has a symmetric derivative for any fixed u 2 W0 ; moreover, Fu0 .u/ W W0 ! L.W0 ; W0 /: (3) For any s > 0 and v 2 W0 , F.sv/ D s 1Cq F.v/, q > 0. (4) The following upper estimate is valid: jF.u/j0 BjujqC1 ; 0 (5) The functional tiable.

B > 0:

.u/ .F.u/; u/0 W W0 ! R1 is continuously Fréchet differen-

Conditions (B). (1) Let the operator B W W1 ! W1 be monotonic and semilinear. (2) The following upper and lower estimates hold: jB.u/j1 Cjuj1

qC1

where C > 0 and c > 0;

;

qC2

.B.u/; u/1 cjuj1

;

q > 0;


75

(3) .B.u/; u/1=.qC2/ is a norm on W1 , which is equivalent to the initial norm of this 1 Banach space. (4) The operator B is positively homogeneous of degree q C 1: B.sv/ D s qC1 B.v/;

s > 0:

(5) The operator B has a symmetric Fréchet derivative for any fixed u 2 W1 : Bu0 .u/ W W1 ! L.W1 ; W1 /: (6) The functional .u/ .B.u/; u/1 W W1 ! R1 is continuously Fréchet differentiable. Note that conditions (A0 ) imply the fact that the value hA0 u; ui1=2 0 is a norm on V0 . Conditions (V1) imply the following. Conditions (V2). (1) hAi .v/; wi D hAi .v/; wii for all v; w 2 V , i D 0; N ; (2) hF.v/; wi D .F.v/; w/0 for all v; w 2 V . Now let us prove the following lemma. Lemma 2.2.10. If an operator A W X ! X is Fréchet differentiable and has a symmetric Fréchet derivative A0u .u/ W X ! L.X; X / and A.su/ D s p1 A.u/ for s 0 and p 2, where X is a Banach space and X is the dual space with the duality bracket h; i, then the functional .u/ hA.u/; ui W X ! R1 is continuously Fréchet differentiable and its Fréchet derivative is 0 f

.u/ D pA.u/

for all u 2 X:

Proof. Let us prove the following equality: A0u .u/u D .p 1/A.u/: Indeed, the following equalities hold: A.u/ D p1 A.u/; A0u .u/ D p1 A0u .u/; A0u .u/ D p2 A0u .u/;

Z 1

Z 1 d 0 A.u/; w D d d Au .u/u; w 8w 2 X; d 0 0 hA.u/ .p 1/1 A0u .u/u; wi D 0

8w 2 X:

76


Therefore, the following operator equality holds: .p 1/A.u/ D A0u .u/u

8u 2 X:

We have .u C h/

.u/ D hA.u C h/; u C hi hA.u/; ui D hA.u/ C A0u .u/h C !.u; h/; u C hi hA.u/; ui D hA.u/; hi C hA0u .u/h C !.u; h/; u C hi D hA.u/; hi C hA0u .u/h; ui C hA0u .u/h; hi C h!.u; h/; u C hi D hA.u/ C A0u .u/u; hi C !.u; h/;

where

!.u; h/ D hA0u .u/h; hi C h!.u; h/; u C hi:

Moreover, j!.u; h/j kA0u .u/hk khk C k!.u; h/k Œkuk C khk c1 khk2 C k!.u; h/k Œkuk C khk: Hence,

j!.u; h/j D 0: khk khk!0 lim

Therefore, the Fréchet derivative of the functional 0 f

.u/ is

.u/ D A.u/ C A0u .u/u D A.u/ C .p 1/A.u/ D pA.u/:

This implies that the functional

.u/ is continuously Fréchet differentiable.

Remark 2.2.11. This auxiliary lemma implies that conditions (A6), (A0 5), (F5), and (B6) are satisfied automatically. Lemma 2.2.12. Let the conditions of Lemma 2.2.10 hold and assume that u.t / 2 C .1/ .Œ0; TI X/ for some T > 0. Then .u/.t / hA.u/; ui 2 C .1/ .Œ0; T/: Proof. First, by the chain rule for the Fréchet derivative and Lemma 2.2.10, we have the equality d D h f0 .u/; u0 i D phA.u/; u0 i: dt Let us consider the function f .t / hA.u/; u0 i


77

and prove that f .t / 2 C.Œ0; T/. Indeed, let t 2 Œ0; T be fixed and t C s 2 Œ0; T; then f .t C s/ f .s/ D hA.u.t C s//; u0 .t C s/i hA.u.t //; u0 .t /i D hA.u.t // C A0u .u.t //Œu.t C s/ u.t / C !.t; s/; u0 .t C s/i hA.u.t //; u0 .t /i D hA.u.t //; u0 .t C s/ u0 .t /i C hA0u .u.t //Œu.t C s/ u.t /; u0 .t C s/i C h!.t; s/; u0 .t C s/i: From this we obtain the following estimates: jf .t C s/ f .t /j kA.u.t //k ku0 .t C s/ u0 .t /k C kA0u .u.t //kL.XIX / ku.t C s/ u.t /kku0 .t C s/k C k!.t; s/k ku0 .t C s/k: Note that ku0 .t C s/k ku0 .t /k C ku0 .t C s/ u0 .t /k c1 ; where c1 > 0 is independent of t; s 2 Œ0; T. Thus we have the estimate jf .t C s/ f .t /j c2 ku0 .t C s/ u0 .t /k C c3 ku.t C s/ u.t /k C c4 k!.t; s/k ; where c2 ; c3 ; c4 2 .0; C1/ depend only on t 2 Œ0; T. Moreover, lim

ku.tCs/u.t/k!0

k!.t; s/k D 0:

Thus, we have lim f .t C s/ D f .t /:

s!0

Therefore, f .t / 2 CŒ0; T. Remark 2.2.13. Note that it can be proved that if u.t / 2 C.Œ0; T/ and the conditions of Lemma 2.2.10 hold, then .u/.t / D hA.u.t //; u.t /i 2 C.Œ0; T/. Lemma 2.2.14. Let the conditions of Lemma 2.2.10 and the conditions (A0 ) and (A) hold. Assume that u.t / 2 C .1/ .Œ0; TI V /. Then X pj 1 1 hA0 u.t /; u.t /i0 C hAj .u.t //; u.t /ij 2 C .1/ .Œ0; T/: 2 pj N

ˆ.t / D

j D1

78


Proof. Indeed, from the conditions (A6) and (A0 5) it follows that the functionals 1 j .u/ hAj .u/; uij W Vj ! R , j D 0; N , are continuously Fréchet differentiable. Then from Lemma 2.2.10 we have 0 jf

.u/ D pj Aj .u/;

j D 0; N :

Hence, from Lemma 2.2.12 we have ˆ0 .t / D hA0 u.t /; u0 .t /i0 C

N X

.pj 1/hAj .u.t //; u0 .t /ij 2 CŒ0; T:

j D1

The lemma is proved. Lemma 2.2.15. If u.t / 2 C .1/ .Œ0; TI V / and the conditions (F) and (B) hold, then the functionals .u/ .F.u/; u/0 W W0 ! R1 and .u/ .B.u/; u/1 W W1 ! R1 ; introduced in the conditions (F5) and (B6) belong to the class C .1/ .Œ0; T/.

.u/.t /; .u/.t / 2

Proof. This follows from Lemmas 2.2.10 and 2.2.14.

2.3

Unique solvability of problem (2.1) in the weak generalized sense and blow-up of its solutions

Definition 2.3.1. A solution satisfying the conditions Z

T 0

N d X .t / hAj .u/; wij .F.u/; w/0 dt D 0 dt

j D0

8w 2 V ;

8 2 L2 .0; T/;

(2.9)

u.0/ D u0 2 V is said to be a weak generalized solution of the abstract Cauchy problem (2.1). We search for a solution of problem (2.9) in the class u.t / 2 L1 .0; TI V /; Aj .u/ 2 L1 .0; TI Vj /;

j D 0; N ;

du 2 L2 .0; TI V0 /; dt N d X d A.u/ D Aj .u/ 2 L2 .0; TI V /; dt dt j D0

79

Section 2.3 Weak generalized solvability of problem (2.1)

and, moreover, A.u/ D

N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /;

j D0

i.e., the operator A.u/ is strongly absolutely continuous and weakly differentiable on the segment Œ0; T. Therefore, because of the properties (A), (A0 ), and (F) and the conditions of Theorem 2.3.2 below, we obtain that A.u/ D

N X

Aj .u/ 2 H1 .0; TI V /;

F.u/ 2 L2 .0; TI V /;

j D0

A.u/ D

N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /:

j D0

Thus, by virtue of the conditions (V) and the conditions of Theorem 2.3.2 below, problem (2.9) is equivalent to the following problem: Z

T 0

N d X .t / hAj .u/; wi hF.u/; wi dt D 0 dt

8w 2 V ;

8 .t / 2 L2 .0; T/;

j D0

u.0/ D u0 2 V ; where h; i is the duality bracket between the Banach spaces V and V . Using the result of Appendix A.12, we conclude that problem (2.9) is equivalent to the following:

Z T d A.u/; v hF.u/; vi D 0 8v 2 L2 .0; TI V /; u.0/ D u0 2 V : dt dt 0 Really, it suffices to verify that for h A.u/, the following relation holds: Z T Z T d dt .t / hh; wi D dt .t /hh0 ; wi 8w 2 V : dt 0 0 To prove this relation, we consider the following difference equality:

h.t C ˛/ h.t / 1 Œhh.t C ˛/; wi hh.t /; wi D ;w 8t; t C ˛ 2 .0; T/: ˛ ˛ .1/ The limit of the right-hand side exists since h 2 Cw .Œ0; TI V / as ˛ ! C0. Thus, the limit of the left-hand side also exists. According to the Lebesgue theorem, we obtain the required result. Now we point at the place of the proof where we need the condition .1/ .Œ0; TI V /: h A.u/ 2 H1 .0; TI V / \ Cw

80


This condition is sufficient for the fact that hh; wi 2 H1 .0; T/ AC.Œ0; T/

8w 2 V ;

and, thus, the following formula of integrating by parts is valid: Z hh; wi.t / D hh.0/; wi C

t

ds 0

d hh.s/; wi ds

8w 2 V :

Indeed, by virtue of the fact that h.t / 2 H1 .0; T/I V /, the following estimates are valid: Z Z

T 0 T

0

Z dt jhh.t /; wij ˇ˝ ˛ˇ dt ˇ h0 .t /; w ˇ

Finally, the condition

Z

T

0 T 0

dt kh.t /k2 0

1=2

2

dt kh .t /k

T1=2 kwk; 1=2 T1=2 kwk:

.1/ .Œ0; TI V / h.t / 2 Cw

is sufficient for the validity of the following equality for almost all t 2 .0; T/: d hh.t /; wi D hh0 .t /; wi 8w 2 V : dt Theorem 2.3.2 (necessary and sufficient condition of the blow-up). Let the conditions (A), (A0 ), and (F) hold. Assume that V V0 ,! W0 , i.e., the inclusion operator is a compact operator. Assume that either 2 < p < q C 2 or p q C 2 and in the latter case, Vj W0 , where p D max pj ; j 21;N

j 2 1; N ;

pj D p:

Let the inequality .F.u0 /; u0 /0 > 0 hold. Then for any u0 2 V , there exists the maximal T0 Tu0 > 0 such that the Cauchy problem (2.1) has a unique weak generalized solution of the class u.t / 2 L1 .0; TI V /; A.u/ D

N X j D0

du .t / 2 L2 .0; TI V0 /; dt

1 Aj .u/ 2 H1 .0; TI V / \ Cw .Œ0; TI V /

8T 2 .0; T0 /:

81


For the function X pj 1 1 hAj .u/; uij ; ˆ.t / hA0 u; ui0 C 2 pj N

j D1

which is positive-definite (by virtue of the conditions (A0 2) and (A3)) and has the sense of the kinetic energy, in the three cases corresponding to possible values of the variable qC2 ; p max pj ; ˛ p j D1;N we have the following two-sided estimates: (1) if ˛ D 1, then T0 D C1 and ˆ0 exp¹E0 t º ˆ.t / ˆ0 exp¹C2 t ºI (2) if ˛ 2 .0; 1/, then T0 D C1 and ˆ0 Œ1 C .1 ˛/E0 ˆ˛1 t 1=.1˛/ ˆ.t / 0 t 1=.1˛/ I ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 0 (3) if ˛ > 1, then lim sup ˆ.t / D C1;

T1 T0 T2 ;

t"T0

where C2

BCjqC2

p p1

.qC2/=p

1 ˆ0 hA0 u0 ; u0 i0 C 2

; N X j D1

1=pj

jvj0 Cj hAj .v/; vij

pj 1 hAj .u0 /; u0 ij ; pj

q=2 T1 D .q=2/1 ˆ0 B

1

;

T2

E0

p D max pj ; j D1;N

.F.u0 /; u0 /0 ; ˆ˛0

1 ˆ0 ; ˛ 1 .F.u0 /; u0 /0

qC2 .qC2/=2

B BC1

;

2

;

C1 is the constant of the best embedding V ,! W0 , and B is the constant from the condition (F4). Remark 2.3.3. Since the solution belongs to the class u.t / 2 L1 .0; TI V /;

du 2 L2 .0; TI V0 /; dt

after a possible change on a set of zero Lebesgue measure, the mapping u.t / W Œ0; T ! V0 becomes continuous. Thus, the initial equation u.0/ D u0 has sense.

82


Proof. We prove Theorem 2.3.2 in several steps: we construct Galerkin approximations, obtain a priori estimates, apply the monotonicity method, prove the uniqueness of solution, and finally prove the blow-up of solution. Step 1. Galerkin approximations. By the separability of the Banach space V , there exists a countable, everywhere dense in V , linearly independent system of functions ¹wi ºm iD1 . We prove the solvability of problem (2.9) by using the Galerkin method and the monotonicity and compactness methods [275]. First, consider the following finite-dimensional approximation of problem (2.9): Z T 0

N d X d hAj .um /; wk ij .F.um /; wk /0 .t / dt D 0 hA0 um ; wk i0 C dt dt j D1

(2.10) for all .t / 2 L2 .0; T/, k D 1; m, um D

m X

cmi .t /wi ;

um0 D

i D1

cmi .0/ D ˛mi ;

m X

cmi .0/wi ;

i D1

um0 ! u0

strongly in V :

If cmk .t / belong to the class C .1/ .Œ0; Tm0 /, by virtue of (2.10) we obtain the relation Z TX m 0

iD1

0 cmi

N X 0 hA0 wi ; wk i0 C hAj;um .um /wi ; wk ij .F.um /; wk /0 .t / dt D 0: j D1

Below, we prove the continuity in t 2 Œ0; T of the functionals 0 fj hAj;u .um /wi ; wk ij : m

Thus, the expression in the brackets belongs to the class CŒ0; T. By the obvious inclusion C01 Œ0; T L2 .0; T/ and the principal lemma of the calculus of variations, we obtain the equality m X iD1

N X 0 0 hA0 wi ; wk i0 C cmi hAj;u .u /w ; w i m i k j D .F.um /; wk /0 ; m

k D 1; m

j D1

(2.11) which holds pointwise for all t 2 Œ0; T. Introduce the notation aik hA0 wi ; wk i0 C

N X

0 hAj;u .um /wi ; wk ij : m

j D1

83


It is obvious that m;m X i;kD1;1

aik i k D hA0 ; i0 C

N X

0 hAj;u .um /; ij hA0 ; i0 ; m

D

j D1

m X

i wi ;

i D1

since A0 is a positive definite operator; therefore, hA0 ; i0 0 and the equality hA0 ; i0 D 0 can hold if and only if D 0. On the other hand,

m X

i wi ;

i D1

and by the linear independence of the system of functions ¹wi ºm i D1 in V , we see that D 0. This implies (see also [335]) that det¹aik ºm;m > 0. D 0 if and only if ¹i ºm i D1 i;kD1;1 0 .u /w ; w i are continuous with Now we prove that the functionals fj hAj;u m i k j m respect to the collection of variables cmi , i D 1; m. Really, let c m1 ; : : : ; c mm be a point of the Euclidean space Rm . Fix arbitrary " > 0. The following inequality holds: jfj .c m1 ; : : : ; c mm / fj .cm1 ; : : : ; cmm /j jfj .c m1 ; : : : ; c mm / fj .cm1 ; c m2 ; : : : ; c mm /j C jfj .cm1 ; c m2 ; : : : ; c mm / fj .cm1 ; cm2 ; c m3 ; : : : ; c mm /j C jfj .cm1 ; cm2 ; c m3 : : : ; c mm / fj .cm1 ; cm2 ; cm3 ; : : : ; c mm /j C C jfj .cm1 ; cm2 ; : : : ; c.m1/m ; c mm / fj .cm1 ; : : : ; cmm /j:

(2.12)

By the condition (A2), there exists ı."/ > 0 such that each summand in the right-hand side of inequality (2.12) under the condition m X

jc mk cmk j ı."/

kD1

is less than "=.m C 1/. Now we prove the Lipschitz continuity of the functionals f0k D .F .um /; wk /0 with respect to the collection of variables ¹cmi ºm i D1 . Let uj D

m X i D1

j cmi wi ;

j D 1; 2:

By the condition (F1), the following inequalities hold: j.F .u1 / F.u2 /; wk /0 j jF.u1 / F.u2 /j0 jwj0 Bju1 u2 j0 B1

m X lD1

1 2 jcml cml j:

84


Thus, the functions f0k D f0k .cm1 ; : : : ; cmm / are Lipschitz-continuous and, therefore, continuous with respect to the set of variables. Note that the inverse matrix of the matrix N X 0 hAj;u .um /wi ; wk ij aik hA0 wi ; wk i0 C m j D1

is continuous with respect to cm D .cm1 ; : : : ; cmm / (see Appendix A.18). Therefore, the system of ordinary differential equations (2.11) is a system of Cauchy– Kovalevskaya type and satisfies the conditions that guarantee its solvability on a certain segment Œ0; Tm0 , Tm0 > 0, in the class cmk .t / 2 C 1 .Œ0; Tm0 /, k D 1; m (see, e.g., [319]). Step 2. A priori estimates Lemma 2.3.4. There exists T > 0 independent of m 2 N such that the sequence ¹um º of Galerkin approximations satisfies the following properties uniformly with respect to m 2 N: um

is bounded in L1 .0; TI V /;

u0m

is bounded in L2 .0; TI V0 /;

A 0 um


Aj .um /

is bounded in L1 .0; TI Vj /;

F.um /

is bounded in L1 .0; TI W0 /.

Proof. Multiplying both parts of (2.11) by cmk .t / and summing over k D 1; m, we obtain hA0 u0m ; um i0

C

N X

0 hAj;u .um /u0m ; um ij D .F.um /; um /0 : m

(2.13)

j D1

On the other hand, by the fact that um D

N X

cmk .t /wk 2 C 1 .Œ0; Tm I V /

kD1

and the conditions (A), the following equalities hold (see, e.g., [260]): d hAj .um /; um ij D pj hAj .um /; u0mt ij ; dt h.Aj .um //0t ; um ij C hAj .um /; u0mt ij D pj hAj .um /; u0mt ij ; h.Aj .um //0t ; um ij D .pj 1/hAj .um /; u0mt ij D

pj 1 d hAj .um /; um ij : pj dt

85


We prove the relation d hAj .um /; um ij D pj hAj .um /; u0mt ij : dt We have

Z Jmj .t /

d Jmj .t / D dt

Z Z

D Z D

1 0

1 0 1 0

ds hAj .sum /; um ij D

ds shA0sum .sum /u0m ; um ij C hAj .sum /; u0m ij

d ds s hAj .sum /; u0m ij C hAj .sum /; u0m ij ds

1

ds 0

1 hAj .um /; um ij ; pj

d ŒshAj .sum /; u0m i D hAj .um /; u0m ij : ds

Here we have used the fact that the Fréchet derivatives of the operators Aj are symmetric by the conditions (A). From this and (2.13) we obtain N X d 1 pj 1 hAj .um /; um ij D .F.um /; um /0 : hA0 um ; um i0 C dt 2 pj

(2.14)

j D1

Integrating both parts over t 2 .0; Tm /, we obtain from (2.14) X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj N

(2.15)

j D1

X pj 1 1 hAj .um0 /; um0 ij C D hA0 um0 ; um0 i0 C 2 pj N

j D1

Z 0

t

ds .F.um /; um /0 :

By the condition (A3), in the Banach space V0 , a norm equivalent to the initial norm can be found: 1=2

kvk0 D hA0 v; vi0 :

(2.16)

On the other hand, V0 ,! W0 and qC2

j.F.um /; um /0 j Bjum j0

:

86


From (2.15) and (2.16) we obtain X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj N

j D1

X pj 1 1 hA0 um0 ; um0 i0 C hAj .um0 /; um0 ij 2 pj N

(2.17)

j D1

Z C B2

.qC2/=2

C1

qC2

.qC2/=2 N X pj 1 1 hA0 um ; um i0 C ds hAj .um /; um ij : 2 pj 0 t

j D1

Thus, we have obtained the first a priori estimate of solutions of the Galerkin approximations (2.11). 0 , summing Now we derive the second a priori estimate. Multiplying (2.11) by cmk over k D 1; m, and integrating both parts of the relation obtained over t 2 .0; Tm / we have Z

t 0

N X 0 0 0 0 ds hA0 um ; um i0 C h.Aj .um // ; um ij C j D1

1 .F.um0 /; um0 /0 qC2

D

1 .F.um /; um /0 : (2.18) qC2

By the fact that .F.u0 /; u0 /0 > 0, there exists a subsequence of the sequence ¹um0 º such that .F .um0 /; um0 /0 > 0. From this and (2.18) we have .F.um /; um /0 > 0:

(2.19)

X pj 1 1 hAj .um /; um ij ; ˆm hA0 um ; um i0 C 2 pj

(2.20)

Introduce the notation N

j D1

ˆm0 ˆm .0/:

(2.21)

Therefore, by (2.14) and (2.19)–(2.21), we have ˆ0m .t / 0. From (2.17) we obtain Z ˆm ˆm0 C B

t 0

ds ˆ.qC2/=2 .s/; m

where qC2 .qC2/=2

B BC1

2

;

(2.22)

87


C1 is the constant of the best inclusion V ,! W0 , and B is the constant from the condition (F4). From (2.22) and the Bihari theorem (see, e.g., [112]) we have ˆm

ˆm0 2=q Œ1 q2 ˆq=2 m0 Bt

:

(2.23)

Since um0 ! u0 strongly in V , ˆm0 C0 is independent of m 2 N. The following two cases for a certain subsequence of the sequence ¹um º are possible: either ˆm0 # ˆ0 or ˆm0 " ˆ0 . Consider the first case where ˆm0 " ˆ0 . In this case, the following equalities hold: 1

q q q=2 Bˆq=2 m0 t 1 Bˆ0 t 2 2 ˆ m C1

8t 2 .0; T1 /;

8t 2 .0; T/;

T1 D B

1 2

q=2

; q 1 2 q=2 T 2 .0; T1 /; T1 D B ; ˆ q 0 ˆ0

(2.24)

where C1 is independent of m 2 N. Consider the case where ˆm0 # ˆ0 . For any m < m, we have ˆm0 > ˆm0 ; 1 2 q=2 q ˆm0 /,

For any t 2 Œ0; B

q q q=2 1 Bˆq=2 m0 t 1 Bˆm0 t: 2 2

we obtain

ˆm

Œ1

C0 : q q=2 2=q ˆ Bt 2 m0

Thus, for any fixed T from the interval T 2 .0; T1 /, we can find m 2 N such that inequality (2.24) holds. By the fact that such pj D maxj D1;N pj > 2 can be found and pj q C 2 and Vj W0 , the following inequality holds: 1=pj


:

From (2.25) we immediately obtain that .qC2/=pj pj qC2 qC2 jvj0 Cj pj 1 .qC2/=pj N X pj 1 1 hAj .v/; vij : hA0 v; vi0 C 2 pj

(2.25)

(2.26)

j D1

Using (2.15), (2.21), (2.25), and (2.26) we conclude that .qC2/=pj Z t pj ˛ qC2 ˆm ˆm0 C BCj ds ˆmj .s/; pj 1 0

(2.27)

88


where ˛j

qC2 : pj

Consider the following two cases: ˛j < 1 and ˛j D 1. Using the Gronwall– Bellman and Bihari theorems (see, e.g., [112]), from (2.27) we obtain 1˛j

ˆm Œˆm0

C .1 ˛j /C2 t 1=.1˛j / ;

ˆm ˆm0 exp¹C2 t º;

˛j 2 .0; 1/;

˛j D 1;

where qC2 C2 BCj

(2.28) (2.29)

pj pj 1

.qC2/=pj :

By the fact that um0 ! u0 strongly in V , we have that ˆm0 C0 is dependent of m 2 N. Thus, we obtain ˆm0 C3

8T 0;

8˛j 2 .0; 1:

(2.30)

From the conditions (A3) and (2.20), (2.24), and (2.30) we have that qC2 .F.um /; um /0 Bjum j0 .qC2/=2 N X pj 1 .qC2/=2 1 BCqC2 2 u ; u i C hA .u /; u i hA 0 m m 0 j m m j 1 2 pj j D1

C3 ˆ.qC2/=2 m

C30 ;

where 0 < C30 < C1 is the constant independent of m 2 N, for any T > 0 in the case where ˛j 2 .0; 1, and for any T 2 .0; T1 /, where T1 is defined in (2.24), in the case where ˛j > 1. Now from (2.18) and the conditions (A2) and (A0 2) we have Z m0

t 0

ds ku0m k20

Z

t 0

ds hA0 u0m ; u0m i0 C40 ;

where 0 < C40 < C1 is the constant independent of m 2 N for any T > 0 in the case where ˛j 2 .0; 1 and for any T 2 .0; T1 /, where T1 is defined in (2.24), in the case where ˛j > 1. Thus, we have obtained the second a priori estimate. Thus, um is bounded in L1 .0; TI V /; u0m is bounded in L2 .0; TI V0 /;

(2.31)

where the inclusions hold for any T > 0 in the case where ˛j 2 .0; 1 and for any T 2 .0; T1 /, where T1 is defined in (2.24), in the case where ˛j > 1.

89


Therefore, by (2.31), there exists a subsequence of the sequence ¹um º such that um * u

-weakly in L1 .0; TI V /;

(2.32)

u0m * u0 weakly in L2 .0; TI V0 /: Note that (2.32) implies

u0m * u0

in D 0 .0; TI V /I

therefore, by the weak convergence in L2 .0; TI V0 /, u0m * ; we see that .t / D u0 .t / for almost all t 2 .0; T/. From (2.32) and the compact inclusion V0 ,! W0 , using Lemma A.15.1 (see Appendix A.15) in which we set W D W0 , we obtain um .s/ ! u.s/

strongly in W0 for almost all s 2 Œ0; T:

On the other hand, by virtue of the condition (F1), we have jF.um / F.u/j0 .R/jum uj0 ! 0

as m ! C1;

(2.33)

F.um /.t / ! F.u/.t / strongly in W0 for almost all t 2 .0; T/:

(2.34)

where R D max¹juj0 ; jum j0 º. From (2.33) we obtain

Step 3. Monotonicity method. Now we apply the monotonicity method. Recall, that on the Step 2, we have obtained the limit inclusions (2.32) and (2.34) and, moreover, A.um / *


A.u/

N X

Aj .u/:

j D0

Lemma 2.3.5. Let A.u/

N X

Aj .u/:

j D0

Then there exists a subsequence of the sequence ¹um º such that A.um / * A.u/ -weakly in L1 .0; TI V / for certain T > 0. Moreover, for a certain subsequence of the sequence ¹um º, um ! u

strongly in Lpj .0; TI Vj /;

j D 0; N :

(2.35)

90


Proof. Rewrite Eq. (2.10) in the following equivalent form: Z hA.um /; wj i D hA.um0 /; wj i C

t

ds .F.um /; wj /0 ;

0

(2.36)

where h; i is the duality bracket between the Banach spaces V and V . Fix j 2 N and pass to the limit as m ! C1 in Eq. (2.36). We obtain Z D

t

ds F.u/ C A.u0 /:

0

(2.37)

Let Z 0 X

0

Z D Z

T 0

T

0

T

dt hA.u / A.v/; u vi Z dt hA.u /; u i Z

dt hA.u /; u i D

T 0

T 0

Z dt hA.u /; vi Z

dt hA.u 0 /; u i C

Z

T

t

dt 0

0

T

dt hA.v/; u vi;

0

ds F.u /.s/; u .t / 0 :

This implies Z 0 lim sup X !C1

dt 0

Z

T 0

T 0

0

Z

C D

Z

T

t

ds .F.u/.s/; u.t //0 Z

dt hA.u0 /; ui

T 0

Z dt h ; vi

dt h A.v/; u vi:

0

T

dt hA.v/; u vi (2.38)

Now we set v D u w for any v; w 2 Lr .0; TI V /, > 0, u 2 L1 .0; TI V / 0 Lr .0; TI V /, ; A.v/ 2 Lr .0; TI V /, r > 1, and r 0 D r=.r 1/. By (2.38), the following inequality holds: Z 0

T

dt h A.u w/; wi 0;

from which, by the semicontinuity of the operators Aj .v/, j D 0; N , we obtain D A.u/:

91


On the other hand, by (2.37), Z T lim sup dt hAj .u /; u ij

(2.39)

!C1 0

N X

lim inf

!C1

kD0;k¤j

N X kD0;k¤j

Z

T 0

Z

T

Z

T

dt hAk .u /; u ik C lim sup

!C1 0

0

Z dt hAk .u/; uik C

T 0

where we have used the fact that 1=pj Z T dt hAj .u/; uij ;

Z dt hA.u/; ui D

Z

0

dt hA.u /; u i

T 0

T

0

dt hAj .u/; uij ;

1=2 dt hA0 u; ui0

;

by the properties (A0 3) and (A3), are norms on the uniformly convex Banach spaces Vj and V0 , respectively. Finally, by the fact that u * u weakly in Lpj .0; TI V /, j D 0; N , and the fact that the operators Aj .v/ generate norms on the uniformly convex Banach spaces Vj , j D 0; N , according to the above rule, we have Z T Z T dt hAj .u /; u ij dt hAj .u/; uij : (2.40) lim inf !C1 0

0

From (2.39) and (2.40) we obtain Z T Z dt hAj .u /; u ij D lim !C1 0

T 0

dt hAj .u/; uij :

(2.41)

Now, by the conditions (A3), we see that Z

1=pj

T

dt hAj .u/; uij

0

is a norm equivalent to the initial norm Z

T

dt 0

p kukj j

1=pj

of the Banach space Lpj .0; TI Vj /. Therefore, (2.41) implies that there exists a subsequence of the sequence ¹um º such that Z

T 0

Z

1=pj dt hAj .um /; um ij

!

0

T

1=pj dt hAj .u/; uij

;

j D 0; N :

92


On the other hand, um * u

weakly in Lpj .0; TI Vj /;

j D 0; N I

therefore, there exists a subsequence of the sequence ¹um º such that um ! u strongly in any Lpj .0; TI Vj /, j D 0; N . Step 4. Passage to the limit and the uniqueness. From the limiting expressions (2.32), (2.34), and (2.35), we obtain the possibility of the passage to the limit as m ! C1 in Eq. (2.36). Let fmk .t / D

N X

hAj .um /; wk ij

j D0

fk .t / D

N X

N X

Z hAj .um0 /; wk ij

j D0

hAj .u/; wk ij

j D0

N X

Z hAj .u0 /; wk ij

j D0

0

t

0

ds .F.um /.s/; wk /0 ;

t

ds .F.u/.s/; wk /0 :

We see that fmk .t / * fk .t / -weakly in L1 .0; T/, i.e., Z T Z T dt fmk .t /g.t / ! dt fk .t /g.t / 8g 2 L1 .0; T/: 0

0

On the other hand, fmk .t / D 0 for all t 2 Œ0; T. Therefore, Z T dt f .t /g.t / D 0 8g 2 L1 .0; T/: 0

Using the obvious inclusions L1 .0; T/ L2 .0; T/ and C01 .Œ0; T/ L1 .0; T/ for any finite 0 < T < C1 and the principal lemma of the calculus of variations, we obtain that f .t / D 0 for almost all t 2 .0; T/. As a result of the passage to the limit above, for almost all t 2 .0; T/ we obtain hA0 u; vi0 C

N X

hAj .u/; vij D hA0 u0 ; vi0 C

j D1

Z C

hAj .u0 /; vij

j D1

t 0

N X

ds .F.u/; v/0 .s/

8v 2 V ;

u.0/ D u0 2 V : On the other hand, u0 2 L2 .0; TI V0 /, u 2 L1 .0; TI V /. We rewrite the latter relation in the following form: N X j D0

hAj .u/; vij D

N X

Z hAj .u0 /; vij C

j D0

t 0

ds .F.u/; v/0 .s/

8v 2 V :

93


Therefore, for almost all t 2 .0; T/, the following equality in the sense of V holds: A.u/ D

N X

Aj .u/ D

j D0

N X j D0

Z Aj .u0 / C

t 0

ds F.u/.s/

(2.42)

for almost all t 2 .0; T/. Note that the right-hand side of (2.42) belongs to the class C.Œ0; TI V /; therefore, the left-hand side, after a redefinition on a set of zero Lebesgue measure set on the interval .0; T/, will also belong to the class C.Œ0; TIV /. We have proved above that A.u/ D

N X

Aj .u/ 2 L1 .0; TI V / L2 .0; TI V /:

j D0

Moreover, from (2.42) we obtain N d X Aj .u/ 2 L2 .0; TI V /: dt j D0

Therefore, A.u/ 2 H1 .0; TI V /. Finally, we prove that N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /;

j D0 .1/

where Cw .Œ0; TI V / denotes the set of all functions demicontinuous on the segment Œ0; T with values in a reflexive Banach space B that have demicontinuous weak derivatives. After a redefinition on a set of zero Lebesgue measure on the interval .0; T/, we obtain that A.u/ 2 C.Œ0; TI V /. Finally, the following expression for the function h.t / D A.u/ holds:

Z tC˛

1 h.t C ˛/ h.t / ds F.u/.s/ F.t /; w J F.u/.t /; w D ˛ ˛ t

for all w 2 V , a.a. t 2 .0; T/. The following estimate for almost all t 2 .0; T/ holds: 1 jJj ˛

Z

tC˛ t

ds kF.u/.s/ F.u/.t /k kwk ! C0

as ˛ ! C0 by virtue of the result of [198, Theorem 3.8.5]. Therefore, we have u.t / 2 C.Œ0; TI V0 / C.Œ0; TI W0 /:

94


Now we prove that F.u/ 2 C.Œ0; TI W0 /. Indeed, by virtue of the inclusion V0 W0 and the conditions (F) we have jF.u/.t / F.u/.t0 /j0 Cju.t / u.t0 /j0

8t; t0 2 Œ0; T:

Hence we obtain the required result. Since W0 V , we conclude F.u/.t / 2 C.Œ0; TI V /. Therefore, .1/ .Œ0; TI V /: A.u/ 2 Cw

Now we consider Eq. (2.10) rewritten in the form Z

T 0

N

d X dt .t / Aj .um /; w hF.um /; wi D 0 dt

j D0

8 .t / 2 L2 .0; T/;

8w 2 V :

By Lemma 2.3.4 we have: F.um / * F.u/ On one hand,

weakly in L2 .0; TI V /:

A.um / * A.u/ weakly in L2 .0; TI V /:

Therefore, d d A.um / ! A.u/ dt dt in the sense of D 0 .0; TI V /. On the other hand, the relation d A.um / D F.um / dt holds pointwise by t 2 Œ0; T in the sense of V . Therefore, d A.um / * dt


Thus, d d A.um / * A.u/ weakly in L2 .0; TI V /: dt dt Passing to the limit as m ! C1 in Eq. (2.10), we obtain that u.t / is a solution of the problem Z

T 0

N d X dt .t / Aj .u/; w hF.u/; wi D 0 dt j D0

which is equivalent, in its turn, to problem (2.9).

8 .t / 2 L2 .0; T/;

8w 2 V ;

95


Z

Now let u be an arbitrary solution of the problem X

N T d dt .t / Aj .u/; w hF.u/; wi D 0 8 .t / 2 L2 .0; T/; dt 0

8w 2 V ;

j D0

belonging to the class u 2 L1 .0; TI V /;

N X

.1/ Aj .u/ 2 H1 .0; TI V / \ Cw .Œ0; TI V /;

j D0

F .u/ 2 L1 .0; TI V /: We have proved above that u satisfies the following relation: Z t N N X X hAj .u/; wij D hAj .u0 /; wij C ds .F.u/; w/0 .s/ j D0

0

j D0

8w 2 V :

Prove that in the class of functions satisfying this relation, the solution of our problem is unique. Indeed, Z t N N X X Aj .u/ D Aj .u0 / C dsF.u/.s/ (2.43) j D0

0

j D0

in the sense of V for almost all t 2 .0; T/. Let u1 and u2 be two generalized solutions in the smoothness class considered that correspond to the same function u0 2 V . Then from (2.43) we obtain the relation Z t N X ds .F.u1 /.s/ F.u2 /.s// Aj .u1 / Aj .u2 / D 0

j D0

for almost all t 2 .0; T/ in the sense of V . Since w D u1 u2 2 L1 .0; TI V /, for almost all t 2 .0; T/ the following expression is defined: Z t N X hA0 w; wi0 C hAj .u1 / Aj .u2 /; wi D ds .F.u1 /.s/ F.u2 /.s/; w.t //0 : 0

j D1

By the monotonicity of the operators Aj .u/, j D 1; N , the condition (A0 2), and the inclusion V0 W0 , we obtain the inequality Z t 2 1 kwk0 .t / m0 jwj0 .t / ds jF.u1 / F.u2 /j0 Z D1 kwk0 .t /

0 t

0

ds kwk0 .s/;

D1 > 0;

from which, by the Gronwall–Bellman theorem, we obtain u1 D u2 almost everywhere on the set .0; T/ .

96


Remark 2.3.6. Note that in the proof of the uniqueness, the presence of the linear operator A0 on time derivative with domain of definition V0 W0 is substantial. If this operator is absent, the nonuniqueness can occur. For completeness of the exposition, we present the following example. Example (example of nonuniqueness). Consider the following problem: @ .div.jrujp2 ru/ jujp2 u/ C jujq u D 0; @t uj@ D 0; u.x; 0/ D 0;

(2.44)

where p > q C 2. We search for nontrivial solutions of the problem (2.44) in the class u 2 C .1/ .Œ0; TI W01;p .//. Let u.x; t / D .t /f .x/. Assume that the following relation holds: d .j jp2 / D j jq ; dt

.0/ D 0;

p > q C 2I

(2.45)

then from (2.44) we obtain p f jf jp2 f C jf jq f D 0;

f j@ D 0:

(2.46)

Solutions of problem (2.45) are as follows: ´ p 2 q 1=.p2q/ .t T0 /1=.p2q/ ; t T0 ; .t / D ˙ p1 0; t 2 Œ0; T0 : Now we prove the solvability of problem (2.46). Using the well-known method of fibering functionals of Pokhozhaev [332] and the Lyusternik–Schnirelman category theory [285], we conclude that problem (2.46) have a countable set of linear independent solutions. Step 5. Blow-up of solutions Lemma 2.3.7. Let X pj 1 1 hA0 u; ui0 C hAj .u/; uij ; 2 pj N

ˆ

j D1

˛

qC2 ; p

p max pj : j D1;N

Then (1) if ˛ 2 .0; 1/, then T0 D C1 and the following inequality holds: t 1=.1˛/ ˆ ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 t 1=.1˛/ I ˆ0 Œ1 C .1 ˛/E0 ˆ˛1 0 0

97


(2) if ˛ D 1, then T0 D C1 and the following inequality holds: ˆ0 exp.E0 t / ˆ ˆ0 exp.C2 t /I (3) if ˛ > 1, then T0 2 ŒT1 I T2 and the following limit relation holds: lim sup ˆ.t / D C1; t"T0

where 2 q=2 1 T1 D ˆ0 B ; q

ˆ0 1 T2 D ; ˛ 1 .F.u0 /; u0 /0 1=pj

jvj0 Cj hAj v; vij E0

.F.u0 /; u0 /0 ; ˆ˛0

ˆ0 ˆ.0/;

C2

qC2 BCj

p p1

˛ ;

; qC2 .qC2/=2

B BC1

2

:

Proof. By the conditions (A2) and the Schwarz inequality [293] for Fréchet deriva0 W Vj ! L.Vj I Vj / of the operators Aj W Vj ! Vj , the following relations tives Aj;u hold: 0 jh.Aj .um //0 ; um ij j D jhAj;u .um /u0m ; um ij j m 0 hAj;u .um /u0m ; u0m ij m

1=2

0 hAj;u .um /um ; um ij m

1=2

D h.Aj .um //0 ; u0m ij1=2 .pj 1/1=2 hAj .um /; um ij1=2 ; 1=2 jhA0 u0m ; um i0 j hA0 u0m ; u0m i1=2 0 hA0 um ; um i0 :

(2.47) (2.48)

Here we have used the equality 0 Aj;v .v/v D .pj 1/Aj .v/:

Indeed, the following equalities hold: 0 0 0 0 Aj .v/ D pj 1 Aj .v/; Aj;v .v/ D pj 1 Aj;v .v/; Aj;v .v/ D pj 2 Aj;v .v/;

Z 1

Z 1 d 0 Aj .v/; w D d d Aj;v .v/v; w 8w 2 Vj ; j D 0; N ; d 0 0 j j 0 .v/v; wij D 0 hAj .v/ .pj 1/1 Aj;v

8w 2 Vj ;

j D 0; N :

Therefore, we have 0 .v/v .pj 1/Aj .v/ D Aj;v

8v 2 Vj ;

j D 0; N :

98


Introduce the notation X pj 1 1 ˆm hA0 um ; um i0 C hAj .um /; um ij : 2 pj N

(2.49)

j D1

From (2.47)–(2.49) we have ˇ2 ˇ2 ˇ ˇ N X ˇ ˇ ˇ ˇd 0 0 ˇ ˇ ˇ ˆm jhA0 u ; um i0 j C jh.Aj .um // ; um ij jˇˇ m ˇ ˇ ˇ dt j D1

N X hA0 u0m ; u0m i0 C h.Aj .um //0 ; u0m ij j D1

N X hA0 um ; um i0 C .pj 1/hAj .um /; um ij j D1

N X 0 0 0 0 h.Aj .um // ; um ij p hA0 um ; um i0 C j D1

X pj 1 1 hA0 um ; um i0 C hAj .um /; um ij p p N

j D1

N X h.Aj .um //0 ; u0m ij p hA0 u0m ; u0m i0 C j D1

N X pj 1 1 hAj .um /; um ij ; hA0 um ; um i0 C 2 pj

(2.50)

j D1

where p D maxj D1;N pj > 2. Recall relations (2.14) and (2.18): N X pj 1 d 1 hAj .um /; um ij D .F.um /; um /0 ; hA0 um ; um i0 C dt 2 pj

(2.51)

j D1

hA0 u0m ; u0m i0 C

N X

h.Aj .um //0 u0m ij D

j D1

1 d .F.um /; um /0 : q C 2 dt

(2.52)

From (2.50)–(2.52) we obtain the following second-order ordinary differential inequality: 2 ˆ00m ˆm ˛ ˆ0m 0;

˛

qC2 ; p

p D max pj > 2: j D1;N

(2.53)

99


From (2.53) we obtain ˆ0m .F.um0 /; um0 /0 Em0 : ˆ˛m ˆ˛m0

(2.54)

Now we consider separately three cases: ˛ > 1, ˛ D 1, and 0 < ˛ < 1. In the case where ˛ > 1, from (2.53) we obtain ˆm ‰m0 T2m

Œˆ1˛ m0

1 ‰m0 ; 1=.˛1/ ŒT2m t 1=.˛1/ .˛ 1/Em0 t ˆ˛=.˛1/ m0

1

.˛ 1/1=.˛1/ .F.um0 /; um0 /1=.˛1/ 0

;

(2.55)

1 ˆm0 : ˛ 1 .F.um0 /; um0 /0

In the case where ˛ D 1, from (2.54) we directly obtain ˆm ˆm0 exp¹Em0 t º:

(2.56)

Similarly, in the case where ˛ 2 .0; 1/, we have 1=.1˛/ ˆm ˆm0 Œ1 C .1 ˛/Em0 ˆ˛1 : m0 t

(2.57)

Now we prove that ˆm .t / ! ˆ.t / for almost all t 2 .0; T/, where X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij : 2 pj N

j D1

On one hand, from (2.51) we obtain Z ˆm D ˆm0 C

t

ds .F.um /; um /0 .s/:

0

By Lemma A.15.1 from Appendix A.15, in which we set W D W0 , we obtain Z t ds .F.u/; u/0 .s/ ˆm ! ˆ0 C 0

as m ! C1. By the definition of weak generalized solutions, in which we assume ´ u.s/; s 2 Œ0; t ; v.s/ D 0; s 2 .t; T; we obtain

Z ˆ D ˆ0 C

t 0

ds .F.u/; u/0 .s/:

100


Therefore, ˆm .t / ! ˆ.t / for almost all t 2 Œ0; T:

(2.58)

By the fact that um0 ! u0 strongly in V , we have X pj 1 1 ˆm0 ! ˆ0 hA0 u0 ; u0 i0 C hAj .u0 /; u0 ij ; 2 pj N

j D1

Em0 ! E0

.F.u0 /; u0 /0 ˆ˛0

as m ! C1. From (2.58), passing to the limit as m ! C1 in Eqs. (2.28), (2.29) and (2.56), (2.57), we obtain ˆ0 exp¹E0 t º ˆ.t / ˆ0 exp¹C2 t º;

˛D

qC2 D 1; p

(2.59)

t 1=.1˛/ ˆ.t / ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 t 1=.1˛/ ; ˆ0 Œ1 C .1 ˛/E0 ˆ˛1 0 0 (2.60) where ˛

qC2 ; p

qC2 C2 BCj 2.qC2/=p ; 1=pj


;

p D max pj : j D1;N

From (2.59) and (2.60) obtain the first two statements of the theorem. Now consider (2.23) and (2.55). Note that ‰m0 ! ‰0 T2m ! T2

ˆ˛=.˛1/ 0

1

.˛ 1/1=.˛1/ .F.u0 /; u0 /1=.˛1/ 0

;

ˆ0 1 ; ˛ 1 .F.u0 /; u0 /0 q=2 1

T1m ! T1 D .q=2/1 ˆ0

B

as m ! C1, where B BCqC2 2.qC2/=2 ; 1 C1 is the constant of the best inclusion V ,! W0 , and B is constant from the condition (F4).

101

Section 2.4 Strong generalized solvability of problem (2.1)

First, we consider inequality (2.55). Without loss of generality, by the passage to a subsequence, we can assume that the sequence T2m > 0 is uniform with respect to m since T2 > 0. Owing to the convergence T2m ! T2 as m ! C1, we can choose a monotonically convergent subsequence; denote it again by T2m . Appropriate subsequences of the sequences ¹u0m º and ¹um º we also denote again by ¹u0m º and ¹um º, respectively. Let T2m " T2 ; then inequality (2.55) holds uniformly with respect to t 2 Œ0; T2m /, m m, for a certain fixed m 2 N. Passing to the limit as m ! C1 for such t in Eq. (2.24), we obtain 1 ‰0 ˆ 1˛ D (2.61) 1=.˛1/ ŒT2 t 1=.˛1/ Œˆ0 .˛ 1/E0 t for t 2 Œ0; T2m /. Hence by the arbitrariness of m 2 N we immediately obtain that (2.61) holds for all t 2 Œ0; T2 / and the blow-up time for solutions of problem (2.9) has an upper estimate: T0 T2 . Next, let T2m # T2 . Assume that T0 > T2 and, moreover, M

sup ˆ.t / < C1: t2Œ0;T2

Then (2.55) holds uniformly with respect to t 2 Œ0; T2 /. Passing to the limit as m ! C1 in inequality (2.55), we obtain the inequality ˆ

Œˆ1˛ 0

‰0 1 D M < C1 1=.˛1/ ŒT2 t 1=.˛1/ .˛ 1/E0 t

for all t 2 Œ0; T2 /. The form of the inequality obtained implies the fact that our assumption is not valid. Hence we immediately obtain that T0 T2 . Similarly, (2.23) implies the fact that ˆ.t /

Œ1

ˆ0 q q=2 2=q 2 ˆ0 Bt

8t 2 .0; T1 /;

T0 T1 :

Lemma 2.3.7 is proved. Theorem 2.3.2 is completely proved.

2.4

Unique solvability of problem (2.1) in the strong generalized sense and blow-up of its solutions

In this section, we consider problem (2.1) under some conditions on its operator coefficients for which it is possible to apply the method of contraction mappings for proving the local solvability. Here, as in Section 2.3, we obtain necessary and sufficient conditions of the blow-up of solutions of problem (2.1) and two-sided estimates of the blow-up time.

102


Definition 2.4.1. A solution of the class u 2 C .1/ .Œ0; TI V / of problem (2.1) satisfying the following conditions, with the time derivative meant in the classical sense, is called a strong generalized solution of problem (2.1): h.A0 u/0 ; vi0 C

N X

h.Aj .u//0 ; vij D .F.u/; v/0

8v 2 V ;

u.0/ D u0 2 V :

j D1

By the properties (A), (A0 ), and (F), under the assumption that the Fréchet deriva0 .u/ W V ! L.V I V / of the operator A is strongly continuous with retive Aj;u j j j j spect to u 2 Vj , we obtain that A0 u 2 C .1/ .Œ0; TI V0 /, Aj .u/ 2 C .1/ .Œ0; TI Vj /, and F .u/ 2 C.Œ0; TI W0 /, and, therefore, A0 u and Aj .u/ belong to the class C .1/ .Œ0; TI V0 /, and F.u/ 2 C.Œ0; TI V0 /. Thus, by the conditions (V) and the conditions of Theorem 2.4.2 below, the problem considered is equivalent to the following problem: h.A0 u/0 ; vi0 C

N X

h.Aj .u//0 ; vi0 D hF.u/; vi0

8v 2 V ;

u.0/ D u0 2 V ;

j D1

where the time derivative is meant in the classical sense and h; i0 is the duality bracket between the Banach spaces V0 and V0 . The following necessary and sufficient condition of the blow-up of solutions of problem (2.1) is valid. Theorem 2.4.2. Let the conditions (A), (A0 ), and (F) hold. Assume that V0 Vj , j D 1; N , V0 W0 . Moreover, assume that either 2 < p < q C 2 or p q C 2, and in the latter case, Vj W0 , where p D max pj ; j 21;N

j 2 1; N ;

pj D p:

Assume that the Fréchet derivatives of the operators Aj W Vj ! Vj satisfy the condition 0 2 BC.Vj I L.Vj ; Vj //; Aj;u i.e., they are continuous and bounded. Moreover, assume that they are monotonic in the following sense 0 0 .u/u1 Aj;u .u/u2 ; u1 u2 ij 0 hAj;u

8u; u1 ; u2 2 Vj ;

j D 1; N :

Let the inequality .F.u0 /; u0 /0 > 0 hold. Then for any u0 2 V0 , there exists maximal T0 Tu0 > 0 such that the Cauchy problem (2.1) has a unique solution in the class u.t / 2 C .1/ .Œ0; T0 /I V0 /:

103


The following two-sided estimates hold: (1) if ˛ 2 .0; 1/, then T0 D C1 and Œˆ1˛ C .1 ˛/E0 t 1=.1˛/ ˆ.t / Œˆ1˛ C .1 ˛/At 1=.1˛/ I 0 0 (2) if ˛ D 1, then T0 D C1 and ˆ0 exp¹E0 t º ˆ.t / ˆ0 exp¹At ºI (3) if ˛ > 1, then T0 2 ŒT1 ; T2 and the following limit relation holds: lim sup kuk0 D C1; t"T0

where X pj 1 1 hAj .u/; uij ; hA0 u; ui0 C 2 pj N

ˆ.t /

ˆ0 ˆ.0/;

j D1

.qC2/=p A CjqC2 ; M2

T1

2 q=2 1 B ; ˆ q 0

E0

.F.u0 /; u0 /0 ; ˆ˛0

B D MCqC2 2.qC2/=2 ; 1

T2 ˆ1˛ .˛ 1/1 E1 0 0 ;

˛

qC2 : p

Proof. We divide the proof into several steps. Step 1. Local solvability Lemma 2.4.3. For any u0 2 V0 , there exists maximal T0 > 0 such that for any T 2 .0; T0 /, a unique solution of the class C .1/ .Œ0; TI V0 / of problem (2.1) exists. Proof. Analyze properties of the operator A.u/ A0 u C

N X

Aj .u/ W V0 ! V0 :

j D1

The operator A./ is radially continuous, monotonic, and coercive. Indeed, by the condition (V2), the following relations hold: hA.u1 / A.u2 /; u1 u2 i0 D hA0 u1 A0 u2 ; u1 u2 i0 C

N X ˝ j D1

˛ Aj .u1 / Aj .u2 /; u1 u2 j

104


for all u1 ; u2 2 V0 . Therefore, by the conditions (A0 1) and (A1), we have hA.u1 / A.u2 /; u1 u2 i0 0 for all u1 ; u2 2 V0 . On the other hand, by the conditions (A0 1) and (A3), we have hA.u/; ui0 D hA0 u; ui0 C

N X

hAj .u/; uij kuk20 C

j D1

N X

p

kukj j

j D1

for any u 2 V0 . Therefore, we immediately obtain hA.u/; ui0 D C1: kuk0 kuk0 !C1 lim

Owing to the equality hA.u1 / A.u2 /; u1 u2 i0 D hA0 u1 A0 u2 ; u1 u2 i0 N X

C

hAj .u1 / Aj .u2 /; u1 u2 ij

j D1

hA0 u1 A0 u2 ; u1 u2 i0 D ku1 u2 k20 ; for the operator A, there exists a Lipschitz-continuous inverse operator A1 W V0 ! V0 with Lipschitz constant equal to 1. In Eq. (2.1), we perform the substitution v D A.u/; then the equation becomes dv D F.A1 .v//; dt

v.0/ D v0 ;

where v0 D A.u0 /. Problem (2.62) is equivalent to the following problem: Z t v.t / D v0 C ds ŒF.A1 .v//.s/:

(2.62)

(2.63)

0

We apply the method of contraction mappings to prove the local solvability of Eq. (2.63) in the class v 2 C .1/ .Œ0; TI V0 /. For this, we introduce a closed, bounded, convex subset of the Banach space L1 .0; TI V0 /: BR D ¹v 2 L1 .0; TI V0 / W kvkT Rº; where

kvkT D ess sup kvk0 : t2.0;T/

Prove that the operator Z H.u/

0

t

ds ŒF.A1 .v//.s/

105


acts from BR into BR and is contractive. It is not difficult to prove that the operator H.v/ acts from BR into BR for sufficiently large R and sufficiently small T. Then we obtain that the operator H.v/ is a contraction. Indeed, the following inequalities hold: kH.v1 / H.v2 /kT T1 .R/kA1 .v1 / A1 .v2 /kT ;

(2.64)

where k kT ess sup k k0 ; t2.0;T/

R D max¹kA1 .v1 /kT ; kA1 .v2 /kT º: By the properties of the operator A1 , from (2.64) we obtain the following inequalities: kH.v1 / H.v2 /kT TC1 .R/kv1 v2 kT :

(2.65)

Under the condition T 1=2Œ1 .R/C1 , inequality (2.65) implies that the operator H.v/ is a contraction on the set BR . Therefore, there exists a unique solution of Eq. (2.63) of the class v 2 L1 .0; TI V0 /. Using the standard algorithm of the extension of solutions of integral equations in time, we see that there exists maximal T0 > 0 such that a unique solution of Eq. (2.63) of the class v 2 L1 .0; T0 I V0 / exists, and either T0 D C1 or T0 < C1, and in the latter case, the following limit relation holds: lim sup kvk0 D C1: t"T0

Finally, from Eq. (2.63), by virtue of smoothing properties of the operator H.v/, we obtain v 2 C .1/ .Œ0; T0 /I V0 /. Note that the relation A.u/ A0 u C

N X

Aj .u/ D v.t / 2 C .1/ .Œ0; T0 /I V0 /

(2.66)

j D1

holds. By virtue of the Browder–Minty theorem, Eq. (2.66) is equivalent to u.t / D A1 .v/ in the class u 2 C .1/ .Œ0; T0 /I V0 /. The latter equation implies the fact that u.t / 2 C.Œ0; T0 /I V0 /. Indeed, ku.t / u.t0 /k0 kA1 .v/.t / A1 .v/.t0 /k0 kv.t / v.t0 /k0 ! C0 as t ! t0 . Therefore, a unique solution of Eq. (2.66) of the class u 2 C.Œ0; T0 /I V0 / exists. Now we prove that the solution belongs to the class C .1/ .Œ0; T0 /I V0 /.

106


First, consider the Fréchet derivative of the operator A: A0u .u/ D A0 C

N X

0 Aj;u .u/:

(2.67)

j D1

By the conditions of the theorem, we have hA0u .u/u1 A0u .u/u2 ; u1 u2 i0 ku1 u2 k20 for any u; u1 ; u2 2 V0 . Therefore, by (2.67) and the Browder–Minty theorem, for any fixed u 2 V0 , the Lipschitz-continuous inverse operator ŒA0u .u/1 2 L.V0 I V0 / exists. Now consider the following equation: A0 u C

n X

Aj .u/ D v.t / 2 C .1/ .Œ0; T0 /I V0 /:

(2.68)

j D1

For further presentation, we must prove that the operator O D I C B; O C

BO D

n X

0 A1 0 Aj;u .u/

j D1 0 .u/ is the Fréchet derivative on a fixed has a bounded inverse operator. Here Aj;u element u 2 C.Œ0; T0 /I V0 / of the operator Aj . Indeed, consider the equation

O ŒI C Bw D f 2 V0 :

(2.69)

We prove that this equation has only a trivial solution. For this, apply the operator A0 to both parts of Eq. (2.69); then we obtain A0u w D A0 w C

n X

Aj;u .u/w D A0 f

8u 2 V0 :

j D1

But we have already proved that for the operator A0u .u/, a Lipschitz-continuous inverse operator ŒA0u .u/1 is defined. Now we apply this operator to both parts of the latter equality. Therefore, for any f 2 V0 , a solution of Eq. (2.69) exists, and it is not difficult to prove that it is unique. Therefore, the inverse operator C 1 exists. In the class u 2 C .1/ .Œ0; T0 /I V0 /, Eq. (2.68) is equivalent to the following problem: N X 0 Aj;u .u/ u0 D v 0 2 C.Œ0; T0 /I V0 /; u0 D A1 v0 : A0 C j D1


107

We apply the operator A1 0 to the latter equation; then we obtain O 0 D A1 v 0 : ŒI C Bu 0 Introduce the operator

(2.70)

O D I C BO W V0 ! V0 ; C

for which, as we have already proved, an inverse linear bounded operator exists. Therefore, why (2.70) is equivalent to the following equation: 0 O 1 A1 u0 D C 0 v :

(2.71)

O 1A1v 0 2 C.Œ0;T0 /IV0 / for fixed u 2 C.Œ0;T0 /IV0 /. It remains to prove that u0 D C 0 Indeed, by (2.71), the following inequalities hold: 0 0 O 1 .t0 /A1 ku0 .t / u0 .t0 /k0 kC 0 Œv .t / v .t0 /k0 0 O 1 .t0 / C O 1 .t //A1 C k.C 0 v .t0 /k0

O 1 .t0 / C O 1 .t /kV !V : Ckv 0 .t / v 0 .t0 /k0 C CkC 0 0

(2.72)

O which is linear for fixed u 2 C.Œ0; T0 /I V0 /, is continuous Note that the operator C, O 1 is linear, and, therefore, by the Banach theorem on inverse mapping, the operator C continuous, and, therefore, bounded by virtue of the linearity. Thus, we can use the O 1 W V0 ! V0 . spectral representation for the linear bounded operator C O First, introduce the resolvent of the operator C: O D .I C/ O 1 : R.; C/ Let be a circle jj D r of sufficiently large radius, which is greater than sup t2Œt0 ";t0 C"

O V !V : kCk 0 0

The introduced value is well-defined since for t 2 Œt0 "; t0 C " Œ0; T0 /, we have sup t2Œt0 ";t0 C"

kuk0 < C1:

O 1 .t0 / O 1 .t / and C Now we can use the spectral representation for the operators C with the same contour introduced above: Z Z 1 1 1 1 1 O O O O 0 //: C .t / D d R.; C.t //; C .t0 / D d 1 R.; C.t 2 i 2 i Obviously, we have O 1 .t0 / D 1 O 1 .t / C C 2 i

Z

O // R.; C.t O 0 //: d 1 ŒR.; C.t

108


Now we use the well-known representation for resolvents of operators: O // R.; C.t O 0 // D R.; C.t O 0 // R.; C.t

C1 X

O / C.t O 0 //R.; C.t O 0 /n Œ.C.t

nD1

under the condition O /kV !V kR.; C.t O 0 //kV !V ı < 1: O 0 / C.t kC.t 0 0 0 0 The following inequality holds: O // R.; C.t O 0 //kV !V kR.; C.t 0 0 O 0 //kV !V kR.; C.t 0 0

C1 X

n O 0 //kn O O kR.; C.t V0 !V0 kC.t / C.t0 /kV0 !V0 :

nD1

Note that O / C.t O 0 / D A1 C.t 0

N X

ŒAj;u .u.t // Aj;u .u.t0 //:

j D1

By the continuity of the Fréchet derivatives Aj;u ./ with respect to u 2 V0 and u 2 C.Œ0; T0 /I V0 /, we have O / C.t O 0 /kV !V kC.t 0 0

N X j D1

kAj;u .u.t // Aj;u .u.t0 //kV0 !V0 ! C0;

O // R.; C.t O 0 //kV !V ! 0; kR.; C.t 0 0 O 0 /1 kV !V ! C0 O /1 C.t kC.t 0 0 as t ! t0 . Therefore, by (2.72), u 2 C .1/ .Œ0; T0 /I V0 /. Note that under the conditions (A7), the operator C.u/ is boundedly Lipschitz-continuous and, therefore, the operator C 1 .u/ is also boundedly Lipschitz-continuous and hence Eq. (2.71) has a local solution in the class u.t / 2 C .1/ .Œ0; T00 /I V0 /. It is not difficult to prove that T0 D T00 . Step 2. A priori estimates and blow-up Lemma 2.4.4. Let X pj 1 1 hAj .u/; uij ; ˆ.t / hA0 u; ui0 C 2 pj N

j D1

˛

qC2 : p

109


Then (1) if ˛ 2 .0; 1/, then T0 D C1 and the following two-sided estimate holds: C .1 ˛/E0 t 1=.1˛/ ˆ.t / Œˆ1˛ C .1 ˛/At 1=.1˛/ I Œˆ1˛ 0 0 (2) if ˛ D 1, then T0 D C1 and the following two-sided estimate holds: ˆ0 exp.E0 t / ˆ.t / ˆ0 exp.At /I (3) if ˛ > 1, then T0 2 ŒT1 ; T2 , and lim ˆ.t / D C1;

t"T0

where 2 q=2 1 1 ˆ ˆ1˛ E1 B ; T2 0 ; q 0 ˛1 0 .F.u0 /; u0 /0 .qC2/=p ; A CjqC2 : E0 M2 ˆ˛0 T1

Proof. Let u 2 C .1/ .Œ0; T0 /I V0 / be the maximal solution of problem (2.1). Then, multiplying both sides of Eq. (2.1) by u.t / and by u0t .t /, after integrating by parts we obtain the following equalities: N X pj 1 d 1 hAj .u/; uij D .F.u/; u/0 ; hA0 u; ui0 C dt 2 pj

(2.73)

j D1

0

0

hA0 u ; u i0 C

N X

h.Aj .u//0 ; u0 ij D

j D1

1 d .F .u/; u/0 : q C 2 dt

Integrating Eq. (2.73) over t 2 .0; T/, we obtain Z t ds .F.u/; u/0 ; ˆ.t / D ˆ0 C

(2.75)

0

where

X pj 1 1 hAj .u/; uij ; ˆ.t / hA0 u; ui0 C 2 pj N

ˆ0 D ˆ.0/:

j D1

By the condition V0 W0 , there exists a constant C1 > 0 such that 1=2

jvj0 C1 hA0 v; vi0 ;

(2.74)

110


and (2.75) implies the inequality Z t Z t qC2 .qC2/=2 ds hA0 v; vi0 ˆ0 C B ds ˆ.qC2/=2 .s/; ˆ.t / ˆ0 C MC1 0

0

MCqC2 2.qC2/=2 : 1

B

According to the Bihari theorem (see [112]), we have ˆ.t /

Œ1

ˆ0 q q=2 2=q 2 ˆ0 Bt

8t 2 Œ0; T1 /;

T1

2 q=2 1 B : ˆ q 0

(2.76)

Therefore, in the case where p < q C 2, by (2.76), T0 T1 . Now let p q C 2. We use the fact that in this case, there exist j 2 1; N and pj D p, pj q C 2, such that Vj W0 . Let Cj be the constant of the inclusion Vj W0 : 1=p

jvj0 Cj hAj .v/; vij j ; Z t .qC2/=pj qC2 ˆ.t / ˆ0 C Cj M ds hAj .v/; vij 0

ˆ0 C CjqC2 M

p p1

.qC2/=p Z

t

ds ˆ.qC2/=p .s/:

0

Let ˛

qC2 ; p

We consider two cases: ˛ D 1 and ˛ 2 .0; 1/. In the first case, according to the Gronwall–Bellman theorem (see [112]) ˆ.t / ˆ0 exp¹At º;

A

CjqC2 M

p p1

.qC2/=p :

(2.77)

In the second case, ˛ 2 .0; 1/, according to the Bihari theorem, we obtain ˆ.t / Œˆ1˛ C .1 ˛/At 1=.1˛/ : 0

(2.78)

Inequalities (2.77) and (2.78) imply that T0 D C1 for ˛ 2 .0; 1. Now from relations (2.73) and (2.74), in just the same way as in Step 5 of the proof of Theorem 2.3.2, we obtain the following second-order ordinary differential inequality: ˆ00 ˆ ˛.ˆ0 /2 0;

˛D

qC2 : p

(2.79)

111


Consider three possible cases: ˛ > 1, ˛ D 1, and ˛ 2 .0; 1/. Let ˛ > 1. Then, integrating the differential inequality (2.79) and using (2.73) and (2.74), we obtain ˆ.t /

Œˆ1˛ 0

and hence

1 ; .˛ 1/E0 1=.˛1/

E0

.F.u0 /; u0 /0 ˆ˛0

.˛ 1/1 E1 T0 T2 ˆ1˛ 0 0 :

Now let ˛ D 1; then (2.79) implies ˆ.t / ˆ0 exp¹E0 t º: Finally, if ˛ 2 .0; 1/, then (2.79) implies ˆ.t / Œˆ1˛ C .1 ˛/E0 t 1=.1˛/ : 0 The lemma is proved. Theorem 2.4.2 is completely proved.

2.5

Unique solvability of problem (2.2) in the weak generalized sense and estimates of time and rate of the blow-up of its solutions

In this section, we assume that the following dense and continuous inclusions hold: ds

ds

ds

ds

ds

ds

W1 V0 W0 H W0 V0 W1 ; i.e., W1 is dense in V0 , V0 is dense in W0 , and W0 is dense in H. Moreover, these inclusions imply the following properties: .F.v/; w/1 D .F.v/; w/0

for all v; w 2 W1

.A0 v; w/1 D hA0 v; wi0

for all v; w 2 W1 :

and

Definition 2.5.1. A solution u.t / of the problem (2.2) satisfying the condition Z T dt .t /ŒhA0 u0 ; wi0 C .B.u/; w/1 .F.u/; w/0 D 0 (2.80) 0

for all w 2 W1 and all .t / 2 L2 .0; T/ is called a weak generalized solution of problem (2.2).

112


In the class u 2 L1 .0; TI W1 /;

u0 2 L2 .0; TI V0 /;

hA0 u; ui0 2 H1 .0; T/;

A0 u 2 L1 .0; TI V0 / \ H1 .0; TI W1 /; under the condition that almost everywhere on the interval .0; T/, the relation d hA0 u; ui0 D 2hA0 u0 ; ui0 ; dt holds, problem (2.80) is equivalent to the following problem: Z 0

T

dt .t / A0 u0 C B.u/ F.u/; w 1 D 0

8w 2 W1 ;

8 .t / 2 L2 .0; T/;

which, in its turn, by virtue of the result of Appendix A.12 is equivalent to the problem Z

T 0

dt .A0 u0 C B.u/ F.u/; v/1 D 0 8v 2 L2 .0; TI W1 /:

The following theorem holds. Theorem 2.5.2. Let the conditions (A0 ), (B), (F) hold and, moreover, u0 2 W1 and V0 ,! W0 , i.e., the embedding operator is a compact operator, and the Banach space W1 is uniformly convex. Let, moreover, the initial function satisfy the condition .F.u0 /; u0 /0 .B.u0 /; u0 /1 > 0: Then there exists maximal T0 > 0 such that a unique strong generalized solution of problem (2.2) of the class du 2 L2 .0; TI V0 / 8T 2 .0; T0 /; dt A0 u 2 L1 .0; TI V0 / \ H1 .0; TI W1 /;

u 2 L1 .0; TI W1 /;

hA0 u; ui0 2 H1 .0; T/; exists and almost everywhere on the interval .0; T/, the following relation holds: d hA0 u; ui0 D 2hA0 u0 ; ui0 : dt Here lim suphA0 u; ui0 D C1 t"T0

113


and the following two-sided estimates hold: (1) for the rate of the blow-up: 1=2 Œqc1 1=q hA0 u; ui0 ŒT0 t 1=q Œqc2 1=q

8t 2 Œ0; T0 /;

(2) for the time of blow-up: .qc1 /1 hA0 u0 ; u0 iq=2 T0 .qc2 /1 hA0 u0 ; u0 iq=2 0 0 where c2

Œ.F.u0 /; u0 /0 .B.u0 /; u0 /1 hA0 u0 ; u0 i.qC2/=2 0 qC2

j.F.v/; v/0 j B jvj0

;

c1 BCqC2 ; 1

1=2 jvj0 C1 hA0 v; vi0 :

;

Remark 2.5.3. The conditions of Theorem 2.5.2 imply the fact that u.t / W Œ0; T ! V0 is a strongly absolutely continuous function on the segment Œ0; T taking its values in V0 . Therefore, the initial condition u.0/ D u0 makes sense. Proof. As above, we divide the proof of the theorem into several steps. Step 1. Galerkin approximations. Consider the following scheme of Galerkin approximations: Z T dt .t /ŒhA0 u0m ; wj i0 C .B.um /; wj /1 .F.um /; wj /0 D 0 0

8 .t / 2 L2 .0; T/; um D

m X

cmk .t /wk ;

8j D 1; m;

cmk .0/ D ˛mk ;

(2.81)

kD1

um0 D

m X

˛mk wk ! u0

strongly in W1 ;

kD1

is a system of linearly independent functions dense in W1 . where ¹wk ºC1 kD1 By the definition of um , in the class cmk .t / 2 C .1/ Œ0; Tm0 / from (2.81) we obtain Z

T

dt .t / 0

X m kD1

dcmk akj C .B.um /; wj /1 .F.um /; wj /0 D 0 dt

8 .t / 2 L2 .0; T/;

akj D hA0 wk ; wj i0 :

In the class cmk .t / 2 C .1/ .Œ0; T/, the expression in the brackets in the integrand belongs to the class CŒ0; T. By virtue of the obvious inclusion C01 .Œ0; T/ L2 .0; T/

114


and the principal lemma of the calculus of variations, we obtain (2.81) the pointwise equality m X kD1

akj

dcmk C .B.um /; wj /1 D .F.um /; wj /0 ; dt

akj hA0 wk ; wj i0 ;

(2.82)

where akj hA0 wk ; wj i0 . Note that the corresponding quadratic form m;m X

akj k j D hA0 ; i0

i;j D1;1

is positive semidefinite byPvirtue of the positive definiteness of the operator A0 ; it vanm ishes if and only if N i D1 i wi . By definition, the system of functions ¹wi ºi D1 m is linearly independent in W1 ; therefore, D 0 if and only if ¹i ºi D1 D 0. Hence, by virtue of the Sylvester criterion (see [335]) we conclude that det¹akj ºm;m > 0. Fik;j D1;1 nally, by virtue of the semicontinuity of the operator B W W1 ! W1 and the bounded Lipschitz continuity of the operator F W W0 ! W0 we conclude that (2.82) is a system of Cauchy–Kovalevskaya type satisfying the conditions of the general theorem on the solvability of systems of ordinary differential equations [319]. Therefore, there exists Tm0 > 0 such that a solution of system (2.82) of the class C .1/ .Œ0; Tm0 /I W1 / can be found. Step 2. A priori estimates Lemma 2.5.4. There exists T > 0 independent of m 2 N and such that for a certain subsequence of the sequence ¹um º, the following properties hold uniformly with respect to m 2 N: um

is bounded in L1 .0; TI W1 /I

u0m

is bounded in L2 .0; TI V0 /I

A0 um


B.um /


F.um / is bounded in L1 .0; TI W0 /: 0 and summing over Proof. Multiplying both sides of Eq. (2.82) by cmj or by cmj j D 1; m, we obtain respectively

1d hA0 um ; um i0 C .B.um /; um /1 D .F.um /; um /0 ; 2 dt 1 d 1 d .B.um /; um /1 D .F.um /; um /0 : hA0 u0m ; u0m i0 C q C 2 dt q C 2 dt

(2.83) (2.84)

115


From (2.84) we obtain Z

t 0

ds hA0 u0m ; u0m i0 C D

1 .B.um /; um /1 qC2

1 1 Œ.B.um0 /; um0 /1 .F.um0 /; um0 /0 C .F.um /; um /0 : qC2 qC2

According to the condition of the theorem .F.u0 /; u0 /0 > .B.u0 /; u0 /1 : Passing to a subsequence if necessary, we obtain .F.um0 /; um0 /0 > .B.um0 /; um0 /1 : In this case, we immediately conclude that Z t 1 1 .B.um /; um /1 .F.um /; um /0 : ds hA0 u0m ; u0m i0 C qC2 qC2 0

(2.85)

Integrating Eq. (2.83) over t 2 Œ0; T, we obtain Z t hA0 um ; um i0 hA0 um0 ; um0 i0 C 2 ds .F.um /; um /0 0

Z

hA0 um0 ; um0 i0 C 2B

t

ds jum jqC2 0

0

hA0 um0 ; um0 i0 C 2BCqC2 1

Z 0

t

ds hA0 um ; um i.qC2/=2 : 0

Let ˆm hA0 um ; um i0 ; then Z ˆm ˆm0 C A

0

t

qC2

ds ˆ.qC2/=2 ; m

A 2BC1

:

(2.86)

According to the Bihari theorem (see [112]) we obtain from (2.86) ˆm

ˆm0 Œ1

q q=2 2=q 2 ˆm0 At

D

. q2 /2=q A2=q ŒT2m t

2=q

;

T2m

2 q=2 1 A : ˆ q m0

Since um0 ! u0 strongly in W1 V0 , we see that T2m ! T2 , where T2

2 q=2 1 ˆ A : q 0

Passing to a subsequence, we can assume that either T2m " T2 or T2m # T2 .

(2.87)

116


First, consider the case where T2m # T2 . Then T2m t T2 t and from (2.87) we obtain 2=q 1 2 A2=q 8t 2 Œ0; T2 /: ˆm q ŒT2 t 2=q Now consider the case where T2m " T2 . Here T2m t T2m t , m > m 2 N, and (2.87) implies 2=q 1 2 A2=q ˆm q ŒT2m t 2=q

8t 2 Œ0; T2m /:

Hence we immediately obtain that for any T 2 .0; T2 /, there exists a constant C.T/ such that ˆm C.T/ < C1: 1=2 On the other hand, jum j0 C0 ˆm C.T/ and (2.85) implies the chain of inequalities Z t 1 .B.um /; um /1 ds hA0 u0m ; u0m i0 C qC2 0 1 1 qC2 .F.um /; um /0 Bjum j0 qC2 qC2 C1 B .qC2/=2 C.T/ < C1 8T 2 .0; T2 /: ˆ qC2 m

Thus, we have proved that u0m


um

is bounded in L1 .0; TI W1 /;

A0 u m


B.um /

is bounded in L1 .0; TI W1 /:

Lemma 2.5.4 is proved. Passing to a subsequence if necessary, we obtain um * u

-strongly in L1 .0; TI W1 /;

u0m * u0

strongly in L2 .0; TI V0 /;

A 0 um * A 0 u

-strongly in L1 .0; TI V0 /;

B.um / *

-strongly in L1 .0; TI W1 /:

Note that um * u

-weakly in L1 .0; TI W1 /

117


implies

u0m * u0

in D 0 .0; TI V /I

therefore, by virtue of the weak convergence in L2 .0; TI V0 /, u0m * ; we conclude that .t / D u0 .t /. Moreover, by virtue of the fact that V0 ,! W0 , the bounded Lipschitz continuity of the operator F W W0 ! W0 W1 , and Lemma A.15.1 from Appendix A.15, where we set W D W0 , we have um .s/ ! u.s/


Therefore, jF .um / F .u/j1 CjF.um / F.u/j0 Cjum uj0 ! 0;

m ! C1:

Step 3. Monotonicity method. Taking into account that B.um / ! -weakly in L1 .0; TI W1 /, We prove the following lemma. Lemma 2.5.5. There exists a subsequence of the sequence ¹um º such that B.um / * B.u/

-weakly in L1 .0; TI W1 /:

Proof. Scheme (2.81) implies d hA0 um ; wj i0 C .B.um /; wj /1 D .F.um /; wj /0 ; dt

j D 1; m:

The results of Step 2 imply hA0 u0m ; wj i0 * hA0 u0 ; wj i0 weakly in L2 .0; T/; .B.um /; wj /1 * . ; wj /1 hA0 um0 ; wj i0 ! hA0 u0 ; wj i0 ;

-weakly in L1 .0; T/; hA0 um .T/; wj i0 ! hA0 u.T/; wj i0 :

Passing to the limit as m ! C1 for fixed j D 1; m, we obtain d hA0 u; wj i0 C . ; wj /1 D .F.u/; wj /0 ; dt

j D 1; C1:

We immediately obtain that, in the sense of L2 .0; TI W1 /, the following relation holds: d A0 u C D F.u/: dt

(2.88)

118


Since the operator B W W1 ! W1 is monotonic, we have Z T X dt .B.u / B.v/; u v/1 8v 2 Lp .0; TI W1 /:

(2.89)

0

According to (2.81) Z T Z T dt .B.u /; u /1 D hA0 u 0 ; u 0 i0 hA0 u .T/; u .T/i0 C dt .F.u /; u /0 : 0

0

Since u ! u weakly in W1 for almost all t 2 .0; T/, by virtue of the compact embedding W1 ,! W0 we have F.u /; u 0 ! .F.u/; u/0 as ! C1 1=2

for almost all t 2 .0; T/. Moreover, by the conditions (A0 ), hA0 u; ui0 the Banach space V0 and hence we have

is a norm of

lim inf hA0 u .T/; u .T/i0 hA0 u.T/; u.T/i0

!C1

since W1 V0 . On the other hand, by virtue of the fact that u 0 ! u0 strongly in W1 , we have hA0 u 0 ; u 0 i0 ! hA0 u0 ; u0 i0 and

Z

T

lim sup !C1 0

Z dt .B.u /; u /1 hA0 u0 ; u0 i0 hA0 u.T/; u.T/i0 C

T 0

dt .F.u/; u/0 : (2.90)

Then, by (2.88) we obtain Z Z T dt . ; u/1 D hA0 u0 ; u0 i0 hA0 u.T/; u.T/i0 C 0

Relations (2.89)–(2.91) imply Z T Z 0 lim sup X dt . ; v/1 !C1

Z

T 0

w2 obtain

0

dt . B.v/; u v/1 0;

Lr .0; TI W1 /, Z 0

0

T

T 0

dt .F.u/; u/0 : Z

dt .B.v/; u v/1 C

v D u w;

(2.91)

T 0

dt . ; u/1 ;

u 2 L1 .0; TI W1 /;

> 0. Since the operator B W W1 ! W1 is semicontinuous, we T

dt . B.u/; w/1 0 8w 2 Lr .0; TI W1 /:

Hence we conclude that D B.u/. Lemma 2.5.5 is proved.


119

Step 4. Passage to the limit and uniqueness. By virtue of limit relations obtained on the Steps 2 and 3, we conclude that there exists a subsequence of the sequence ¹um º such that we can pass to the limit as m ! C1 in Eq. (2.81) and obtain Z

T 0

dt .t /ŒhA0 u0 ; vi0 C .B.u/; v/1 .F.u/; v/0 D 0

(2.92)

for all .t / 2 L2 .0; T/ and all v 2 W1 . From (2.81), by the principal lemma of the calculus of variations, we have Z t ds Œ.F.um /; wj /0 .B.um /; wj /1 : hA0 um ; wj i0 D hA0 um0 ; wj i0 C 0

Passing to the limit in the latter equality, we obtain that for almost all t 2 .0; T/, the following relation holds: Z hA0 u; wj i0 D hA0 u0 ; wj i0 C Therefore,

Z A0 u D A0 u0 C

t

0

t 0

ds Œ.F.u/; wj /0 .B.u/; wj /1 :

ds ŒF.u/.s/ B.u/.s/:

Arguing as on Step 4 of the proof of Theorem 2.3.2, we can show that A0 u 2 H1 .0; TI W1 /: Now we prove that hA0 u; ui0 2 H1 .0; T/ and that the relation

d hA0 u; ui0 D 2hA0 u0 ; ui0 dt holds almost everywhere on interval .0; T/. Indeed, by virtue of the fact that u 2 L2 .0; TI V0 / and u0 2 L2 .0; TI V0 / we have hA0 u; ui0 2 L2 .0; T/;

hA0 u0 ; ui0 2 L2 .0; T/:

Consider the divided difference 1 ŒhA0 u.t C h/; u.t C h/i0 hA0 u.t /; u.t /i0 h 1 1 D hA0 u.t C h/ A0 u.t /; u.t C h/i0 C hA0 u.t /; u.t C h/ u.t /i0 h h DJ

120


for all t; tCh 2 .0; T/. Note that, according to the conditions (A0 ), the operator A0 is symmetric and hence the following relation for J holds: JD

1 1 hA0 u.t C h/ A0 u.t /; u.t C h/i0 C hA0 u.t C h/ A0 u.t /; u.t /i0 : h h

Introduce the notation B.u/ D F.u/ B.u/: Note that in the sense of L2 .0; TI W1 /, the following relation holds: A0 u0 D B.u/: In the class u0 2 L2 .0; TI V0 / we have: A0 u0 2 L2 .0; TI V0 /. Therefore, B.u/ 2 L2 .0; TI V0 /. Rewrite the equality for J in the following form:

Z tCh 1 ds B.u/.s/; u.t C h/ u.t / JD h t 0 Z tCh

1 C2 ds ŒB.u/.s/ B.u/.t /; u.t / C 2hB.u/.t /; u.t /i0 : h t 0 Since B.u/ 2 L2 .0; TI V0 / and, moreover, u 2 C.Œ0; TI V0 /, the first two summands in the expression for J for almost all t 2 .0; T/ vanish as h ! 0. On the other hand, we have B.u/ D A0 u0 in the sense of L2 .0; TI V0 /. Thus, we conclude that for almost all t 2 .0; T/, the following relation holds: d hA0 u; ui0 D 2hA0 u0 ; ui: dt From this equality in the class u0 2 L2 .0; TI V0 /, u 2 L2 .0; TI V0 / we obtain that hA0 u; ui0 2 H1 .0; T/. Now we can rewrite Eq. (2.92) in the following equivalent form: Z

T 0

dt .t /.A0 u0 C B.u/ F.u/; w/1 D 0 8 .t / 2 L2 .0; T/;

8w 2 W1 ;

which is equivalent, in its turn, to the following: Z

T 0

dt .A0 u0 C B.u/ F.u/; v/1 D 0 8v 2 L2 .0; TI W1 /:


121

Now let u1 and u2 be two weak generalized solutions of the latter problem of the class du 2 L2 .0; TI V0 / 8T 2 .0; T0 /; dt A0 u 2 L1 .0; TI V0 / \ H1 .0; TI W1 /;

u 2 L1 .0; TI W1 /;

hA0 u; ui0 2 H1 .0; T/; and almost everywhere on interval .0; T/, the following relation holds: d hA0 u; ui0 D 2hA0 u0 ; ui0 ; dt which corresponds to the same initial function u0 2 W1 . Let ´ u1 u2 ; s 2 Œ0; t I v.s/ D 0; s 2 Œt; T/: Then by virtue of the monotonicity of B.u/ we obtain the equality m0 ku1 u2 k20 .t / hA0 u1 A0 u2 ; u1 u2 i0 Z t 2 ds jF.u1 / F.u2 /j0 ju1 u2 j0 0

Z C

t 0

ds ku1 u2 k20 .s/:

According to the Gronwall–Bellman theorem, we conclude that u1 D u2 almost everywhere .0; T/. Step 5. Blow-up of solutions of problem (2.2). Now we prove a result on the properties of Galerkin approximations um needed in the what follows. Lemma 2.5.6. There exists a subsequence of the sequence ¹um º such that um ! u strongly in LqC2 .0; TI W1 /: Proof. By the lower semicontinuity of a norm of a reflexive Banach space and the property (B2) of the operator B W W1 ! W1 , we conclude that since u * u weakly in LqC2 .0; TI W1 /, Z T Z T .B.u /; u /1 dt dt .B.u/; u/1 : lim inf !C1 0

0

On the other hand, we have proved on Step 4 that Z T Z lim sup dt .B.u /; u /1 !C1 0

0

T

dt .B.u/; u/1 :

122


Therefore, Z lim

!C1 0

T

Z dt .B.u /; u /1 D

T

dt .B.u/; u/1 :

0

(2.93)

By the conditions (B), the norm Z

1=.qC2/

T 0

dt .B.v/; v/1

is equivalent to the initial norm of the uniformly convex Banach space LqC2.0;TIW1/. Since u * u -weakly in L1 .0; TI W1 /, u * u weakly in LqC2 .0; TI W1 /. Then by (2.93) we conclude that u ! u strongly in LqC2 .0; TI W1 /. Let ˆm hA0 um ; um i0 ;

ˆm0 ˆm .0/;

ˆ hA0 u; ui0 ;

ˆ0 ˆ.0/:

Prove that ˆm ! ˆ for almost all t 2 .0; T/. The strong convergence um ! u in LqC2 .0; TI W1 / LqC2 .0; TI V0 / implies Z T Z T dt hA0 um ; um i0 ! dt hA0 u; ui0 : (2.94) 0

0

On the other hand, Z hA0 um ; um i0 .T/ D

T

0

dt .B.um /; um /1 Z

C hA0 um0 ; um0 i0 C

T 0

dt .F.um /; um /0 :

(2.95)

The right-hand side of Eq. (2.95) converges by (2.93); therefore, the left-hand side also converges for almost all T 2 .0; T0 / to a certain function .T/: ˆm .T/ ! .T/ 2 L1 .0; T0 /

for all T 2 .0; T0 /:

According to the Lebesgue theorem on the passage to the limit under the integral sign (see [129]) we have Z T Z T dt hA0 um ; um i ! dt .t / under m ! C1: 0

0

Therefore, by virtue of (2.94) Z 0

T

dt Œ.t / ˆ.t / D 0:

Therefore, .T/ D ˆ.T/ for almost all T 2 .0; T0 /. Lemma 2.5.6 is proved.

123


Lemma 2.5.7. For any u0 2 W0 , under the condition .F.u0 /; u0 /0 .B.u0 /; u0 /1 > 0 there exists T0 > 0 such that lim ˆ.t / D C1;

t"T0

where ˆ.t / hA0 u; ui0 . Proof. From relations (2.83) and (2.84), we obtain for ˆm .t / the ordinary differential inequality ˆ00m ˆm ˛.ˆ0m /2 0;

˛

qC2 ; 2

(2.96)

which implies

ˆ0m ˆ˛m

0

0;

ˇ ˆ0m ˆ0m ˇˇ ˛ ˇ : ˆ˛m ˆm0 tD0

(2.97)

Since E0 .F .u0 /; u0 /0 .B.u0 /; u0 /1 > 0, passing to a subsequence if necessary, we obtain Em0 .F.um0 /; um0 /0 .B.um0 /; um0 /1 > 0: Then (2.97) implies 2=q 1 q 1C2=q 2=q ˆm ˆm0 Em0 ; T1m ˆm0 .˛ 1/1 E1 m0 : 2 ŒT1m t 2=q

(2.98)

Passing to a subsequence, we can assume that either T1m " T1 or T1m # T1 , where ˆ0 ˆ.0/;

ˆ.t / hA0 u; ui0 ; T1

E0 .F.u0 /; u0 /0 .B.u0 /; u0 /1 ;

2 ˆ0 : q .F.u0 /; u0 /0 .B.u0 /; u0 /1

First, consider the case where T1m " T1 ; then T1m t T1 t and for any m > m 2 N, t 2 Œ0; Tm /, by virtue of (2.98) the following inequality holds:

q 2=q Dm0 2=q 1C2=q 8t 2 .0; T /; D Em0 ˆm0 : (2.99) ˆm m0 1m 2=q 2 ŒT1 t Passing to the limit as m ! C1 uniformly with respect to t 2 .0; T1m /, by virtue of the fact that ˆm .t / ! ˆ.t / for almost all t 2 .0; T1m /, from (2.99) we obtain

q 2=q D0 2=q 1C2=q ˆ.t / ; D E0 ˆ0 8t 2 .0; T1m /: (2.100) 0 2 ŒT1 t 2=q

124


Hence, by the arbitrariness of m 2 N, we obtain that inequality (2.100) holds for any t 2 .0; T1 /. Now let T1m # T1 . In this case, we have ˆm .t /

Dm0

8t 2 Œ0; T1m /:

ŒT1m t 2=q

Passing to the limit as m ! C1 in the latter inequality, we obtain ˆ.t /

D0

8T 2 Œ0; T1 /:

ŒT1 T2=q

(2.101)

Now assume that T0 > T1 and, therefore, ˆ.t / C.t / 8t 2 .T1 ; T0 /: However, by (2.100) and (2.101), we conclude that there exists t 2 .0; T1 / such that the contrary inequality holds. Therefore, T0 T1 . Lemma 2.5.7 is proved. Step 6. Estimates of the rate and time of the blow-up of solutions to problem (2.2). Introduce the notation Um .s; x/

um .sm C t ; x/ ; Mm .t /

m

1

Mm .t / hA0 um ; um i0 ; 1=2

q ; Mm .t /

where t is a fixed time moment of the interval Œ0; T0 /. It is not difficult to show that the introduced function Um .s; x/ is a solution of the following problem: d hA0 Um ; wj i0 C .B.Um /; wj /1 D .F.Um /; wj /0 ; j D 1; m; ds m X t ˛mk um0 Um0 ; Um0 D wk ; Um m Mm .t / Mm .t / kD1

um0 D

m X

˛mk wk ! u0

(2.102)

strongly in W1 :

kD1

Lemma 2.5.8. For any u0 2 W1 , the following two-sided estimates of the rate of the blow-up of solutions to problem (2.2) holds: Œqc1 1=q M.t /ŒT0 t 1=q Œqc2 1=q

8t 2 Œ0; T0 /;

where c2

Œ.F.u0 /; u0 /0 .B.u0 /; u0 /1 .qC2/=2 hA0 u0 ; u0 i0 qC2


;

;

qC2

c1 BC1

1=2

jvj0 C1 hA0 v; vi0 :

;

125


Proof. Prove that there exist c1m ; c2m 2 .0; C1/ such that 0 < c2m

d 1=2 ˇ hA0 .Um /; Um i0 ˇsD0 c1m < C1: ds

(2.103)

Note that for problem (2.102), the energy relation (2.83) implies the following inequalities: d 1=2 ˇ hA0 Um ; Um i0 ˇsD0 j.F.Um /; Um /jsD0 ds BCqC2 hA0 Um ; Um i.qC2/=2 jsD0 BCqC2 c1m ; 1 1 0 where

qC2


;

1=2 jvj0 C1 hA0 v; vi0 :

Thus, d 1=2 ˇ hA0 Um ; Um i0 ˇsD0 c1m : ds

(2.104)

Now let 'm hA0 Um ; Um i0 . Similarly to previous arguments, we can easily show that for the function 'm .s/, the following inequality of the form (2.96) holds: d 2 'm d'm 2 qC2 ˛ 0; ˛ 'm : (2.105) 2 ds ds 2 Integrating the differential inequality (2.105) over s 2 .t =; t /, we obtain ˇ Œ.F.um0 /; um0 /0 .B.um0 /; um0 /1 d'm 1 ˇˇ 2 : (2.106) ˇ ˛ ds 'm sDt hA0 um0 ; um0 i.qC2/=2 0 Now we use the fact that according to the condition of Theorem 2.5.2 .F.u0 /; u0 /0 .B.u0 /; u0 /1 > 0 and hence there exists a subsequence of the sequence ¹um0 º such that .F.um0 /; um0 /0 .B.um0 /; um0 /1 > 0: From (2.106) we have d Œ.F.um0 /; um0 /0 .B.um0 /; um0 /1 1=2 ˇ c2m > 0: hA0 Um ; Um i0 ˇsD0 .qC2/=2 ds hA0 um0 ; um0 i0 Inequality (2.103) is equivalent to the following two-sided inequality: 0 < c2m

1 dMm .t / c1m < C1; qC1 Mm .t / dt

q > 0;

(2.107)

126


where c2m

Œ.F.um0 /; um0 /0 .B.um0 /; um0 /1 hA0 um0 ; um0 i.qC2/=2 0 qC2

j.F .v/; v/0 j B jvj0

;

;

c1m BCqC2 ; 1

1=2 jvj0 C1 hA0 v; vi0 :

Integrating (2.107) over t from t to Tm0 , lim sup Mm .t / D C1; t "Tm0

we obtain Œqc1m 1=q Mm .t /.Tm0 t /1=q Œqc2m 1=q ;

t 2 Œ0; Tm0 /:

(2.108)

Here the following lower and upper estimates of the blow-up time Tm0 are valid: A1m Tm0 A2m ; where A1m A2m

1 qCqC2 BhA0 um0 ; um0 iq=2 1 0

;

hA0 um0 ; um0 i0 : q Œ.F.um0 /; um0 /0 .B.um0 /; um0 /1

It is not difficult to prove that there exist constants 0 < A1 ; A2 < C1 independent of m 2 N such that 0 < A1 Tm0 A2 < C1: Let T0 lim infm!C1 Tm0 . Then there exists a subsequence of the sequence ¹Tm0 º such that either Tm0 " T0 or Tm0 # T0 . First, we consider the case where Tm0 " T0 . For any fixed m 2 N, we can pass to the limit as m ! C1 in inequality (2.108) uniformly with respect to t 2 Œ0; Tm0 : Œqc1 1=q M.t /ŒT0 t 1=q Œqc2 1=q

8t 2 Œ0; Tm0 /;

(2.109)

where c2

Œ.F.u0 /; u0 /0 .B.u0 /; u0 /1 .qC2/=2

hA0 u0 ; u0 i0

qC2


;

;

c1 BCqC2 ; 1 1=2

jvj0 C1 hA0 v; vi0 :

Hence, by virtue of the arbitrariness of m 2 N, we obtain that inequality (2.109) holds uniformly with respect to t 2 Œ0; T0 /. Now we consider the case where Tm0 # T0 . Then, passing to the limit as m ! C1 in Eq. (2.108) uniformly with respect to t 2 Œ0; T0 /, we obtain (2.109). Lemma 2.5.8 Theorem 2.5.2 is completely proved.

Section 2.6 Strong generalized solvability of problem (2.2) for B 0

2.6

127

Strong solvability of problem (2.2) in the case where B 0

Definition 2.6.1. A solution of problem (2.2) belonging to the class u 2 C.1/.Œ0;TIV0 / and satisfying the conditions h.A0 u/0 ; vi0 D .F.u/; v/0

8v 2 V0 ;

8t 2 Œ0; T;

where the time derivative is meant in the classical sense, is called a strong generalized solution of problem (2.2). Since A0 u 2 C .1/ .Œ0; TI V0 /; F.u/ 2 C.Œ0; TI W0 / C.Œ0; TI V0 /; by the conditions of Theorem 2.6.2 below, problem (2.2) is equivalent to the problem h.A0 u/0 ; vi0 D hF.u/; vi0

8v 2 V0 ;

8t 2 Œ0; T;

where the time derivative is meant in the classical sense and h; i0 denotes the duality bracket between the Banach spaces V0 and V0 . The following theorem holds. Theorem 2.6.2. Let the conditions (A0 ) and (F) hold and V0 W0 . Then for any u0 2 V0 , there exists T0 > 0 such that the Cauchy problem (1.1)–(1.2) with B D 0 has a unique solution of the class C .1/ .Œ0; T0 /I V0 /. If, moreover, .F.u0 /; u0 /0 > 0 for given u0 2 V0 , ku0 k0 > 0, then the solution existence time 0 < T0 < C1 and lim ku.t /k0 D C1:

t"T0

(2.110)

Moreover, there exist positive constants C1 and C2 such that the following lower and upper estimates of the time and rate of the blow-up of the solution holds: C1 .T0 t /1=q kuk0 C2 .T0 t /1=q : Proof. Step 1. Local solvability Lemma 2.6.3. For any u0 2 V0 , under the condition .F.u0 /; u0 /0 > 0, there exists maximal T0 > 0 such that a unique solution of problem (2.2) with B D 0 exists in the class u 2 C .1/ .Œ0; TI V0 / for any T 2 .0; T0 /.

128


Proof. Consider the following problem, which equivalent to (2.2) in the class C .1/ .Œ0; TI V0 /: du G.u/.t / D 0; dt

u.0/ D u0 2 V0

8t 2 Œ0; T;

u 2 C .1/ .Œ0; TI V0 /; (2.111)

where G.u/ D A1 0 F.u/.t /, u 2 C.Œ0; TI V0 /. The following relations hold: lim kG.u/.t / G.u/.t0 /k0 m1 0 lim jF.u/.t / F.u/.t0 /j0

t!t0

t!t0

m1 lim 0 .R/ t!t

0

ju.t / u.t0 /j0

C lim ku.t / u.t0 /k0 D 0; t!t0

where R D max¹ku.t0 /k; ku.t /kº. Therefore, G W C.Œ0; TI V0 / ! C.Œ0; TI V0 /: Introduce the set BR D ¹u.t / 2 C.Œ0; TI V0 / W kukT Rº; where kvkT sup kvk0 : t2Œ0;T

We prove that the operator Z U.u/ u0 C

t 0

ds G.u/.s/;

t 2 Œ0; T;

(2.112)

acts from BR into BR and is a contraction on BR . Indeed, since G W C.Œ0; TI V0 / ! C.Œ0; TI V0 /; it is obvious that U W C.Œ0; TI V0 / ! C .1/ .Œ0; TI V0 /: On the other hand, Z kU.u/kT ku0 k0 C kG.u/k0 .t /

T 0

m1 0 kF.u/k0

ds kG.u/k0 .s/ ku0 k0 C T sup kG.u/k0 .t /; t2Œ0;T

C m1 0 jF.u/j0

1 m1 .CR/kuk0 : 0 C.juj0 /juj0 Cm0


Therefore,

129

kUkT ku0 k C m1 0 C.CR/RT:

Let ku0 k0 R=2 and T < m0 =.2C.CR//; then for sufficiently large R > 0 and small T > 0, we have kUkT R: Thus, U W BR ! BR . Now we prove that the operator U defined by formula (2.112) is a contraction on BR . Let u; v 2 BR ; then we have Z T dt kG.u/ G.v/k0 kU .u/ U.v/kT 0

m0 1

Z 0

T

dt kF .u/ F.v/k0 .t / m1 0 C.CR/Tku vkT :

From the latter inequality under T < m0 =.C2.CR// we obtain 1 kU.u/ U.v/kT < ku vkT : 2 Thus, the operator U is a contraction on BR . Therefore, by the theorem on contraction mappings, for any u0 2 V0 , there exists T > 0 such that a unique solution of problem (2.111) of the class C .1/ .Œ0; TI V0 / exists. Using the standard algorithm of the extension of solutions on time, we obtain that T0 > 0 and a unique solution of problem (2.2) with B D 0 of the class C .1/ .Œ0;T0 /IV0 / exists; moreover, either T0 < C1, and then lim sup kuk0 D C1; t"T0

or T0 D C1. Indeed, consider the norm .T/ ku.t /kT . The function .T/ monotonically increase. Therefore, as T " T0 , the function .T/ has either a finite or infinite limit. Assume that .T/ has a finite limit as T " T0 . Consider the following auxiliary integral equation of the form (2.112): Z t 0 u.x; t / D u.x; T / C ds G.u/.s/: 0

The norm kruk22 .T0 / is uniformly bounded on T0 2 .0; T0 /; this allows choosing T 2 .0; T0 / such that for any T0 2 .0; T0 /, the integral equation has a unique solution of the class u.x; t / 2 C .1/ .Œ0; T I V0 /. Let T0 D T0 T =2; we denote by v.x; t / the corresponding solution of the latter integral equation and define b u.x; t / on the segment Œ0; T0 C T =2 as follows: b u.x; t / D ¹u.x; t /; t 2 Œ0; T0 I v.x; t T0 /; t 2 ŒT0 ; T0 C T =2º:

130


By construction, b u.x; t / is a solution of problem (1.1)–(1.2) on the segment Œ0; T0 C T =2 and, by the local solvability, is an extension of the function u.x; t /. This contradicts the maximality of the segment Œ0; T0 and hence lim ku.t /kT D C1:

T"T0

This immediately implies that lim sup ku.t /k0 D C1: t"T0

Thus, for any u0 2 V , there exists T0 > 0 such that a unique maximal solution of problem (2.2) with B D 0 of the class u.t / 2 C .1/ .Œ0; T0 /I V0 / exists. Lemma 2.6.3 is proved. Step 2. Blow-up of solutions of problem (2.2) with B D 0 for finite time. Estimates of blow-up time and rate Lemma 2.6.4. For any u0 2 V0 , there exists T0 > 0 such that lim ˆ.t / D C1;

t"T0

ˆ.t / hA0 u; ui0 :

Moreover, the following two-sided estimate for the blow-up rate holds: C1 .T0 t /1=q kuk0 C2 .T0 t /1=q ; where 1=2

C1 D M0

1=2

Œq.m0

2 1=q B1 /m1 ; 0 B1

1=2 .qC2/=2 1=q : C2 D m0 Œq.F .u0 /; u0 /0 hA0 u0 ; u0 i0

Proof. By the conditions (A0 ) and (F) and Eq. (2.2), the following relations hold: 1d hA0 u; ui0 D .F.u/; u/0 ; 2 dt 1 d hA0 u t ; u t i0 D .F.u/; u/0 : q C 2 dt hA0 u t ; ui0 D

(2.113) (2.114)

Introduce the notation '.t / hA0 u; ui0 ; then from (2.113), (2.114), and the Schwarz inequality (see [293]) by a direct calculation we obtain ' 00 .t /'.t /

qC2 0 .' .t //2 0; 2

and in the class u.t / 2 C .1/ .Œ0; T0 /I V0 / we have '.t / 2 C .1/ Œ0; T0 /.

(2.115)


131

On the other hand, from Eq. (2.114) we see that the right-hand side of Eq. (2.113) is continuously differentiable with respect to t 2 Œ0; T0 /. Thus, in the class u.t / 2 C .1/ .Œ0; T0 /I V0 / we obtain '.t / 2 C .2/ Œ0; T0 /: By the condition of the theorem, the inequality .F.u0 /; u0 /0 > 0, and relations (2.113)–(2.114) we have Z 0 < .q C 2/

t 0

hA0 us ; us i0 ds C .F .u0 /; u0 /0 D .F.u/; u/0 ; Z

hA0 u; ui0 D hA0 u0 ; u0 i0 C 2

0

(2.116)

t

ds .Fu; u/0 > 0:

(2.117)

From (2.116) and (2.117) we obtain that '.t / > 0 for t 2 Œ0; T0 /. Assume that T0 D C1; then u.t / 2 C .1/ .Œ0; C1/I V0 /. Here '.t / 2 C .2/ Œ0; C1/. Inequality (2.115) implies

'0 '˛

0

0;

˛D

qC2 : 2

Hence we obtain the inequalities q=2

'0

' q=2 .t / q.F.u0 /; u0 /0 .hA0 u0 ; u0 i0 /.qC2/=2 t; q=2

'0

q.F.u0 /; u0 /0 .hA0 u0 ; u0 i0 /.qC2/=2 t:

The latter inequality means that t cannot take arbitrarily large values. This contradiction proves that T0 < C1. The first statement of the lemma has been proved. Now, using the technique proposed in [354], we obtain estimates of time and rate of the blow-up of solutions to problem (2.2). Introduce the function v .s/ D N 1 .t /u.s C t /;

D N q .t /;

N.t / D hA0 u; ui1=2 0 .t /:

It is not difficult to verify that the function v .s/ is a solution of the following problem: A0

dv .s/ D F.v /.s/; ds

v .t =/ D u0 N.t /1 :

(2.118)

We prove that there exist positive constants c1 and c2 such that 0 < c2

d 1=2 hA0 v ; v i0 .0/ c1 < C1: ds

(2.119)

132


By (2.118), the following inequalities hold: d hA0 v ; v i0 .s/ D 2.F.v /; v /0 ds 2.jv j0 /jv j20 2.B1 kv k0 /B21 kv k20 1 2 2.B1 m1=2 hA0 v ; v i1=2 0 0 /m0 B1 hA0 v ; v i0 ;

(2.120)

d hAv ; v i0 .0/ 2.B1 m0 1=2 /B21 m0 1 ; ds d 1=2 hAv ; v i0 .0/ c1 ; ds 1=2

where c1 D .B1 m0 /B21 m1 0 . Let ' .s/ D hA0 v ; v i0 .s/; then, subject to (2.115), we obtain '00 .s/' .s/

qC2 0 .' .s//2 0: 2

This implies .qC2/=2

'0 .s/='˛ .s/ '0 .s/='˛ .s/jsDt = D 2.F.u0 /; u0 /0 hA0 u0 ; u0 i0

;

where ˛ D .q C 2/=2. Thus, we have d 1=2 .qC2/=2 D c2 : hA0 v ; v i0 .0/ .F .u0 /; u0 /0 hA0 u0 ; u0 i0 ds

(2.121)

Relations (2.120) and (2.121) imply inequality (2.119), which is equivalent to the following inequality: 0 < c2 N 0 .t /=N qC1 .t / c1 < C1:

(2.122)

Integrating inequality (2.122) from t to T0 and taking into account (2.110), we obtain Œqc1 1=q N.t /.T0 t /1=q Œqc2 1=q ;

1=2

N.t / D hA0 u; ui0 ;

(2.123)

where 1=2 c1 D .B1 m0 /B21 m1 0 ;

c2 D .F.u0 /; u0 /0 hA0 u0 ; u0 i0.qC2/=2 :

On the other hand, we have 1=2 m1=2 0 kuk0 .t / N.t / M0 kuk0 .t /:

From this and (2.123) we obtain C1 .T0 t /1=q kuk0 C2 .T0 t /1=q ;

(2.124)

133

Section 2.7 Examples

where 1=2

C1 D M0

1=2

Œq.m0

2 1=q B1 /m1 ; 0 B1

1=2 .qC2/=2 1=q : C2 D m0 Œq.F .u0 /; u0 /0 hA0 u0 ; u0 i0

From (2.124) we obtain the following lower and upper estimates for the blow-up time: q

q

q

q

C1 ku0 k0 T0 C2 ku0 k0 : Lemma 2.6.4 is proved. Theorem 2.6.2 is completely proved.

2.7

Examples

Now we consider examples (2.3)–(2.8) of strongly nonlinear Sobolev-type equations that satisfy the conditions for operator coefficients of Eqs. (2.1), (2.2), under which the theorems proved above are valid. In these examples, standard Sobolev embedding theorems are used; appropriate results can be found in [108–111]. In all examples, H D L2 ./. Example 2.7.1. n X @ div.jrujpj 2 ru/ C jujq u D 0; u C @t j D1

uj@ D 0;

u.x; 0/ D u0 .x/;

where RN is a bounded domain with smooth boundary of the class C .2;ı/ , ı 2 .0; 1, and the operator coefficients have the form A0 u u W H01 ./ ! H1 ./; 1;pj

Aj .u/ div.jrujpj 2 ru/ W W0

0

./ ! W 1;pj ./;

0

F .u/ jujq u W LqC2 ./ ! Lq ./; pj 2; V0 H01 ./;

q > 0;

pj0 1;pj

Vj W0

pj ; pj 1

./; 0

q0

qC2 ; qC1

W0 LqC2 ./;

V0 H1 ./; Vj W 1;pj ./; W0 L.qC2/=.qC1/ ./; \ n 1;pj 1 V H0 ./ \ W0 ./ : j D1

134


The embedding V0 H01 ./ W0 LqC2 ./ is valid under the following conditions: 4 for N 3; 0 < q < C1 for N D 1; 2: 0 0;

W0 LqC2 ./;

0

0

V0 H1 ./; V1 Lp1 ./; W0 Lq ./: The embedding V0 ,! W0 is valid under the condition 0 0 for N D 1; 2 and 0 < q < 4=.N 2/ for N 3. Finally, the upper inequality in the case (3) holds under the same conditions for q. Proof. Let um be Galerkin approximations defined in (2.137). Since u0m * u0 strongly in W01;p ./ and um * u

-weakly in L1 .0; TI W0

1;p

.//;

passing in (2.141) and (2.142) to the limit as m ! C1, subject to the fact that E.t / lim inf Em .t / m!C1

160


(see [436]), we obtain 1

C .1 ˛/B2 t 1˛ ; E.t / ŒE1˛ 0 E.t / E0 exp ¹B2 t º ;

˛ < 1;

˛ D 1:

(2.163) (2.164)

Thus, by (2.163) and (2.164) we have obtained upper estimates in the first two items of the theorem. 1;p Now we consider the case where ˛ > 1. Since u0m ! u0 strongly in W0 ./ 1 ./, by a result of [214] we have H0 E0m ! E0 : In this case, from the numerical sequence E0m , we can select either a monotonically nondecreasing or monotonically increasing subsequence. We denote the required subsequence of the sequence E0m and the corresponding subsequences of the sequences u0m and um by the same symbol. Consider inequality (2.143). Assume that E0m is a monotonically nonincreasing ¹nº sequence and introduce the notation ¹E0m º for the sequence obtained from E0m by eliminating first n 2 N summands. Then inequality (2.143) holds uniformly with ¹nº 1 1 1˛ respect to m for all t 2 Œ0; T¹nº 1 /, where T1 .˛ 1/ B2 E0n . Since um * u

-weakly in L1 .0; TI W01;p .//;

for all such t we can pass in (2.143) to the limit as m ! C1 and, subject to the result of [436] we obtain that t 1=.˛1/ E.t / E0 Œ1 .˛ 1/B2 E˛1 0

for t 2 Œ0; T¹nº 1 /:

(2.165)

¹nº

By the arbitrariness of n 2 N and the fact that T1 " T1 as n " C1, we conclude that (2.165) holds for t 2 Œ0; T1 / and, moreover, T0 T1 . Now let E0m be a monotonically nondecreasing sequence. Then, repeating the previous arguments, we obtain that (2.165) holds for all t 2 Œ0; T1 / and T0 T1 . For further consideration, we must prove that Em .t / ! E.t / for almost all t 2 Œ0; T and all T 2 .0; T0 /. We use again Lemma A.15.1 from Appendix A.15, where we set W D LqC2 ./, and obtain that um .s/ ! u.s/

strongly in LqC2 ./ for almost all s 2 Œ0; T:

Next, arguing in the standard way, we obtain the required statement.

Section 2.8 Initial-boundary-value problem with double nonlinearity

By (2.139) and (2.144), since um .t / 2 C .1/ .Œ0; TI W01;p .//, we have p N 1d p 1 d X @um qC2 2 krum k2 C @x D kum kqC2 .s/; 2 dt p dt i p kru0m k22

C .p 1/

N Z X i D1

161

(2.166)

i D1

ˇ ˇ ˇ ˇ ˇ @um ˇp2 ˇ @u0m ˇ2 ˇ ˇ ˇ ˇ D 1 d kum kqC2 : (2.167) dx ˇ qC2 ˇ ˇ @xi @xi ˇ q C 2 dt

Equations (2.166) and (2.167) imply the fact that Em .t /; E0m .t / 2 ACŒ0; T. The following inequalities hold: ˇ ˇZ ˇ ˇ2 ˇ dx ru0 ; rum ˇ kru0 k2 krum k2 ; (2.168) m m 2 2 ˇ ˇ

ˇZ ˇ ˇ ˇp2 ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ @um ˇp1 ˇ @u0m ˇ ˇ @um @u0m ˇˇ ˇ dx ˇ @um ˇ ˇ ˇ ˇ ˇ dx ˇ (2.169) ˇ ˇ @x ˇ ˇ @x ˇ @xi @xi ˇ @xi ˇ i i ˇ ˇ ˇ ˇ Z ˇ @um ˇp2 ˇ @u0m ˇ2 1=2 @um p=2 ˇ ˇ ˇ ˇ dx ˇ : ˇ @x ˇ @xi p @xi ˇ i

By the Schwarz inequality (see, e.g., [293]) we have ˇ Z ˇ ˇ ˇ N X ˇ @um ˇp2 ˇ @u0m ˇ2 1=2 @um p=2 ˇ ˇ ˇ ˇ dx ˇ @x ˇ @x ˇ @x ˇ i p i i iD1

ˇ N Z ˇ ˇ ˇ X N ˇ @um ˇp2 ˇ @u0m ˇ2 1=2 @um p 1=2 X ˇ ˇ ˇ dx ˇˇ : (2.170) @x ˇ @x ˇ @xi ˇ i p i i D1

i D1

By (2.166)–(2.170), the following inequalities hold: jhum ; Œum p um 0t ij2

ˇ ˇX ˇ ˇ Z ˇ @um ˇp2 @um @u0m ˇ2 ˇN ˇ ˇ ˇ C .p 1/ ˇ dx ˇ @xi ˇ @xi @xi ˇ i D1 ˇN Z ˇ ˇ ˇ X ˇ ˇ @um ˇp2 @um @u0m ˇ 0 ˇ ˇ ˇ ˇ C 2.p 1/j.rum ; rum /jˇ dx ˇ @xi ˇ @xi @xi ˇ jhru0m ; rum ij2

2ˇ

i D1

ˇ N Z ˇ ˇ ˇ N X ˇ @um ˇp2 ˇ @u0m ˇ2 @um p X ˇ ˇ ˇ ˇ C .p 1/ dx ˇ @x ˇ @x ˇ @xi ˇ i p i i D1 i D1 p 1=2 X N @um C 2.p 1/kru0m k2 krum k2 @x i p i D1 ˇ ˇp2 ˇ 0 ˇ2 1=2 X N Z ˇ @um ˇ ˇ @um ˇ ˇ ˇ ˇ dx ˇˇ ˇ ˇ @x ˇ @xi i kru0m k22 krum k22

i D1

2

162


N X @um p 2 krum k2 C .p 1/ @x i p i D1 ˇ ˇ ˇ ˇ N Z X ˇ @um ˇp2 ˇ @u0m ˇ2 0 2 ˇ ˇ ˇ dx ˇˇ krum k2 C .p 1/ ˇ @x ˇ @xi ˇ i i D1

ˇ ˇ ˇ ˇ ˇ @um ˇp2 ˇ @u0m ˇ2 ˇ ˇ ˇ ˇ C .p 1/ dx ˇ p ˇ @x ˇ @xi ˇ i i D1 p N 1 p 1 X @um : krum k22 C 2 p @xi p

kru0m k22

N Z X

i D1

From (2.166), (2.167), and the latter inequality we obtain d Em 2 qC2 d 2 Em E ˛ 0; ˛ D : m 2 dt dt p

(2.171)

Now we consider three cases: 0 < ˛ < 1, ˛ D 1, and 1 < ˛. As above, we can assume that the sequence E0m is monotonic and positive; then Em > 0 for all m 2 N. In the first case where 0 < ˛ < 1, (2.171) implies 0 0 qC2 ku0m kqC2 E0m Em 0; ; E˛m E˛m E˛0m (2.172) qC2 1 1˛ k ku 0m qC2 Em .t / E1˛ t : 0m C .1 ˛/ E˛0m In the second case, where ˛ D 1, from (2.171) we obtain 0 0 2 E0m 00 Em Em Em 0; ln 0; Em ² qC2 ³ ku0m kqC2 t : Em .t / E0m exp E0m In the third case, where ˛ > 1, (2.171) implies 0 0 qC2 ku0m kqC2 E0m Em 0; ; E˛m E˛m E˛0m 1=.˛1/ ku0m kqC2 qC2 Em E0m 1 .˛ 1/ t : E0m

(2.173)

(2.174)

1;p By the strong convergence u0m ! u0 in W0 ./ H01 ./ and by virtue of the embedding W01;p ./ LqC2 ./ we obtain

E0m ! E0 ;

ku0m kqC2 ! ku0 kqC2 :

Section 2.8 Initial-boundary-value problem with double nonlinearity

163

On the other hand, we have showed that Em .t / ! E.t / for almost all t 2 Œ0; T0 /. Therefore, passing in Eqs. (2.172) and (2.173) to the limit as m ! C1, we obtain qC2 1 ku0 kqC2 1˛ 1˛ E.t / E0 C .1 ˛/ t for ˛ 2 .0; 1/; E˛0 ² qC2 ³ ku0 kqC2 E.t / E0 exp t for ˛ D 1: E0 Introduce the notation

qC2

F0m D

ku0m kqC2

: E0m Without loss of generality, by the passage to a subsequence, we can assume that E0m > 0 uniformly with respect to m since E0 > 0. By the convergence F0m ! F0 qC2 as m ! C1, where F0 ku0 kqC2 =E0 , we can select a monotonically convergent subsequence; we denote it also by F0m . The corresponding subsequences of the sequences ¹u0m º and ¹um º we also denote by ¹u0m º and ¹um º. Rewrite (2.174) taking into account the notation introduced: Em E0m Œ1 .˛ 1/F0m t 1=.˛1/ :

(2.175)

Let F0m be a monotonically nondecreasing sequence; then inequality (2.175) holds uniformly with respect to m for all t 2 Œ0; .˛ 1/1 F1 /, where F D supm2N F0m . Passing to the limit as m ! C1, for such t in inequality (2.175) we obtain E E0 Œ1 .˛ 1/F0 t 1=.˛1/

under t 2 Œ0; .˛ 1/1 F1 /:

(2.176)

But F D F0 by virtue of the fact that F0m is a monotonically nondecreasing sequence qC2 converging to F0 ku0 kqC2 =E0 . Hence from (2.176) we obtain that q2

T0 .˛ 1/1 ku0 kqC2 E0 T2 : ¹nº Now let F0m be a monotonically nondecreasing sequence. Denote by ¹F0m º the sequence obtained from the sequence ¹F0m º by removing the first n 2 N terms. Assume that q2 T0 > .˛ 1/1 ku0 kqC2 E0 T2

(2.177)

and, moreover, let M

sup E.t / < C1: t2Œ0;T2

Then (2.175) holds uniformly with respect to t 2 Œ0; .˛ 1/1 F1 0n /. Passing in inequality (2.175) to the limit as m ! C1, by virtue of the assumption on T0 , we obtain the following inequality: E0 Œ1 .˛ 1/F0 t 1=.˛1/ M < C1;

t 2 Œ0; .˛ 1/1 F1 0n /:

(2.178)

164


However, by the fact that F0n # F0 as n ! C1, there exist n 2 N and tn 2 .0; .˛ 1/1 F1 0n / such that (2.178) does not hold. Therefore, our assumption (2.177) is invalid. Hence we obtain that q2 T0 .˛ 1/1 ku0 kqC2 E0 T2 :

Theorem 2.8.5 is proved.

2.9

Problem for a strongly nonlinear equation of type (2.2) with inferior nonlinearity

First, we derive the problem, which will be studied below. Let 2 RN , N 1, be a bounded domain with smooth boundary of the class @ 2 C .2;ı/ , ı 2 .0; 1. The domain is filled with a semiconductor. One of the fundamental properties of semiconductors consists of the fact that nonstationary processes observed in them are described by systems of quasi-stationary field equations, the continuity equation, and constitutive equations. In their description, a substantial role is played by the explicit form of constitutive equations that link the intensity E of the electric field and the electric displacement D, on one hand, and the intensity E of the electric field and the current density J if the in semiconductor, on the other hand. The required system of equations in a certain Cartesian coordinates in the general case has the form (see [233]) div D D 4 e n;

E D r';

@n D div J C Q; @t

Ji D

3 X

D D E C 4 P;

(2.179)

i; j D 1; 3;

(2.180)

ij Ej ;

j D1

where P is the polarization vector. For certain models, the following phenomenological expression of the polarization vector through the potential ' of the electric field is valid: div P D k1 j'jq ';

k1 > 0:

(2.181)

Moreover, assume that the conductivity tensor ij is nonlinear with respect to the components of the intensity Ej of the electric field (see [256]): ij D ıij jEj jp2 ;

p > 2:

(2.182)

Finally, assume that in the semiconductor there are sources of free-charge current whose distribution in the self-consistent electric field has the form Q D k2 j'jp2 ';

k2 > 0:

(2.183)

165

Section 2.9 Problem with inferior nonlinearity

System (2.179)–(2.183) implies the following strongly nonlinear pseudoparabolic equation: @ .' 4 k1 j'jq '/ C 4 ep ' C 4 ek2 j'jp2 ' D 0: @t

(2.184)

Under the assumption that the semiconductor is grounded and there is the initial distribution of electric field, we obtain the following problem for Eq. (2.184) in the dimensionless variables: @ .u jujq u/ C p u C jujp2 u D 0; @t u.x; 0/ D u0 .x/;

uj@ D 0;

p > 2;

q 0;

> 0;

ˇ ˇ N X @ ˇˇ @u ˇˇp2 @u : p u @xi ˇ @xi ˇ @xi

(2.185)

i D1

2.9.1 Unique weak solvability of problem (2.185) Let @ 2 C .2;ı/ , ı 2 .0; 1, be the boundary of a bounded domain and a natural number s 2 be such that .s 1/=N 1=2 1=p. Consider the following problem on eigenfunctions and eigenvalues: ˇ @n wk ˇˇ s s .1/ wk k wk D 0; D 0; n D 1; s 1: (2.186) @xin ˇ@ By the definition of s, the following continuous embedding is valid: 1;p

H0s ./ W0

./;

p > 2: 1;p

Eigenfunctions of problem (2.186) form a Galerkin basis in W0 ./. We denote by h; i0 the duality bracket between the Hilbert spaces H01 ./ and H1 ./, by h; i the duality bracket between the Banach spaces W01;p ./ and 0 W 1;p ./, and by .; / the inner product in the sense of L2 ./. We state a definition of weak generalized solutions. Definition 2.9.1. A solution of the problem Z

T

dt .t / 0

d d hu; wi0 .jujq u; w/ C hp u; wi C .jujp2 u; w/ D 0 dt dt 8 .t / 2 L2 .0; T/;

8w 2 W01;p ./; 1;p

u.0/ D u0 2 W0

./;

is called a weak generalized solution of problem (2.185).

166


We search for a solution of the given problem in the following class: u 2 L1 .0; TI W0

1;p

0

.//;

p u 2 L1 .0; TI W 1;p .//;

u0 2 L2 .0; TI H01 .//; 0

u jujq u 2 H1 .0; TI W 1;p .//:

In the considered smoothness class, our problem is equivalent to the problem Z

T 0

d q p2 dt .t / u; w D 0 .u juj u/ C p u C juj dt 8 .t / 2 L2 .0; T/;

8w 2 W01;p ./;

which, in its turn, is equivalent to the problem Z

T

dt 0

d .u jujq u/ C p u C jujp2 u; v D 0 dt

1;p

8v 2 L2 .0; TI W0

.//:

1;p Remark 2.9.2. As w 2 W0 ./, we can take elements wj 2 H0s ./, j D 1; C1, since their linear combinations are dense in W01;p ./.

The following theorem is valid. 1;p

Theorem 2.9.3. Let u0 .x/ 2 W0 ./. If either N D 1; 2 for q 0 or N 3 for 0 q < 4=.N 2/ and 2 p < 2N=.N 2/, then there exist maximal T0 D T0 .u0 ; q; p/ > 0 and a function u.x; t / W .0; T0 / ! R such that for any T 2 .0; T0 /, the following inclusions hold: u 2 L1 .0; TI W0

1;p

.//;

u0 2 L2 .0; TI H01 .//; 0

u jujq u 2 H1 .0; TI W 1;p .//: In the case where q D 0, the function u possesses the following properties: hu u; ui0 2 H1 .0; T/I for almost all t 2 .0; T/, the following relation holds: d hu u; ui0 D 2hu0 u0 ; ui0 ; dt where u0 denotes the time derivative; @ .u jujq u/ C p u C jujp2 u D 0 @t

167

Section 2.9 Problem with inferior nonlinearity 0

in the sense of L2 .0; TI W 1;p .//, p 0 D p=.p 1/; ˇ ˇ N X @ ˇˇ @u ˇˇp2 @u : p u @xi ˇ @xi ˇ @xi i D1

Moreover, u.0/ D u0 almost everywhere in . Under the condition q D 0, the function u.x; t / is unique. Remark 2.9.4. Since, under the conditions of Theorem 2.3.2, u 2 L1 .0; TI W01;p .//;

u0 2 L2 .0; TI H01 .//;

we obtain (see [275]) that u.x/.t / W Œ0; T ! H01 ./ is a strongly continuous vectorvalued function. Therefore, the initial condition makes sense. Proof. To prove the local solvability of the problem, we use the Galerkin method together with the monotonicity and compactness methods (see [275]). We search for an approximate solution of problem (2.185) in the following form: Z T dt .t /Œhu0m ; wj i0 ..jum jq um /0 ; wj / 0

C hp um ; wj i C .jum jp2 um ; wj / D 0;

(2.187)

for all .t / 2 L2 .0; T/, j D 1; m, where um D

m X

cmk .t /wk ;

u0m um .0/ D

kD1

m X

cmk .0/wk ! u0

kD1

in W01;p ./;

cmk .t / 2 C .1/ Œ0; Tm ;

hence in the class cmk .t / 2 C .1/ Œ0; Tm0 /, for certain Tm0 > 0, from (2.187) we obtain the pointwise equation ˇ N ˇ X ˇ @um ˇp2 @um @wj 0 q 0 ˇ ˇ .rum ; rwj / C .q C 1/.jum j um ; wj / C ; ˇ @x ˇ @xi @xi i i D1

D .jum j

p2

um ; wj /;

j D 1; m;

8t 2 Œ0; Tm : (2.188)

From (2.188) we obtain N m X X

0 Œ.rwk ; rwj / C .q C 1/.jum jq wk ; wj /cmk kD1 iD1 ˇ N ˇ X ˇ @um ˇp2 @um @wj ˇ ˇ D .jum jp2 um ; wj /: C ; ˇ @x ˇ @xi @xi i i D1

Now as a basis in duce the notation

1;p W0 ./,

we choose eigenfunctions of problem (2.186). Intro-

akj .rwk ; rwj / C .q C 1/.jum jq wk ; wj /:

168


We have m;m X

akj k j D krk22 C .q C 1/.jum jq ; 2 /;

k;j D1;1

m X

D

wk k ;

lD1

m X

k wk :

kD1

P The norm krk2 vanishes in the class H01 ./ if and only if D m kD1 k wk D 0. 1;p C1 m Since ¹wk ºkD1 is a basis in W0 ./, D 0 if and only if ¹k ºkD1 D 0; therefore, > 0. by virtue of the Sylvester criterion (see, e.g., [335]) we obtain det¹akj ºm;m k;j D1;1 Note that the inverse matrix of the matrix akj .rwk ; rwj / C .q C 1/.jum jq wk ; wj / is continuous with respect to cm D .cm1 ; : : : ; cmm / (see Appendix A.18). General results on nonlinear systems of ordinary differential equations (see, e.g., [319]) guarantee the existence of a solution of problem (2.188) on a certain interval Œ0; Tm , Tm > 0, in the sense of ckm 2 C .1/ Œ0; Tm . Next, we obtain a priori estimates, which imply Tm D T, where T is independent 1;p of m. Therefore, ckm 2 C .1/ Œ0; T, um 2 C .1/ .Œ0; TI W0 .//. Multiplying both sides of Eq. (2.188) by cmj and summing over j D 1; m, we obtain N X @um p 1 d qC1 d qC2 p (2.189) krum k22 C kum kqC2 C @x D kum kp : 2 dt q C 2 dt i p i D1

Note that

X N @v @x i D1

p 1=p ;

kvkpp

i p

N X @v C @x

i p

i D1

1;p

are equivalent on the space v 2 W0 1;p the space W0 ./ with the norm

p 1=p

./ (see, e.g., [275]). Therefore, we consider

X N @v p 1=p : @x i p i D1

Relation (2.189) implies qC1 1 qC2 krum k22 C kum kqC2 C 2 qC2 D

Z

t

ds 0

N X @um p @x i D1

i

p

1 qC1 qC2 krum0 k22 C kum0 kqC2 C 2 qC2

Z 0

t

ds kum kpp ds;

> 0:

169


By the conditions on p, q, and N and by virtue of Sobolev embedding theorems (see, e.g., [26]), the inclusion H01 ./ Lp ./ holds. Now we have kum kp B1 krum k2 ; qC1 1 1 qC2 krum k22 C kum kqC2 krum0 k22 C 2 qC2 2 Z t C Bp ds 1 0

qC1 qC2 (2.190) kum0 kqC2 qC2 p=2 qC1 qC2 krum k22 C 2 : kum kqC2 qC2

Therefore, by (2.190) Z Em .t / Em0 C B2 Em .t / D

t 0

ds Ep=2 m .s/;

p=2 B2 D Bp ; 12

(2.191)

1 qC1 qC2 krum k22 C kum kqC2 : 2 qC2

On the other hand, if p q C 2, then by the boundedness of we have kum kp B3 kum kqC2 : Therefore, by (2.190) and (2.191) qC1 1 qC1 1 qC2 qC2 krum k22 C kum kqC2 krum0 k22 C kum0 kqC2 2 qC2 2 qC2 p=.qC2/ Z t qC1 qC21 qC2 q C 2 2 CBp ds k k ; (2.192) ku C kru m qC2 m 2 3 qC2 qC1 qC12 0 p=.qC2/ Z t p p qC2 ˛ ; B4 D B3 Em .t / Em0 C B4 ds Em .s/; ˛ D : qC2 qC1 0 By the conditions for p > 2 and q 0 we have ˛ 2 .0; 1. Using the Gronwall–Bellman and Bihari lemmas (see, e.g., [112]), from (2.191) and (2.192) we obtain 1=.1˛/ Em .t / ŒE1˛ m0 C .1 ˛/B4 t

qC2 qC1 i2=.p2/ h

p .p2/=2 Em .t / Em0 1 1 B2 Em0 t 2

Em .t / Em0 exp¹B5 t º;

B5 D Bp 3

for p < q C 2;

(2.193)

for p D q C 2;

(2.194)

for p > q C 2;

(2.195)

170


p=2 . Now we multiply Eq. (2.188) by c 0 and sum over j D 1; m. After B2 D Bp 12 mj integrating over s 2 Œ0; t we obtain

Z

t

kru0m k22

ds 0

C

Z C .q C 1/

dx jum j

q

.u0m /2

p N 1 X @um C 1 kum0 kp D 1 kum kp C 1 p p p @xi p p p p i D1

@um0 p @x : (2.196) i p

By the inclusion H01 ./ Lp ./ from (2.196) we obtain Z

t

ds 0

kru0m k22

Z C .q C 1/

dx jum j

q

.u0m /2

p N @u 1 X m C p @xi p

i D1

p C1 C Bp 1 krum k2 ; (2.197) p where u0m ! u0 strongly in W01;p ./. The following inequality holds: N X @um0 p @x C1 ; i D1

i

p

where C1 is independent of m. Hence from (2.193)–(2.195) we obtain the existence of T0 > 0 such that the following inequality holds: krum k2 A.t /;

t 2 Œ0; T;

8T 2 .0; T0 /;

where A is independent of m. Here in the cases (2.193) and (2.194), T0 D C1. Thus, (2.193)–(2.197) imply the fact that there exists maximal T0 T0 .u0 ; q; p/ such that for any T 2 .0; T0 /, the following inclusions hold: 1;p

bounded in L1 .0; TI W0

um u0m

bounded in

@ .jum jq=2 um / bounded in L2 .Q/; @t q

.//;

L2 .0; TI H01 .//; Q .0; T/; qC2 q D ; qC1 p p0 D : p1

0 L1 .0; TI Lq .//;

jum j um

bounded in

jum jp2 um

bounded in L1 .0; TI Lp .//;

(2.198)

0

0

From (2.198) it follows that Tm D T, where T is independent of m 2 N. Therefore, um D

m X kD1

1;p

ckm .t /wk 2 C .1/ .Œ0; TI W0

.//:

171


From inclusions (2.198) and results of [275] we conclude that there exists a subsequence of the sequence ¹uºm (we denote it again by ¹uºm ) such that um * u


u0m * u0

weakly in L2 .0; TI H01 .//;

um * u

-weakly in L1 .0; TI H1 .//;

p um *

-weakly in L1 .0; TI W 1;p .//;

jum jq um * wq

-weakly in L1 .0; TI Lq .//;

0

(2.199)

qC2 ; qC1 p : p0 D p1

0

q0 D

0

jum jp2 um * wp -weakly in L1 .0; TI Lp .//; Note that (2.199)2 implies the existence of a weak limit: u0m * u0

1;p

in D 0 .0; TI W0

.//I

therefore, by the weak convergence u0m * in L2 .0; TI H01 .//, we conclude that .t / D u0 .t /. Hence we have um * u

weakly in H1 .Q/;

Q D .0; T/ I

then, by the compact inclusion H1 .Q/ ,! L2 .Q/, we conclude that um ! u strongly in L2 .Q/ and, therefore, almost everywhere (see, e.g., [168]). Then, by [275, Lemma 1.3] we obtain jum jq um * jujq u

0

-weakly in L1 .0; TI Lq .//;

q0 D

qC2 ; qC1

and thus wq D jujq u. Finally, 0

jum jp2 um * jujp2 u -weakly in L1 .0; TI Lp .//;

p0 D

and hence wp D jujp2 u. From (2.198) we obtain jum jq=2 um 2 H1 .0; TI L2 .//;

um 2 H1 .0; TI H01 .//:

Taking into account the fact that 1;p 0

p um * -weakly in L1 .0; TI W0

.//;

p ; p1

172


we prove that D p u. From (2.188) we obtain hu0m ; wj i h.jum jq um /0 ; wj i C hp um ; wj i D hjum jp2 um ; wj i:

(2.200)

Now we can pass to the limit in (2.200) as m ! C1 and fixed j . Consider each summand in (2.200) separately. Embeddings (2.198) and (2.199) imply hu0m ; wj i * hu0 ; wj i

weakly in L2 .0; T/;

hp um ; wj i * h ; wj i

-weakly in L1 .0; T/;

hjum jp2 um ; wj i * hjujp2 u; wj i -weakly in L1 .0; T/; hjum jq um ; wj i * hjujq u; wj i

-weakly in L1 .0; T/;

um0 * u0

weakly in H1 ./;

um .T/ * u.T/

weakly in H1 ./;

jum0 jq um0 * ju0 jq u0

weakly in Lq ./;

jum jq um .T/ * jujq u.T/

weakly in Lq ./

(2.201)

0 0

and, therefore, d h.u jujq u/ C jujp2 u; wj i D 0: dt Hence by virtue of (2.201) we obtain d .u jujq u/ C jujp2 u D 0: dt

(2.202)

By the monotonicity of the operator A.v/ D p v with respect to the duality bracket 0 h; i between the Banach spaces W01;p ./ and W 1;p ./, p 0 D p=.p 1/, the following inequality holds: Z X

0

T

dt hA.u / A.v/; u vi 0 8v 2 Lp .0; TI W01;p .//:

According to (2.188) Z

T 0

Z

T

1 1 dt ku kpp C kru 0 k22 kru .T/k22 2 2 0 qC1 qC1 qC2 qC2 C ku 0 kqC2 ku .T/kqC2 : qC2 qC2

hA.u /; u i dt D

Since u * u weakly in W01;p ./ for almost all t 2 .0; T/, by the inclusions 1;p

W0

./ H01 ./ ,! LqC2 ./

173


we have (see, e.g., [436]) lim inf kru .T/k2 kru.T/k2 ;

!C1

qC2 qC2 lim inf ku .T/kqC2 ku.T/kqC2 :

!C1

Hence Z

1 1 qC1 qC2 dt ku kpp C kru0 k22 kru.T/k22 C ku0 kqC2 2 2 qC2 Z T Z T qC1 qC2 dt h ; vi dt hA.v/; u vi: ku.T/kqC2 qC2 0 0

lim sup X lim

T

!C1 0

!C1

Let us prove the following limit relation: Z T Z dt hju jp2 u ; u i D lim !C1 0

0

Indeed, Z Z T p2 dt hju j u ; u i D 0

T

T

0

dt kukpp :

Z p2

dt hju j

u ; u ui C

T 0

dt hju jp2 u ; ui:

Consider all summands separately. First, Z T dt jhju jp2 u ; u uij 0

Z

T

0

Z

0

Z dt kju j T

p2

u kp=.p1/ ku ukp D 1=p Z

dt ku ukpp

T 0

T 0

dt ku kpp1 ku ukp

.p1/=p dt ku kpp

! C0

as ! C1 since u 2 L1 .0; TI H01 .// L1 .0; TI Lp .// and u ! u strongly in Lp .Q/, Q D .0; T/ . Moreover, the following limit relation holds: Z T Z T dt hju jp2 u ; ui ! dt kukpp 0

0

since 0

-weakly in L1 .0; TI Lp .//; p u 2 L1 .0; TI Lp .//; p 0 D : p1

ju jp2 u * jujp2 u

174


Therefore, Z lim sup X

!C1

T 0

1 1 kukpp dt C kru0 k22 kru.T/k22 2 2

qC1 qC1 qC2 qC2 ku0 kqC2 ku.T/kqC2 qC2 qC2 Z T Z T dt h ; vi dt hA.v; u v/i: C

0

0

1;p L1 .0; TI W0 .//

and u0 2 L2 .0; TI H01 .// and by the conditions Note that u 2 for p, q, and N , the following inclusions hold: W01;p ./ H01 ./ ,! LqC2 ./ \ Lp ./: Multiply both sides of Eq. (2.202) by u and integrate by parts; this is possible by virtue of the following estimates: Z

T 0

dt jhu0 ; uij

Z

T 0

dt ku0 k21

1=2 Z

t2.0;T/

Z

T 0

q

Z

0

dt jhŒjuj u ; uij .q C 1/

T

0

Z .q C 1/ Z

T 0

Z dt jh ; uij

T

0

0

Z

T1=2 sup kuk2C1 .t /

0

T

T 0

1=2 dt kuk2C1

dt ku0 k21

1=2 ;

dt ku0 kqC2 kukqC2 qC1

T

dt ku0 k2qC2

1=2 Z

T 0

2.qC1/

1=2

dt kukqC2

dt k kW 1;p0 ./ kukW 1;p ./ 0

T sup k kW 1;p0 ./ kukW 1;p ./ ; t2.0;T/

0

where k kC k kH1 and k k k kH1 . Thus, from (2.202) we obtain 0

Z

T 0

Z

T

1 1 dt kukpp C kru0 k22 kru.T/k22 2 2 0 qC1 q C 1 qC2 qC2 C ku0 kqC2 ku.T/kqC2 : qC2 qC2

dt h ; ui D

On the other hand, by the definition of X we obtain Z T 1;p dt h A.v/; u vi 0 8v 2 Lp .0; TI W0 .//: 0

;

175


We set v D u w, > 0, and w 2 Lp .0; TI W01;p .//; then Z

T 0

dt h A.u w/; wi 0I

as ! C0, by virtue of the Lebesgue theorem we obtain Z T 1;p dt h A.u/; wi 0 8w 2 Lp .0; TI W0 .//: 0

Therefore, D p u. Now we can pass in Eq. (2.187) to the limit as m ! C1. Indeed, d d weakly in L2 .0; T/; .rum ; rwj / * .ru; rwj / dt dt d d .jum jq um ; wj / * .jujq u; wj / weakly in L2 .0; T/; dt dt .jum jp2 um ; wj / * .jujp2 u; wj / -weakly in L1 .0; T/; and ˇ ˇ N ˇ N ˇ X X ˇ @um ˇp2 @um @wj ˇ @u ˇp2 @u @wj ˇ ˇ ˇ ˇ * ; ; ˇ @x ˇ ˇ @x ˇ @xi @xi @xi @xi i i i D1

iD1

-weakly in L1 .0; T/. As a result, from (2.187) we obtain Z

T

dt .t / 0

d d hu; wi hjujq u; wi C hp u; wi C hjujp2 u; wi D 0 dt dt 8 .t / 2 L2 .0; T/;

1;p

8w 2 W0

./: (2.203)

Note that for .t / D 1, (2.187) implies the equation Z t q q ds Œp um C jum jp2 um um jum j um D um0 jum0 j um0 0

0

for almost all t 2 .0; T/ in the sense of W 1;p ./. We can pass in the given equality to the limit as m ! C1 and obtain Z t ds Œp u C jujp2 u u jujq u D u0 ju0 jq u0 0

in the sense of W

1;p 0

./ for almost all t 2 .0; T/. The latter equality implies 0

u jujq u 2 H1 .0; TI W 1;p .//:

176


Thus, from (2.203) we obtain the equivalent problem

Z T d q p2 dt .t / u; w D 0 Œu juj u C p u C juj dt 0 1;p

for all .t / 2 L2 .0; T/ and all w 2 W0 ./, which, in its turn, is equivalent to the problem

Z T d q p2 dt u; v D 0 8v 2 L2 .0; TI W01;p .//: Œu juj u C p u C juj dt 0 Thus, the solvability is proved. Let q D 0; then we can prove the validity of the inclusion hu u; ui0 2 H1 .0; T/; and, for almost all t 2 .0; T/, of the relation d hu u; ui0 D 2hu0 u0 ; ui0 : dt Prove the uniqueness. Let uj .t /, j D 1; 2, be two generalized solutions of the problem

Z T d Œu u C p u C jujp2 u; v D 0 8v 2 L2 .0; TI W01;p .// dt dt 0 with q D 0 in the classes formulated in the conditions of Theorem 2.9.3 with the same 1;p 1;p initial condition u0 2 W0 ./ and w D u1 u2 2 L2 .0; TI W0 .//. Let ´ u1 u2 ; s 2 Œ0; t ; v.s/ D 0; s 2 Œt; T; 0

1;p then, by virtue of the monotonicity of the operator p W W0 ./ ! W 1;p ./, we obtain the inequality

kru1 ru2 k22 C ku1 u2 k22 Z t Z ds dx Œju1 jp2 u1 ju2 jp2 u2 Œu1 u2 : (2.204) 0

From (2.204) we obtain kru1

ru2 k22

Z

2.p 1/

ds

Z C

0

t

0

Z 2.p 1/

Z

t

0

t

dx max¹ju1 jp2 ; ju2 jp2 ºju1 u2 j2

ds Œku1 kpp2 C ku2 kpp2 ku1 u2 kp2

ds kru1 ru2 k22 :


177

Hence by the Gronwall–Bellman theorem (see [112]) we obtain krwk2 D 0 8 t 2 Œ0; T/ which proves the uniqueness of Problem A under the condition q D 0.

2.9.2 Solvability in a finite cylinder and blow-up for a finite time Introduce the notation 1 qC1 qC2 Em .t / krum k22 C kum kqC2 ; Em0 D Em .0/; 2 qC2 qC1 1 qC2 E0 D E.0/; kukqC2 ; E.t / kruk22 C 2 qC2 N N X X @um0 p @u0 p p ; F ku k Fm0 kum0 kpp 0 0 p @x @x ; i p i p i D1 i D1 ˛ p qC2 q C 2 p p ; B5 Bp ; B2 B1 2 2 ; B4 B3 3 qC1 qC1 p ˛ ; kvkp B1 krvk2 ; kwkp B3 kwkqC2 : qC2 The following theorem holds. Theorem 2.9.5. Let all the conditions of Theorem 2.9.3 hold and, moreover, the additional condition F0 > 0 is fulfilled for the validity of lower estimates. Then there exist a weak generalized solution of problem (2.185) and constants B2 , B4 , and B5 such that (1) if q C 2 > p, then T0 D C1 and 1 1 F0 1˛ 1˛ E0 C .1 ˛/ ˛ t E.t / ŒE1˛ C .1 ˛/B4 t 1˛ I 0 E0 (2) if p D q C 2, then T0 D C1 and ² ³ F0 E0 exp t E.t / E0 exp ¹B5 t º I E0 (3) if p > q C 2 (i.e., ˛ > 1), then lim sup t"T0 2 p2

T1

N X

kruk22 D C1;

i D1

2 E0 E˛ T0 .˛ 1/1 0 T2 : p 2 B2 F0

178


Proof. We need an auxiliary result on the sequence ¹um º of Galerkin approximations. Lemma 2.9.6. There exists a subsequence of the sequence ¹um º such that @um @u ! @xi @xi

strongly in Lp .Q/;

Q .0; T/ :

Proof. Introduce the notation ˇ ˇ @ ˇˇ @v ˇˇp2 @v Ai .v/ ; @xi ˇ @xi ˇ @xi Z T Z T Z T N X @u p dt hA .u /; u i dt C lim sup dt hA.um /; um i: lim sup i m m @x j p m!C1 0 m!C1 0 0 j D1;i ¤j

Then the following inequality holds: Z lim sup m!C1 0

T

Z dt hA.um /; um i

0

T

1 1 dt kukpp C kru0 k22 kru.T/k22 2 2

qC1 qC1 qC2 qC2 ku0 kqC2 ku.T/kqC2 qC2 qC2 Z T Z T N X @u p D dt hp u; ui D dt @x : i p 0 0 C

i D1

Therefore, Z

T

lim sup m!C1 0

Z T @um p dt dt @xi p 0

@u p @x : i p

(2.205)

On the other hand, @um @u * @xi @xi

weakly in Lp .Q/;

Q .0; T/ :

Therefore, Z T @um p dt dt @xi p 0

@u p @x : i p

(2.206)

Relations (2.205) and (2.206) imply Z T Z T @um p lim dt D dt m!C1 0 @xi p 0

@u p @x : i p

(2.207)

Z lim inf

m!C1 0

T

179


In its turn, (2.207) implies the fact that there exists a subsequence of the sequence ¹um º such that @u @um ! @xi @xi

strongly in Lp .Q/;

i D 1; N ;

Q .0; T/ :

(2.208)

Lemma 2.9.6 is proved. Now we establish the following inequalities: ˇZ ˇ2 ˇ ˇ ˇ dx hrum ; ru0 iˇ krum k2 kru0 k2 ; mt ˇ 2 mt 2 ˇ ˇ2 ˇZ Z ˇ ˇ ˇ dx jum jq um u0 ˇ kum kqC2 dx jum jq .u0mt /2 mt ˇ qC2 ˇ

(2.209)

and ˇ2 ˇZ ˇ ˇ q 0 ˇ um .um C jum j um / dx ˇ t ˇ ˇ

j.rum ; ru0mt /j2 C .q C 1/2 j.jum jq um ; u0mt /j2 C 2.q C 1/j.rum ; ru0mt /j j.jum jq um ; u0mt /j krum k22 kru0mt k22 C .q C 1/2 .jum jq ; .u0mt /2 /kum kqC2 q .qC2/=2 C 2.q C 1/krum k2 kru0mt k2 .jum jq ; .u0mt /2 /kum kqC2 Z qC2 kru0mt k22 C .q C 1/ dx jum jq .u0mt /2 Œkrum k22 C .q C 1/kum kqC2 qC2

.q C 2/

kru0mt k22

Z

C .q C 1/

dx jum j

q

.u0mt /2

1 qC1 qC2 2 krum k2 C kum kqC2 : 2 qC2

(2.210)

Relations (2.189) and (2.196) imply the equalities N X @um p qC1 d 1d qC2 2 p krum k2 C kum kqC2 C @x D kum kp ; 2 dt q C 2 dt i p kru0m k22

Z C .q C 1/

(2.211)

i D1

dx jum j

q

.u0m /2

N X @um p 1 d p kum kp D @x : p dt i p i D1

(2.212)

180


Assume that ku0 kpp

N X @u0 p > @x : i

i D1

p

Since um0 ! u0 strongly in H01 ./, there exists a subsequence of the sequence ¹um0 º such that N X @um0 p p kum0 kp > @x : i

i D1

p

From (2.209)–(2.212) we obtain Em

d 2 Em p 2 dt qC2

d Em dt

2

0:

(2.213)

Let um be a sequence of Galerkin approximations defined in (2.188). Since u0m * u0 strongly in H01 ./; um * u


passing in (2.193) and (2.194) to the limit as m ! C1 and taking into account the fact that E.t / lim infm!C1 Em .t /, we obtain 1

E.t / ŒE1˛ C .1 ˛/B4 t 1˛ ; 0 E.t / E0 exp ¹B5 t º ;

˛ < 1;

˛ D 1:

(2.214) (2.215)

Now we consider the case where ˛ > 1. Since u0m ! u0 strongly in H01 ./, we obtain that E0m ! E0 . In this case, we can select from the numerical sequence ¹E0m º either a monotonically nondecreasing subsequence or a monotonically nonincreasing subsequence; we denote the required subsequence of the sequence ¹E0m º and the corresponding subsequences of the sequences ¹u0m º and ¹um º by the same symbol. Consider inequality (2.195). Assume that ¹E0m º is a monotonically nonincreasing sequence. Introduce the nota¹nº tion ¹E0m º for the sequence obtained from ¹E0m º by eliminating the first n 2 N sum¹nº mands; then inequality (2.195) holds uniformly with respect to m for all t 2 Œ0; T1 /, ¹nº .p2/=2 . Since where T1 2=.p 2/B1 2 E0n um * u


for all such t we can pass in (2.195) to the limit as m ! C1 and obtain i h

p .p2/=2 2=.p2/ E.t / E0 1 1 B2 E0 t for p > q C 2; 2 p

where B2 D B1 2p=2 .

(2.216)

181


By the arbitrariness of n 2 N and by the fact that T¹nº " T1 as n " C1, we 1 conclude that (2.207) holds for t 2 Œ0; T1 / and, moreover, T0 T1 . Now let ¹E0m º be a monotonically nondecreasing sequence. Then, repeating the previous arguments, we obtain that (2.216) holds for all t 2 Œ0; T1 / and T0 T1 . Thus, estimates (2.214)–(2.216) are obtained for any possible values of ˛. For further consideration, we must prove that for almost all t 2 Œ0; T, for all T 2 .0; T0 /, we have Em .T/ ! E.T/. For this, we note that we have obtained (2.208) above and, therefore, by the inclusion Lp .Q/ L2 .Q/, Q D .0; T/ for all T 2 Œ0; T0 /, we have Z T Z T ds Em .s/ ! ds E.s/ 8T 2 .0; T0 /: (2.217) 0

0

On the other hand, if the conditions q > 0 and p > 2 (for N D 1; 2) or 0 < q < 4=.N 2/ and p < 2N=.N 2/ (for N 3) hold, then we have the compact inclusions H1 .Q/ LqC2 .Q/ and H1 .Q/ Lp .Q/, and the sequence um weakly converges in H1 .Q/. Therefore, um ! u strongly in LqC2 .Q/ \ Lp ./, Q D .0; T/ for all T 2 .0; T0 /. Since the right-hand side of the expression for Em .T/ converges for any T 2 .0; T0 / to a certain function of .T/, from the proof of Theorem 2.9.3 we have Em .T/ C < C1, where C is independent of m and T. By the Lebesgue theorem on the passage to the limit under the Lebesgue integral sign, we have: .T/ 2 L1 .0; T0 / and Z T Z T Em .s/ds ! ds.s/ 8T 2 .0; T0 /: (2.218) 0

0

Comparing (2.217) with (2.218), we conclude that Z T ds ŒE.s/ .s/ D 0 8T 2 .0; T0 /;

(2.219)

0

where E.T/ 2 L1 .0; T0 /, .T/ 2 L1 .0; T0 /, therefore, by virtue of (2.219) we obtain that .T/ D E.T/; for almost all T 2 .0; T0 /: Thus, we conclude that, under the conditions q > 0 and p > 2, the sequence Em .T/ converges almost everywhere to E.T/ for all T 2 .0; T0 /. Consider the ordinary differential inequality (2.213) in the following three cases: 0 < ˛ < 1, ˛ D 1, and 1 < ˛. As above, without loss of generality, we can assume that the sequence E0m is monotonic and positive; then Em > 0 for all m 2 N. Moreover, let F0 > 0. Then there exists a subsequence of the sequence F0m such that Fm0 > 0. In the first case (0 < ˛ < 1), (2.213) implies

E0m E˛m

0 0;

E0m Fm0 ˛ ˛ ; Em Em0

Em .t /

E1˛ m0

Fm0 C .1 ˛/ ˛ t Em0

1 1˛

:

(2.220)

182


In the second case (˛ D 1), from (2.213) we obtain 0 E0m 00 0 2 Em Em .Em / 0; ln 0; Em ² ³ Fm0 t : Em .t / E0m exp E0m Finally, in the third case (˛ > 1), (2.213) implies 0 0 E0m Fm0 Em 0; ˛ ; ˛ Em E˛m Em0 Fm0 1=.˛1/ Em E0m 1 .˛ 1/ ˛ t : E0m

(2.221)

(2.222)

By the strong convergence u0m ! u0 in H01 ./ and by the inclusion H01 ./ LqC2 ./ \ Lp ./; we obtain E0m ! E0 ;

ku0m kqC2 ! ku0 kqC2 :

On the other hand, we have established that Em .t / ! E.t / for almost all t 2 .0; T0 /. Therefore, passing in inequalities (2.220) and (2.221) to the limit as m ! C1, we obtain 1 F0 1˛ 1˛ ; ˛ 2 .0; 1/; E.t / E0 C .1 ˛/ ˛ t E0 ² ³ F0 E.t / E0 exp t ; ˛ D 1: E0 Introduce the notation Km0 D Fm0 =E˛m0 . Without loss of generality, by the passage to a subsequence, we can assume that the sequence Em0 > 0 uniformly with respect to m since E0 > 0. By the convergence of the sequence Km0 ! K0 as m ! C1, we can select a monotonically convergent subsequence (we denote this subsequence again by Km0 ); the corresponding subsequences of the sequences ¹u0m º and ¹um º we denote again by ¹u0m º and ¹um º. Rewrite (2.222) with account of the notation introduced: Em Em0 Œ1 .˛ 1/Km0 t 1=.˛1/ :

(2.223)

Let ¹Km0 º be a monotonically nondecreasing sequence; then inequality (2.223) holds uniformly with respect to m for all t 2 Œ0; .˛ 1/1 K1 /, where K D supm2N Km0 : For such t , passing in inequality (2.223) to the limit as m ! C1, we obtain E E0 Œ1 .˛ 1/K0 t 1=.˛1/

for

t 2 Œ0; .˛ 1/1 K1 /:

(2.224)

183


But K D K0 , and by fact that K0m is a monotonically nondecreasing sequence converging to K0 F0 =E˛0 , we obtain that T0 .˛ 1/1 F0 =E˛0 T2 : ¹nº Now let ¹Km0 º be a monotonically nonincreasing sequence. Denote by ¹Km0 º the sequence obtained from the sequence ¹Km0 º by eliminating the first n 2 N summands. Assume that

T0 > .˛ 1/1

F0 T2 E˛0

(2.225)

and, moreover, let M sup t2.0;T2 / E.t / < C1. Then (2.223) holds uniformly with respect to t 2 Œ0; .˛ 1/1 K1 0n /. Passing in Eq. (2.223) to the limit as m ! C1, by the assumption on T0 , we obtain the inequality E0 Œ1 .˛ 1/K0 t 1=.˛1/ M < C1 for t 2 Œ0; .˛ 1/1 K1 0n /:

(2.226)

However, by the fact that K0n # K0 as n ! C1, there exist n 2 N and tn 2 .0; .˛ 1/1 K1 0n / such that (2.226) loses a sense. Therefore, our assumption (2.225) does not hold. Hence we obtain that T0 .˛ 1/1

E˛0 T2 F0

(2.227)

for ˛ > 1. The latter result means that under ˛ > 1, the blow-up of a solution occurs for the finite time T0 , for which the explicit upper estimate (2.227) has been expressed by the problem data.

2.9.3 Rate of the blow-up of solutions In the case where q D 0, we can obtain optimal two-sided estimates for the rate of blow-up of solutions. Theorem 2.9.7. Let all the conditions of Theorem 2.9.3 hold and, moreover, q D 0 and F0 > 0. Then there exists T0 > 0 such that Œ.p 2/c2 2=.p2/ Œ.p 2/c1 2=.p2/ 2 2 kruk C kuk ; 2 2 .T0 t /2=.p2/ .T0 t /2=.p2/

(2.228)

where c1 D

Bp 1;

c2 D

p ku0 kp

PN

Œkru0 k22 C

@u0 p i D1 k @xi kp ; ku0 k22 p=2

B1 is the best constant of embedding H01 ./ in Lp ./.

(2.229)

184


Proof. Introduce the notation Um .s; x/

um .sm C t ; x/ 1 ; m q1 ; Mm .t / M .t / q Mm .t / krum k22 C kum k22 ;

q1 D p 2;

where t is the fixed time moment of the interval Œ0; T0 /. It is not difficult to verify that the introduced function Um .s; x/ is a solution of the following problem:

@Um ; rwj r @s

@Um ; wj C @s

ˇ N ˇ X ˇ @Um ˇq1 @Um @wj ˇ ˇ C D .jUm jq1 Um ; wj /; ˇ @x ˇ @x ; @x

t Um Um0 ; m um0 D

m X

i D1

Um0

˛mk wk ! u0

i

i

i

j D 1; m;

(2.230)

m X ˛mk um0 D wk ; Mm .t / Mm .t / kD1

strongly in W01;p ./:

kD1

Prove that there exist c1m ; c2m 2 .0; C1/ such that 0 < c2m

ˇ d ŒkrUm k22 C kUm k22 1=2 ˇsD0 c1m < C1: ds

(2.231)

Note that for problem (2.230), the energy equality (2.211) holds, from which we obtain ˇ d ŒkrUm k22 C kUm k22 1=2 ˇsD0 kUm kpp jsD0 ds 2 2 p=2 Bp jsD0 D Bp 1 ŒkrUm k2 C kUm k2 1 c1m ; kvkp < B1 krvk2 ;

d ŒkrUm k22 C kUm k22 1=2 jsD0 c1m : ds

(2.232)

Now let 'm krUm k22 C kUm k22 : For the function 'm .s/, the following equality of the form (2.213) holds: d 2 'm d'm 2 p 'm ˛ 0; ˛ : ds 2 ds 2

(2.233)

Integrating the differential inequality (2.233) over s 2 .t =; t /, we obtain P ˇ p @um0 p kum0 kp N d'm 1 ˇˇ i D1 k @xi kp 2 : (2.234) ˛ ˇ ds 'm Œkrum0 k22 C kum0 k22 p=2 tDs

185


Now we use the fact that, according to the condition of Theorem 2.9.5, ku0 kpp

N X @u0 p @x > 0 i

i D1

p

and, therefore, there exists a subsequence of the sequence ¹um0 º such that kum0 kpp

N X @um0 p @x > 0: i

i D1

p

From (2.234) we obtain the inequality P p @um0 p kum0 kp N d i D1 k @xi kp 2 2 1=2 c2m > 0: ŒkrUm k2 C kUm k2 ds Œkrum0 k22 C kum0 k22 p=2 Thus, inequality (2.231) is proved by virtue of (2.232). Inequality (2.231) is equivalent to 0 < c2m

d Mm .t / 1 c1m < C1; Mm .t /p1 dt

where c2m

p kum0 kp

(2.235)

PN

Œkrum0 k22 C

c1m Bp 1;

p > 2;

@um0 p i D1 k @xi kp ; kum0 k22 p=2

B1 W kvkp krvk2 :

Integrating (2.235) by t from t to Tm0 , where lim sup Mm .t / D C1; t "Tm0

we obtain Œ.p 2/c1m 1=.p2/ Mm .t /.Tm0 t /1=.p2/ Œ.p 2/c2m 1=.p2/ (2.236) for t 2 Œ0; Tm0 /. Here the following lower and upper estimates for the blow-up time Tm0 hold: A1m Tm0 A2m ; where A1m A2m

1

.p .p

; 2 2 .p2/=2 2/Bp 1 Œkrum0 k2 C kum0 k2 krum0 k22 C kum0 k22 : P p @um0 p 2/Œkum0 kp N i D1 k @xi kp

186


It is easy to prove that there exist positive constants A1 and A2 independent of m 2 N such that 0 < A1 Tm0 A2 < C1: Let T0 lim infm!C1 Tm0 . Then there exists a subsequence of the sequence ¹Tm0 º such that either Tm0 " T0 or Tm0 # T0 . First, consider the case where Tm0 " T0 . Then for any fixed m 2 N, we can pass in inequality (2.236) to the limit as m ! C1 uniformly with respect to t 2 Œ0; Tm0 and obtain Œ.p 2/c1 1=.p2/ M.t /ŒT0 t 1=.p2/ Œ.p 2/c2 1=.p2/

(2.237)

for all t 2 Œ0; Tm0 /, where c1

p B1 ;

c2

p ku0 kp

PN

Œkru0 k22 C

@u0 p i D1 k @xi kp : ku0 k22 p=2

In the case where Tm0 # T0 , we can pass in inequality (2.236) to the limit as m ! C1 uniformly with respect to t 2 Œ0; T0 / and obtain (2.237) again. Therefore, (2.228) holds with the constants of (2.229). Remark 2.9.8. Note that a technique similar to that described above can be used in the case where by p , p > 2, we mean p v div.jrvjp2 rv/. Moreover, from the standpoint of physics of semiconductors, the natural value is p D 4, and the corresponding equation has the form @ .u jujq u/ C div.jruj2 ru/ C u3 D 0; @t

> 0:

Also we note that the additional condition on F0

ku0 kpp

N X @u0 p @x > 0 i D1

i

p

used in Theorems 2.9.3 and 2.9.5 for results on the blow-up is, in a sense, optimal since solutions of the problem div.jrujp2 ru/ C jujp2 u D 0;

u 2 W01;p ./;

p > 2;

(2.238)

of eigenfunctions and eigenvalues are stationary and, therefore, global-on-time solutions of this problem. For initial functions u0 ¤ 0 on a positive-measure set, that satisfy problem (2.238) under certain > 0, the obvious equality F0 D 0 holds. Problem (2.238) is studied in [274, 327, 328] (see also references therein).

Section 2.10 Problem for a semilinear equation of the form (2.2)

2.10

187

Problem for a semilinear equation of the form (2.2)

In this section, we analyze the blow-up for a finite time of classical solutions of initialboundary-value problems for a semilinear equation of combined type:

@u C ujujq D 0; @t u.x; t / D 0;

in .0; T;

q > 0;

on @ Œ0; T/;

u.x; 0/ D u0 .x/;

x 2 ;

(2.239) (2.240) (2.241)

where RN , N 2, is a bounded simply-connected domain with piecewise smooth boundary @ of class A.1;/ and @2x1 C C@2xN is the Laplace operator.

2.10.1 Blow-up of classical solutions We search for a classical solution of problem (2.239)–(2.241) in the following class of real functions: u.x; t / 2 C .0;1/ . Œ0; T/;

@u.x; t / 2 C .2;0/ . .0; T//: @t

(2.242)

Note that on the given class of functions u.x; t /, the operators and @=@t do not commute, generally speaking, although the operator @t@ is defined in the classical sense. Moreover, if u.x; t / 2 C .0;1/ . Œ0; T/, then u0 .x/ 2 C0 ./ and the following condition holds: ˇ @u.x; t / ˇˇ D 0; t 2 Œ0; T /: @t ˇ@ Denote by G.x; y/ the Green function of the first boundary-value problem for the Laplace operator in the bounded domain RN with piecewise smooth boundary @ of class A.1;/ . In the class of functions (2.242), problem (2.239)–(2.241) is equivalent to the following Cauchy problem for the integro-differential equation: Z @u dy G.x; y/.ujujq /.y; t /; D @t (2.243) u.x; 0/ D u0 .x/; which, in its turn, is equivalent to the integral equation Z tZ dy d G.x; y/.ujujq /.y; /: u.x; t / D u0 .x/ C 0

(2.244)

188


Lemma 2.10.1. If u.x; t / 6 0 is a solution of problem (2.239)–(2.241) of class (2.242), where u.x; t / 2 C .1/ ŒŒ0; TI H01 ./, then this solution blows up for a finite time. Proof. Multiply both sides of Eq. (2.239) by u.x; t / and integrate over the domain ; owing to the boundary condition (2.240) we obtain d kruk22 .t / D 2kukqC2 qC2 .t /: dt

(2.245)

Now multiplying both sides of Eq. (2.239) by u t .x; t / and integrating over the domain , owing to the boundary condition (2.240) in the considered class of functions we obtain kru t k22 .t / D

1 d qC2 kukqC2 .t /: q C 2 dt

(2.246)

On the other hand, by the Cauchy–Bunyakovskii inequality, the following inequality holds: kru t k22 kruk22 jhru t ; ruij2 ; which, by relations (2.245)–(2.246) for the function '.t / D kruk22 , implies the following inequality: q C 2 d' 2 d 2' ' 0: (2.247) dt 2 2 dt From (2.247) we obtain ' 00 .' 0 /2 qC2 ˛ 0; ˛ D ; '˛ ' ˛C1 2 0 0 qC2 2ku0 kqC2 '0 ' 0; ; '˛ '˛ kru0 kqC2 1 '0q=2

1 ' q=2 .t /

2 qC2 ku0 kqC2 t: q kru0 kqC2 2

The latter inequality immediately implies the fact that the classical solution in the considered class does not exist globally on time. Now prove the nonexistence of global-on-time solution of problem (2.239)–(2.241). Theorem 2.10.2. For any function u0 .x/ 2 C .0;˛/ ./ \ C0 ./, ˛ 2 .0; 1/, which is not identically equal to zero, there exists Tu0 > 0 such that a unique solution of problem (2.239)–(2.241) of the class (2.242) with T D Tu0 exists. Moreover, if u0 .x/ 2 C .0;˛/ ./ \ C0 ./ \ H01 ./;

(2.248)

189


then Tu0 < C1 and the following relation holds: sup ju.x; t /j D C1:

lim

(2.249)

T"Tu0 Œ0;T

Proof. Consider the Banach space B C. Œ0; T/ with the norm k k D sup j j Œ0;T

and the recurrent sequence A.uk /; ukC1 .x; t / D u0 .x/ C b

k 1;

u1 .x; t / D u0 .x/; where b A.u/

Z tZ 0

(2.250)

dy d G.x; y/.ujujq /.y; /:

Z

Note that kb A.u/k cTkukqC1 ;

c D sup

dy G.x; y/:

x2

Moreover, in the same way as in [275, 276], we can show that if kuk M and kvk M, then kb A.u/ b A.v/k .q C 1/Mq cTku vk:

(2.251)

Now we estimate the norm of ukC1 .x; t / using the recurrent sequence (2.250): kukC1 k ku0 k C T ckuk kqC1 : qC1 Let 1 D ku0 k and kC1 D 1 CcTk , k 1; then it is easy to prove by induction that kuk k k . Let

T < q q .q C 1/q1 ku0 kq c 1 I

(2.252)

then the sequence ¹k º is strictly bounded by the value M D q 1 .q C 1/ku0 k. Since A.ukC1 / b A.uk /; ukC2 ukC1 D b by (2.251) we obtain kukC2 ukC1 k rkukC1 uk k;

where r D c .q C 1/ T

qC1 q

q

ku0 kq < 1

190


by the choice of T. Therefore, C1 X

kukC1 uk k < C1:

kD1

Hence the convergence of the sequence ¹uk .x; t /º in the Banach space C. Œ0; T/ is proved. Denote the limit of the sequence by u.x; t / 2 C. Œ0; T/. Inequality (2.251) implies the convergence in B of the sequence ¹b A.uk /º to the function b A.u/. Therefore, we conclude that u.x; t / is a solution of the integral equation (2.244) in the class C. Œ0; T/ with T > 0, which satisfies condition (2.252). Now we prove that u.x; t / 2 C.Œ0; TI C .0;˛/ .//. Indeed, for N 2 we have G.x; y/ D g.x; y/ C .x y/; ´ 1 jx yj2N ; if N > 2; /!N .x y/ D N.2N 1 if N D 2; 2 log jx yj; and g.x; y/ is a harmonic function in , continuous till the boundary, d is the diameter of the domain , and K is a compact set. Hence we obtain Z dy Œjrg.x; y/j C jr.x y/j jŒu˛IK j1 Œu0 ˛IK C T d 1˛ kukqC1 sup x2K

Œu0 ˛IK C C.K/Td

1˛

kukqC1

8K :

Since u.x; t / 2 C.Œ0; TI C .0;˛/ ./ \ C0 .//, the right side of the integral equation (2.244) belongs to the class (2.242) with certain T satisfying condition (2.250). Therefore, u.x; t / also belongs to the class (2.242). Introduce the notation uT1 .x/ D u.x; T1 / 2 C./ and consider the auxiliary problem (2.243) for t T1 D T, where the function u0 .x/ is replaced by the function uT1 .x/. Using the scheme given above, as a result we obtain a monotonically increasing sequence ¹Tn º, where for any given n 2 N, a solution of the integral equation (2.244) exists and belongs to the class (2.242) with T D Tn . Since a monotonic sequence has either finite or infinite limit, we denote the limit of the sequence ¹Tn º by Tu0 . Now we prove the uniqueness of a solution of the integral equation (2.244) for t 2 Œ0; Tu0 /. Let u1 .x; t / and u2 .x; t / be two solutions of the integral equation (2.244) that belong to the Banach space B on a certain segment Œ0; T0 (T0 > 0) and correspond to the initial function u0 .x/ 2 C0 ./. The difference v.x; t / D u1 .x; t / u2 .x; t / satisfies the integral equation Z tZ v.x; t / D dy d G.x; y/Œ.u1 ju1 jq /.y; / .u2 ju2 jq /.y; /: (2.253) 0


191

Introduce the function MŒu.T/ sup ju.x; t /j:

(2.254)

Œ0;T

Since L.T0 / max¹MŒu1 .T0 /; MŒu2 .T0 /º < C1 and, moreover, L.T/ L.T0 / for T T0 (this follows from the definition (2.254) of the function MŒu.T/, by (2.251) from the integral equation (2.253) we obtain MŒv.T/ .q C 1/ c T L.T0 /q MŒv.T/

for T 2 .0; T0 /:

The latter inequality yields MŒv.T/ 0 for 0 < T < ..q C 1/cL.T0 /q /1 . Thus, v.x; t / 0 for t 2 .0; ..q C 1/cL.T0 /q /1 /. Next, applying the algorithm of the extension of solutions in the same way as in the proof of the solvability, we conclude that v.x; t / 0 for t 2 Œ0; Tu0 /. Now we prove the nonexistence of global-on-time solution of problem (2.239)– (2.241) in the class (2.242) under condition (2.248) and also relation (2.249). Let a function u0 .x/, distinct from identical zero, belongs to the class (2.246). Assume that the monotonic sequence ¹Tn º has an infinite limit. Since the initial function u0 .x/ belongs to the class (2.246), we obtain from the integral equation (2.244) that the required solution u.x; t / satisfies all the conditions of Lemma 2.10.1. Therefore, Tu0 < C1. To prove relation (2.249) we note that MŒu.T/ is a monotonically nondecreasing function defined on the set T 2 Œ0; Tu0 /. Therefore, as T " Tu0 , it has either a finite or infinite limit. Assume that the limit is finite; then the solution of the integral equation of B can be extended for t > Tu0 . The contradiction obtained shows that relation (2.247) is valid. Remark 2.10.3. In the case of initial functions u0 .x/ 2 C .0;˛/ ./ \ C0 ./ of constant signs, it is especially easy to prove the nonexistence of global solutions by using the technique originally proposed by Kaplan (see [216]). Let 1 .x/ be an eigenfunction corresponding to the first eigenvalue 1 of the first boundary-value problem for the Laplace operator in the domain RN with piecewise smooth boundary @ of class A.1;/ . We normalize 1 .x/ to the unit: Z dx 1 .x/ D 1:

To fix an idea, take a function u0 .x/ 0, which does not vanish identically (the case of nonpositive u0 .x/ can be considered similarly). Let u.x; t / be the corresponding solution of the integral equation (2.244) of the class (2.242); this solution exists and is unique on the interval Œ0; Tu0 / by Theorem 2.10.2. Note that the form of the corresponding recurrent sequence ¹uk .x; t /º (see (2.248)) implies that u.x; t / 0.

192


Introduce the function

Z Y .t /

u.x; t /

1 .x/ dx:

(2.255)

Multiply both sides of Eq. (2.243) by the function 1 .x; t / and integrate over the domain using the fact that G.x; y/ is the Green function of the first boundaryvalue problem for the Laplace operator in and u.x; t / is nonnegative: Z 1 d Y .t / dy 1 .y/uqC1 .y; t /: (2.256) D dt 1 Now, using the Jensen integral inequality (see [445]) for convex functions (q > 0), the normalization condition 1 .x/ 0, and definition (2.255), we obtain from (2.256) the ordinary differential inequality d Y .t / 1 Y .t /qC1 ; dt 1

q > 0:

(2.257)

Integrating (2.257) over t 2 Œ0; T, T < Tu0 , we obtain q C 1 1=q q T ; Y .T/ Y0 1

T 2 Œ0; Tu0 /:

Hence we immediately obtain that Tu0 < C1. Now we prove a comparison criterion. Lemma 2.10.4. Let functions u01 .x/ and u02 .x/ belong to the class C0 ./ and u01 .x/ u02 .x/. Then the corresponding solutions of the integral equation (2.244) satisfies the inequality u1 .x; t / u2 .x; t / on the set Œ0; minŒT01 ; T02 /, where T01 and T02 are the corresponding “existence” times in the sense of Theorem 2.10.2. Proof. Let ¹u1k .x; t /º and ¹u2k .x; t /º be the recurrent sequences corresponding to the initial functions u01 .x/ and u02 .x/, defined by (2.248), and converging to solutions u1 .x; t / and u2 .x; t / of Eq. (2.244) in the Banach space B on a certain segment Œ0; T. By virtue of the facts that u01 .x/ u02 .x/ and G.x; y/ is positive in and the function z q is monotonic, we obtain u1 .x; t / u2 .x; t / for .x; t / 2 Œ0; T/. Next, applying the extension algorithm, we obtain the required inequality on the existence interval of both solutions. Lemma 2.10.4 is proved. Now we consider a specific character of the blow-up of solutions of problem (2.239)–(2.241). Consider the following auxiliary stationary problem on free oscillations: f .x/ C qf qC1 .x/ D 0; f .x/j@ D 0:

.q > 1/; (2.258)


193

Note that in the general case problem (2.258) was studied by Pokhozhaev [327]. Precisely, for N 3 and 0 < q < 4=.N 2/ or for N D 2 and q > 0, Theorems 1 and 3 of [327] are valid. Theorem 2.10.5 (see [327]). Let N > 2. Assume that there exists a function v.x/ 2 H01 .G/ for which Z F .v/ dx D ¤ 0; e G where

Z F .u/ D

u

f .t / dt: 0

Let a function f .u/ 2 C .0;˛/ .R1 / satisfy the condition jf .u/j A C Bjujm ;

m
0:

Thus, > 0 and a solution f .x/ 2 C 2 ./ \ C0 ./ of problem (2.258) exists and f .x/ 0. The following theorem holds. Theorem 2.10.7. Let N D 2 and q > 0 or N 3 and 0 < q < 4=.N 2/. If for certain 1 > 0 and 2 > 0, the inequality f 1 .x/ u0 .x/ f 2 .x/ holds, u0 .x/ 2 C .0;˛/ ./ \ C0 ./, then 2 Tu0 1 and the following inequalities hold: u.x; t / u.x; t /

1 1 t 2 2 t

1=q f 1 .x/;

t 2 Œ0; Tu0 /;

f 2 .x/;

t 2 Œ0; 2 /;

1=q

where f .x/ is a solution of problem (2.258), u.x; t / is the solution of the integral equation (2.244) corresponding to u0 .x/, and Tu0 is the solution-existence time in the sense of Theorem 2.10.2. Proof. By Theorem 2.10.2, a solution of problem (2.239)–(2.241) with the initial function f .x/ is unique. On the other hand, one can easily verify that the solution has the form 1=q f .x/: u .x; t / D t Now, applying Lemma 2.10.4, we obtain Theorem 2.10.7. Theorem 2.10.8. Let u0 .x/ 2 C .0;˛/ ./ \ C0 ./ \ H01 ./. If N D 2 and q > 0 or N 3 and 0 < q < 4=.N 2/, then there exists T0 D T0 .u0 / > 0 such that qC2

ŒqB1 .q; /1=q B2 .q; u0 / kruk2 ; 1=q .T0 t / .T0 t /1=q

(2.259)


195

where B1 .q; / is the constant of inclusion H01 ./ in LqC2 ./, B2 .q; u0 / q

qC2

ku0 kqC2

1=q ;

kru0 kqC2 2

and lim

sup ju.x; t /j D C1:

T"T0 Œ0;T

Proof. We use the technique proposed by Rossi (see [354]). Introduce the auxiliary function v .s; x/ D

1 u.s C t ? ; x/; M.t ? /

D

1 ; M.t ? /q

M.t ? / D kruk2 .t ? /: (2.260)

One can directly verify that by (2.260), v .s; x/ satisfies all the conditions of the following problem:

@v C v jv jq D 0; @s v .s; x/j@ D 0; ? 1 t u0 .x/: v ; x D M.t ?/

(2.261)

We prove that there exist c1 and c2 such that 0 < c2

d krv k.0/ c1 < C1: ds

(2.262)

First, we obtain an upper estimate. To this end, we multiply the first equation of system (2.261) by v and integrate over the domain taking into account the boundary condition. We obtain d qC2 qC2 krv k22 .s/ D 2kv kqC2 .s/ 2ŒB1 .q; /qC2 krv k2 .s/: ds Hence, for s D 0 we have d krv k.0/ ŒB1 .q; /qC2 D c1 ; ds where B1 .q; / is the constant of the best embedding H01 ./ in LqC2 ./ for N 2 or 0 < q < 4=.N 2/ and N 3. Now we prove a lower estimate. Let ' .s/ D krv k22 . Then the following inequality holds (see Lemma 2.10.1: d 2 ' .s/ q C 2 d' .s/ 2 ' .s/ 0: ds 2 2 ds

196


From this we obtain ˇ qC2 ku0 kqC2 '0 .s/ ˇ ˇ D 2 ; qC2 ' .s/˛ ' .s/˛ ˇsDt ? = kru0 k2 '0 .s/

˛D

qC2 : 2

Hence, using the equality ' .0/ D 1 we obtain for s D 0 qC2 ku0 kqC2 d D c2 > 0: krv k.0/ ds kru0 kqC2 2

Thus, inequality (2.262) is proved. Inequality (2.262) is equivalent to the following: 0 < c2

M0 .t ? / c1 < C1: M.t ? /qC1

Integrating this by t ? from t to T0 , where lim sup M.t / D C1; t"T0

we obtain

Œqc1 1=q M.t /.T0 t /1=q Œqc2 1=q :

The last assertion of Theorem 2.10.8 follows from the structure of the integral equation (2.243). Note that relation (2.259) allows one to obtain an estimate for the solution-existence time T0 : kru0 k22 1 T : 0 qB1qC2 kru0 kq2 qku0 kqC2 2

2.11

On sufficient conditions of the blow-up of solutions of the Boussinesq equation with sources and nonlinear dissipation

In this section, we consider the blow-up of solutions of the following initial-boundary value problem: @ .u u/ C .jujq1 u/ C jujq2 u D 0; q1 ; q2 > 0; @t uj@ D 0; u.x; 0/ D u0 .x/ 2 H01 ./;

(2.263) (2.264)

where R3 is a bounded domain with smooth boundary @ 2 C 2;ı , ı 2 .0; 1.

Section 2.11 Blow-up in generalized Boussinesq equations

197

Kozhanov [243] studied the blow-up of classical solutions of problem (2.263)– (2.264) under the condition q2 > q1 in the case where is a ball by using the comparison criterion. In what follows, the reader will see that in the case q1 D q2 , without changing technique, one can consider Eq. (2.263) with arbitrary positive constant coefficients. In this section, we prove that weakened solutions of problem (2.263)–(2.264) blow up for a finite time in the case q1 D q2 .

2.11.1 Local solvability of strong generalized solutions Let A C I W H01 ./ ! H1 ./: For the operator A, there exists a Lipschitz continuous inverse operator with Lipschitz constant equal to 1. Indeed, the following relations are valid: hAu1 Au2 ; u1 u2 i D kru1 ru2 k22 C ku1 u2 k22 ku1 u2 k2C1 ; (2.265) where h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./. From (2.265) we obtain kA1 z1 A1 z2 kC1 kz1 z2 k1

8z1 ; z2 2 H1 ./:

We denote by u0 the classical time derivative and by h; i the duality bracket between the Hilbert spaces H01 ./ and H1 ./. We state a definition of a strong generalized solution of problem (2.263)–(2.264). Definition 2.11.1. A solution of problem (2.263)–(2.264) belonging to the class C .1/ .Œ0; TI H01 .// and satisfying the conditions hu0 u0 C .jujq1 u/ C jujq2 u; wi D 0 8w 2 H01 ./;

8t 2 Œ0; T;

u.0/ D u0 2 H01 ./; is called a strong generalized solution of this problem.

Definition 2.11.2. A strong generalized solution of the class C .1/ .Œ0; TI C .1/ ./ \ C0 .// is called a weakened solution of problem (2.263)–(2.264).

(2.266) (2.267)

198


We search for a solution of problem (2.263)–(2.264) in the “weakened” sense, i.e., in the class C .1/ .Œ0; TI C .1/ ./ \ C0 .//. In the strong generalized sense, from (2.266)–(2.267) we obtain the following abstract Cauchy problem for the operator equation: Au0 C Af1 .u/ D f .u/;

u.0/ D u0 2 H01 ./;

(2.268)

where f1 .u/ jujq1 u;

f .u/ jujq1 u C jujq2 u:

Since we search for a solution of problem (2.268) in the “weakened” sense, the following embeddings hold: u0 2 C.Œ0; TI C .1/ ./ \ C0 .// C.Œ0; TI H01 .//; jujqi u 2 C .1/ .Œ0; TI C .1/ ./ \ C0 .// C .1/ .Œ0; TI H01 .// C .1/ .Œ0; TI H1 .//;

i D 1; 2:

Hence from (2.268) we obtain the following equivalent equation: u0 C f1 .u/ D A1 f .u/;

(2.269)

whose solution belongs to the class C .1/ .Œ0; TI C .1/ ./ \ C0 .//. Note that the restriction of the operator A1 W H1 ./ ! H01 ./ to the class v.x/ 2 C .1/ ./ coincides with the integral operator Z dy G.x; y/v.y/; A1 v D

where G.x; y/ is the Green function of the first boundary-value problem for the operator C I in the domain . Thus, from (2.269) in the weakened sense we come to the following integral equation: Z u D u0

0

t

Z ds f1 .u/.s/ C

Z

t

ds 0

dy G.x; y/f .u/ H.u/:

(2.270)

Theorem 2.11.3. Let 1 q1 < C1 and 0 < q2 < C1. Then for any u0 2 C .1/ ./ \ C0 ./, there exist maximal T0 > 0 such that a unique weakened solution of problem (2.263)–(2.264) exists and either T0 D C1 or T0 < C1, and in the latter case, the following limit relation holds: lim sup sup jrx uj D C1: t"T0

x2


199

Proof. We use the method of contraction mappings to prove the local-on-time solvability of the given equation in the class L1 .0; TI C .1/ ./ \ C0 .//. In the Banach space L1 .0; TI C .1/ ./ \ C0 .//; introduce the following closed, convex, bounded subset: ® ¯ BR u 2 L1 .0; TI C .1/ ./ \ C0 .// W kukT ess sup sup jrx u.x; t /j R : t2.0;T/ x2

Introduce the notation kvk D sup jrx v.x/j: x2

We prove that the operator H.u/ defined by (2.270) acts from BR into BR and is a contraction on BR . Indeed, kH.u/kT ku0 k C T1 .R/kukT C T Œ1 .R/ C 2 .R/ kukT ; where i .R/ D CRqi ;

i D 1; 2;

R D kukT :

Now we prove that H.u/ is a contraction on BR . Indeed, kH.u1 / H.u2 /kT T Œ1 .R/ C 2 .R// ku1 u2 kT : Therefore, for sufficiently small T > 0 and large R > 0 under the condition T

1 1 2 1 .R/ C 2 .R/

we obtain that

1 kH.u1 / H.u2 /kT ku1 u2 kT : 2 Using the standard algorithm of the extension solution on time, we obtain the existence of maximal T0 > 0 such that the solution of the class L1 .0; TI C .1/ ./ \ C0 .// is unique for any T 2 .0; T0 /, and either T0 D C1 or T0 < C1, and in the latter case we have lim sup kuk D C1: t"T0

Using the smoothing-on-time property of the operator H.u/, we obtain H.u/ W L1 .0; TI C .1/ ./ \ C0 .// ! AC.Œ0; TI C .1/ ./ \ C0 .//; H.u/ W AC.Œ0; TI C .1/ ./ \ C0 .// ! C .1/ .Œ0; TI C .1/ ./ \ C0 .// for any T 2 .0; T0 /.

200


2.11.2 Blow-up of solutions Using the results of the previous subsection, we see that problem (2.263)–(2.264) in the weakened sense is equivalent to the following abstract Cauchy problem: du C jujq1 u D dt

Z dy G.x; y/f .u/;

u0 D u.0/ 2 C .1/ ./ \ C0 ./; (2.271)

where f .u/ jujq1 u C jujq2 u. The following theorem holds. Theorem 2.11.4. Let q q1 D q2 1. Then for any u0 2 C .1/ ./ \ C0 ./ under the condition Z Z 2 dx dy G.x; y/.ju0 jq u0 /.x/.ju0 jq u0 /.y/ > ku0 k2qC2 2qC2

time 0 < T0 T2 and the following limit relation holds: lim sup kukqC2 D C1; t"T0

where 1 ˆ.0/ 2q C 2 ; ˛ ; 0 ˛ 1 ˆ .0/ qC2 1 qC2 ku0 kqC2 ; ˆ.0/ D qC2 Z Z 2qC2 0 ˆ .0/ 2 dx dy G.x; y/.ju0 jq u0 /.y/.ju0 jq u0 / ku0 k2qC2 : T2

Under the condition 1 1, where 1 is the first eigenvalue of the Laplace operator in the bounded domain with smooth boundary, the following a priori estimate holds: kukqC2 ku0 kqC2 : Proof. Multiplying both sides of Eq. (2.271) by jujq u and integrating over the domain , we obtain the first energy equality 1 d qC2 2qC2 kukqC2 C kuk2qC2 q C 2 dt Z Z D2 dx dy G.x; y/.jujq u/.x; t /.jujq u/.y; t / F.t /: (2.272)

201


Multiply both sides of Eq. (2.271) by the function .jujq u/0 and integrating over the domain , we obtain the second energy equality Z 1d 1 2qC2 .q C 1/ dxjujq .u0 /2 C (2.273) kuk2qC2 D F0 .t /: 2 dt 2 Introduce the energy functional 1 qC2 kukqC2 : qC2

ˆ.t /

(2.274)

The following inequalities hold: 0 2 ˆ D

Z

dx jujq uu0

2

Z

dx jujq .u0 /2

where

Z J .q C 1/

Z

dx jujqC2

qC2 Jˆ.t /; qC1 (2.275)

dx jujq .u0 /2 :

(2.276)

Now, using the energy equalities (2.272) and (2.273), we obtain an upper estimate for J. We have 1 J D ˆ00 : 2

(2.277)

Relations (2.274)–(2.277) imply the inequality ˆˆ00 ˛.ˆ0 /2 0;

˛

2q C 2 > 1: qC2

We impose the condition ˆ0 .0/ > 0; taking (2.272) into account, we have Z Z 2qC2 2 dx dy G.x; y/.ju0 jq u0 /.y/.ju0 jq u0 / > ku0 k2qC2 :

(2.278)

(2.279)

We give an example of initial functions satisfying this inequality. Note that, since u0 2 C .1/ ./ \ C0 ./, we can represent ju0 jq u0 2 H01 ./ by a series expansion with respect to eigenfunctions of the following problem: Z k C k k D 0; k 2 H01 ./; dx k .x/ l .x/ D ıkl ; (2.280)

v0 ju0 jq u0 D

C1 X kD1

˛k k .x/;

C1 X kD1

˛k2 < C1:

(2.281)

202


Substituting the function v0 D ju0 jq u0 (see (2.281)) in condition (2.279) and taking (2.280) into account, we obtain the following equivalent inequality: C1 X kD1

C1 X 2 2 ˛k > ˛k2 : 1 C k

(2.282)

kD1

Assume that all the coefficients ˛k vanish for k > m 2 N and ˛m ¤ 0. Now we require that m < 1; then, owing to (2.282), we conclude that ˇ m ˇq=.1Cq/ m X ˇX ˇ ˛k k .x/ˇˇ ˛k k .x/ˇˇ (2.283) u0 D kD1

kD1

satisfies condition (2.279). Integrating Eq. (2.278), owing to condition (2.279) we obtain the inequality ˆ.t /

Œˆ1˛ 0

1 ; .˛ 1/‰0 t 1=.˛1/

‰0

ˆ0 .0/ : ˆ˛ .0/

(2.284)

Inequality (2.284) yields the following upper estimate of the solution-existence time: T0 T2 , where 1 ˆ.0/ : (2.285) T2 ˛ 1 ˆ0 .0/ Thus, by (2.283) and (2.285), we arrive at the first conclusion of the theorem. Note an interesting fact. For any initial function u0 2 C .1/ ./ \ C0 ./ and 1 1, the inequality kukqC2 ku0 kqC2 holds on the interval Œ0; T0 / where a weakened solutions exists. Indeed, consider the first energy equality (2.272) rewritten in the form Z Z dˆ dx dy G.x; y/.jujq u/.x; t /.jujq u/.y; t / kuk2qC2 (2.286) D2 2qC2 : dt Since u 2 C .1/ .Œ0; T0 /I C .1/ ./ \ C0 .//, for any t 2 Œ0; T0 / we have v jujq u 2 H01 ./ and, therefore, vD

C1 X

˛k .t / k .x/;

kD1

C1 X

˛k2 .t / < C1;

kD1

where k are defined in (2.280). Substituting (2.287) in (2.286) we obtain C1 X 2 dˆ 1 ˛k2 .t /: dt 1 C k kD1

Now we impose the condition 1 1. Then for all t 2 Œ0; T0 /, we have kukqC2 ku0 kqC2 : Theorem 2.11.4 is proved.

(2.287)

Section 2.12 Blow-up for a Rosenau-type equation

2.12

203

Sufficient conditions of the blow-up of solutions of initial-boundary-value problems for a strongly nonlinear pseudoparabolic equation of Rosenau type

In this section, we consider the following initial-boundary-value problem describing quasistationary processes in semiconductors in the presence of negative differential conductivity, strong spatial dispersion, and nonlinear anisotropic dependence of the dielectric conductivity on the electric field: ˇ ˇ N X @ @ ˇˇ @u ˇˇpj 2 @u 2 u u C div.jrujp2 ru/ D 0; (2.288) @t @xj ˇ @xj ˇ @xj j D1 ˇ @u ˇˇ D 0; u.x; 0/ D u0 .x/; (2.289) uj@ D @n ˇ@ where x D .x1 ; x2 ; : : : ; xN / 2 RN , N 1, @ 2 C 4;ı , ı 2 .0; 1, pj 3, p > 2. Note that this problem does not satisfy conditions for operator coefficients introduced above since in Eq. (2.288), under the sign of time derivative, we have, generally speaking, a sum of monotonic, homogenous, nonlinear operators that are not operators of norm type. Therefore, unfortunately, the technique used above does not allow one to obtain necessary and sufficient conditions of the blow-up of solutions, except for the single simplest case where pj D p > 2 for all j D 1; N .

2.12.1 Local solvability of the problem in the strong generalized sense We state a definition of a strong generalized solution. Definition 2.12.1. A strong generalized solution of problem (2.288)–(2.289) is a solution of the class C .1/ .Œ0; TI H02 .// satisfying the following conditions: hD.u/; wi D 0 8w 2 H02 ./; 8t 2 Œ0; T; u.0/ D u0 2 H02 ./; ˇ ˇ N X (2.290) @ @ ˇˇ @u ˇˇpj 2 @u 2 u u C div.jrujp2 ru/; D.u/ ˇ ˇ @t @xj @xj @xj j D1

where h; i is the duality bracket between the Hilbert spaces H02 ./ and H2 ./. Introduce the notation p D maxj D1;N pj , pQ D max¹p; pº. The following theorem holds. Theorem 2.12.2. Let u0 .x/ 2 H02 ./. Moreover, assume that either N 2 or pQ 2N.N 2/1 for N 3. Then there exists maximal T0 D T0 .u0 / > 0

204


such that a unique strong generalized solution of problem (2.288)–(2.289) of the class C .1/ .Œ0; T0 /I H02 .// exists and either T0 D C1 or T0 < C1, and in the latter case, the following limit relation holds: lim sup kuk2 D C1:

(2.291)

t"T0

Proof. Introduce the notation (2.292) A0 u 2 u u W H02 ./ ! H2 ./; ˇpj 2 ˇ 0 @u @ ˇˇ @u ˇˇ 1;p W H02 ./ W0 j ./ ! W 1;pj ./ H2 ./; Aj .u/ ˇ ˇ @xj @xj @xj (2.293) 1;p

F .u/ div.jrujp2 ru/ W H02 ./ W0

0

./ ! W 1;p ./ H2 ./; p p0 D : (2.294) p1

We denote by k kC2 the norm of the Hilbert space H02 ./ and by k k2 the norm of the Hilbert space H2 ./. Using the introduced notation (2.292)–(2.294), we rewrite problem (2.290) in the form N X d A0 u C Aj .u/ D F.u/; dt

u.0/ D u0 2 H02 ./:

(2.295)

j D1

Introduce the operator A.u/ D A0 u C

N X

Aj .u/ W H02 ./ ! H2 ./:

(2.296)

j D1

Note that hA.u1 / A.u2 /; u1 u2 i D ku1 u2 k22 C kru1 ru2 k22 ˇ ˇ ˇ ˇ N Z X ˇ @u1 ˇpj 2 @u1 ˇ @u2 ˇpj 2 @u2 @u1 @u2 ˇ ˇ ˇ ˇ C dx ˇ ; : @xj ˇ @xj ˇ @xj ˇ @xj @xj @xj j D1

Therefore, the operator A./ defined by (2.296) has a Lipschitz-continuous inverse operator with Lipschitz constant equal to 1. Similarly, we can prove that the operator A0 W H02 ./ ! H2 ./ has a Lipschitz continuous inverse operator with Lipschitz constant equal to 1.

205


Now consider the operator ˇ ˇ 0 @ ˇˇ @u ˇˇpj 2 @u 1;p Aj .u/ W W0 j ./ ! W 1;pj ./; ˇ ˇ @xj @xj @xj

j D 1; N ;

in the case where pj > 2. The Fréchet derivative of the operator Aj .u/ has the form Aj0 .u/h

ˇ ˇ @ ˇˇ @u ˇˇpj 2 @h : D .pj 1/ @xj ˇ @xj ˇ @xj

(2.297)

Now we prove that the Fréchet derivative Aj0 ./ is a strongly continuous and bounded mapping H02 ./ ! L.H02 ./; H2 .//. Indeed, by (2.297) we have kAj0 .u/ Aj0 .un /kH2 ./!H2 ./ D 0

C

sup khkC2 D1

sup khkC2 D1

kAj0 .u/h Aj0 .un /hk Z

Jn;j .h/ D .pj 1/

ˇ ˇ @h dx ˇˇ @x

j

ˇp 0 ˇ j ˇ ˇ

W

kAj0 .u/h Aj0 .un /hk2

1;p 0 j

./

DC

sup khkC2 D1

Jnj .h/; (2.298)

ˇˇ ˇ 0! 0 ˇˇ @u ˇˇpj 2 ˇˇ @u ˇˇpj 2 ˇpj 1=pj ˇˇ ˇ n ˇ ˇ ˇˇ : (2.299) ˇˇ ˇ ˇ ˇ ˇ @xj ˇ @xj

From (2.299) we obtain an upper estimate for Jn;j .h/: Z Jn;j .h/ .pj 1/

ˇ ˇ @h dx ˇˇ @x

j

ˇpj 1=pj ˇ ˇ ˇ

ˇˇ ˇ ! Z ˇˇ @u ˇˇpj 2 ˇˇ @u ˇˇpj 2 ˇpj =.pj 2/ .pj 2/=pj ˇˇ n ˇ ˇ ˇ ˇ dx ˇˇ ˇ : ˇ ˇ @xj ˇ ˇ @xj ˇ

(2.300)

Consider the operators ˇ ˇ ˇ @u ˇpj 2 ˇ : fj D ˇˇ @xj ˇ

(2.301)

Operators (2.301) are generated by Carathéodory functions; it is easy to verify that the Nemytskii operators (2.301) act from Lpj ./ into Lpj =.pj 2/ ./ for pj > 2. Therefore, according to M. A. Krasnoselskii’s theorem, the operator fj W Lpj ./ ! Lpj =.pj 2/ ./ is bounded and strongly continuous, i.e., Z

ˇˇ ˇ ˇˇ @u ˇˇpj 2 ˇˇ @u ˇˇpj 2 ˇpj =.pj 2/ ˇˇ n ˇ ˇ ˇ dx ˇˇ ˇˇ ! C0 ˇ ˇ ˇ ˇ ˇ @xj @xj

206

Chapter 2 Blow-up of solutions of nonlinear equations of Sobolev type 1;pj

as un ! u strongly in H02 ./ W0 we conclude that

./. Therefore, by virtue of (2.298)–(2.300)

kAj0 .u/ Aj0 .un /kH2 ./!H2 ./ ! C0 0

as un ! u strongly in H02 ./ Now let pj D 2; then

1;p W0 j ./.

Aj .u/ D

@2 u @xj2

and the Fréchet derivative has the form Aj0 .u/h D Aj .h/ D

@2 h ; @xj2

and since it is independent of u 2 H02 ./, obviously, the operators are strongly continuous and bounded by u 2 H02 ./. Now consider the operator F.u/. The following inequalities hold: Z ˇp=.p1/ p=.p1/ ˇ p2 p2 ˇ ˇ dx jrv1 j rv1 jrv2 j rv2 kF.v1 / F .v2 /k2 C

C.R/krv1 rv2 kp ; where R D max¹krv1 kp ; krv2 kp º;

.R/ D CRp2 ;

p > 2:

Therefore, kF.v1 / F.v2 /k2 .R/kv1 v2 kC2 ; p2

and R D max¹kv1 kC2 ; kv2 kC2 º. where .R/ D CR Consider problem (2.295). We search for v in the class C .1/ .Œ0; TI H2 .//. By virtue of the properties proved above and the Browder–Minty theorem, for the operator A W H02 ./ ! H2 ./; a Lipschitz-continuous inverse operator A1 W H2 ./ ! H02 ./ exists. Thus, u D A1 .v/ and problem (2.295) is equivalent to the problem dv D F.A1 .v//; dt

v.0/ D v0 D A.u0 / 2 H2 ./:

(2.302)

in the class v 2 C .1/ .Œ0; TI H2 .//. From (2.302) we obtain the following integral equation: Z t v.t / D v0 C ds Q.v/.s/; Q.v/ D F.A1 .v//: (2.303) 0

207


We search for solutions of the integral equation (2.303) in the class v 2 L1 .0; TI H2 .// for sufficiently small T > 0. We use the method of contraction mappings. In the Banach space L1 .0; TI H2 .//, introduce a closed convex bounded subset ® ¯ (2.304) BR v 2 L1 .0; TI H2 .// W kvkT ess sup kvk2 R : t2.0;T/

Prove that the operator Z H.v/ D v0 C

t 0

ds Q.v/

(2.305)

acts from BR into BR and is a contraction on BR for sufficiently small T > 0 and sufficiently large R > 0. Thus, let kv0 k2 R=2; then kH.v/kT

R C TkvkT .R/; 2

where .R/ D CRp2 , R kvkT . Therefore, kH.v/kT R under the condition T .2.R//1 . Now we prove the contraction property of the operator H.v/ defined by (2.305) on the set BR (see (2.304)). Indeed, kH.v1 / H.v2 /kT TkQ.v1 / Q.v2 /kT T.R/kv1 v2 kT ; R D max¹kv1 kT ; kv2 kT º under the condition T < 1=Œ2.R/. Therefore, the operator H is a contraction on BR for sufficiently large R > 0 and sufficiently small T > 0. Therefore, there exists a unique solution of the integral equation (2.303) of the class L1 .0; TI H2 .//. Using the standard extension algorithm for solution of the integral equation (2.303) on time, we conclude that there exists T0 > 0 such that either T0 D C1 or T0 < C1, and in the latter case the limit relation lim sup kvk2 D C1 t"T0

holds. Note that the explicit form of the operator H.v/ implies H.v/ W L1 .0; TI H2 .// ! AC.Œ0; TI H2 .//; H.v/ W AC.Œ0; TI H2 .// ! C .1/ .Œ0; TI H2 .//:

(2.306)

208


Therefore, there exists a unique solution of problem (2.302) of the class C .1/ .Œ0; T0 /I H2 .// and either T0 D C1 or T0 < C1, and in the latter case, the limit relation (2.306) holds. Consider the equation A.u/ D v 2 C .1/ .Œ0; TI H2 .//:

(2.307)

The properties of the operator A imply u D A1 .v/:

(2.308)

By virtue of (2.308), the following inequalities hold: ku.t / u.t0 /kC2 kA1 .v/.t / A1 .v/.t0 /kC2 kv.t / v.t0 /k2 ! C0 as t ! t0 , t; t0 2 Œ0; T0 /. Therefore, u.x; t / 2 C.Œ0; T0 /I H02 .//. Now we prove that, in fact, the function u from (2.308) belongs to the class C .1/ .Œ0; T0 /I H02 .//: Indeed, by virtue of the Fréchet differentiability of the operator A in the class u.x; t / 2 C .1/ .Œ0; T0 /I H02 .//, we obtain from (2.307) that A0u .u/u0 D v 0 2 C.Œ0; T0 /I H2 .//:

(2.309)

We write Eq. (2.309) in a more detailed form: ŒA0 C

A01;u .u/u0

0

Dv;

A01;u .u/

D

N X

Aj0 .u/:

(2.310)

j D1

By virtue of the properties of the operator A0 , Eq. (2.310) is equivalent to 0 0 1 0 ŒI C A1 0 A1;u .u/u D A0 v :

(2.311)

Now we prove that the linear (under fixed u 2 C.Œ0; T0 /I H02 .//) operator 0 O D I C A1 C 0 A1;u .u/

(2.312)

is invertible and bounded. Indeed, the boundedness follows from the properties of the operator A0 , which has a Lipschitz-continuous inverse operator, and the operator A01;u is bounded as an operator H02 ./ ! L.H02 ./I H2 .//. Now we prove that the operator C defined in (2.312) is invertible. Consider the equation O D f 2 H02 ./: Cw

(2.313)

209


Apply the operator A0 to both sides of Eq. (2.313); we obtain ŒA0 C A01;u .u/w D A0 f: Prove that the operator E D A0 C A01;u .u/ is invertible. Indeed, we have hEw1 Ew2 ; w1 w2 i D kw1 w2 k22 C krw1 rw2 k22 ˇ ˇ ˇ ˇ Z N X ˇ @u ˇpj 2 ˇ @w1 @w2 ˇ2 ˇ ˇ ˇ ˇ C .pj 1/ dx ˇ ˇ @x @x ˇ @xj ˇ j j j D1

kw1 w2 k2C2 : Therefore, by the Browder–Minty theorem, the operator E has a Lipschitz continuous inverse with Lipschitz constant equal to 1. Thus, Eq. (2.313) has a solution in the class H02 ./ for any f 2 H02 ./. Moreover, it is easy to prove the uniqueness of solution of problem (2.313). Therefore, the inverse operator C 1 is defined. Therefore, from (2.310) and (2.311) we obtain 0 O 1 A1 u0 D C 0 v :

(2.314)

O 1 A1 v 0 2 C.Œ0; T0 /I H2 .// for Now we must prove only the fact that u0 D C 0 0 2 fixed u 2 C.Œ0; T0 /I H0 .//. Indeed, by (2.314), the following inequalities hold: 0 0 O 1 .t0 /A1 ku0 .t / u0 .t0 /kC2 kC 0 Œv .t / v .t0 /kC2 0 O 1 .t0 / C O 1 .t //A1 C k.C 0 v .t0 /kC2

(2.315)

O 1 .t0 / C O 1 .t /k 2 Ckv 0 .t / v 0 .t0 /k2 C CkC H ./:!H2 ./ : 0

0

O is continuous Note that the linear (under fixed u 2 C.Œ0; T0 /I H02 .//) operator C O 1 and, therefore, by virtue of the Banach inverse-mapping theorem, the operator C is linear and continuous and, therefore, it is bounded by virtue of linearity. Thus, we O 1 W H2 ./ ! can use the spectral representation for the linear bounded operator C 0 2 ./. H0 O First, introduce the resolvent of the operator C: O D .I C/ O 1 : R.; C/ Let be a circle jj D r with sufficiently large radius, greater than sup t2Œt0 ";t0 C"

O 2 kCk H ./!H2 ./ : 0

0

The value introduced is well-defined since, for a given t 2 Œt0 "; t0 C " Œ0; T0 /, the following inequality holds: sup t2Œt0 ";t0 C"

kukC2 < C1:

210


O 1 .t / and C O 1 .t0 / Now we can use the spectral representation for the operators C with the same contour introduced above: Z Z O //; C O 1 .t0 / D 1 O 0 //: O 1 .t / D 1 d 1 R.; C.t d 1 R.; C.t C 2 i 2 i Obviously, we have O 1 .t / C O 1 .t0 / D 1 C 2 i

Z

O // R.; C.t O 0 //: d 1 ŒR.; C.t

Now we use the well-known representation for operator resolvents: O 0 // O // R.; C.t O 0 // D R.; C.t R.; C.t

C1 X

O / C.t O 0 //R.; C.t O 0 /n Œ.C.t

nD1

under the condition O /k 2 O O 0 / C.t kC.t H ./!H2 ./ kR.; C.t0 //kH2 ./!H2 ./ ı < 1: 0

0

0

0

The following inequality holds: O O // R.; C.t O 0 //k 2 kR.; C.t H ./!H2 ./ kR.; C.t0 //kH2 ./!H2 ./ 0

C1 X nD1

Note that

0

0

0

O 0 //kn 2 O / C.t O 0 /kn 2 kR.; C.t kC.t : H ./!H2 ./ H ./!H2 ./ 0

0

0

0

O / C.t O 0 / D A1 ŒA0 .u.t // A0 .u.t0 //: C.t 0 1;u 1;u

By the continuity of the Fréchet derivatives A01;u with respect to u 2 H02 ./ and by virtue of the fact that u 2 C.Œ0; T0 /I H02 .//, we have 0 0 O / C.t O 0 /k 2 kC.t H ./!H2 ./ kA1;u .u.t // A1;u .u.t0 //kH2 ./!H2 ./ ! C0; 0

0

0

O // R.; C.t O 0 //k 2 kR.; C.t H ./!H2 ./ ! 0; 0

O / kC.t

1

O 0/ C.t

1

0

kH2 ./!H2 ./ ! C0 0

0

as t ! t0 . Therefore, by (2.315) we have u 2 C .1/ .Œ0; T0 /I H02 .//. Note that, under the condition pj 3, the operator C.u/ is boundedly Lipschitzcontinuous and, therefore, the operator C 1 .u/ and hence the equation (2.314) has a local solution in the class u.t / 2 C .1/ .Œ0; T00 /I H02 .//. In is easy to prove that T0 D T00 . Theorem 2.12.2 is proved.

211


2.12.2 Blow-up of strong solutions of problem (2.288)–(2.289) and solvability in any finite cylinder In the previous subsection, we proved the unique solvability of problem (2.288)– (2.289) in the strong generalized sense. In this subsection, we obtain a sufficient condition of the blow-up of strong generalized solutions and also a sufficient condition of its solvability in any finite cylinder. Introduce the notation pj N X pj 1 1 @u 1 2 2 ; (2.316) ˆ.t / kuk2 C kruk2 C 2 2 pj @xj pj j D1

J.t / ku0 k22 C kru0 k22 C

N X

Z .pj 1/

j D1

ˆ0 ˆ.0/;

ˇ ˇ @u dx ˇˇ @x

j

ˇpj 2 ˇ 0 ˇ2 ˇ ˇ @u ˇ ˇ ˇ ˇ ˇ ˇ @x ˇ ;

(2.317)

j

p D max pj : j D1;N

The following theorem holds. Theorem 2.12.3. Let all the conditions of Theorem 2.12.2 hold. Then the following properties are valid: (1) if pj p for all j D 1; N , then T0 D C1; (2) if ˛ D p=p > 1, then T0 2 ŒT1 ; T2 , where T1 D C1 2

2 1p=2 ; ˆ p2 0

T2 D

1 ˆ0 ; ˛ 1 kru0 kpp 1;p

C1 is the constant of the best embedding H02 ./ W0

p=2 C2 D Cp ; 12

./:

krvkp C1 kvk2 for all v 2 H02 ./. Proof. In problem (2.290), we take as w a function u 2 C .1/ .Œ0; T0 /I H02 .//; then after integrating by parts we obtain the first energy equality d ˆ.t / D krukpp : dt

(2.318)

Now in problem (2.290), we take as w a function u0 2 C.Œ0; T0 /I H02 .//; then after integrating by parts we obtain the second energy equality JD

1 d krukpp : p dt

(2.319)

212


The following auxiliary estimates hold: ˇ ˇZ ˇ ˇ ˇ dx u0 uˇ ku0 k2 kuk2 ; ˇ ˇ

ˇZ ˇ ˇ ˇ ˇ dx .ru0 ; ru/ˇ kru0 k2 kruk2 ; ˇ ˇ ˇZ ˇ ˇ ˇ ˇpj 2 ˇ ˇ ˇ ˇ ˇ Z Z 0ˇ ˇ ˇ @u ˇ ˇ @u ˇpj 2 ˇ @u0 ˇ2 1=2 ˇ @u ˇpj 1=2 @u @u ˇ dx ˇ ˇ ˇ ˇ ˇ ˇ ˇ dx ˇˇ dx ˇˇ : ˇ ˇ @x ˇ ˇ @x ˇ @xj @xj ˇ @xj ˇ @xj ˇ j j (2.320)

The following relation is valid: 0 2

.ˆ / D

Z

0

Z

dx u u C

Z

0

dx .ru ; ru/ C

ˇ ˇ @u dx ˇˇ @x

ˇpj 2 ˇ @u @u0 2 ˇ : ˇ @xj @xj j

Hence by (2.320) we obtain the inequality .ˆ0 /2 pˆ.t /J:

(2.321)

By (2.318), (2.319), and (2.321) we obtain the following second-order ordinary differential inequality: ˆˆ00 ˛.ˆ0 /2 0;

˛D

p : p

(2.322)

We consider three cases: ˛ > 1, ˛ D 1, and ˛ 2 .0; 1/. Relation (2.318) implies that the inequality ˆ.t / > 0 holds on the interval Œ0; T0 / under the condition ˆ0 > 0. Consider the case where ˛ > 1. From (2.322) we obtain the lower inequality ˆ.t /

Œˆ1˛ 0

1 : 1=.˛1/ .˛ 1/kru0 kpp ˆ˛ 0 t

(2.323)

Hence we obtain T0 T2 D

1 ˆ0 : ˛ 1 kru0 kpp

In the case where ˛ D 1, from (2.322) we obtain the lower inequality ² p ³ kru0 kp t : ˆ.t / ˆ0 exp ˆ0

(2.324)

(2.325)

Consider the case where ˛ 2 .0; 1/. From (2.322) we obtain p kru0 kp 1=.1˛/ t : ˆ.t / ˆ0 1 C .1 ˛/ ˆ0

(2.326)

213


By (2.323) and (2.324), we obtain the second conclusion of the theorem. Inequalities (2.325) and (2.326) yield the lower estimate of the growth rate of the function ˆ.t /. Now let pj p for all j D 1; N . From the first energy equality we obtain Z ˆ.t / ˆ0 C C.p/

t

ds

N Z X

dx

j D1

0

@u @xj

p :

(2.327)

Moreover, the following estimates hold: kvj kp Cj kvj kpj ;

j D 1; N ;

vj D

@u : @xj

Then from (2.327) we obtain the inequality Z ˆ.t / ˆ0 C

t

ds 0

N X j D1

C.p/Cjp

pj pj 1

p=pj

ˆp=pj :

Thus, we obtain the following integral inequality: Z ˆ.t / ˆ0 C where

t

ds 0

N X

Aj ˆ˛j ;

(2.328)

j D1

p=pj pj p Aj D ; ˛j D : pj 1 pj Consider the following ordinary differential equation: C.p/Cjp

N X d‰ D Aj ‰ ˛j ; dt

‰.0/ D ˆ0 ;

‰ > 0;

(2.329)

j D1

where p Aj D C.p/Cj

pj pj 1

p=pj ;

˛j D

p : pj

We can prove that the inequality ˆ.t / ‰.t / holds on the interval Œ0; T0 /. Let ˆ0 1; then (2.329) implies the equality Z ‰.t/ 1 P ds D t; ˛ Aj s C j ¤j Aj s ˛j ˆ0

(2.330)

where ˛ D maxj D1;N ˛j D ˛j and ‰.t / ˆ0 . From (2.330) we obtain the following inequality (s 1, ˛ ˛j ): Z

‰.t/

ˆ0

ds s ˛

N X j D1

Aj t:

(2.331)

214


Consider two cases: ˛ D 1 and ˛ 2 .0; 1/. In the first case, from (2.331) and (2.328) we obtain the following inequality: ˆ.t / ‰.t / ˆ0 exp¹At º;

AD

N X

Aj :

(2.332)

j D1

In the case ˛ 2 .0; 1/, from (2.331) and (2.328) we obtain ˆ.t / ‰.t /

ˆ01˛

C .˛ 1/

N X

1=.1˛/ Aj t

:

(2.333)

j D1

Now let 0 < ˆ0 < 1; then (2.325) and (2.326) imply the fact that for a finite time T00 > 0, we have ˆ00 ˆ.T00 / 1. From the first energy inequality we conclude that ˆ.t / ˆ.T00 / 1. Thus, we obtain the following inequalities: ˆ.t / ‰.t / ˆ00 exp¹A.t T00 /º;

AD

N X

Aj ;

(2.334)

j D1

for ˛ D 1 and t T00 . However, for ˛ 2 .0; 1/ we conclude that 1=.1˛/ N X 1˛ ˆ.t / ‰.t / ˆ00 C .˛ 1/ Aj .t T00 /

(2.335)

j D1

for t T00 . Thus, from (2.332)–(2.335) we obtain the first conclusion of the theorem. Now we obtain the lower estimate for the time of the blow-up of a solution. Indeed, the first energy equality (2.318) yields the integral inequality Z t ds krukpp .s/: (2.336) ˆ.t / ˆ0 C 0

1;p W0 ./

Note that the embedding H02 ./ with the best constant of embedding C1 holds: krvkp C1 kvk2 : Then, owing to the definition of ˆ.t /, from (2.336) we obtain Z t ˆ.t / ˆ0 C C2 ds ˆp=2 .s/;

(2.337)

0

p where C2 D C1 2p=2 . From (2.337), by the Gronwall–Bellman and Bihari theorems (see [112]) we obtain the upper estimate

ˆ.t /

1 1p=2 Œˆ0

C2 .p 2/21 t 2=.p2/

:

(2.338)


215

From (2.338) we obtain the following lower estimate for the time of the blow-up of a solution: 2 1p=2 : ˆ0 T0 T1 D C1 2 p2 Theorem 2.12.3 is proved.

2.12.3 Physical interpretation The result of Theorem 2.12.3 implies the fact that if the main “role” is played by the summand div.jrujp2 ru/, i.e., p > pj for all j D 1; N , then the blow-up of solutions of problem (2.288)–(2.289) occurs. When pj p for all j D 1; N , the solvability in any finite cylinder Q D .0; T/ takes place. Thus, we have analyzed the problem on disruption initiation in semiconductors within the framework of the model given.

Chapter 3

Blow-up of solutions of strongly nonlinear Sobolev-type wave equations and equations with linear dissipation

In this chapter, we consider two abstract Cauchy problems for first-order ordinary differential equations with nonlinear operator coefficients. As applications, we give examples of strongly nonlinear pseudoparabolic equations in bounded domains with smooth boundaries. For these problems, we obtain sufficient, close to necessary conditions of the blow-up of solutions for a finite time and conditions of the global solvability. In particular, under certain conditions on nonlinear operators, we prove the solvability in any finite cylinder and the blow-up of solutions for a finite time under certain conditions on the norm of an initial function in a certain Banach space, the conditions having the sense of sufficiently large initial function. We discuss problems for pseudoparabolic equations – wave equations or equations with linear dissipation. Results presented in this chapter were obtained in [238] (see also [267]).

3.1

Formulation of problems

The aim of this section is to obtain optimal results like existence/nonexistence theorems for strongly nonlinear Sobolev-type equations in the abstract formulation, for Cauchy problems with operator coefficients in Banach spaces: N X d Aj .u/ C Lu D F.u/; u.0/ D u0 ; (3.1) A0 u C dt j D1

N X d Aj .u/ C DP .u/ D F.u/; A0 u C dt

u.0/ D u0 ;

(3.2)

j D1

and also to obtain lower and upper estimates for the times of the blow-up of solutions to problems (3.1) and (3.2). By optimal results, we mean theorems which assert that for some input parameters of the problem, the blow-up of solutions occurs and under other conditions, we have the global solvability. Therefore, we have intentionally restricted the class of Sobolev-type equations considered to the class for which we can obtain optimal results. Moreover, we consider only problems for which the uniqueness of solution meant in one or another sense can be proved.

Section 3.2 Preliminary definitions and conditions and auxiliary lemma

217

We give some examples of certain model, three-dimensional, strongly nonlinear Sobolev-type equations derived in the first chapter: N X @ pj 2 div.jruj ru/ u C jujq u D 0; u C @t

(3.3)

@ .u jujq1 u/ C u C jujq u D 0; @t

(3.4)

@ .u C div.jrujp2 ru/ jujq1 u/ u C jujq u D 0; @t

(3.5)

j D1

@ .2 u C u C div.jrujp1 2 ru// C u div.jrujp2 2 ru/ D 0; @t N X @u @ div.jrujpj 2 ru/ C u C u3 D 0; u C @t @x1

(3.6) (3.7)

j D1

@ @jujq2 C1 .u jujq1 u/ C C juj2q2 u D 0; @t @x1 @u @ div.jruj2 ru/ D 0; .2 u C u C div.jrujp1 2 ru// C u @t @x1 @ .2 u C u/ div.jruj2 ru/ @t @ @ @ @u @u @u @u @u @u Cˇ1 C ˇ2 C ˇ3 D 0; @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2

(3.8) (3.9)

(3.10)

where ˇ1 Cˇ2 Cˇ3 D 0, jˇ1 jCjˇ2 jCjˇ3 j > 0, pj > 2, q; q1 ; q2 0, p; p1 ; p2 > 2, and ˇ 2 Œ0; 1/.

3.2

Preliminary definitions and conditions and auxiliary lemma

We assume that the conditions (V), (A), (A0 ), and (F) from Section 2.2 hold. Let an operator L be defined on the space W1 with values in W1 : L W W1 ! W1 ;

i D 1; N I

we also assume that the set of values of the operator L1 coincides with the Banach space W1 . Assume that, in addition to the conditions of Section 2.2, the following conditions hold.

218

Chapter 3 Blow-up in wave and dissipative equations

Conditions (L). (1) .Lu Lv; u v/1 d1 ju vj21 for all u; v 2 W1 , d1 > 0. (2) The operator L is symmetric. (3) The operator L W W1 ! W1 satisfies the condition jLuj1 D1 juj1 , where D1 > 0 is a constant. 1=2

(4) .Lu; u/1 is a norm on W1 , which is equivalent to the initial norm of this Banach space. (5) The functional .u/ .Lu; u/1 W W1 ! R1 is Fréchet continuously differentiable.

Conditions (DP). (1) The operator D W W3 ! W4 V0 is linear and bounded. (2) The operator P .u/ W V0 W2 ! W3 is boundedly Lipschitz-continuous: jP .u1 / P .u2 /j3 2 .R/ju1 u2 j2 ; where 2 .R/ is a continuous increasing function. (3) The following inequality holds: .qC2/=2

jP .u/j3 B2 juj2

:

(4) For any u 2 V0 , the following relation holds: hDP .u/; ui0 D 0: 1=2 The condition (L) implies the fact that .Lu; u/1 is a norm on the space W1 . Moreover, under the conditions (V1), the following properties hold:

.F.u/; w/0 D hF.v/; wi

for all v; w 2 V ;

.Lu; w/1 D hLv; wi

for all v; w 2 V ;

hDP .v/; wi0 D hDP .v/; wi for all v; w 2 V : In what follows, we will need the following auxiliary statement. Proposition 3.2.1. For any u; v 2 V0 W0 W2 , under the additional condition j.F .v/; v/0 j c > 0 on the unit sphere jvj0 D 1, the following inequality holds: 1=2

jhDP .u/; vi0 j ChA0 v; vi0

j.F.u/; u/0 j1=2 ;

C > 0:

(3.11)

219


Proof. In what follows, we denote by C various constants. The following inequalities hold: jhDP .u/; vi0 j kDP .u/k0 kvk0 ChA0 v; vi0 jDP .u/j4 1=2

1=2 .qC2/=2 ChA0 v; vi1=2 0 jP .u/j3 ChA0 v; vi0 juj2 1=2

.qC2/=2

ChA0 v; vi0 juj0

1=2

ChA0 v; vi0

j.F.u/; u/0 j1=2 :

The proof is complete. Without loss of generality, we can set C D 1 in (3.11). Remark 3.2.2. Lemma 2.2.10 formulated and proved Section 2.2 of Chapter 2 implies that condition (L5) is satisfied automatically. Lemma 3.2.3. If u.t / 2 C .1/ .Œ0; TI V /, then the functional .u/ .Lu; u/1 W W1 ! R1 satisfying the conditions (L) belongs to the class .u/.t / 2 C .1/ .Œ0; T/. Proof. Lemma lem3.2.3 follows from Lemmas 2.2.10 and 2.2.12.

3.3


Definition 3.3.1. A solution satisfying the conditions X Z T N d .t / hAj .u/; wij .F.u/; w/0 C .Lu; w/1 dt D 0 dt 0 j D0

8w 2 V ;

8 2 L2 .0; T/;

(3.12)

u.0/ D u0 2 V : is called a weak generalized solution of the abstract Cauchy problem (3.1). We search for a solution of problem (3.12) in the class u.t / 2 L1 .0; TI V /;

u0 .t / 2 L2 .0; TI V0 /;

Lu 2 L1 .0; TI W1 /; F.u/ 2 L1 .0; TI W0 /; Aj .u/ 2 L1 .0; TI Vj /;

j D 0; N ;

N d d X Aj .u/ 2 L2 .0; TI V /; A.u/ D dt dt j D0

A.u/ D

N X j D0

.1/ Aj .u/ 2 Cw .Œ0; TI V /;

220


i.e., A.u/ is strongly absolutely continuous and weakly differentiable on the segment Œ0; T. Therefore, by virtue of the properties (A), (A0 ), and (F) and the conditions of Theorem 3.3.2 below we obtain A.u/ D

N X

Aj .u/ 2 H1 .0; TI V /;

F.u/ 2 L2 .0; TI V /;

j D0

A.u/ D

N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /:

j D0

Thus, by virtue of the conditions (V) and the conditions of Theorem 3.3.2 below, problem (3.12) is equivalent to the problem Z

T 0

N d X .t / hAj .u/; wi C hLu; wi hF.u/; wi dt D 0; dt j D0

u.0/ D u0 2 V

8w 2 V ;

(3.13)

8 .t / 2 L2 .0; T/;

where h; i is the duality bracket between the Banach spaces V and V . Using Appendix A.12, we conclude that problem (3.12) is equivalent to the following problem: Z

T

dt 0

d A.u/; v C hLu; vi hF.u/; vi D 0 8v 2 L2 .0; TI V /; dt

(3.14)

u.0/ D u0 2 V : The following theorem holds. Theorem 3.3.2. Let the conditions (A), A0 , (F), and (L) hold. Assume that V0 ,! W0 and V0 ,! W1 , i.e., the embedding operator is a compact operator. Moreover, assume that either 2 < p < q C 2 or p q C 2 and, in the latter case, Vj W0 , where p D max pj ; j 21;N

j 2 1; N ;

pj D p:

Then for any u0 2 V , there exists maximal T0 Tu0 > 0 such that the Cauchy problem (3.1) has a unique solution in the class u.t / 2 L1 .0; TI V /; A.u/ D

N X j D0

u0 .t / 2 L2 .0; TI V0 /

8T 2 .0; T0 /;

.1/ Aj .u/ 2 H1 .0; TI V / \ Cw .Œ0; TI V /:

(3.15)

221


Here for the function X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij ; 2 pj N

(3.16)

j D1

which, by virtue of the conditions (A0 2) and (A3), is positively defined and has the sense of kinetic energy, the following estimates depending on possible values of the variable qC2 ; p max pj ; ˛ p j D1;N hold: (1) if ˛ D 1, then T0 D C1 and ˆ.t / ˆ0 exp¹C2 t ºI

(3.17)

(2) if ˛ 2 .0; 1/, then T0 D C1 and t 1=.1˛/ I ˆ.t / ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 0 (3) if ˛ > 1 and

(3.18)

p q 2p .F.u0 /; u0 /0 > .Lu0 ; u0 /1 C ˆ0 ; qC2p qC2 .Lu0 ; u0 /1 ; .F.u0 /; u0 /0 > 2

then there exists T0 > 0 such that lim ˆ.t / D C1;

t"T0

T1 T0 T2 ;

(3.19)

where T1 D

2 q=2 1 B ; ˆ q 0

1 1 T2 D ˆ1˛ A0 ; 0

1=pj

jvj0 Cj hAj v; vij ˆ0 ˆ.0/;

˛1

˛ C2 BCjqC2 2 ;

;

pCqC2 ; 2p

1 1 1=2 A0 Œ.˛1 1/2 ˆ2˛ Œ.F.u0 /; u0 /0 .Lu0 ; u0 /1 2 .˛1 1/ˇˆ22˛ ; 0 0

ˇ

q2 qC2p

qC2 in the case where q C 2 > p max pj ; B BC1 2.qC2/=2 ; j D1;N

where C1 is the constant of the best embedding V ,! W0 and B is the constant from the condition (F4).

222


Remark 3.3.3. The fact that a solution belongs to the class u.t / 2 L1 .0; TI V /;

u0 .t / 2 L2 .0; TI V0 /

implies that, after a possible change on a set of zero Lebesgue measure, the mapping u.t / W Œ0; T ! V0 becomes continuous. Therefore, the initial condition u.0/ D u0 makes sense. Proof. Step 1. Galerkin approximations. By virtue of the separability of V , there exists a countable, everywhere dense in V , linearly independent system of functions ¹wi ºm iD1 . We prove the solvability of problem (3.12) by using the Galerkin method and the monotonicity and compactness methods [275]. First, consider the following finite-dimensional approximation of problem (3.12): Z

T

dt .t / 0

N d d X hAj .um /; wk ij hA0 um ; wk i0 C dt dt j D1 C .Lum ; wk /1 .F.um /; wk /0 D 0; k D 1; m;

(3.20)

8 .t / 2 L2 .0; T/; um D

m X

cmi .t /wi ;

um0 D

i D1

m X

cmi .0/wi ;

cmi .0/ D ˛mi ;

i D1

um0 ! u0

strongly in V :

From (3.20) we obtain that in the class cmk 2 C .1/ Œ0; Tm , the following pointwise equality holds: m X i D1

N X 0 0 cmi hAj;u .u /w ; w i hA0 wi ; wk i0 C m i k j m j D1

C .Lum ; wk /1 D .F.um /; wk /0 ;

k D 1; m:

(3.21)

Introduce the notation aik hA0 wi ; wk i0 C

N X


(3.22)

j D1

Obviously, m;m X i;kD1;1


N X


j D1

D

m X i D1

i wi ;

223


since A0 is a positive definite operator and, therefore, hA0 ; i0 0 and the equality hA0 ; i0 D 0 holds if and only if D 0. On the other hand,

m X

i wi ;

(3.23)

i D1

and by the linear independence of the system of functions ¹wi ºm i D1 in V we conclude that D 0 if and only if ¹i ºm D 0. Hence (see [335]) we obtain that i D1 > 0. det¹aik ºm;m i;kD1;1 0 .um /wi ; wk ij are continuous with Now we prove that the functionals fj hAj;u m respect to the set of variables cmi , i D 1; m. Indeed, let c m1 ; : : : ; c mm be a certain point of the Euclidean space Rm . Fix an arbitrary " > 0. The following inequality holds: jfj .c m1 ; : : : ; c mm / fj .cm1 ; : : : ; cmm /j jfj .c m1 ; : : : ; c mm / fj .cm1 ; c m2 ; : : : ; c mm /j C jfj .cm1 ; c m2 ; : : : ; c mm / fj .cm1 ; cm2 ; c m3 ; : : : ; c mm /j C jfj .cm1 ; cm2 ; c m3 ; : : : ; c mm / fj .cm1 ; cm2 ; cm3 ; : : : ; c mm /j C C jfj .cm1 ; cm2 ; : : : ; c.m1/m ; c mm / fj .cm1 ; : : : ; cmm /j:

(3.24)

By the condition (A2), there exists ı."/ > 0 such that each summand in the right-hand side of inequality (3.24) under the condition m X

jc mk cmk j ı."/

kD1

is less than the value of "=.m C 1/. Now we prove the Lipschitz continuity of the functionals f0k D .F .um /; wk /0 with respect to the set of variables ¹cmi ºm i D1 . Let uj D

m X i D1

j cmi wi ;

j D 1; 2:

By the condition (F1), the following inequalities hold: j.F.u1 / F.u2 /; wk /0 j jF.u1 / F.u2 /j0 jwj0 Bju1 u2 j0 B1

m X lD1

1 2 jcml cml j:

224


Therefore, the functions f0k D f0k .cm1 ; : : : ; cmm / are Lipschitz continuous and continuous with respect to the set of variables. Note that the inverse matrix to matrix (3.22) is continuous with respect to cm D .cm1 ; : : : ; cmm / (see Appendix A.18). Therefore, system (3.21) of ordinary differential equations is a system of Cauchy– Kovalevskaya type and satisfies the conditions that guarantee the solvability on a certain segment Œ0; Tm , Tm > 0, in the class cmk .t / 2 C 1 .Œ0; Tm /, k D 1; m (see, e.g., [319]). Step 2. A priori estimates Lemma 3.3.4. There exists T > 0 independent of m 2 N and such that for the sequence ¹um º of Galerkin approximations, the following properties hold uniformly with respect to m 2 N: um

is bounded in L1 .0; TI V /I

u0m


A 0 um


Aj .um / is bounded in L1 .0; TI Vj /I F.um /


Lum

is bounded in L1 .0; TI W1 /:

(3.25)

Proof. Multiplying both sides of Eq. (3.21) by cmk .t / and summing ober k D 1; m, we obtain hA0 u0m ; um i0 C

N X

0 hAj;u .um /u0m ; um ij C .Lum ; um /1 D .F.um /; um /0 : m

j D1

(3.26) On the other hand, by the fact that um D

N X


kD1

and the conditions (A2) and (A4), the following equalities hold: d hAj .um /; um ij D pj hAj .um /; u0mt ij ; dt h.Aj .um //0t ; um ij C hAj .um /; u0mt ij D pj hAj .um /; u0mt ij ; h.Aj .um //0t ; um ij D .pj 1/hAj .um /; u0mt ij D

(3.27)


225


We prove the relations d hAj .um /; um ij D pj hAj .um /; u0mt ij : dt Indeed, we have Z Jmj .t / d Jmj .t / D dt

Z Z

D Z D

1 0 1 0 1 0



ds ŒshA0sum .sum /u0m ; um ij C hAj .sum /; u0m ij


1

ds 0


Here we have used the fact that the Fréchet derivatives of the operators Aj are symmetric by virtue of the conditions (A). Hence from (3.26) we obtain N X pj 1 d 1 u ; u i C hA .u /; u i hA0 m m 0 j m m j C .Lum ; um /1 D .F.um /; um /0 : dt 2 pj j D1 (3.28) From (3.28), integrating over t 2 .0; Tm /, we obtain X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj N

j D1

X pj 1 1 D hA0 um0 ; um0 i0 C hAj .um0 /; um0 ij 2 pj j D1 Z t C ds Œ.F.um /; um /0 .Lum ; um /1 : N

(3.29)

0

By the condition (A4), we can choose in the Banach space V0 a norm equivalent to the initial norm: 1=2 kvk0 D hA0 v; vi0 :

(3.30)

On the other hand, V0 ,! W0 ;

qC2


:

226


From (3.29), by virtue of (3.30), we obtain X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj N

j D1

X pj 1 1 hA0 um0 ; um0 i0 C hAj .um0 /; um0 ij 2 pj N

(3.31)

j D1

Z C BC1

qC2 .qC2/=2

2

X pj 1 1 ds hAj .um /; um ij hA0 um ; um i0 C 2 pj 0 t

N

.qC2/=2 :

j D1

0 , summing Now we derive the second a priori estimate. Multiplying (3.21) by cmk over k D 1; m, and integrating both sides over t 2 .0; Tm /, we obtain

Z

t 0

C

N X ds hA0 u0m ; u0m i0 C h.Aj .um //0 u0m ij j D1

1 1 .F.um0 /; um0 /0 .Lum0 ; um0 /1 qC2 2 1 1 D .F.um /; um /0 .Lum ; um /1 : qC2 2

(3.32)

If 1 1 .F.u0 /; u0 /0 .Lu0 ; u0 /1 > 0; qC2 2

(3.33)

then there exists a subsequence of the sequence ¹um0 º such that 1 1 .F.um0 /; um0 /0 .Lum0 ; um0 /1 > 0: qC2 2 Indeed, ¹um0 º strongly converges in the Banach space V , which is continuously embedded in the Banach spaces W0 and W1 . Therefore, owing to the bounded Lipschitzcontinuity of the operator F W W0 ! W0 and the continuity of the linear operator L W W1 ! W1 , we obtain the following limit relations: .F.um0 /; um0 /0 ! .F.u0 /; u0 /0 as m ! C1; .Lum0 ; um0 /1 ! .Lu0 ; u0 /1

as m ! C1:

By the positive definiteness of the operator L we have .F.um0 /; um0 /0 0:

227


Hence from (3.33) we obtain .F.um /; um /0 .Lum ; um /1 > 0:

(3.34)

X pj 1 1 hAj .um /; um ij ; ˆm hA0 um ; um i0 C 2 pj

(3.35)

Introduce the notation N

j D1

ˆm0 ˆm .0/: From (3.28) subject to (3.34) we obtain that ˆ0m .t / > 0. From (3.29) and (3.35) we obtain Z t ds ˆ.qC2/=2 .s/; ˆm ˆm0 C B m

(3.36)

0

where

B BCqC2 2.qC2/=2 ; 1

C1 is the constant of the best embedding V ,! W0 and B is the constant from the condition (F4). From (3.36) and the Bihari theorem (see, e.g., [112]) we have ˆm

Œ1

ˆm0 : q q=2 2=q 2 ˆm0 Bt

(3.37)

Since um0 ! u0 strongly in V , ˆm0 C0 and C0 is independent of m 2 N. For a certain subsequence of the sequence ¹um º, the following two cases are possible: either ˆm0 # ˆ0 or ˆm0 " ˆ0 . In the first case where ˆm0 " ˆ0 , the following inequalities hold: q q q=2 q=2 1 Bˆm0 t 1 Bˆ0 t 2 2 ˆ m C1

8t 2 .0; T1 /; T1 D B

8t 2 .0; T1 /;

1 2

q=2

; ˆ q 0 1 2 q=2 T1 D B ; ˆ q 0

where C1 is dependent of m 2 N. Now consider the case where ˆm0 # ˆ0 . For any m < m, we have ˆm0 > ˆm0 ; 1 2 q=2 q ˆm0 /,

For any t 2 Œ0; B

q q q=2 1 B ˆq=2 m0 t 1 B ˆm0 t: 2 2

we obtain

ˆm

Œ1

C0 : q q=2 2=q 2 ˆm0 Bt

(3.38)

228


Therefore, for any fixed T from the interval T 2 .0; T1 /, there exists m 2 N such that inequality (3.37) holds. By virtue of the fact that there exist pj D maxj D1;N pj > 2 and pj q C 2 such that Vj W0 , the following inequality holds: 1=pj


:

(3.39)

From (3.39) we directly obtain that qC2

jvj0

CjqC2

pj pj 1

.qC2/=pj

X pj 1 1 hAj .v/; vij hA0 v; vi0 C 2 pj N

.qC2/=pj :

(3.40)

j D1

Owing to (3.35), (3.39), and (3.40) we obtain the inequality ˆm ˆm0 C

BCjqC2

pj pj 1

.qC2/=pj Z

t

0

ds ˆ˛m .s/;

˛

qC2 : pj

(3.41)

Consider separately the cases where ˛ < 1 and ˛ D 1. Using the Gronwall–Bellman and Bihari theorems (see [112]), from (3.41) we obtain 1=.1˛/ ; ˛ 2 .0; 1/; ˆm Œˆ1˛ m0 C .1 ˛/C2 t

(3.42)


(3.43)

where C2

BCjqC2

˛ D 1; pj pj 1

.qC2/=pj :

By virtue of the fact that um0 ! u0 strongly in V , we see that ˆm0 C0 and is independent of m 2 N. Therefore, we obtain ˆm0 C3

8T 0;

8˛ 2 .0; 1:

(3.44)

The conditions (A) and (3.44) imply qC1 jF.um /j0 Bjum j0

BCqC2 2.qC2/=2 1

X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj

C3 ˆ.qC2/=2 C30 ; m

N

j D1

.qC2/=2

229


where 0 < C30 .T/ < C1 is a constant independent of m 2 N. Here the inequality above holds for any T > 0 in the case where ˛j 2 .0; 1 and for any T 2 .0; T1 /, where T1 is defined in (3.38), in the case ˛j > 1. Now from (3.32) and the conditions (A2) and (A0 2) we obtain that Z t Z t 0 2 ds kum k0 ds hA0 u0m ; u0m i0 C40 ; m0 0

0

where 0 < C40 < C1 is a constant independent of m 2 N. Here the inequality above holds for any T > 0 in the case ˛j 2 .0; 1 and for any T 2 .0; T1 /, where T1 is defined in (3.38), in the case ˛j > 1. Thus, um


u0m


A0 u m


Aj .um / is bounded in L1 .0; TI Vj /; F.um /


Lum


(3.45)

where the inclusions hold for any T > 0 in the case ˛ 2 .0; 1 and for any T 2 .0; T1 /, where T1 is defined in (3.38), in the case ˛ > 1. Therefore, by (3.45) there exists a subsequence of the sequence ¹um º such that um * u


u0m * u0 weakly in L2 .0; TI V0 /:

(3.46)

Note that (3.46)2 implies u0m * u0

in D 0 .0; TI V0 /I

therefore, by the weak convergence u0m * in L2 .0; TI V0 /, we conclude that .t / D u0 .t /. Now we apply the result of Lemma A.15.1 (see Appendix A.15), where we set W D W0 ; we obtain u.s/ ! u.s/


On the other hand, by the condition (F1), we have jF.um / F.u/j0 .R/jum uj0 ! 0

as m ! C1;

(3.47)

230


where R D max¹juj0 ; jum j0 º. From (3.47) we obtain the strong convergence F.um /.t / ! F.u/.t / strongly in W0 for almost all t 2 .0; T/:

(3.48)

Note that by the fact that V0 ,! W1 and according to Lemma A.15.1 (see Appendix A.15), where we set W D W1 , we obtain u.s/ ! u.s/


Hence from the condition (L3) we obtain the strong convergence Lum ! Lu in W1 for almost all t 2 .0; T/. Thus, the required a priori estimates have been proved. Step 3. Monotonicity method. Now we use the monotonicity method. Recall that in Step 2, we have obtained limit relations (3.46) and (3.48) and, moreover, A.um / *

-weakly in

L1 .0; TI V /;

N X

A.u/

Aj .u/:

(3.49)

j D0

Lemma 3.3.5. Let A.u/ sequence ¹um º such that

PN

j D0 Aj .u/.

Then there exists a subsequence of the

A.um / * A.u/ -weakly in L1 .0; TI V / for certain T > 0. Moreover, for a certain subsequence of the sequence ¹um º, um ! u strongly in Lpj .0; TI Vj /; um ! u strongly in L2 .0; TI V0 /: Proof. Rewrite Eq. (3.20) in the equivalent form Z t ds Œ.F.um /; wj /0 .Lum ; wj /1 ; hA.um /; wj i D hA.um0 /; wj i C

(3.50)

0

where h; i is the duality brackets between the Banach spaces V and V . Fix j 2 N and pass in Eq. (3.50) to the limit as m ! C1; then we obtain Z t D ds ŒF.u/ Lu C A.u0 /: (3.51) 0

Let

Z

0 X

0

Z D

T

0

T

dt hA.u / A.v/; u vi Z dt hA.u /; u i

T 0

Z dt hA.u /; vi

0

T

dt hA.v/; u vi:

231


Then Z T 0

Z dt hA.u /; u i D

T 0

Z

C Hence we obtain

Z

0 lim sup X

dt hA.u 0 /; u i Z

T

dt 0

0

Z

T

t

dt

!C1

0

Z

C Z D

0 T

0 T

0

t

ds Œ.F.u /.s/; u .t //0 .Lu .s/; u .t //1 :

ds Œ.F.u/.s/; u.t //0 .Lu.s/; u.t //1 Z

dt hA.u0 /; ui

T

0

Z dt h ; vi

T

dt hA.v/; u vi

0

dt h A.v/; u vi:

(3.52)

Now we set v D u w for any v; w 2 Lr .0; TI V /, > 0, u 2 L1 .0; TI V / 0 Lr .0; TI V /, ; A.v/ 2 Lr .0; TI V /, r > 1, r 0 D r=.r 1/. By (3.52), the following inequality holds: Z T dt h A.u w/; wi 0; 0

from which, by the semilinearity of operators Aj .v/, j D 0; N , we obtain D A.u/: On the other hand, by (3.51) we have the inequality Z T dt hAj .u /; u ij lim sup

(3.53)

!C1 0

N X

lim inf

!C1

kD0;k¤j

Z

N X kD0;k¤j

T 0

Z 0

T

Z dt hAk .u /; u ik C lim sup

T

!C1 0

Z dt hAk .u/; uik C

T 0

dt hA.u /; u i

Z dt hA.u/; ui D

0

T

dt hAj .u/; uij ;

where we have used the fact that each of the operators Aj .v/, j D 0; N , by virtue of the conditions (A0 ) and (A), generates norms of the uniformly convex Banach spaces L2 .0; TI V0 / and Lpj .0; TI Vj /, and these norms are equivalent to the initial norms by virtue of the conditions (A0 ) and (A) by means of the following formulas: Z

T 0

Z

1=2 dt hA0 u; ui0

T

; 0

1=pj dt hAj .u/; uij

:

232


Finally, by the weak convergence u * u in Lpj .0; TI V /, j D 0; N , and the fact that the operators Aj .v/, j D 0; N , generate norms of uniformly convex Banach spaces according to the rules above, we have Z T Z T dt hAj .um /; um ij dt hAj .u/; uij : (3.54) lim inf m!C1 0

0

Inequalities (3.53) and (3.54) imply the fact that Z T Z dt hAj .um /; um ij D lim m!C1 0

Now, by the conditions (A), 1=pj Z T dt hAj .u/; uij ;

Z

are norms equivalent to the initial norms 1=pj Z T pj dt kukj ;

Z

0

T 0

T 0

dt hAj .u/; uij :

1=2 dt hA0 u; ui0

1=2

T

dt

0

(3.55)

0

kuk20

of the Banach spaces Lpj .0; TI Vj /. Therefore, (3.55) implies the existence of a subsequence of the sequence ¹um º such that 1=pj 1=pj Z T Z T dt hAj .um /; um ij ! dt hAj .u/; uij ; j D 1; N ; 0

Z

T 0

0

1=2 dt hA0 um ; um i0

Z !

T

0

1=2 dt hA0 u; ui0

:



j D 0; N I

therefore, there exists a subsequence of the sequence ¹um º such that um ! u strongly in any Lpj .0; TI Vj /, j D 0; N . Step 4. Passage to the limit and the uniqueness. The limit relations (3.46), (3.48), and (3.49) imply the possibility of the passage in Eq. (3.50) to the limit as m ! C1. Hence we obtain that for almost all t 2 .0; T/, the following equality holds: hA0 u; vi0 C

N X


j D1

hAj .u0 /; vij

j D1

Z C

N X

t 0

ds Œ.F.u/; v/0 .s/ .Lu; v/1 .s/

233


for all v 2 V and u.0/ D u0 2 V . On the other hand, u0 2 L2 .0; TI V0 / and u 2 L1 .0; TI V /. Rewrite the latter equality in the form N X

hAj .u/; vij D

j D0

N X

Z hAj .u0 /; vij C

j D0

t 0

ds Œ.F.u/; v/0 .s/ .Lu; v/1 .s/

for all v 2 V and almost all t 2 .0; T/. Therefore, for almost all t 2 .0; T/, the following equality holds in the sense of V : A.u/ D

N X

N X

Aj .u/ D

j D0

Z Aj .u0 / C

j D0

t 0

ds ŒF.u/.s/ Lu.s/ :

(3.56)

We have proved above that A.u/ D

N X

Aj .u/ 2 L1 .0; TI V / L2 .0; TI V /:

j D0

Moreover, (3.56) implies N d X Aj .u/ 2 L2 .0; TI V /: dt j D0

Therefore, A.u/ 2 H1 .0; TI V /. Finally, by reasons similar to that used in Step 4 of the proof of Theorem 2.3.2, we can prove that N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /:

j D0

Consider Eq. (3.20) rewritten in the form Z

T

dt .t / 0

N d X Aj .um /; w hF .um /; wi C hLum ; wi D 0 dt j D0

for all .t / 2 L2 .0; T/ and all w 2 V . By Lemma 3.3.4 we obtain that F .um / * F .u/; Moreover,

Lum ! Lu


A.um / * A.u/ weakly in L2 .0; TI V /:

Therefore, d d A.um / ! A.u/ dt dt

234


in the sense of D 0 .0; TI V /. On the other hand, the following pointwise equality holds: d A.um / D F.um / Lum : dt Therefore, d A.um / ! dt


Thus, d d A.um / ! A.u/ dt dt


Passing in Eq. (3.20) to the limit as m ! C1, we obtain that u.t / is a solution of the problem Z

T

dt .t / 0

N d X Aj .u/; w hF.u/; wi C hLu; wi D 0 dt j D0

for all .t / 2 L2 .0; T/ and all w 2 V , which is equivalent to problem (3.12). Now let u be an arbitrary solution of the problem Z

T

dt .t / 0

N d X Aj .u/; w hF.u/; wi C hLu; wi D 0 dt j D0

for all .t / 2 L2 .0; T/ and all w 2 V of the class N X

u 2 L1 .0; TI V /;

.1/ Aj .u/ 2 H1 .0; TI V / \ Cw .Œ0; TI V /;

j D0

Lu; F.u/ 2 L1 .0; TI V /: Note that our solution satisfies the relation A.u/ D

N X j D0

Aj .u/ D

N X j D0

Z Aj .u0 / C

0

t

ds ŒF.u/.s/ Lu.s/ :

(3.57)

We prove that in the class of the functions satisfying this equality, our solution is unique. Let u1 and u2 be two weak generalized solutions of problem (3.12) of the smoothness class mentioned in the conditions of Theorem 3.3.2 with the same initial function u0 2 V . Then for the difference w D u1 u2 of these solutions, by virtue of (3.57),

235


we obtain the relation hA0 w; wi0 C

hAj .u1 / Aj .u2 /; wi

j D1

Z D

N X

t 0

ds Œ.F.u1 /.s/ F.u2 /.s/; w.t //0 .Lu1 .s/ Lu2 .s/; w.t //1 :

Hence by the monotonicity of the operators Aj .u/, the condition (A0 2), and the embeddings V0 W0 and V0 W1 , we obtain the inequality Z t ds jF.u1 / F.u2 /j0 .s/ kwk20 .t / Cjwj0 .t / 0

Z

C Cjwj1 .t / Z Ckwk0 .t /

0

t

0 t

ds jLu1 Lu2 j1 .s/

ds kwk0 .s/;

C > 0;

from which, by virtue of the Gronwall–Bellman theorem (see [112]), we obtain that u1 D u2 almost everywhere on the set .0; T/ . Step 5. Blow-up of solutions Lemma 3.3.6. Let X pj 1 1 hAj .u/; uij ; hA0 u; ui0 C 2 pj N

ˆ

˛

j D1

qC2 ; p

p max pj :

Then (1) for ˛ 2 .0; 1/, the following inequality holds: ˆ ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 t 1=.1˛/ I 0 (2) for ˛ D 1, the following inequality holds: ˆ ˆ0 exp.C2 t /I (3) for ˛ > 1, if the following condition holds: p q 2p .F.u0 /; u0 /0 > .Lu0 ; u0 /1 C ˆ0 ; qC2p qC2 .F.u0 /; u0 /0 > .Lu0 ; u0 /1 ; 2

j D1;N

236


there exists T0 2 ŒT1 ; T2 such that the limit relation lim ˆ.t / D C1

t"T0

holds, where T1 D

2 q=2 1 B ; ˆ q 0

1 1 T2 D ˆ1˛ A0 ; 0

1=pj


qC2

C2 BCj 2˛ ; ;

pCqC2 ; 2p

˛1

1 1 1=2 Œ.F.u0 /; u0 /0 .Lu0 ; u0 /1 2 .˛1 1/ˇˆ22˛ ; A0 Œ.˛1 1/2 ˆ2˛ 0 0

ˇ

q2 qC2p

qC2 in the case q C 2 > p max pj ; B BC1 2.qC2/=2 : j D1;N

Proof. By the condition (A2), the Schwarz inequality (see [293]) is valid for the 0 Fréchet derivatives Aj;u W Vj ! L.Vj I Vj / of the operators Aj W Vj ! Vj : 0 .um /u0m ; um ij j jh.Aj .um //0 ; um ij j D jhAj;u m 0 hAj;u .um /u0m ; u0m ij m

1=2

0 hAj;u .um /um ; um ij m

1=2

D h.Aj .um //0 ; u0m ij1=2 .pj 1/1=2 hAj .um /; um ij1=2 ; 1=2 jhA0 u0m ; um i0 j hA0 u0m ; u0m i1=2 0 hA0 um ; um i0 :

(3.58) (3.59)

Here we have used the relation 0 Aj;v .v/v D .pj 1/Aj .v/:

Prove it. Indeed, Aj .v/ D pj 1 Aj .v/;

0 0 Aj;v .v/ D pj 1 Aj;v .v/;

0 0 .v/ D pj 2 Aj;v .v/; j D 0; N ; Aj;v

Z 1

Z 1 d 0 d d Aj;v .v/v; w 8w 2 Vj ; Aj .v/; w D d 0 0 j j 0 .v/v; wij D 0 hAj .v/ .pj 1/1 Aj;v

Therefore,

0 .pj 1/Aj .v/ D Aj;v .v/v

8w 2 Vj ;

8v 2 Vj ;

j D 0; N ;

j D 0; N :

j D 0; N :

237


Introduce the notation X pj 1 1 hAj .um /; um ij : ˆm hA0 um ; um i0 C 2 pj N

(3.60)

j D1

From (3.58)–(3.60) we have ˇ2 ˇ2 ˇ ˇ N X ˇ ˇ ˇ ˇd 0 ˇ ˇ ˆm ˇ ˇjhA0 u0 ; um i0 j C jh.A .u // ; u i j j m m j m ˇ ˇ ˇ ˇ dt j D1

N X 0 0 0 0 hA0 um ; um i0 C h.Aj .um // ; um ij j D1

N X .pj 1/hAj .um /; um ij hA0 um ; um i0 C j D1



j D1


N X 1 pj 1 hA0 um ; um i0 C hAj .um /; um ij ; 2 pj

(3.61)

j D1

if p pj > 2, where p D maxj D1;N pj > 2. By (3.28) and (3.32), the following energy equalities hold: N X pj 1 d 1 hAj .um /; um ij C .Lum ; um /1 D .F.um /; um /0 ; hA0 um ; um i0 C dt 2 pj j D1

(3.62) hA0 u0m ; u0m i0 C

N X

h.Aj .um //0 u0m ij

j D1

D

1d 1 d .Lum ; um /1 C .F .um /; um /0 : 2 dt q C 2 dt

(3.63)

238


Let q C 2 > p and .F.u0 /; u0 /0 >

qC2 .Lu0 ; u0 /1 I 2

then there exists a subsequence of the sequence ¹um0 º such that .F.um0 /; um0 /0 >

qC2 .Lum0 ; um0 /1 : 2

Indeed, ¹um0 º strongly converges in the Banach space V , which is continuously embedded in Banach spaces W0 and W1 . Therefore, owing to the bounded Lipschitz– continuity of the operator F W W0 ! W0 and the continuity of the linear operator L W W1 ! W1 , the following limit relation holds: .F.um0 /; um0 /0 ! .F.u0 /; u0 /0

as m ! C1;

.Lum0 ; um0 /1 ! .Lu0 ; u0 /1

as m ! C1:

Hence from (3.63) we obtain .F .um /; um /0 >

qC2 .Lum ; um /1 ; 2

.F.um /; um /0 > .Lum ; um /1 :

By (3.62) we obtain ˆ0m .t / 0. Proposition 3.3.7. Let the conditions (L) and (A0 ) hold and, moreover, V0 W1 . Then the following inequality holds: .Lu; u/1 ChA0 u; ui0

8u 2 V0

for certain C > 0. Proof. Indeed, let u 2 V0 W1 ; then the following inequalities hold: .Lu; u/1 D1 juj21 D1 C1 kuk20 D1 C1 m1 0 hA0 u; ui0 D ChA0 u; ui0 : The proof is complete. Without loss of generality, we can assume that the constant C in Proposition 3.3.7 is equal to 1. Indeed, introduce the new operators e L C1 L;

e F C1 F

and, moreover, perform the change e t Ct . Then for the operators introduced, Proposition 3.3.7 with the constant C D 1 holds.

239


Note that, by virtue of Proposition 3.3.7 and the Cauchy and Schwarz inequalities (see [293]), the following inequalities hold: hA0 u0m ; u0m i0

C

N X

h.Aj .um //0 ; u0m ij

j D1

.Lum ; u0m /1 C

1 d .F.um /; um /0 q C 2 dt

1 q ˆ00m .Lum ; u0m /1 qC2 qC2 1 q " q 1 ˆ00 C .Lu0m ; u0m /1 C .Lum ; um /1 qC2 m qC22 q C 2 2" 1 q " q 1 ˆ00m C hA0 u0m ; u0m i0 C hA0 um ; um i0 qC2 qC22 q C 2 2" 1 q 1 q ˆ00m C ˆm C "hA0 u0m ; u0m i0 qC2 qC2" 2.q C 2/ D

X q h.Aj .um //0 ; u0m ij : " 2.q C 2/ N

C

j D1

Hence we conclude that 1"

q 2.q C 2/

N X 0 h Aj .um / ; u0m ij hA0 u0m ; u0m i0 C j D1

q 1 1 ˆ00 C ˆm qC2 m qC2"

qC2 : 8" 2 0; 2 q

From (3.61) and (3.63) we obtain the second-order ordinary differential inequality (see [210]): (3.64) ˆ00m ˆm ˛1 ."/.ˆ0m /2 C ˇ."/ˆ2m 0; qC2 q q ˛1 ."/ 1" ; ˇ."/ D ; p D max pj > 2; p 2.q C 2/ " j D1;N which implies 1 1˛1 ."/ 0: .ˆ1˛1 ."/ /00 C ˇ."/ˆm 1 ˛1 ."/ m Introduce the notation 1 ."/ Zm ˆ1˛ : m

240


We emphasize the following properties of the function Zm .t /: 1 ."/ 0 Z0m .1 ˛1 ."//ˆ˛ ˆm 0 m

Zm 0; under the condition

qC2p ; " 2 0; 2 q

q C 2 > p;

since we have proved above that ˆ0m .t / 0. Indeed, by (3.62) d ˆm D .F.um /; um /0 .Lum ; um /1 dt and, moreover, by (3.32), .F.um /; um /0 .Lum ; um /1 > 0: The following inequalities hold: Z00m ."/Zm ;

.˛1 ."/ 1/ˇ."/;

..Z0m /2 /0 ."/.Z2m /0 ; 2˛1 ."/

Bm0 .˛1 ."/ 1/2 ˆm0

Z00m Z0m ."/Zm Z0m ;

.Z0m /2 Bm0 C ."/Z2m ;

Œ.F.um0 ; um0 //0 .Lum0 ; um0 /1 2

22˛1 ."/ .˛1 ."/ 1/ˇ."/ˆm0 :

For further consideration, we must analyze the condition Bm0 > 0 and choose " 2 .0; 2.q C 2 p/=q/, q C2 > p, in such a way that we could obtain optimal condition for the value of the initial function u0 . To this end, rewrite the condition mentioned as follows: ˛1 ."/ 1 1=2 Œ.F.um0 /; um0 /0 .Lum0 ; um0 /1 > ˆm0 : ˇ."/ Now consider the function ˛1 ."/ 1 f ."/ ˇ."/

qC2p for " 2 0; 2 ; q

q C 2 > p:

This function reaches its maximum at the point "0

qC2p ; q

f ."0 / D

Thus, the condition on um0 takes the form

.q C 2 p/2 : q 2 2p

p q 2p .F.um0 /; um0 /0 > .Lum0 ; um0 /1 C ˆm0 : qC2p

241


This condition holds trivially for a certain subsequence of the sequence ¹um0 º under the condition p q 2p ˆ0 : .F.u0 /; u0 /0 > .Lu0 ; u0 /1 C qC2p Indeed, ¹um0 º strongly converges in the Banach space V , which is continuously embedded in Banach spaces W0 and W1 . Therefore, owing to the bounded Lipschitz continuity of the operator F W W0 ! W0 and the continuity of the linear operator L W W1 ! W1 , the following limit relation holds: .F.um0 /; um0 /0 ! .F.u0 /; u0 /0 as m ! C1; .Lum0 ; um0 /1 ! .Lu0 ; u0 /1

as m ! C1;

X pj 1 1 D hA0 um0 ; um0 i0 C hAj .um0 /; um0 ij 2 pj N

ˆm0

j D1

X pj 1 1 ! ˆ0 D hA0 u0 ; u0 i0 C hAj .u0 /; u0 ij 2 pj N

as m ! C1:

j D1

The conditions (A3), (A4), (A0 3), and (A0 4) now imply that the functionals 1=2

hA0 u; ui0

W V0 ! R1 ;

1=pj

hAj .u/; uij

W Vj ! R1

are norms. Thus, consider the following differential inequality obtained from (3.64) with " D "0 .q C 2 p/=q/: .Z0m /2 A2m0 C .Zm /2 ;

(3.65)

where 22˛1 1=2 2 1 Am0 Œ.˛1 1/2 ˆ2˛ ; m0 Œ.F .um0 /; um0 /0 .Lum0 ; um0 /1 .˛1 1/ˇˆm0

.˛1 1/ˇ;

˛1 D

qC2Cp ; 2p

ˇD

q2 : qC2p

Owing to the fact that Z0m 0, (3.65) implies the inequality Zm .t / Zm .0/ Am0 t:

(3.66)

Inequality (3.66) implies the lower estimate for the function ˆm .t /: ˆm .t /

‰m0 ŒTm2 t 1=.˛1 1/

;

1=.˛1 C1/

‰m0 D Am0

;

1 1 Tm2 D ˆ1˛ m0 Am0 : (3.67)

242


Now we prove that ˆm .t / ! ˆ.t / for almost all t 2 .0; T/, where X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij : 2 pj N

j D1

Indeed, by virtue of Lemma A.15.1 (see Appendix A.15), where we first set W D W0 and then W D W1 , we obtain um .s/ ! u.s/

strongly in W0 for almost all s 2 Œ0; T;

um .s/ ! u.s/


On the other hand, from (3.62) we obtain Z t ˆm D ˆm0 C ds Œ.F.um /; um /0 .s/ .Lum ; um /1 .s/ 0

and, therefore,

Z ˆ m ! ˆ0 C

t 0

ds Œ.F.u/; u/0 .s/ .Lu; u/1 .s/ : ´

Now we set v.s/ D

u.s/; s 2 Œ0; t ; 0; s 2 .t; T;

in the definition of a weak generalized solution, and we obtain that Z t ds Œ.F.u/; u/0 .s/ .Lu; u/1 .s/ : ˆ D ˆ0 C 0

Hence we directly obtain that ˆm .t / ! ˆ.t /

(3.68)

for almost all t 2 .0; T/. Owing to the fact that um0 ! u0 strongly in V , we have as m ! C1 X pj 1 1 hAj .u0 /; u0 ij ; ˆm0 ! ˆ0 hA0 u0 ; u0 i0 C 2 pj N

j D1

‰m0 ! ‰0

A01=.1˛1 / :

From (3.68), passing in inequalities (3.31) and (3.32) to the limit as m ! C1, we obtain that qC2 ˛D D 1; (3.69) ˆ.t / ˆ0 exp¹C2 t º; p t 1=.1˛/ ; ˆ.t / ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 0

˛ < 1;

(3.70)

243


where ˛

qC2 ; p

qC2 C2 BCj 1=pj


;

p p1

.qC2/=p ;

p D max pj : j D1;N

Thus, (3.69) and (3.70) imply the first two statements of the lemma. Now consider inequalities (3.26) and (3.67). Note that as m ! C1 1=.1˛1 /

‰m0 ! ‰0 A0

;

1 1 T2m ! T2 ˆ1˛ A0 ; 0

q=2 1

T1m ! T1 D .q=2/1 ˆ0

B

;

B BCqC2 2.qC2/=2 ; 1 where C1 is the constant of the best embedding V ,! W0 and B is the constant from the condition (F4). First, consider inequality (3.67). Without loss of generality, by the passage to a subsequence, we can consider the sequence T2m bounded away from zero T2m > 0 uniformly with respect to m since T2 > 0. By the convergence of the sequence T2m ! T2 as m ! C1, we can choose a monotonic converging subsequence (we denote it again by T2m ). We also denote the corresponding subsequences of the sequences ¹u0m º and ¹um º by the same symbols ¹u0m º and ¹um º. Let T2m " T2 ; then inequality (3.67) holds uniformly with respect to t 2 Œ0; T2m /, m m, for certain fixed m 2 N. Passing in the inequality (3.67) to the limit as m ! C1 for such t , we obtain ˆ

‰0 ŒT2 t 1=.˛1/

;

(3.71)

for t 2 Œ0; T2m /. Hence by the arbitrariness of m 2 N we directly obtain that (3.71) holds for all t 2 Œ0; T2 / and the blow-up time of solutions for problem (3.12) has an upper estimate T0 T2 . Now let T2m # T2 . Assume that T0 > T2 and, moreover, M

sup ˆ.t / < C1: t2Œ0;T2

Then (3.67) holds uniformly with respect to t 2 Œ0; T2 /. Passing in inequality (3.27) to the limit as m ! C1, by the assumption relative to T0 , we obtain the inequality ˆ

‰0 M < C1 for t 2 Œ0; T2 /: ŒT2 t 1=.˛1/

244


However, the form of the obtained inequality implies the fact that our assumption does not hold. Hence we directly obtain that T0 T2 . Similarly, inequality (3.26) implies ˆ.t /

Œ1

ˆ0 q q=2 2=q 2 ˆ0 Bt

8t 2 .0; T1 /;

T0 T1 :

The proof of Proposition 3.3.7 is complete. Theorem 3.3.2 is completely proved.

3.4


Definition 3.4.1. A solution of problem (3.1) of the class C .1/ .Œ0; TI V0 / satisfying the conditions h.A0 u/0 ; vi0 C

N X

h.Aj .u//0 ; vij C .Lu; v/1 D .F.u/; v/0 ;

j D1

u.0/ D u0 2 V0

8v 2 V0 ;

where the time derivative is meant in the classical sense, is called a strong generalized solution of problem (3.1). 0 .u/ W V ! L.V I V / of the operator We require that the Fréchet derivative Aj;u j j j Aj be strongly continuous with respect to u 2 Vj and V0 Vj , j D 1; N . In the considered class, the embeddings

A0 u 2 C .1/ .Œ0; TI V0 /; Aj .u/ 2 C .1/ .Œ0; TI Vj /; Lu 2 C.Œ0; TI W1 /;

F.u/ 2 C.Œ0; TI W0 /

hold and, therefore, A0 u and Aj .u/ belong to the class C .1/ .Œ0; TI V0 /; on the other hand, Lu and F.u/ belong to the class C.Œ0; TI V0 /. Thus, by the conditions (V) and the conditions of Theorem 3.4.2 below, the problem is equivalent to the problem h.A0 u/0 ; vi0 C

N X

h.Aj .u//0 ; vi0 C hLu; vi0 D hF.u/; vi0 ;

j D1

u.0/ D u0 2 V0

8v 2 V0 ;

where the time derivative is meant in the classical sense and h; i0 is the duality bracket between the Banach spaces V0 and V0 .

245


The following theorem holds. Theorem 3.4.2. Let the conditions (A), (A0 ), (F), and (L) hold. Assume that V0 Vj , j D 1; N , V0 W0 , and V0 W1 . Moreover, assume that either 2 < p < q C2 or p q C 2 and, in the latter case, Vj W0 , where j 2 1; N ;

p D max pj ; j 21;N

pj D p:

Let the Fréchet derivatives of the operators Aj W Vj ! Vj be strongly continuous and bounded, i.e., 0 2 BC.Vj I L.Vj ; Vj //; Aj;u and, moreover, let them be monotonic: 0 0 hAj;u .u/u1 Aj;u .u/u2 ; u1 u2 ij 0 8u; u1 ; u2 2 Vj ;

j D 1; N :

Then for any u0 2 V0 , there exists maximal T0 Tu0 > 0 such that the Cauchy problem (3.1) has a unique solution in the class u.t / 2 C .1/ .Œ0; T0 /I V0 /: The existence time T0 > 0 of the solution possesses the following properties: (1) if ˛ 2 .0; 1/, then T0 D C1 and ˆ.t / Œˆ1˛ C .1 ˛/At 1=.1˛/ I 0 (2) if ˛ D 1, then T0 D C1 and ˆ.t / ˆ0 exp¹At ºI (3) if ˛ > 1 and the conditions p q 2p ˆ0 ; .F.u0 /; u0 /0 > .Lu0 ; u0 /1 C qC2p qC2 .Lu0 ; u0 /1 ; .F.u0 /; u0 /0 > 2 hold, then there exists T0 2 ŒT1 ; T2 such that lim ˆ.t / D C1;

t"T0

246


where X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij ; 2 pj N

ˆ0 ˆ.0/;

j D1

2 q=2 1 1 1 B ; T2 ˆ1˛ A0 ; ˆ 0 q 0 .qC2/=p p qC2 A Cj M ; p1

T1

1 1 1=2 Œ.F.u0 /; u0 /0 .Lu0 ; u0 /1 2 .˛1 1/ˇˆ22˛ ; A0 Œ.˛1 1/2 ˆ2˛ 0 0

B D MCqC2 2.qC2/=2 ; 1 ˛1 D

qC2Cp ; 2p

D .˛1 1/ˇ;

ˇD

q2 : qC2p

Proof. Step 1. Local solvability Lemma 3.4.3. For any u0 2 V0 , there exists maximal T0 > 0 such that a unique solution of the class C .1/ .Œ0; TI V0 / of problem (3.1) exists for any T 2 .0; T0 /. Proof. Consider problem (3.1). Analyze the properties of the operator A.u/ A0 u C

N X

Aj .u/ W V0 ! V0 :

j D1

The operator A./ is radially continuous, monotonic, and coercive. Indeed, by the conditions (V2), the following relations hold: hA.u1 / A.u2 /; u1 u2 i0 D hA0 u1 A0 u2 ; u1 u2 i0 C

N X

hAj .u1 / Aj .u2 /; u1 u2 ij

j D1

for all u1 ; u2 2 V0 . Therefore, by the conditions (A0 1) and (A1), hA.u1 / A.u2 /; u1 u2 i0 0 for all u1 ; u2 2 V0 . On the other hand, by the conditions (A0 1) and (A3), we have hA.u/; ui0 D hA0 u; ui0 C

N X j D1

hAj .u/; uij kuk20 C

N X

p

kukj j

j D1

247


for any u 2 V0 . Hence we directly obtain that hA.u/; ui0 D C1: kuk0 kuk0 !C1 lim

For the operator A.u/ A0 u C

N X

Aj .u/ W V0 ! V0 ;

j D1

we have hA.u1 / A.u2 /; u1 u2 i0 D hA0 u1 A0 u2 ; u1 u2 i0 C

N X

hAj .u1 / Aj .u2 /; u1 u2 ij

j D1

hA0 u1 A0 u2 ; u1 u2 i0 D ku1 u2 k20 : Thus, for the operator A, a Lipschitz-continuous inverse operator A1 W V0 ! V0 exists with Lipschitz constant equal to 1. We perform the substitution v D A.u/ in (3.1); then the equation takes the following equivalent form: dv D F.A1 .v// LA1 .v/; dt

v.0/ D v0 ;

(3.72)

where v0 D A.u0 /. Problem (3.72) is equivalent to the following integral equation: Z t ds ŒF.A1 .v// LA1 .v/.s/: (3.73) v.t / D v0 C 0

We apply the method of contraction mappings to prove the local solvability of Eq. (3.73) in the class v 2 C .1/ .Œ0; TI V0 /. In the Banach space L1 .0; TI V0 /, introduce the closed, bounded, convex subset BR D ¹v 2 L1 .0; TI V0 / W kvkT Rº; where

kvkT D ess sup kvk0 : t2.0;T/

Prove that the operator Z H.u/

t 0

ds ŒF.A1 .v// LA1 .v/.s/

acts from BR into BR and is a contraction. It is easy to prove that the operator H.v/ acts from BR into BR for sufficiently large R > 0 and sufficiently small T > 0. We

248


need prove the contraction property of the operator H.v/ W BR ! BR . Indeed, the following inequalities hold: kH.v1 / H.v2 /kT T1 .R/kA1 .v1 / A1 .v2 /kT C TCkA1 .v1 / A1 .v2 /kT ;

(3.74)

where k kT ess sup k k0 ; t2.0;T/

R D max¹kA1 .v1 /kT ; kA1 .v2 /kT º: By the properties of the operator A1 , (3.74) implies the inequality kH.v1 / H.v2 /kT TC Œ1 C 1 .R/ kv1 v2 kT :

(3.75)

Under the condition T 1=2Œ1 C 1 .R/1 C1 , inequality (3.75) implies the contraction property of the operator H.v/ on the set BR . Therefore, there exists a unique solution of Eq. (3.73) of the class v 2 L1 .0; TI V0 /. Using the standard extensionin-time algorithm for solutions of integral equations, we conclude that there exist maximal T0 > 0 and a unique solution of Eq. (3.73) of the class v 2 L1 .0; T0 I V0 /, where either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim sup kvk0 D C1 t!T0

holds. Finally, from (3.73), by virtue of the smoothing properties of the operator H.v/, we obtain that v 2 C .1/ .Œ0; T0 /I V0 /. Note that A.u/ A0 u C

N X

Aj .u/ D v.t / 2 C .1/ .Œ0; T0 /I V0 /:

(3.76)

j D1

Equation (3.76) in the class u 2 C .1/ .Œ0; T0 /I V0 / is equivalent to u.t / D A1 .v/: The latter equation implies the fact that u.t / 2 C.Œ0; T0 /I V0 /. Indeed, ku.t / u.t0 /k0 kA1 .v/.t / A1 .v/.t0 /k0 kv.t / v.t0 /k0 ! C0 as t ! t0 . Therefore, there exists a unique solution of Eq. (3.76) of the class u.t / 2 C.Œ0; T0 /I V0 /.

249


Now we prove that, in fact, this solution belongs to the class u 2 C .1/ .Œ0; T0 /I V0 /. First, consider the Fréchet derivative of the operator A: A0u .u/ D A0 C

N X

0 Aj;u .u/:

(3.77)

j D1

By the conditions of the theorem and (3.77) we have hA0u .u/u1 A0u .u/u2 ; u1 u2 i ku1 u2 k20 for any u; u1 ; u2 2 V0 . Therefore, for any fixed u 2 V0 , by virtue of the Browder– Minty theorem, there exists a Lipschitz-continuous inverse operator ŒA0u .u/1 2 L.V0 I V0 /. Consider the equation A0 u C

N X

Aj .u/ D v.t / 2 C .1/ .Œ0; T0 /I V0 /:

(3.78)

j D1

For further consideration, we must prove that the operator O D I C B; O C

BO D

N X

0 A1 0 Aj;u .u/

j D1 0 .u/ is the Fréchet derivative on the fixed has a bounded inverse operator, where Aj;u element u 2 C.Œ0; T0 /I V0 / of the operator Aj . Indeed, consider the equation

O ŒI C Bw D f 2 V0 :

(3.79)

Prove that this equation has only a trivial solution. To this end, we apply the operator A0 to both sides of the equation and obtain A0u w D A0 w C

N X

Aj;u .u/w D A0 f

8u 2 V0 :

j D1

But we have proved above that for the operator A0u .u/, the Lipschitz-continuous inverse operator ŒA0u .u/1 is defined. Now we apply this operator to both sides of Eq. (3.79). Therefore, a solution of problem (3.79) exists for any f 2 V0 . It is easy to prove the uniqueness. Therefore, the inverse operator C 1 exists. In the class u 2 C .1/ .Œ0; T0 /I V0 /, Eq. (3.78) is equivalent to the following problem: N X 0 A0 C Aj;u .u/ u0 D v 0 2 C.Œ0; T0 /I V0 /; u D A1 v: j D1

250


We apply the operator A1 0 to the latter equation and obtain O 0 D A1 v 0 : ŒI C Bu 0 Introduce the operator

(3.80)

O D I C BO W V0 ! V0 : C

For this operator, an inverse linear bounded operator is defined. Therefore, (3.80) is equivalent to the equality 0 O 1 A1 u0 D C 0 v :

(3.81)

O 1 A1 v 0 2 C.Œ0; T0 /I V0 / for fixed u 2 We have only to prove that u0 D C 0 C.Œ0; T0 /I V0 /. Indeed, by virtue of (3.81), the following inequalities hold: 0 0 O 1 .t0 /A1 ku0 .t / u0 .t0 /k0 kC 0 Œv .t / v .t0 /k0 0 O 1 .t0 / C O 1 .t //A1 C k.C 0 v .t0 /k0

O 1 .t0 / C O 1 .t /kV !V : Ckv 0 .t / v 0 .t0 /k0 C CkC 0 0

(3.82)

O is continuous and, Note that the linear (under fixed u 2 C.Œ0; T0 /I V0 /) operator C O 1 is linear, contherefore, by the inverse-mapping Banach theorem, the operator C tinuous, and bounded by virtue of the linearity. Thus, we can use the spectral repreO 1 W V0 ! V0 . sentation for the linear bounded operator C O First, introduce the resolvent of the operator C: O D .I C/ O 1 : R.; C/ Let be a circle jj D r with sufficiently large radius, greater than sup t2Œt0 ";t0 C"

O V !V : kCk 0 0

The introduced variable is well defined since, under mentioned t 2 Œt0 "; t0 C " Œ0; T0 /, the inequality kuk0 < C1 sup t2Œt0 ";t0 C"

O 1 .t / and holds. Now we can use the spectral representation for the operators C 1 O C .t0 / with the same contour introduced above: Z Z 1 1 1 1 1 O O O 0 //: O d R.; C.t //; C .t0 / D d 1 R.; C.t C .t / D 2 i 2 i Obviously, we have O 1 .t / C O 1 .t0 / D 1 C 2 i

Z

O // R.; C.t O 0 //: d 1 ŒR.; C.t

251


Now we use the well-known representation for the resolvents of operators: O // R.; C.t O 0 // D R.; C.t O 0 // R.; C.t

C1 X

O / C.t O 0 //R.; C.t O 0 /n Œ.C.t

nD1

under the condition O /kV !V kR.; C.t O 0 //kV !V ı < 1: O 0 / C.t kC.t 0 0 0 0 The following inequalities hold: O // R.; C.t O 0 //kV !V kR.; C.t 0 0 O 0 //kV !V kR.; C.t 0 0

C1 X

n O 0 //kn O O kR.; C.t V0 !V0 kC.t // C.t0 //kV0 !V0 :

nD1

Note that O / C.t O 0 / D A1 C.t 0

N X

ŒAj;u .u.t // Aj;u .u.t0 //:

j D1

By the continuity of the Fréchet derivatives Aj;u with respect to u 2 V0 and by the fact that u 2 C.Œ0; T0 /I V0 / we have O / C.t O 0 /kV !V kC.t 0 0

N X j D1

kAj;u .u.t // Aj;u .u.t0 //kV0 !V0 ! C0;

O // R.; C.t O 0 //kV !V ! 0; kR.; C.t 0 0 O 0 /1 kV !V ! C0 O /1 C.t kC.t 0 0 as t ! t0 . Therefore, by (3.82) we have u 2 C .1/ .Œ0; T0 /I V0 /. Note that under the condition (A7), the operator C.u/ is boundedly Lipschitzcontinuous and, therefore, the operator C 1 .u/ is also boundedly Lipschitz-continuous (see Appendix A.17) and hence Eq. (3.81) has a local solution in the class u.t / 2 C .1/ .Œ0; T00 /I V0 /. It is easy to prove that T0 D T00 . Step 2. A priori estimates and blow-up Lemma 3.4.4. Let X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij ; 2 pj N

j D1

˛

qC2 : p

252


Then (1) for ˛ 2 .0; 1/, the inequality C .1 ˛/At 1=.1˛/ ˆ.t / Œˆ1˛ 0 holds; (2) for ˛ D 1, the inequality ˆ.t / ˆ0 exp.At /I holds; (3) for ˛ > 1, under the conditions p q 2p ˆ0 ; .F.u0 /; u0 /0 > .Lu0 ; u0 /1 C qC2p qC2 .F.u0 /; u0 /0 > .Lu0 ; u0 /1 ; 2 there exists T0 2 ŒT1 ; T2 such that lim ˆ.t / D C1;

t"T0

where 2 q=2 1 1 1 B ; T2 ˆ1˛ A0 ; ˆ 0 q 0 .qC2/=p p qC2 ; A Cj M p1

T1

1 1 1=2 A0 Œ.˛1 1/2 ˆ2˛ Œ.F .u0 /; u0 /0 .Lu0 ; u0 /1 2 .˛1 1/ˇˆ22˛ ; 0 0

˛1 D

qC2Cp ; 2p

ˇD

q2 : qC2p

Proof. Let u 2 C .1/ .Œ0; T0 /I V0 / be a solution of problem (3.1). Multiplying both sides of Eq. (3.1) by u.t / or by u0t .t / and integrating by parts, we obtain the following equalities: N X d 1 pj 1 hA0 u; ui0 C hAj .u/; uij D .F.u/; u/0 .Lu; u/1 ; dt 2 pj

(3.83)

j D1

hA0 u0 ; u0 i0 C

N X 0 h Aj .u/ ; u0 ij D j D1

1 d 1d .F.u/; u/0 .Lu; u/1 : (3.84) q C 2 dt 2 dt

253


Integrating (3.83) over t 2 .0; T/, we obtain Z t ˆ.t / D ˆ0 C ds Œ.F.u/; u/0 .Lu; u/1 ;

(3.85)

0

where

X pj 1 1 hAj .u/; uij ; ˆ.t / hA0 u; ui0 C 2 pj N

ˆ0 D ˆ.0/:

j D1

Since, according to the assumption V0 W0 , there exists a constant C1 > 0 such that jvj0 C1 hA0 v; vi1=2 0 : From (3.85) we obtain the inequality Z t Z t qC2 ds hA0 v; vi.qC2/=2 ˆ C B ds ˆ.qC2/=2 .s/; ˆ.t / ˆ0 C MC1 0 0 0

0

B

MCqC2 2.qC2/=2 : 1

According to the Bihari theorem (see [112]) we have ˆ.t /

ˆ0

8t 2 Œ0; T1 /;

2=q Œ1 q2 ˆq=2 0 Bt

T1

2 q=2 1 B : ˆ q 0

(3.86)

Therefore, in the case where p < q C 2 from (3.86) we obtain T0 T1 . Now let p q C 2. We use the fact that there exists j 2 1; N (pj D p, pj q C 2) such that Vj W0 . Let Cj be the constant of embedding Vj W0 : 1=p

jvj0 Cj hAj .v/; vij j ; Z t .qC2/=pj ˆ.t / ˆ0 C CjqC2 ds hAj .v/; vij M 0

ˆ0 C CjqC2 M

p p1

.qC2/=p Z

t

ds ˆ.qC2/=p .s/:

0

We separately consider the cases ˛ D 1 and ˛ 2 .0; 1/, where ˛

qC2 : p

In the first case, according to the Gronwall–Bellman theorem (see [112]), we have ˆ.t / ˆ0 exp¹At º;

qC2

A Cj M

p p1

.qC2/=p :

(3.87)

254


In the second case where ˛ 2 .0; 1/, by the Bihari theorem we obtain ˆ.t / Œˆ1˛ C .1 ˛/At 1=.1˛/ : 0

(3.88)

From (3.87) and (3.88) we obtain that T0 D C1 in the case ˛ 2 .0; 1. Now from relations (3.83) in the same way as in Step 5 we derive the following second-order ordinary differential inequality with ˆ0 .t / 0 (see [210]): 2 ˆ00 ˆ ˛1 ˆ0 C ˇˆ2 0;

˛1 D

qC2Cp ; 2p

ˇD

q2 : qC2p

(3.89)

Integrating the differential inequality (3.89) and taking into account (3.83), (3.84), and the inequalities p q 2p qC2 .Lu0 ; u0 /1 ; ˆ0 ; .F.u0 /; u0 /0 > .F .u0 /; u0 /0 > .Lu0 ; u0 /1 C qC2p p we have

A01=.1˛1 / ˆ ŒT2 t 1=.˛1 1/ Lemma 3.4.4 is proved.

8t 2 Œ0; T2 /;

1 1 T2 ˆ1˛ A0 : 0

Theorem 3.4.2 is completely proved.

3.5


Definition 3.5.1. A solution of problem problem (3.2) satisfying the conditions Z

T

0

N d X .t / hAj .u/; wij .F.u/; w/0 C hDP .u/; wi0 dt D 0 dt

j D0

8w 2 V ;

8 2 L2 .0; T/;

u.0/ D u0 2 V is called a weak generalized solution of the abstract Cauchy problem (3.2). We search for a solution of problem (3.90) in the class u.t / 2 L1 .0; TI V /;

u0 .t / 2 L2 .0; TI V0 /;

DP .u/ 2 L1 .0; TI W4 /; F.u/ 2 L1 .0; TI W0 /; Aj .u/ 2 L1 .0; TI Vj /;

N d X d A.u/ D Aj .u/ 2 L2 .0; TI V /; dt dt j D0

(3.90)


255


N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /;

j D0

i.e., A.u/ is strongly absolutely continuous and weakly differentiable on the segment Œ0; T. Therefore, by the properties (A), (A0 ), and (F) and by virtue of the assumptions of Theorem 3.5.2 below we obtain that A.u/ D

N X

Aj .u/ 2 H1 .0; TI V /;

F.u/ 2 L2 .0; TI V /;

j D0

A.u/ D

N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /:

j D0

Thus, by the conditions (V) and the assumptions of Theorem 3.5.2 below, problem (3.1) is equivalent to Z

T 0

N d X .t / hAj .u/; wi C hDP .u/; wi hF.u/; wi dt D 0 dt j D0

8w 2 V ;

8 .t / 2 L2 .0; T/;

u.0/ D u0 2 V ; where h; i is the duality bracket between the Banach spaces V and V . Using results of Appendix A.12, we conclude that problem (3.90) is equivalent to Z

T

dt 0

d A.u/; v C hDP .u/; vi hF.u/; vi D 0 dt 8v 2 L2 .0; TI V /; u.0/ D u0 2 V :

The following theorem holds. Theorem 3.5.2. Let the conditions (A), (A0 ), (F), and (DP) hold. Assume that V0 ,! W0 , i.e., the embedding operator is a compact operator. Moreover, require that W0 W2 . Assume that either 2 < p < qC2 or p qC2, and in the latter case, Vj W0 , where p D max pj ; j 2 1; N ; pj D p: j 21;N

Let .F .v/; v/0 c > 0 for all v 2 W0 , jvj0 D 1. Then for any u0 2 V , there exists maximal T0 Tu0 > 0 such that the Cauchy problem (3.2) has a unique solution in

256


the class u.t / 2 L1 .0; TI V /;

du.t / 2 L2 .0; TI V0 / dt

8T 2 .0; T0 /;

.1/ .Œ0; TI V /: A.u/ 2 H1 .0; TI V / \ Cw

Here for the function X pj 1 1 hAj .u/; uij ; hA0 u; ui0 C 2 pj N

ˆ.t /

j D1

which is positive definite by virtue of the conditions (A0 2) and (A3) and has the sense of kinetic energy, the following estimates depending on possible values of the variable ˛

qC2 ; p

pj D p max pj ; j D1;N

hold: (1) if ˛ D 1, then T0 D C1 and ˆ.t / ˆ0 exp¹C2 t ºI (2) if ˛ 2 .0; 1/, then T0 D C1 and t 1=.1˛/ I ˆ.t / ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 0 (3) if ˛ > 1 and the condition X pj 1 1 .q C 2 p/2 hAj .u0 /; u0 ij < .F.u0 /; u0 /0 hA0 u0 ; u0 i0 C 2 pj p .q C 2/2 N

j D1

holds, then there exists T0 2 ŒT1 ; T2 such that lim ˆ.t / D C1;

t"T0

where T1 D

2 q=2 1 B ; ˆ q 0

1 ˆ0 T2 D ln 1 ; ˛1 1 ˆ1 1=pj


.q C 2/2 2.q C 2 p/

˛1

pCqC2 ; 2p

qC2 C2 BCj 2˛ ;

;

ˆ1 D

.F.u0 /; u0 /0 ; ˆ˛0 1

qC2 in the case q C 2 > p max pj ; B BC1 2.qC2/=2 ; j D1;N

C1 is the constant of the best embedding V ,! W0 , and B is the constant from the condition (F4).

257


Remark 3.5.3. Since u.t / 2 L1 .0; TI V /;

u0 .t / 2 L2 .0; TI V0 /;

we obtain that, after a possible change on a set of zero Lebesgue measure, the mapping u.t / W Œ0; T ! V0 becomes strongly continuous. Therefore, the initial condition u.0/ D u0 makes sense. Proof. Step 1. Galerkin approximations. By the separability of V , there exists a countable, everywhere dense in V , linearly independent system of functions ¹wi ºm i D1 . We prove the solvability of problem (3.90) by the Galerkin method and the monotonicity and compactness methods [275]. First, consider the following finite-dimensional approximation of problem (3.90): Z

T

dt .t / 0

N d d X hAj .um /; wk ij hA0 um ; wk i0 C dt dt j D1

C hDP .um /; wk i0 .F.um /; wk /0 D 0;

(3.91)

k D 1; m 8 .t / 2 L2 .0; T/; um D

m X

cmi .t /wi ;

um0 D

i D1

m X

cmi .0/wi ;

cmi .0/ D ˛mi ;

i D1

um0 ! u0

strongly in V :

As in the proof of Step 1 in Theorem 2.3.2 of Chapter 2, we can prove that in the class cmk .t / 2 C .1/ Œ0; Tm0 / for Tm0 > 0 from (3.91) we obtain the pointwise equality N d d X hA0 um ; wk i0 C hAj .um /; wk ij C hDP .um /; wk i0 D .F.um /; wk /0 : dt dt j D1 (3.92)

The proof of the local solvability of system of ordinary differential equations (3.92) is exactly the same as the corresponding proof of Step 1 of Theorem 2.3.2. Therefore, the system of ordinary differential equations (3.92) is a system of Cauchy–Kovalevskaya type and satisfies the conditions that guarantee the solvability on a certain segment Œ0; Tm , Tm > 0, in the class cmk .t / 2 C 1 .Œ0; Tm /, k D 1; m (see, e.g., [319]). Step 2. A priori estimates Lemma 3.5.4. There exists T > 0 independent of m 2 N such that for the sequence of Galerkin approximations ¹um º, the following properties hold uniformly with respect

258


to m 2 N: um


u0m


A 0 um


Aj .um /

is bounded in L1 .0; TI Vj /I

F.um /


DP .um /

is bounded in L1 .0; TI V0 /:

Proof. Multiplying both sides of Eq. (3.92) by cmk .t /, and summing over k D 1; m, and taking into account the condition (DP4), we obtain hA0 u0m ; um i0

C

N X

0 hAj;u .um /u0m ; um ij D .F.um /; um /0 : m

(3.93)

j D1

On the other hand, by the fact that um D

N X


kD1

and the conditions (A2) and (A4), the following equalities hold: d hAj .um /; um ij D pj hAj .um /; u0mt ij ; dt h.Aj .um //0t ; um ij C hAj .um /; u0mt ij D pj hAj .um /; u0mt ij ; h.Aj .um //0t ; um ij D .pj 1/hAj .um /; u0mt ij D


Prove the equalities d hAj .um /; um ij D pj hAj .um /; u0mt ij : dt Indeed, we have

Z

Jmj .t / d Jmj .t / D dt

Z Z

D Z D

1 0 1 0 1 0


ds ŒshA0sum .sum /u0m ; um ij C hAj .sum /; u0m ij d ds s hAj .sum /; u0m ij C hAj .sum /; u0m ij ds

1

ds 0



259


Here we have used the fact that the Fréchet derivatives of the operators Aj are symmetric by virtue of the conditions (A). Hence from (3.93) we obtain N X d 1 pj 1 hA0 um ; um i0 C hAj .um /; um ij D .F.um /; um /0 : dt 2 pj

(3.94)

j D1

From (3.94), integrating both sides over t 2 .0; Tm /, we obtain X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj N

(3.95)

j D1

X pj 1 1 hA0 um0 ; um0 i0 C hAj .um0 /; um0 ij C 2 pj N

D

j D1

Z

t

0

ds .F.um /; um /0 :

By the condition (A0 3), we can choose in the Banach space V0 a norm equivalent to the initial norm: 1=2 kvk0 D hA0 v; vi0 :

(3.96)

On the other hand, qC2

V0 ,! W0 ;


:

From (3.95) owing to (3.96) we obtain X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj N

j D1

X pj 1 1 hA0 um0 ; um0 i0 C hAj .um0 /; um0 ij C BC1 qC2 2.qC2/=2 2 pj N

j D1

Z

t 0

X pj 1 1 ds hAj .um /; um ij hA0 um ; um i0 C 2 pj N

.qC2/=2 : (3.97)

j D1

0 , summing Now we obtain the second a priori estimate. Multiplying (3.92) by cmk over k D 1; m, and integrating both sides over t 2 .0; Tm /, we obtain

Z

t 0

N X 0 0 0 0 ds hA0 um ; um i0 C h.Aj .um // ; um ij C j D1

D

1 .F.um /; um /0 qC2

1 .F.um0 /; um0 /0 qC2

Z 0

t

ds hDP .um /; u0m i0 : (3.98)

260


From (3.98), Proposition 3.2.1 for C D 1, and the Cauchy inequality we obtain " 1 jhDP .um /; u0m i0 j hA0 u0m ; u0m i0 C .F.u/; u/0 ; 2 2" Z t N X 1 " 0 0 0 0 1 ds hA0 um ; um i0 C h.Aj .um // ; um ij C .F.um0 /; um0 /0 2 0 qC2 j D1 Z 1 1 t .F.um /; um /0 C ds .F.um /; um /0 8" 2 .0; 2/: (3.99) qC2 2" 0 Next, by virtue of (3.97)–(3.99), in the exactly same way as in Lemma 2.3.4, we obtain um


u0m


A 0 um


Aj .um /


F.um /


(3.100)

DP .um / is bounded in L1 .0; TI V0 /; where the inclusions hold for any T > 0 in the case ˛j 2 .0; 1, and for any T 2 .0; T1 /, where T1 is defined in (3.27), in the case ˛j > 1. Lemma 3.5.4 is proved. Therefore, by (3.100), there exists a subsequence of the sequence ¹um º such that um * u


u0m * u0 weakly in L2 .0; TI V0 /:

(3.101)


in D 0 .0; TI V /I

therefore, by the weak convergence u0m *

in L2 .0; TI V0 /

we conclude that .t / D u0 .t / for almost all t 2 .0; T/. Now we apply Lemma A.15.1 (see Appendix A.15), where we set W D W0 : Then we obtain that um .s/ ! u.s/


261


On the other hand, by the condition (F1), we have jF.um / F.u/j0 .R/jum uj0 ! 0

as m ! C1;

(3.102)

where R D max¹juj0 ; jum j0 º. From (3.102) we obtain the strong convergence F.um /.t / ! F.u/.t /

in W0

(3.103)

for almost all t 2 .0; T/. Note that, since V0 ,! W2 , by Lemma A.15.1 (see Appendix A.15), where we set W D W2 , we obtain um .s/ ! u.s/


Hence from the condition (DP3) we obtain the strong convergence DP .um / ! DP .u/

in V0 for almost all t 2 .0; T/:

Step 3. Monotonicity method. Now we apply the monotonicity method. Recall that in Step 2 we have obtained limit inclusions (3.100) and, moreover, A.um / *

-weakly in

L1 .0; TI V /;

A.u/

N X

Aj .u/:

(3.104)

j D0

Lemma 3.5.5. Let A.u/ sequence ¹um º such that

PN

j D0 Aj .u/.

Then there exists a subsequence of the

A.um / * A.u/ -weakly in L1 .0; TI V / for certain T > 0. Moreover, for a certain subsequence of the sequence ¹um º we have um ! u

strongly in Lpj .0; TI Vj /;

um ! u

strongly in L2 .0; TI V0 /:

Proof. Rewrite Eq. (3.92) in the equivalent form Z t ds Œ.F.um /; wj /0 hDP .um /; wj i0 ; hA.um /; wj i D hA.um0 /; wj i C 0

(3.105) where h; i is the duality bracket between the Banach spaces V and V . For fixed j 2 N, we pass in Eq. (3.105) to the limit as m ! C1 and obtain Z t ds ŒF.u/ DP .u/ C A.u0 /: (3.106) D 0

262


Let

Z 0 X

0

Z D Z

T 0

T

T

0


dt hA.u /; u i D

T 0

Z

0 lim sup X

Z

Z

T

dt 0

0

Z

T

dt

!C1

0

Z D

T 0

T 0

0

Z

C

0

Z dt hA.u /; vi

T

dt hA.v/; u vi;

0

dt hA.u 0 /; u i

C Hence obtain that

T

t

t

ds Œ.F.u /.s/; u .t //0 hDP .u /.s/; u .t /i0 :

ds Œ.F.u/.s/; u.t //0 hDP .u/.s/; u.t /i0 Z

dt hA.u0 /; ui

T 0

Z dt h ; vi

0

T

dt hA.v/; u vi

dt h A.v/; u vi:

(3.107)

Now we set v D u w for any v; w 2 Lr .0; TI V /, > 0, u 2 L1 .0; TI V / 0 Lr .0; TI V /, ; A.v/ 2 Lr .0; TI V /, r > 1, and r 0 D r=.r 1/. By (3.107), the following inequality holds: Z T dt h A.u w/; wi 0; 0

from which, by the semicontinuity of the operators Aj .v/, j D 0; N , we obtain D A.u/: On the other hand, by (3.106), in the same way as in the proof of the second part of Lemma 3.3.5 we can prove that there exists a subsequence of the sequence ¹um º such that um ! u strongly converges in each Lpj .0; TI Vj /, j D 0; N . Step 4. Passage to the limit and the uniqueness. The limit expressions (3.101), (3.103), and (3.104) imply the possibility of passing to the limit as m ! C1 in Eq. (3.105). As a result of this passage to the limit, we obtain the following equality for almost all t 2 .0; T/: hA0 u; vi0 C

N X


j D1

hAj .u0 /; vij

j D1

Z C

N X

t 0

ds Œ.F.u/; v/0 .s/ hDP .u/; vi0 .s/

263


for all v 2 V , u.0/ D u0 2 V . On the other hand, u0 2 L2 .0; TI V0 / and u 2 L1 .0; TI V /. Rewrite the latter equality in the following form: N X

hAj .u/; vij D

j D0

N X

Z hAj .u0 /; vij C

j D0

t

0

ds Œ.F.u/; v/0 .s/ hDP .u/; vi0 .s/ (3.108)

for all v 2 V and almost all t 2 .0; T/. Therefore, from (3.108) we obtain that for almost all t 2 .0; T/, the following equality holds in the sense V : A.u/ D

N X

Aj .u/ D

j D0

N X

Z Aj .u0 / C

j D0

t 0

ds ŒF.u/.s/ DP .u/.s/ :

(3.109)

Earlier we have proved that A.u/ D

N X

Aj .u/ 2 L1 .0; TI V / L2 .0; TI V /:

j D0

Moreover, (3.109) implies N d X Aj .u/ 2 L2 .0; TI V /: dt j D0

Therefore, A.u/ 2 H1 .0; TI V /. Finally, in the same way as in Step 4 of the proof of Theorem 2.3.2, we can prove that N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /:

j D0

Now we consider Eq. (3.92) rewritten in the form Z

T 0

N

d X dt .t / Aj .um /; w hF.um /; wi C hDP .um /; wi D 0 dt

j D0

for all .t / 2 L2 .0; T/ and all w 2 V . By Lemma 3.5.4 we see that F.um / * F .u/ and DP .um / ! DP .u/ weakly in L2 .0; TI V /. Moreover, A.um / * A.u/ weakly in L2 .0; TI V /: Therefore, d d A.um / ! A.u/ dt dt

264


in the sense of D 0 .0; TI V /. Moreover, the following pointwise equality holds: d A.um / D F.um / DP .um /: dt Therefore, d A.um / * dt


Thus, d d A.um / * A.u/ weakly in L2 .0; TI V /: dt dt Passing in Eq. (3.92) to the limit as m ! C1 we obtain that u.t / is a solution of the problem Z

T

dt .t / 0

N

d X Aj .u/; w hF.u/; wi C hDP .u/; wi D 0 dt j D0

for all .t / 2 L2 .0; T/ and all w 2 V , which is equivalent to problem (3.90). We can prove the uniqueness of problem (3.90) in the same way that the uniqueness of problem (3.2). Step 5: Blow-up of solutions Lemma 3.5.6. Let X pj 1 1 hAj .u/; uij ; ˆ hA0 u; ui0 C 2 pj N

˛

j D1

.F.v/; v/0 c > 0;

8v 2 W0 ;

qC2 ; p

p max pj ; j D1;N

jvj0 D 1:

Then (1) for ˛ 2 .0; 1/, the inequality t 1=.1˛/ ˆ ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 0 holds; (2) for ˛ D 1, the inequality

ˆ ˆ0 exp.C2 t /

holds; (3) for ˛ > 1, under the condition X pj 1 1 .q C 2 p/2 .F.u0 /; u0 /0 ; hA0 u0 ; u0 i0 C hAj .u0 /; u0 ij < 2 pj p .q C 2/2 N

j D1

265


there exists T0 2 ŒT1 ; T2 such that lim ˆ.t / D C1;

t"T0

where 2 q=2 1 1 ˆ0 T1 D ˆ0 B ; T2 D ln 1 ; q ˛1 1 ˆ1 ˛ p 1=p qC2 ; jvj0 Cj hAj v; vij j ; C2 BCj p1 .F.u0 /; u0 /0 pCqC2 ˆ0 ˆ.0/; ˛1 ; ; ˆ1 D 2p ˆ˛0 1

.q C 2/2 2.q C 2 p/

qC2 in the case q C 2 > p max pj ; B BC1 2.qC2/=2 : j D1;N

Proof. By the condition (A2), the Schwarz inequality (see [293]) for the Fréchet 0 derivatives Aj;u W Vj ! L.Vj I Vj / of the operators Aj W Vj ! Vj holds: 0 jh.Aj .um //0 ; um ij j D jhAj;u .um /u0m ; um ij j m 0 .um /u0m ; u0m ij hAj;u m

1=2

0 hAj;u .um /um ; um ij1=2 m

D h.Aj .um //0 ; u0m ij1=2 .pj 1/1=2 hAj .um /; um ij1=2 ; (3.110) 1=2 jhA0 u0m ; um i0 j hA0 u0m ; u0m i1=2 0 hA0 um ; um i0 :

(3.111)

Here we have used the following equality: 0 .v/v D .pj 1/Aj .v/: Aj;v


0 0 Aj;v .v/ D pj 1 Aj;v .v/;

0 0 .v/ D pj 2 Aj;v .v/; j D 0; N ; Aj;v

Z

Z 1 1 d 0 d d Aj;v .v/v; w 8w 2 Vj ; Aj .v/; w D d 0 0 j j 0 hAj .v/ .pj 1/1 Aj;v .v/v; wij D 0

8w 2 Vj ;

j D 0; N :

Therefore, the following operator equality holds: 0 .v/v .pj 1/Aj .v/ D Aj;v

8v 2 Vj ;

j D 0; N ;

j D 0; N :

266



(3.112)

j D1

From (3.110)–(3.112) we have ˇ2 ˇ ˇ2 ˇ N X ˇ ˇ ˇd ˇ 0 ˇ ˇ ˆm ˇ ˇjhA0 u0 ; um i0 j C jh.A .u // ; u i j j m m j ˇ m ˇ ˇ dt ˇ j D1

N X 0 0 0 0 hA0 um ; um i0 C h.Aj .um // ; um ij j D1

N X .pj 1/hAj .um /; um ij hA0 um ; um i0 C j D1



j D1


N X pj 1 1 hA0 um ; um i0 C hAj .um /; um ij ; 2 pj

(3.113)

j D1

where p D maxj D1;N pj > 2. By relations (3.94) and (3.98) we have N X pj 1 d 1 hAj .um /; um ij D .F.um /; um /0 ; hA0 um ; um i0 C dt 2 pj

(3.114)

j D1

hA0 u0m ; u0m i0 C

N X


j D1

D hDP .um /; u0m i0 C The following statement is valid.

1 d .F.um /; um /0 : q C 2 dt

(3.115)

267


Proposition 3.5.7 (see [210, Lemma 1.1]). Assume that ˆ.t / 2 C .2/ Œ0; T0 / for certain maximal T0 > 0 in the sense that either T0 D C1 or T0 < C1 and then lim ˆ.t / D C1;

(3.116)

t"T0

where ˆ.t / > 0 and ˆ0 .t / > 0 for t 2 Œ0; T0 /. Moreover, let the following secondorder ordinary differential inequality holds for t 2 Œ0; T0 /: ˆ00 ˆ ˛1 .ˆ0 /2 C ˆˆ0 0;

t 2 Œ0; T0 /;

˛1 > 1;

> 0;

(3.117)

and the following inequality holds: ˆ0 < Then T0 T2 ,

˛1 1 ˆ1 ;

ˆ0 ˆ.0/;

ˆ1 ˆ0 .0/:

1 ˆ0 ; T2 ln 1 ˛1 1 ˆ1

(3.118)

(3.119)

and the limit relation (3.116) is valid. Proof. Divide both sides of inequality (3.117) by the function ˆ1C˛ ; after some transformations we obtain 0 0 ˆ0 ˆ C 0; ˛1 > 1; > 0: ˆ˛1 ˆ˛1 Introduce the notation

ˆ0 I ˆ˛1 then for the introduced function, the following inequalities hold: ƒ.t /

ƒ0 .t / C ƒ.t / 0;

ƒ.t / ƒ0 exp. t /;

ˆ0 ƒ0 exp. t /; ˆ˛1

1 .ˆ1˛1 /0 ƒ0 exp. t /; 1 ˛1 ˛1 1 1 ƒ0 Œ1 exp. t /; ˆ1˛1 ˆ1˛ 0 1 ˆ 1˛ : 1 ˛ Œˆ0 1 ƒ0 1 Œ1 exp. t /1=.˛1 1/

(3.120)

Now we require that inequality (3.118) be valid; then (3.120) cannot be valid for all t 2 R1C . There exists T0 T2 , where T2 is defined by formula (3.119), such that the limit relation (3.116) holds. Proposition 3.5.7 is proved.

268


Note that, by Corollaries 3.2.1 and 3.3.7 and the Cauchy and Schwarz inequalities (see [293]), the following inequalities hold: hA0 u0m ; u0m i0 C

N X


j D1

jhDP .um /; u0m i0 j C

1 d .F.um /; um /0 q C 2 dt

1 1=2 1=2 ˆ00m C hA0 u0m ; u0m i0 .F.um /; um /0 qC2 1 " 1 ˆ00m C hA0 u0m ; u0m i0 C .F.um /; um /0 qC2 2 2" N X 1 1 0 " 00 0 0 0 0 h.Aj .um // ; um ij : ˆ C ˆ C hA0 um ; um i0 C q C 2 m 2" m 2

j D1

Hence we conclude that N X 1 1 " hA0 u0m ; u0m i0 C h.Aj .um //0 ; u0m ij ˆ00m C ˆ0m (3.121) 1 2 qC2 2" j D1

for all " 2 .0; 2/. From (3.113)–(3.115) and (3.121) we obtain the second-order ordinary differential inequality (see [210]) ˆ00m ˆm ˛1 ."/.ˆ0m /2 C ."/ˆ0m ˆm 0; qC2 " qC2 1 ; ."/ D ; p D max pj > 2; ˛1 ."/ p 2 2" j D1;N

(3.122)

from which, by Proposition 3.5.7 we obtain that, under the condition ˆm0
T02 and, moreover, M

sup

Z.t/2Œ0;T02

ˆ.t / < C1:

Then (3.131) holds uniformly with respect to Z.t / 2 Œ0; T02 /. Passing in inequality (3.131) to the limit as m ! C1, we obtain by the assumption relative to T0 the following inequality: ˆ

ŒT02

‰0 M < C1 for Z.t / 2 Œ0; T02 /: Z.t /1=.˛1 1/

However, the form of the obtained inequality implies the fact that our assumption is not valid. Hence we directly obtain that Z.T0 / T02 . Similarly, from (3.26) we obtain that ˆ0 8t 2 .0; T1 /; T0 T1 : ˆ.t / q q=2 Œ1 2 ˆ0 Bt 2=q Lemma 3.5.6 is proved. Theorem 3.5.2 is completely proved.


3.6

273


Definition 3.6.1. A solution of problem (3.2) of the class C .1/ .Œ0; TI V0 / satisfying the conditions N X 0 h Aj .u/ ; vij C hDP .u/; vi0 D .F.u/; v/0 ; h.A0 u/ ; vi0 C 0

j D1

u.0/ D u0 2 V0

8v 2 V0 ;

where the time derivative is meant in the classical sense, is called a strong generalized solution of problem (3.2). 0 .u/ W V ! L.V I V / of the operator We require that the Fréchet derivative Aj;u j j j Aj be strongly continuous with respect to u 2 Vj and, moreover, the embedding V0 Vj , j D 1; N , holds. Hence we obtain that in the considered class the following embeddings hold:

A0 u 2 C .1/ .Œ0; TI V0 /; Aj .u/ 2 C .1/ .Œ0; TI Vj /; DP .u/ 2 C.Œ0; TI W4 /; F.u/ 2 C.Œ0; TI W0 / and A0 u and Aj .u/ belong to the class C .1/ .Œ0; TI V0 /, while DP .u/ and F.u/ belong to the class C.Œ0; TI V0 /. Thus, by the conditions (V) and the conditions of Theorem 3.6.2 below, the problem is equivalent to the problem h.A0 u/0 ; vi0 C

N X

h.Aj .u//0 ; vi0 C hDP .u/; vi0 D hF.u/; vi0 ;

j D1

u.0/ D u0 2 V0

8v 2 V0 ;

where the time derivative is meant in the classical sense and h; i0 is the duality bracket between the Banach spaces V0 and V0 . The following theorem holds. Theorem 3.6.2. Let the conditions (A), (A0 ), (F), and (DP) hold. Assume that V0 Vj , j D 1; N , and V0 W0 W2 . Moreover, assume that either 2 < p < q C 2 or p q C 2, and in the latter case, Vj W0 , where p D max pj ; j 21;N

j 2 1; N ;

pj D p:

274


Let the Fréchet derivatives of the operators Aj W Vj ! Vj satisfy the condition 0 ./ 2 BC.Vj I L.Vj ; Vj //; Aj;u

i.e., they are continuous and bounded and, moreover, let them be monotonic in the sense that 0 0 hAj;u .u/u1 Aj;u .u/u2 ; u1 u2 ij 0

8u; u1 ; u2 2 Vj ;

j D 1; N :

Moreover, let .F.v/; v/0 c > 0 for all v 2 W0 ; jvj0 D 1. Then for any u0 2 V0 , there exists maximal T0 Tu0 > 0 such that the Cauchy problem (3.2) has a unique solution in the class u.t / 2 C .1/ .Œ0; T0 /I V0 /: and the following assertions hold: (1) if ˛ 2 .0; 1/, then T0 D C1 and C .1 ˛/At 1=.1˛/ I ˆ.t / Œˆ1˛ 0 (2) if ˛ D 1, then T0 D C1 and ˆ.t / ˆ0 exp¹At ºI (3) if ˛ > 1 and the condition X pj 1 .q C 2 p/2 1 hAj .u0 /; u0 ij < .F.u0 /; u0 /0 hA0 u0 ; u0 i0 C 2 pj p.q C 2/2 N

j D1

holds, then there exists T0 2 ŒT1 ; T2 such that lim ˆ.t / D C1;

t"T0

where

.qC2/=p N X 1 pj 1 p qC2 ˆ.t / hA0 u; ui0 C hAj .u/; uij ; A Cj M ; 2 pj p1 j D1 2 q=2 1 1 ˆ0 T1 ˆ0 B ; T2 D ln 1 ; q ˛ 1 1 ˆ1 1=pj


˛1

pCqC2 ; 2p

.q C 2/2 2.q C 2 p/

ˆ1 D

;

.F.u0 /; u0 /0 ; B MCqC2 2.qC2/=2 ; 1 ˆ˛0 1

in the case q C 2 > p max pj ; j D1;N

C1 is the constant of the best embedding V ,! W0 and B is the constant from the condition (F4).

275


Proof. Step 1. Local solvability Lemma 3.6.3. For any u0 2 V0 , there exists maximal T0 > 0 such that a unique solution of problem (3.2) of the class C .1/ .Œ0; TI V0 / exists for any T 2 .0; T0 /. Proof. The proof is exactly the same as the proof of Lemma 3.4.3. Step 2. A priori estimates and the blow-up Lemma 3.6.4. Let X pj 1 1 hAj .u/; uij ; ˆ.t / hA0 u; ui0 C 2 pj N

˛

j D1

.F .v/; v/0 c > 0

8v 2 W0 ;

qC2 ; p

jvj0 D 1:

Then (1) if ˛ 2 .0; 1/, then T0 D C1 and ˆ.t / Œˆ1˛ C .1 ˛/At 1=.1˛/ I 0 (2) if ˛ D 1, then T0 D C1 and ˆ.t / ˆ0 exp.At /I (3) if ˛ > 1 and X pj 1 .q C 2 p/2 1 hA0 u0 ; u0 i0 C hAj .u0 /; u0 ij < .F.u0 /; u0 /0 ; 2 pj 2p.q C 2/2 N

j D1

then T0 2 ŒT1 ; T2 and

lim ˆ.t / D C1;

t"T0

where .qC2/=p N X 1 pj 1 p hAj .u/; uij ; A CjqC2 ; hA0 u; ui0 C M 2 pj p1 j D1 2 q=2 1 1 ˆ0 ; T1 ˆ0 B ; T2 D ln 1 q ˛1 1 ˆ1 pCqC2 .F.u0 /; u0 /0 ; B MCqC2 2.qC2/=2 ; ˆ0 ˆ.0/; ˛1 ; ˆ1 D ˛1 1 2p ˆ0 ˆ.t /

1=pj

jvj0 Cj hAj v; vij

;

.q C 2/2 2.q C 2 p/

for q C 2 > p max pj ; j D1;N

C1 is the constant of the best embedding V ,! W0 and B is the constant from the condition (F4).

276


Proof. Let u 2 C .1/ .Œ0; T0 /I V0 / be a solution of problem (3.2). Multiplying both sides of (3.2) by u.t / or by u0 .t / and integrating by parts, we obtain the following inequalities: N X pj 1 d 1 hAj .u/; uij D .F.u/; u/0 ; hA0 u; ui0 C dt 2 pj

(3.132)

j D1

hA0 u0 ; u0 i0 C

N X

h.Aj .u//0 ; u0 ij D

j D1

1 d .F.u/; u/0 hDP .u/; u0 i0 : (3.133) q C 2 dt

Integrating Eq. (3.132) over t 2 .0; T/, we obtain Z t ds .F.u/; u/0 ; ˆ.t / D ˆ0 C

(3.134)

0

where

X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij ; 2 pj N

ˆ0 D ˆ.0/:

j D1

According to the condition V0 W0 , there exists a constant C1 > 0 such that 1=2

jvj0 C1 hA0 v; vi0 I hence from (3.134) we obtain the inequality Z t Z t qC2 .qC2/=2 ds hA0 v; vi0 ˆ0 C B ds ˆ.qC2/=2 .s/; ˆ.t / ˆ0 C MC1 0

0

MCqC2 2.qC2/=2 : 1

B

By the Bihari theorem (see [112]) we have ˆ.t /

ˆ0 Œ1

q q=2 2=q 2 ˆ0 Bt

8t 2 Œ0; T1 /;

T1

2 q=2 1 B : ˆ q 0

(3.135)

Therefore, in the case p < q C 2, from (3.135) we obtain T0 T1 . Now let p q C 2. We use the fact that there exists j 2 1; N (pj D p) such that pj q C 2 and Vj W0 . Let Cj be the constant of the embedding Vj W0 : 1=p

jvj0 Cj hAj .v/; vij j ; Z t .qC2/=pj qC2 ds hAj .v/; vij ˆ.t / ˆ0 C Cj M 0

qC2

ˆ0 C Cj M

p p1

.qC2/=p Z

t 0

ds ˆ.qC2/=p .s/:

277


Consider separately the cases where ˛ D 1 or ˛ 2 .0; 1/. In the first case, by the Gronwall–Bellman theorem (see [112]) we have .qC2/=p p qC2 : ˆ.t / ˆ0 exp¹At º; A Cj M p1

(3.136)

In the second case where ˛ 2 .0; 1/, by the Bihari theorem (see [112]) we obtain ˆ.t / Œˆ1˛ C .1 ˛/At 1=.1˛/ : 0

(3.137)

From (3.136) and (3.137) we obtain that T0 D C1 in the case where ˛ 2 .0; 1. Now from relations (3.132) and (3.133) in the same way as in Step 5 we derive the following second-order ordinary differential inequality (see [210]): ˆ00 ˆ ˛1 .ˆ0 /2 C ˆˆ0 0;

t 2 Œ0; T0 /;

where

pCqC2 ; 2p We require the validity of the condition ˛1

˛D

qC2 > 1; p

> 0; (3.138)

.q C 2/2 : 2.q C 2 p/

X pj 1 .q C 2 p/2 1 hAj .u0 /; u0 ij < .F.u0 /; u0 /0 : (3.139) hA0 u0 ; u0 i0 C 2 pj p .q C 2/2 N

j D1

From the condition .F.u/; u/0 0 for all u 2 W0 and from the equality ˆ0 .t / D .F.u/; u/0 (see (3.132)) we obtain ˆ0 .t / 0. Then, owing to (3.138), Proposition 3.5.7 implies the inequality ˆ.t /

ŒT02

‰0 ; Z.t /1=.˛1 1/

(3.140)

where

˛1 1 ‰0 D 0

1=.1˛1 / ;

Z.t / 1 exp. t /;

˛1 1 1 D ; 0 .F.u0 /; u0 /0 0 D ; ˆ˛0 1

T02

1 ˆ1˛ 0

where Z.C1/ D 1, and by condition (3.139), T02 < 1. From (3.140) we immediately obtain that Z.T0 / < T02 . Lemma 3.6.4 is proved. Theorem 3.6.2 is completely proved.

278

3.7


Examples

In this section, we consider certain initial-boundary-value problems for Eqs. (3.3)– (3.11). In these examples, we use standard Sobolev embedding theorems that can be found, for example, in [108–111]. In all examples, H D L2 ./. Example 3.7.1. n X @ pj 2 u C div.jruj ru/ u C jujq u D 0; @t j D1

uj@ D 0;

u.x; 0/ D u0 .x/;


Aj .u/ div.jrujpj 2 ru/ W W0 0

F .u/ jujq u W LqC2 ./ ! Lq ./; pj > 2; V0 H01 ./;

q > 0; 1;pj

Vj W0

pj0

0

./ ! W 1;pj ./;

Lu u W L2 ./ ! L2 ./;

pj ; pj 1

q0

W0 LqC2 ./;

./;

qC2 ; qC1 W1 L2 ./;

0

V0 H1 ./; Vj W 1;pj ./; W0 L.qC2/=.qC1/ ./; W1 L2 ./; \ n 1;p W0 j ./ : V H01 ./ \ j D1

The embedding V0 H01 ./ W0 LqC2 ./ holds under the following conditions: 0 qC2p 2 under the condition p < q C 2. One can verify that the operator coefficients satisfy all the conditions of Theorem 3.3.2. Example 3.7.2. @ .u C div.jrujp2 ru/ jujq1 u/ u C jujq u D 0; @t uj@ D 0; u.x; 0/ D u0 .x/; where RN is a bounded domain with smooth boundary of the class C .2;ı/ , ı 2 .0; 1, and the operator coefficients have the form A0 u u W H01 ./ ! H1 ./;

Lu u W L2 ./ ! L2 ./;

0

0

A2 .u/ jujp2 2 u W Lp2 ./ ! Lp2 ./; F.u/ jujq u W LqC2 ./ ! Lq ./; A1 .u/ div.jrujp1 2 ru/ W W0

1;p1

p1 p > 2; p10 V0 H01 ./;

p1 ; p1 1

0

./ ! W 1;p1 ./;

p2 q1 C 2; p20

V1 W01;p1 ./;

p2 ; p2 1

q1 > 0; q0

qC2 ; qC1

V2 Lp2 ./;

0

0

W1 L2 ./;

W1 L2 ./;

W0 LqC2 ./;

V0 H1 ./; V1 W 1;p1 ./; V2 Lp2 ./; W0 L.qC2/=.qC1/ ./; 1;p

V H01 ./ \ W0

./ \ Lq1 C2 ./:

The embedding V0 H01 ./ W0 LqC2 ./ holds under the conditions 0 .Lu0 ; u0 /1 C .Lu0 ; u0 /1 ˆ0 ; .F.u0 /; u0 /0 > qC2p 2 under the condition p < q C 2. One can verify that the operator coefficients and the corresponding Banach spaces satisfy all the conditions of Theorem 3.3.2. Example 3.7.3. @ .2 u C u C div.jrujp2 ru// C u div.jrujq ru/ D 0; @t ˇ @u ˇˇ D 0; u.0; x/ D u0 .x/; uj@ D @n ˇ @

where RN is a bounded domain with smooth boundary @ of the class C .4;ı/ , ı 2 .0; 1, and the operator coefficients have the form A0 u 2 u u W H02 ./ ! H2 ./; 1;qC2

F.u/ div.jrujq ru/ W W0

1;p

A1 .u/ div.jrujp2 ru/ W W0

0

./ ! W 1;q ./; 0

./ ! W 1;p ./;

Lu D u W H01 ./ ! H1 ./; V0 H02 ./;

1;qC2

W0 W0

V0 H2 ./; W0 W

1;q 0

V H02 ./ \ W01;p ./;

1;p

./;

W1 H01 ./;

./;

W1 H1 ./; V1 W 1;p ./;

q0

V1 W0

./; 0

qC2 ; qC1

p0 D

p ; p1

p 3:

The embedding V ,! W0 holds if 0 .Lu0 ; u0 /1 C qC2p

.F.u0 /; u0 /0 >

qC2 .Lu0 ; u0 /1 2

under the condition p D p < q C 2. Thus, all the conditions of Theorem 3.3.2 hold. Example 3.7.4. @ .u jujp1 2 u/ C u C jujq u D 0; @t uj@ D 0; u.x; 0/ D u0 .x/; where RN is a bounded domain with smooth boundary of the class C .2;ı/ , ı 2 .0; 1, and the operator coefficients have the form A0 u u W H01 ./ ! H1 ./; 0

F .u/ jujq u W LqC2 ./ ! Lq ./; V0 H01 ./;

0

A1 .u/ jujp1 2 u W Lp1 ./ ! Lp1 ./;

V1 Lp1 ./;

Lu u W H01 ./ ! H1 ./; W0 LqC2 ./;

0

0

V0 H1 ./; V1 Lp1 ./; W0 Lq ./; p1 D q1 C 2;

q1 1;

p10

p1 ; p1 1

q0

W1 H01 ./; W1 H1 ./;

qC2 ; qC1

q > 0:

The embedding V0 W0 holds under the condition 0 .Lu0 ; u0 /1 C qC2p

.F.u0 /; u0 /0 >

qC2 .Lu0 ; u0 /1 p

under the condition p D p1 < q C 2. Now we verify the monotonicity of the Fréchet derivative of the operator A1 .u/ D jujq1 u in the sense of Theorem 3.4.2. Indeed, the

282


Fréchet derivative has the form 0

A01;u .u/h .q1 C 1/jujq1 h W Lq1 C2 ./ ! Lq1 ./; q10

q1 C 2 ; q1 C 1

8h 2 Lq1 C2 ./;

hA01;u .u/u1 A01;u .u/u2 ; u1 u2 i1 D .q1 C 1/hjujq1 u1 jujq1 u2 ; u1 u2 i1 Z D .q1 C 1/ dx .jujq1 u1 jujq1 u2 /.u1 u2 / 0 8u; u1 ; u2 2 Lq1 C2 ./:

Now we prove that under the condition krun ruk2 ! 0 as n ! C1, the Fréchet derivatives converge in the uniform operator topology L.H01 ./; H1 .//. Indeed, kA01;u .u/ A01;u .un /kH1 !H1 D 0

sup krhk2 D1

C

sup krhk2 D1

kA01;u .u/h A01;u .un /hk1

kA01;u .u/h A01;u .un /hk.q1 C2/=.q1 C1/ Z

sup krhk2 D1

ˇ ˇ q1 C2 dx ˇŒjujq1 jun jq1 hˇ q1 C1

.q1 C1/=.q1 C2/ :

Consider the following integral: Z ˇ ˇ q1 C2 Jn D dx ˇŒjujq1 jun jq1 hˇ q1 C1 Z ˇ q1 C2 q1 C2 ˇ dx ˇjujq1 jun jq1 ˇ q1 C1 jhj q1 C1

Z

ˇ ˇ.q C2/=q1 dx ˇjujq1 jun jq1 ˇ 1

1 C2/=.q1 C1/ krhk.q 2

Z C

Z

q1 =.q1 C1/ Z

dx jhj


ˇ ˇ.q C2/=q1 dx ˇjujq1 jun jq1 ˇ 1 .q C2/q1 =.q1 C1/

C max¹kruk2 1

q1 C2

1=.q1 C1/

q1 =.q1 C1/

q1 =.q1 C1/ .q C2/q1 =.q1 C1/

; krun k2 1

º C:

Note that, since kru run k2 ! C0 as n ! C1, by the condition q > 0, un ! u strongly in LqC2 ./. Consider the function f .x; u/ D jujq1 W Lq1 C2 ./ ! L

q1 C2 q1

./:

283


One can verify that all the conditions of the Krasnoselskii theorem hold (see Appendix A.15). According to the definition, the continuity of the operator f W Lq1 C2 ./ ! L.q1 C2/=q1 ./ means that the following limit relation hold: “ dx jf .x; u/ f .x; un /j

lim

n!C1

q1 C2 q1

D0

under the strong convergence un ! u in LqC2 ./. Therefore, by the Krasnoselskii theorem we have: Jn ! C0 as n ! C1 uniformly with respect to h on the sphere krhk2 D 1. Thus, all the conditions of Theorem 3.4.2 hold. Example 3.7.5. n X @u @ pj 2 div.jruj ru/ C u C u3 D 0; u C @t @x1 j D1

uj@ D 0;

u.x; 0/ D u0 .x/;


Aj .u/ div.jrujpj 2 ru/ W W0 F .u/ u3 W L4 ./ ! L4=3 ./; Dv

P .u/ u2 W L4 ./ ! L2 ./;

@v W L2 ./ ! H1 ./; @x1

V0 H01 ./;

1;pj

Vj W0

0

./ ! W 1;pj ./;

pj > 2; ./;

W0 L4 ./;

0

V0 H1 ./; Vj W 1;pj ./; W0 L4=3 ./; \ n 1;pj 1 W0 ./ ; V H0 ./ \ j D1

W2 L4 ./;

W3 L2 ./;

W4 H1 ./:

The embedding V0 H01 ./ W0 L4 ./ holds under the condition N 4. Finally, the embedding V ,! W0 D W2 holds under the conditions n \ j D1

1;pj

W0

./ ,! LqC2 ./;

284


which, in its turn, holds under the conditions N p max pj ; j D1;n

4
p: Np

We also require the validity of the following conditions for the initial functions u0 2 V: X pj 1 .4 p/2 1 .F.u0 /; u0 /0 hAj .u0 /; u0 ij < hA0 u0 ; u0 i0 C 2 pj 16p N

j D1

under the conditions p < 4. One can verify that the operator coefficients satisfy all the conditions of Theorem 3.5.2. Example 3.7.6. @u @ div.jruj2 ru/ D 0; .2 u C u C div.jrujp2 ru// C u @t @x1 ˇ @u ˇˇ uj@ D D 0; u.0; x/ D u0 .x/; p 3; @n ˇ@ where RN is a bounded domain with smooth boundary @ of the class C .4;ı/ , ı 2 .0; 1, and the operator coefficients have the form A0 u 2 u u W H02 ./ ! H2 ./; F.u/ div.jruj2 ru/ W W01;4 ./ ! W 1;4=3 ./; 1;p


0

./ ! W 1;p ./;

P .u/ D u2 W H01 ./ L4 ./ ! L2 .//; Dv D

@v W L2 ./ ! H1 ./; @x1

V0 H02 ./;

V1 W01;p ./;

W0 W01;4 ./;

0

V0 H2 ./; W0 W 1;4=3 ./;

V1 W 1;p ./;

W2 L4 ./;

W4 H1 ./;

W3 L2 ./; 1;p

V H02 ./ \ W0

./:

The embedding V ,! W0 W1 holds under the condition N 3. We also require the validity of the condition for the initial function u0 2 H02 ./: X pj 1 1 .4 p/2 .F.u0 /; u0 /0 hAj .u0 /; u0 ij < hA0 u0 ; u0 i0 C 2 pj 16p N

j D1

under the condition p < 4. Thus, all the conditions of Theorem 3.5.2 hold.

285


Example 3.7.7. @ .2 u C u/ div.jruj2 ru/ @t @u @u @u @u @u @u @ @ @ C ˇ2 C ˇ3 D 0; C ˇ1 @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2 ˇ @u ˇˇ uj@ D D 0; u.x; 0/ D u0 .x/; @n ˇ @

where ˇ1 C ˇ2 C ˇ3 D 0, jˇ1 j C jˇ2 j C jˇ3 j > 0, RN is a bounded domain with smooth boundary @ of the class C .4;ı/ , ı 2 .0; 1, and the operator coefficients have the form A0 u 2 u u W H02 ./ ! H2 ./; F .u/ div.jruj2 ru/ W W01;4 ./ ! W 1;4=3 ./; P .u/ D ˇ1

@u @u @u @u @u @u e1 C ˇ2 e2 C ˇ3 e3 W @x2 @x3 @x3 @x1 @x1 @x2 W01;4 ./ ! L2 ./ L2 ./ L2 ./;

Dv div.v/ W L2 ./ L2 ./ L2 ./ ! H1 ./; V0 H02 ./;

W0 W01;4 ./;

W2 W01;4 ./;

W3 L2 ./ L2 ./ L2 ./; W4 H1 ./;

V0 H2 ./;

W0 W 1;4=3 ./;

V H02 ./: Moreover, the embedding V ,! W0 W1 holds. We also require the validity of the condition for the initial function u0 2 H02 ./: X pj 1 1 .4 p/2 .F.u0 /; u0 /0 : hAj .u0 /; u0 ij < hA0 u0 ; u0 i0 C 2 pj 16p N

j D1

Thus, all the conditions of Theorem 3.6.2 hold. Example 3.7.8. @jujq2 C1 @ .u jujp1 2 u/ C C juj2q2 u D 0; @t @x1 uj@ D 0;

u.x; 0/ D u0 .x/;

286


where RN is a bounded domain with smooth boundary @ of the class C .2;ı/ , ı 2 .0; 1, and the operator coefficients have the form 0

A0 u u W H01 ./ ! H1 ./;

A1 .u/ jujp1 2 u W Lp1 ./ ! Lp1 ./; 2q2 C 2 ; 2q2 C 1 @v Dv W L2 ./ ! H1 ./; @x1

00

F .u/ juj2q2 u W L2q2 C2 ./ ! Lq2 ./;

q200 D

P .u/ jujq2 C1 W L02q2 C2 ./ ! L2 ./;

W0 L2q2 C2 ./;

V0 H01 ./;

V1 Lp1 ./;

V0 H1 ./;

V1 Lp1 ./; W0 Lq2 ./;

0

00

W2 L2q2 C2 ./; W3 L2 ./; p1 D q1 C 2;

q1 1;

p10

p1 ; p1 1

W4 H1 ./; q200

2q2 C 2 ; 2q2 C 1

q2 > 0:

The embedding V0 W0 holds under the condition 0 < q2

1 for N 3I N 2

0 < q2 < C1 for N D 1; 2:

This means that V H01 ./ \ Lq1 C2 ./ ,! W0 LqC2 ./: Moreover, W0 D W2 . We also require that X pj 1 .q C 2 p /2 1 hAj .u0 /; u0 ij < .F.u0 /; u0 /0 ; hA0 u0 ; u0 i0 C 2 pj p .q C 2/2 N

j D1

q D 2q2 , p D p1 , under the condition p D p1 < 2q2 C 2. Now we verify the monotonicity of the Fréchet derivative of the operator A1 .u/ D jujq1 u in the sense of Theorem 3.4.2. Indeed, the Fréchet derivative has the form 0

A01;u .u/h .q1 C 1/jujq1 h W Lq1 C2 ./ ! Lq1 ./; q10

q1 C 2 ; q1 C 1

8h 2 Lq1 C2 ./;

hA01;u .u/u1 A01;u .u2 /u2 ; u1 u2 i1 D .q1 C 1/hjujq1 u1 jujq1 u2 ; u1 u2 i1 Z D .q1 C 1/ dx .jujq1 u1 jujq1 u2 /.u1 u2 / 0 8u; u1 ; u2 2 Lq1 C2 ./;

where q1 D p1 2.

287


Now we prove that under the condition krun ruk2 ! 0 as n ! C1, the convergence of the Fréchet derivatives in the uniform operator topology L.H01 ./; H1 .// occurs. Indeed, kA01;u .u/ A01;u .un /kH1 !H1 D 0

sup krhk2 D1


kA01;u .u/h A01;u .un /hk.q1 C2/=.q1 C1/ Z

C

sup krhk2 D1

sup krhk2 D1


.q1 C1/=.q1 C2/ :

Consider the following integral: Z Jn D Z

ˇ ˇ q1 C2 dx ˇŒjujq1 jun jq1 hˇ q1 C1 ˇ q1 C2 q1 C2 ˇ dx ˇjujq1 jun jq1 ˇ q1 C1 jhj q1 C1

Z

ˇ.q C2/=q1 ˇ dx ˇjujq1 jun jq1 ˇ 1

1 C2/=.q1 C1/ krhk.q 2

Z C

Z

q1 =.q1 C1/ Z

q1 C2

dx jhj



1=.q1 C1/

q1 =.q1 C1/

q1 =.q1 C1/

1 C2/q1 =.q1 C1/ C max¹kruk2.q1 C2/q1 =.q1 C1/ ; krun k.q º C: 2

Note that, since kru run k2 ! C0 as n ! C1, under the condition 0 < q1 4=.N 2/ for N 3 and 0 < q1 for N D 1; 2 we have un ! u strongly in Lq1 C2 ./. Consider the function f .x; u/ D jujq1 W Lq1 C2 ./ ! L

q1 C2 q1

./:

One can verify that all the conditions of the Krasnoselskii theorem hold (see Appendix A.15). According to the definition, the continuity of the operator f W Lq1 C2 ./ !

288


L.q1 C2/=q1 ./ means that the limit relation “ dx jf .x; u/ f .x; un /j

lim

n!C1

q1 C2 q1

D0

holds under the condition of the strong convergence un ! u in Lq1 C2 ./. Therefore, by virtue of the Krasnoselskii theorem (see Appendix A.15) we have: Jn ! C0 as n ! C1 uniformly with respect to h on the sphere krhk2 D 1. Thus, all the conditions of Theorem 3.6.2 hold.

3.8

On certain initial-boundary-value problems for quasilinear wave equations of the form (3.2)

In this section, we obtain sufficient conditions of the blow-up of solutions of the following first initial-boundary-value problems: @ @u C u3 D 0; uj@ D 0; u.x; 0/ D u0 .x/I .u u/ C u @t @x1 @u @ div.jruj2 ru/ D 0; .2 u C u/ C u @t @x 1 ˇ @u ˇˇ uj@ D D 0; u.x; 0/ D u0 .x/I @n ˇ@

(3.141) (3.142)

@ .2 u C u/ div.jruj2 ru/ @t @u @u @u @u @u @u @ @ @ C ˇ2 C ˇ3 D 0; (3.143) Cˇ1 @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2 ˇ1 C ˇ2 C ˇ3 D 0; jˇ1 j C jˇ2 j C jˇ3 j > 0; ˇ @u ˇˇ uj@ D D 0; u.x; 0/ D u0 .x/; @n ˇ@ where the surface-simply-connected, bounded domain 2 R3 has a smooth boundary of the class @ 2 C .2;ı/ in the case (3.141), and in the cases (3.142) and (3.143), of one of the class @ 2 C .4;ı/ , where ı 2 .0; 1, .x1 ; x2 ; x3 / 2 .

3.8.1 Local solvability in the strong generalized sense of problems (3.141)–(3.143) In this subsection, we prove the local-on-time solvability in the strong generalized sense of each of problems (3.141)–(3.143).

289

Section 3.8 Blow-up in quasilinear wave equation

First, introduce the notation: A1 u u W H01 ./ ! H1 ./;

(3.144)

A2 u 2 u u W H02 ./ ! H2 ./; 1 @u2 W H01 ./ L4 ./ ! H1 ./; 2 @x1 @u @u @u @u @ @ C ˇ2 F2 .u/ ˇ1 @x1 @x2 @x3 @x2 @x3 @x1 @u @u @ W H02 ./ ! H2 ./; C ˇ3 @x3 @x1 @x2

F1 .u/

(3.145)

(3.146)

F3 .u/ u3 W H01 ./ L4 ./ ! L4=3 ./ H1 ./; F4 .u/ div.jruj2 ru/ W H02 ./

W01;4 ./

(3.147)

! W 1;4=3 ./ H2 ./: (3.148)

Owing to notation (3.144)–(3.148), we represent problems (3.141)–(3.143) as Cauchy problems for abstract first-order differential equations with operator coefficients in Banach spaces: du D F3 .u/ F1 .u/; dt du A2 D F4 .u/ F1 .u/; dt du D F4 .u/ F2 .u/; A2 dt A1

u.0/ D u0 2 H01 ./;

(3.149)

u.0/ D u0 2 H02 ./;

(3.150)

u.0/ D u0 2 H02 ./:

(3.151)

Introduce the following norms and duality brackets: kvkCs

is a norm in H0s ./;

kvks

is a norm in Hs ./;

kvkp

is a norm in Lp ./;

h; is

is the duality bracket between H0s ./ and Hs ./;

.; /p

is the duality bracket between Lp ./ and Lp ./;

s > 0; p 2 Œ1; C1; 0

p0

p : p1

The ordinary differential equations (3.149)–(3.151) can be reduced to the following first-order abstract ordinary differential equation with operator coefficients in the Banach space V0 : A

du D F.u/; dt

u.0/ D u0 2 V0 :

(3.152)

290


We impose the following conditions on the operator coefficients: A W V0 ! V0 ; kAv1 Av2 k0 mkv1 v2 k0 ;

F W V0 ! V0 ;

kF.v1 / F.v2 /k0 .R/kv1 v2 k0 ;

where kvk0 is a norm in V0 and kvk0 is a norm in V0 . Lemma 3.8.1. For any u0 2 V0 , there exists maximal T0 > 0 such that a unique solution of problem (3.152) of the class u 2 C .1/ .Œ0; TI V0 / exists for any T 2 .0; T0 /. Proof. Consider the following problem, which is equivalent to (3.152) in the class C .1/ .Œ0; TI V0 /: du D G.u/.t / 8t 2 Œ0; T; dt u.0/ D u0 2 V0 ;

(3.153)

u 2 C .1/ .Œ0; TI V0 /; where G.u/ D A1 F.u/.t /, u 2 C.Œ0; TI V0 /. The following relations holds: lim kG.u/.t / G.u/.t0 /k0 m1 lim kF.u/.t / F.u/.t0 /k0

t!t0

t!t0

1

m

.ku.t0 /k0 / lim ku.t / u.t0 /k0 t!t0

C lim ku.t / u.t0 /k0 D 0: t!t0

Therefore, G W C.Œ0; TI V0 / ! C.Œ0; TI V0 /. Introduce the set BR D ¹u.t / 2 C.Œ0; TI V0 / W kukT Rº; where kvkT sup kvk0 : t2Œ0;T

Prove that the operator Z U.u/ u0 C

t 0

ds G.u/.s/;

t 2 Œ0; T;

acts from BR into BR and is a contraction on BR . Indeed, since G W C.Œ0; TI V0 / ! C.Œ0; TI V0 /; it is obvious that U W C.Œ0; TI V0 / ! C .1/ .Œ0; TI V0 /:

(3.154)

291


On the other hand, Z kU.u/kT ku0 k0 C kG.u/k0 .t / m

1

T 0

ds kG.u/k0 .s/ ku0 k0 C T sup kG.u/k0 .t /; t2Œ0;T

kF.u/k0

m

1

.kuk0 /kuk0 m

1

.R/kuk0 :

Therefore, kUkT ku0 k C m1 .R/RT: Let ku0 k0 R=2, T < m0 =.2.R//; then for sufficiently large R > 0 and small T > 0, kUkT R: Thus, U W BR ! BR . Now we prove that the operator U defined in (3.154) is a contraction on BR . Let u; v 2 BR ; then we have Z kU.u/ U.v/kT

0

T

dt kG.u/ G.v/k0

m1

Z

T 0

dt kF.u/ F.v/k0 .t / m1 .R/Tku vkT :

From the latter inequality for T < m=.2.R// we obtain 1 kU .u/ U.v/kT < ku vkT : 2 Thus, the operator U is a contraction on BR . Therefore, by the theorem on contraction mappings, for any u0 2 V0 , there exists T > 0 such that a unique solution of problem (3.153) of the class C .1/ .Œ0; TI V0 / exists. Using the standard extension-in-time algorithm for solutions we obtain that a unique solution of problem (3.152) of the class C .1/ .Œ0; T0 /I V0 / exists; moreover, either T0 < C1 and then lim sup kuk0 D C1; t"T0

or T0 D C1. Indeed, consider the norm .T/ ku.t /kT . As a function of T, the function .T/ monotonically increases. Therefore, as T " T0 , the function .T/ has either a finite or infinite limit. Assume that .T/ has a finite limit as T " T0 . Consider the auxiliary integral equation of the form (3.154): 0

u.x; t / D u.x; T / C

Z

t 0

ds G.u/.s/:

292


The norm kruk22 .T0 / is uniformly bounded on T0 2 .0; T0 /; therefore, we can choose T 2 .0; T0 / such that for any T0 2 .0; T0 /, the integral equation has a unique solution of the class u.x; t / 2 C .1/ .Œ0; T I V0 /. Let T0 D T0 T =2. Denote by v.x; t / the corresponding solution of the latter integral equation and define b u.x; t / on the segment Œ0; T0 C T =2 as follows: b u.x; t / D ¹u.x; t /; t 2 Œ0; T0 I v.x; t T0 /; t 2 ŒT0 ; T0 C T =2º: According to the construction, b u.x; t / is a solution of problem (3.154) on the segment Œ0; T0 C T =2 and, by the local uniqueness, it is an extension of the function u.x; t /. This fact contradicts the maximality of the segment Œ0; T0 . This contradiction proves that lim sup ku.t /kT D C1: T"T0

Hence we directly obtain that lim ku.t /k0 D C1:

t"T0

Thus, for any u0 2 V , there exists T0 > 0 such that a unique solution of problem (3.152) of the class u.t / 2 C .1/ .Œ0; T0 /I V0 / exists. The following theorem is valid. Theorem 3.8.2. For any initial functions u0 .x/ 2 H01 ./ for problem (3.149) or u0 .x/ 2 H02 ./ for problems (3.150) and (3.151), there exist maximal time moments T0 > 0 (being the own for each of problems (3.149)–(3.151)) such that unique solutions of problems (3.149)–(3.151) exist; these solutions belong to the class C .1/ .Œ0; T0 /I H01 .// for problem (3.149) or to the class C .1/ .Œ0; T0 /I H02 .// for problems (3.150) and (3.151). Proof. To prove this theorem, we must prove the coercivity of the operators A1 W H01 ./ ! H1 ./ and A2 W H02 ./ ! H2 ./ and the bounded Lipschitz continuity of the operators Fj .u/, j D 1; 4, in the corresponding spaces. We prove the coercivity of the operators A1 and A2 . Indeed, the following inequalities hold: hA1 v1 A1 v2 ; v1 v2 i1 D .rv1 rv2 ; rv1 rv2 /2 kv1 v2 k2C1

8v1 ; v2 2 H01 ./; (3.155)

hA2 v1 A2 v2 ; v1 v2 i2 D .rv1 rv2 ; rv1 rv2 /2 C .v1 v2 ; v1 v2 /2 D kv1 v2 k2C1 C kv1 v2 k2C2 kv1 v2 k2C2

(3.156)

293


for all v1 ; v2 2 H02 ./. The obtained inequalities imply the existence of the inverse 1 1 1 2 2 operators A1 1 W H ./ ! H0 ./, A2 W H ./ ! H0 ./ and by the Lipschitz continuity of the inverse operators with Lipschitz constants equal to 1. Now we prove the bounded Lipschitz continuity of each of the operators Fj .u/, j D 1; 4. First, consider the operator F1 .u/ defined by the formula (3.145): Z ˇ 1=2 ˇ 2 2 2 2 ˇ2 ˇ kF1 .v1 / F1 .v2 /k1 Ckv1 v2 k2 C dx v1 v2 (3.157) C

2 Z X i D1

dx vi4

1=4

kv1 v2 k4 RCkv1 v2 kC1

for all vi 2 H01 ./, kvi kC1 R, where C denotes various constants independent of R: kvkC1 R. Hence we directly obtain that kF1 .v1 / F1 .v2 /k2 1 .R/kv1 v2 kC2

(3.158)

for all vi 2 H02 ./, kvi kC2 R, 1 .R/ D CR. Now consider the operator F2 .u/. The following inequality holds: @v1 @v1 @v2 @v2 C C @v1 @v1 @v2 @v2 kF2 .v1 / F2 .v2 /k2 C @x @x @x2 @x3 1 @x3 @x1 @x3 @x1 1 2 3 @v1 @v1 @v2 @v2 D J1 C J2 C J3 : C C @x @x @x2 @x1 1 2 1 For example, consider J1 (it is quite obvious that J2 and J3 can be estimated similarly). Indeed, @v1 @v1 @v2 @v2 J1 D C @x2 @x3 @x2 @x3 1 @v1 @v1 @v2 @v2 @v1 @v2 C C C @x2 @x3 @x3 1 @x3 @x2 @x2 1 ˇ ˇ ˇ ˇ Z Z ˇ @v1 ˇ2 @v1 @u2 2 1=2 ˇ @v2 ˇ2 @v1 @u2 2 1=2 ˇ ˇ ˇ C dx ˇˇ C C dx ˇ @x ˇ @x @x2 ˇ @x3 @x3 @x2 3 2 C

2 X iD1

krvi k4 krv1 rv2 k4 C

2 X

kvi kC2 kv1 v2 kC2 CRkv1 v2 kC2

i D1

for all vi 2 H02 ./, kvi kC2 R. Thus, we conclude that the following inequality holds: kF2 .v1 / F2 .v2 /k2 2 .R/kv1 v2 kC2 ; where vi 2 H02 ./, 2 .R/ D CR, kvi kC2 R.

(3.159)

294


Now consider the operator F3 .u/. The following inequalities hold: kF3 .v1 / F3 .v2 /k1 Ckv13 v23 k4=3 C C

2 Z X i D1

C

2 Z X i D1

C

8=3

dx jv13 v23 j4=3

jv1 v2 j

4=3

3=.4r1 /Z

dx jvi j

dx jvi j

4

1=2 Z

3=4

3=4

r1 8=3

2 Z X i D1

dx jvi j

Z

dx jv1 v2 j

dx jv1 v2 j

4

3=.4r2 /

r2 4=3

1=4

3 .R/kv1 v2 kC1 for all vi 2 H01 ./, kvi kC1 R, 3 .R/ D CR2 . Hence we obtain that kF3 .v1 / F3 .v2 /k1 3 .R/kv1 v2 kC1

(3.160)

for all vi 2 H01 ./, kvi kC1 R, 3 .R/ D CR2 . Consider the operator F4 .u/. The following inequalities hold: kF4 .v1 / F4 .v2 /k2 C C

Z

2 Z X i D1

C

ˇ4=3 ˇ dx ˇjrv1 j2 rv1 jrv2 j2 rv2 ˇ

2 Z X i D1

dx jrvi j

8=3

dx jrvi j

4

kF4 .v1 / F4 .v2 /k2 4 .R/kv1 v2 kC2

jrv1 rv2 j

1=2

4=3

3=4

3=4

kv1 v2 k4 ; (3.161)

for all vi 2 H02 ./, kvi kC2 R, 4 .R/ D CR2 . Thus, we have proved the bounded Lipschitz continuity of the operators Fj .u/, j D 1; 4, in the corresponding Hilbert spaces H01 ./ and H02 ./. Now we consider the abstract first-order ordinary differential equation (3.152) with operator coefficients in the Banach space V0 . Note that properties (3.155)–(3.161) for the operator coefficients of problems (3.149)–(3.151) imply the fact that these problems can be considered as Eq. (3.152). Thus, for problems (3.149)–(3.151), Lemma 3.8.1 is valid.

295


3.8.2 Blow-up of solutions In this subsection, we prove that T0 > 0 from Theorem 3.8.2 is finite. Thus, we prove that for each of problems (3.141)–(3.143), the limit relation lim kukCs D C1

t"T0

holds, where s D 1 for problem (3.141) and s D 2 for problems (3.142) and (3.143). Lemma 3.8.3 (see [210, Lemma 1.1]). Assume that ˆ.t / 2 C .2/ Œ0; T0 / under certain maximal T0 > 0 in the sense that either T0 D C1 or T0 < C1. Then lim ˆ.t / D C1;

(3.162)

t"T0

where ˆ.t / > 0 and ˆ0 .t / > 0 for t 2 Œ0; T0 /. Moreover, let the following secondorder ordinary differential inequality hold for t 2 Œ0; T0 /: ˆ00 ˆ ˛.ˆ0 /2 C ˆˆ0 0;

t 2 Œ0; T0 /;

˛ > 1;

> 0;

(3.163)

and the inequality ˛1 ˆ1 ; holds. Then T0 T2 , where ˆ0
1; > 0: ˆ˛ ˆ˛ Introduce the notation ˆ0 ƒ.t / ˛ I ˆ then the following inequalities hold: ƒ0 .t / C ƒ.t / 0;

ƒ.t / ƒ0 exp. t /;

ˆ0 ƒ0 exp. t /; ˆ˛

1 .ˆ1˛ /0 ƒ0 exp. t /; 1˛ ˛1 Œ1 exp. t / ; ƒ0 ˆ1˛ ˆ1˛ 0 1 : ˆ 1˛ Œˆ0 ƒ0 ˛1 Œ1 exp. t /1=.˛1/

(3.166)

296


We also require the validity of inequality (3.164); then (3.166) cannot hold for all t 2 R1C , i.e., there exists T0 T2 , where T2 is defined by formula (3.165), such that the limit relation (3.162) holds. Now we prove the principal result of this section. Theorem 3.8.4. Let the following conditions for problems (3.141), (3.142), and (3.143) hold: (1) ku0 k44 > kru0 k22 C ku0 k22 for problem (3.141), (2) kru0 k44 > C41 ./Œku0 k22 C kru0 k22 for problem (3.142), (3) kru0 k44 > 4ˇ 2 Œkru0 k22 C ku0 k22 for problem (3.143), where C1 ./ is the constant of the best embedding kvk4 C1 ./krvk4 ;

ˇ

3 X

jˇi j:

i D1

Then for any u0 2 H01 ./ in the case of problem (3.141) and for any u0 2 H02 ./ in the case of problems (3.142) and (3.143), T0 described in Theorem 3.3.2 is finite and, therefore, the following limit relations are valid: lim kukCs D C1;

t"T0

where s D 1 for problem (3.141) and s D 2 for problems (3.142) and (3.143). For the blow-up times of solutions, the following two-sided estimates hold: T1 T0 T2 , where for T1 and T2 the following relations are valid: (1) for problem (3.141), kru0 k22 C ku0 k22 T2 D ln 1 ; ku0 k44 T1 D

1 2C42 ./

kru0 k22

1 I C ku0 k22

(2) for problem (3.142), 2 2 1 4 ku0 k2 C kru0 k2 ; T2 D 4 ln 1 C1 kru0 k44 C1 T1 D

1 2C43 ./

kru0 k22

1 I C ku0 k22

297


(3) for problem (3.143), T2 D T1 D

2 2 1 2 ku0 k2 C kru0 k2 ln 1 4ˇ ; 4ˇ 2 kru0 k44 1

2C43 ./

kru0 k22

1 ; C ku0 k22

where C2 ./ is the best constant of the embedding kvk4 C2 ./krvk2 for any v 2 H01 ./ and C3 ./ is the best constant of the embedding krvk4 C3 ./kvk2 for any v 2 H02 ./. Proof. First, we prove the result of Theorem 3.8.4 for problem (3.141). Theorem 3.8.2 implies the fact that for problem (3.141), under the condition u0 2 H01 ./, there exists maximal T0 > 0 such that a unique solution u.x; t / 2 C .1/ .Œ0; TI H01 .// exists for any T 2 .0; T0 /. Therefore, we can multiply Eq. (3.141) by u.x; t / and next by u0 .x; t / relative to the duality bracket between the Hilbert spaces H01 ./ and H1 ./. Then, integrating by parts, we obtain the following energy equalities: 1d Œkruk22 C kuk22 D kuk44 ; 2 dt Z 1 @2 u 2 1 d 0 2 0 2 dx kru k2 C ku k2 D u C kuk44 : 2 @x1 @t 4 dt

(3.167) (3.168)

Introduce the notation ˆ1 .t / kruk22 C kuk22 :

(3.169)

The following inequalities hold: ˇZ ˇ ˇ ˇ 0 0 ˇ2 1 ˇˇ dˆ1 ˇˇ2 ˇˇ ˇ dx ru ; ru C u ; u ˇˇ 4 ˇ dt ˇ .kru0 k22 C ku0 k22 /.kruk22 C kuk22 / ˆ1 .kru0 k22 C ku0 k22 /:

(3.170)

On the other hand, (3.167)–(3.169) imply 1 kru0 k22 C ku0 k22 ˆ001 C 8 1 00 ˆ1 C 8

1 " Œkru0 k22 C ku0 k22 ˆ001 C 1 4 8

1 11 "kru0 k22 C kuk44 4 4" " 1 Œkru0 k22 C ku0 k22 C ˆ01 ; 4 8" 1 0 ˆ (3.171) 8" 1

298


for any " 2 .0; 4/. From (3.170) and (3.171) we have the following ordinary differential inequalities: 1 0 4" 0 2 1 00 jˆ1 j ˆ1 ˆ C ˆ ; 16 8 1 8" 1 1 0 4" 0 2 00 jˆ1 j ˆ1 ˆ1 C ˆ1 I 2 " hence we directly obtain that (see [210]) ˆ1 ˆ001 ˛1 .ˆ01 /2 C 1 ˆ1 ˆ01 0; ˛1 D

4" 1 ; 1 D ; 2 "

" 2 .0; 2/: (3.172)

We see that the function ˆ1 .t / under the auxiliary conditions ˆ10
0 such that a unique solution u.x; t / 2 C .1/ .Œ0; T0 /I H02 .// exists. Therefore, we can multiply Eq. (3.142) by u.x; t / and next by u0 .x; t / relative to the duality bracket between the

299


Hilbert spaces H02 ./ and H2 ./. Integrating by parts, we obtain the following energy equalities: 1d Œkruk22 C kuk22 D kruk44 ; 2 dt Z 1d 1 @2 u 2 0 2 0 2 4 kruk4 u : kru k2 C ku k2 D dx 4 dt 2 @x1 @t

(3.176) (3.177)

Owing to (3.176), Eq. (3.176) implies the inequality 1d kruk44 C 4 dt 1 d kruk44 C 4 dt

kru0 k22 C ku0 k22

1 1 " kru0 k22 C ku0 k22 C kuk44 (3.178) 4 4" 1 1 "Œkru0 k22 C kuk22 C C41 ./kruk44 ; 4 4"

where " 2 .0; 4/ and C1 is the best constant of embedding kvk4 C1 ./krvk4 . Introduce the notation ˆ2 .t / kruk22 C kuk22 : From (3.176)–(3.178) we obtain the following ordinary differential inequality for the function ˆ2 .t /: 4" 0 2 1 00 C41 ./ 0 .ˆ2 / ˆ2 ˆ2 C ˆ2 : 16 8 8" Hence we obtain the differential inequality (see [210]) 2 ˆ2 ˆ002 ˛2 ˆ02 C 2 ˆ2 ˆ02 0;

˛2

4" ; 2

2

C41 ./ : "

(3.179)

Now, arguing as in the case of the differential inequality (3.172), we come to the condition of Lemma 3.8.3: ˆ20
C41 ./Œku0 k22 C kru0 k22 : Lemma 3.8.3 and (3.181) imply the condition T0 T2 , 2 2 1 4 ku0 k2 C kru0 k2 T2 D 4 ln 1 C1 : kru0 k44 C1 Now consider problem (3.143). Theorem 3.8.4 implies the fact that for problem (3.143) under the condition u0 2 H02 ./, there exists maximal T0 > 0 such that a unique solution u.x; t / 2 C .1/ .Œ0; TI H02 .// exists for any T 2 .0; T0 /. Therefore, we can multiply Eq. (3.143) by u.x; t / and next by u0 .x; t / relative to the duality bracket between the Hilbert spaces H02 ./ and H2 ./. Integrating by parts, we obtain the following energy equalities: 1d (3.182) Œkruk22 C kuk22 D kruk44 ; 2 dt Z 1d @2 u @u @u dx (3.183) ku0 k22 C kru0 k22 D kruk44 ˇ1 4 dt @x1 @t @x2 @x3 Z Z @2 u @u @u @2 u @u @u dx ˇ3 dx : ˇ2 @x2 @t @x3 @x1 @x3 @t @x1 @x2 From (3.182) and (3.183) we obtain the following inequalities: ku0 k22 C kru0 k22

X 1d jˇi jkru0 k2 kruk24 kruk44 C 4 dt 3

i D1

X 1d " jˇi j Œkru0 k22 C ku0 k22 kruk44 C 4 dt 2 3

i D1

1 X jˇi jkruk44 ; C 2" 3

(3.184)

i D1

where " 2 .0; 2ˇ 1 / and ˇ D

P3

i D1 jˇi j

> 0. Introduce the notation

ˆ3 .t / kruk22 C kuk22 : From (3.182)–(3.184) we obtain the following ordinary differential inequality for the function ˆ3 .t / (see [210]): ˆ3 ˆ003 ˛3 .ˆ03 /2 C 3 ˆ3 ˆ03 0;

˛3 D 2 "ˇ;

3 D 2ˇ=":

(3.185)

301


As above, using Lemma 3.8.3, from (3.185) we obtain the optimal condition for the initial function kru0 k44 > 4ˇ 2 Œkru0 k22 C ku0 k22 and the upper estimate for the blow-up time T0 T2 : ku0 k22 C kru0 k22 1 : T2 D 2 ln 1 4ˇ 2 4ˇ kru0 k44 Now we obtain lower estimates bounds for the blow-up times of solutions of problems (3.141)–(3.143). For problem (3.141), the following energy equality holds: 1d .kruk22 C kuk22 / D kuk44 : 2 dt Hence for the function ˆ1 .t / kruk22 C kuk22 we obtain the inequality

Z

ˆ1 .t / ˆ1 .0/ C

2C42 ./

t 0

ds ˆ21 .s/;

where C2 is the best constant of the embedding kvk4 C2 ./krvk2 for any v 2 H01 ./. From this inequality we obtain ˆ1 .t /

ˆ1 .0/ 1 2C42 ./ˆ1 .0/t

and, therefore, in this case T1 D

1 2C42 ./

kru0 k22

1 : C ku0 k22

For problems (3.142) and (3.143), the energy equality 1d .kuk22 C kruk22 / D kruk44 2 dt holds and then for the function ˆ2 .t / kuk22 C kruk22 we have the inequality

Z

ˆ2 .t / ˆ2 .0/ C 2C43 ./

t 0

ds ˆ22 .s/;

where C3 ./ is the best constant of the embedding krvk4 C3 ./kvk2 for any v 2 H02 ./. Hence we directly obtain that T1 D Theorem 3.8.4 is proved.

1 2C43 ./

kru0 k22

1 : C ku0 k22

302


3.8.3 Breakdown of weakened solutions of problem (3.141) To prove the breakdown of solutions of problem (3.141) for a three-dimensional equation of Benjamin–Bona–Mahony type we must prove that a unique solution of problem (3.141) of the class C .1/ .Œ0; T0 /I C .1/ .// exists for certain T0 > 0. Then by the results of Section 3.3 we obtain that 0 < T0 T0 T2 < C1 and, therefore, sup

lim

T"T 0 t2Œ0;T ;x2

jru.x; t /j D C1;

(3.186)

i.e., at a certain time moment T0 > 0, the value max jru.x; t /j x2

becomes infinite. In this subsection, by a solution of problem (3.141) we mean a so-called weakened solution, i.e., a solution of a certain integral equation related to problem (3.141); the form of such a solution will be written below. A classical solution of problem (3.141) is weakened, but, obviously, the opposite statement is not always valid. We give a definition of a weakened solution of problem (3.141). Definition 3.8.5. A solution of the class u.x; t / 2 C .1/ .Œ0; TI C .1/ ./ \ C0 .// of the double integral equation Z t Z 1 @u2 .y; s/ 3 u.x; t / D u0 .x/ C ds dy G.x; y/ C u .y; s/ ; 2 @y1 0

(3.187)

for certain T > 0, for all t 2 Œ0; T, where G.x; y/ is the Green function of the first boundary-value problem for the operator C I in the domain , is called a weakened solution of problem (3.141). Obviously, we obtain that for the existence of a weakened solution, the condition u0 .x/ 2 C .1/ ./ \ C0 ./ for the initial function is necessary. Now we find relations between strong generalized solutions of problem (3.141) and weakened solutions in the sense of the definition stated. As we have found in Section 3.2, a strong generalized solution of problem (3.141) is equivalent to a solution of the integral operator equation of the form Z t ds A1 F.u/.s/; u0 2 H01 ./; u.t / D u0 C 0

where the operator A1 W H1 ./ ! H01 ./ is inverse to the operator u C u W H01 ./ ! H1 ./. Thus, it is clear that the restriction of the operator A1 to the

303


Banach space C./, where has smooth boundary @ 2 C .2;ı/ , ı 2 .0; 1, coincides with the integral operator whose kernel is the Green function G.x; y/. Thus, we conclude that a weakened solution of problem (3.141) is a strong generalized solution of problem (3.141). We will need this result for the proof of the breakdown of weakened solutions of problem (3.141). Theorem 3.8.6. For any u0 2 C .1/ ./ \ C0 ./, there exists T0 > 0 such that a unique weakened solution of the class u.x; t / 2 C .1/ .Œ0; T0 /I C .1/ ./ \ C0 .// exists. Moreover, either T0 D C1 or T0 < C1 and then the limit relation (3.186) holds. Proof. By the fact that the Green function of the operator C I has the form G.x; y/ D where

.x; y/ C

1 exp.jx yj/ ; 4 jx yj

.x; y/ 2 C .1/ . /, the following relation holds: Z sup jry G.x; y/j dy D C < C1:

(3.188)

(3.189)

x2

Therefore, we can integrate by sides of the integro-differential equation (3.187) and obtain Z t Z Z Z 1 t 3 ds dy G.x; y/u .y; s/ ds dy Gy0 1 .x; y/u2 : u.x; t / D u0 .x/ C 2 0 0 (3.190) Introduce the operator Z U.u/ u0 C

t 0

Z

1 ds dy G.x; y/u .y; s/ 2

Z

Z

t

3

ds 0

dy Gy0 1 .x; y/u2 : (3.191)

Prove that by (3.189), operator (3.191) acts from C.QT / into C.QT /, where QT D .0; T/ for any T > 0 and kvkT;0 D

sup

jvj

t2Œ0;T ;x2

is a norm in C.QT /. Introduce the notation Z t Z ds dy Gy0 1 .x; y/u2 ; U1 .u/ D 0

Z U2 .u/ D

Z

t

dy G.x; y/u3 .y; s/:

ds 0

304


Prove that Ui .u/ W C.QT / ! C.QT /, i D 1; 2. First, consider the operator U1 .u/. By (3.188),

1 @ 4 @y1

0 y1 .x; y/

exp.jx yj/ jx yj

2 C. /;

D

1 x1 y1 exp.jx yj/ 4 jx yj3 1 x1 y1 exp.jx yj/I C 4 jx yj2

then U1 .u/ can be represented in the form U1 .u/ D I1 C I2 C I3 ; where

Z I1 D

ds 0

1 I2 D 4 I3 D

Z

t

1 4

Z

0 2 y1 .x; y/u .y; s/;

dy t

Z

ds Z

0

Z

t

exp.jx yj/ x1 y1 2 u .y; s/; jx yj2 jx yj

dy

exp.jx yj/ x1 y1 2 u .y; s/: jx yj jx yj

ds 0

dy

Prove that I1 .x; t / 2 C.QT /. Indeed, for any .x; t / and .x0 ; t0 / from QT , we have ˇZ t ˇ Z ˇ ˇ 0 2ˇ ˇ ds dy j y1 .x; y/ju ˇ jI1 .x; t / I1 .x0 ; t0 /j ˇ

t0

Z

C

t0 0

ˇZ ˇ ds ˇˇ dy Œ

0 y1 .x; y/

ˇ ˇ

0 2ˇ y1 .x0 ; y/u ˇ :

Hence by the continuity of y0 1 .x; y/ 2 C. / we directly obtain that each of the summands is less than "=2 under the condition jx x0 j C jt t0 j ı."/. Therefore, I1 .x; t / 2 C.QT /. Now we prove that I2 .x; t / 2 C.QT /. Indeed, for any .x; t / and .x0 ; t0 / from QT , we have ˇ ˇZ Z ˇ 1 ˇˇ t exp.jx yj/ jx1 y1 j 2 ds dy u .y; s/ˇˇ jI2 .x; t / I2 .x0 ; t0 /j ˇ 2 4 t0 jx yj jx yj Z t0 ˇZ ˇ 1 exp.jx0 yj/ x10 y1 C ds ˇˇ dy 4 0 jx0 yj2 jx0 yj ˇ ˇ exp.jx yj/ x1 y1 2 ˇ u .y; s/ ˇ jx yj2 jx yj D I21 C I22 :


305

Consider the relations for I21 and I22 separately. First, I21 is less than "=3 under the condition jx x0 j C jt t0 j ı."/. To analyze I22 , we assume that u.x; t / 2 C.QT / and ju.x; t /j M.T/. Next we use the scheme of the proof of the continuity of a potential-type integral, (see, e.g., [421]). We have Z t0 Z exp.jx0 yj/ exp.jx yj/ ds dy C jI22 .x; x0 ; t; t0 /j M.T/ jx0 yj2 jx yj2 0 .x0 / ˇ Z t0 Z ˇ exp.jx0 yj/ x10 y1 C M.T/ ds dy ˇˇ jx0 yj2 jx0 yj 0 n .x0 / ˇ exp.jx yj/ x1 y1 ˇˇ ; jx yj2 jx yj ˇ where .x0 / D ¹x 2 W jx x0 j º. Since the function under the sign of integral has an integrable singularity, the first term is less than "=3 because of choice of the small parameter . In the second summand, the function under the sign of integral is uniformly continuous with respect to .x; y/ in the domain jx x0 j =2, jy x0 j , y 2 , and vanishes for x D x0 . Therefore, this integral is less than "=3 for all jx x0 j C jt t0 j ı."/ and sufficiently small ı."/. Therefore, I2 .x; t / 2 C.QT /. Clearly, the inclusion I3 .x; t / 2 C.QT / can be proved similarly. Thus, we have proved that U1 .u/ W C.QT / ! C.QT /: Now we consider the operator Z U2 .u/ D

Z

t

dy G.x; y/u3 .y; s/:

ds 0

Using the properties of the Green function and potential-type integrals, we obtain that U2 W C.QT / ! C.QT /: Therefore, the operator U.u/.x; t / defined by formula (3.191) acts from C.QT / into C.QT /. Let BR be a closed, bounded, convex subset of the Banach space C.QT / defined as follows: ® ¯ (3.192) BR v.x; t / 2 C.QT / W kvkT max jv.x; t /j R : .x;t/2QT

Prove that for sufficiently small T > 0 and sufficiently large R > 0, the operator U acts from BR into BR and is a contraction on BR . Indeed, kU.u/kT ku0 kT C CTkuk3T C CTkuk2T ku0 kT C CT.R3 C R2 / R

306


under the conditions ku0 kT R=2 and 0 < T 21 .CR3 C CR2 /1 . Thus, for sufficiently small T > 0 and sufficiently large R > 0, the operator U acts from BR into BR . Now we prove that the operator defined on BR for sufficiently small T > 0 and sufficiently large R > 0 is a contraction. Indeed, 1 kU.u1 / U.u2 /kT T CR2 C CR ku1 u2 kT ku1 u2 kT 2 under the condition 0 < T 21 .CR2 C CR/1 . Therefore, the operator U .u/ is a contraction on BR . Thus, under the condition u0 2 C0 ./, a unique solution of the integral equation of the class C.Œ0; TI C0 .// exists. Now we assume, that u0 2 C .1/ ./ \ C0 ./. Prove that a unique solution of the integral equation (3.190) of the class C.Œ0; TI C .1/ ./ \ C0 .// exists. To this end, consider the Banach space C.Œ0; TI C .1/ .// with the norm kvkT;1

3 X @v kvkT;0 C @x

i

i D1

;

kvkT;0 max jv.x; t /j:

(3.193)

.x;t/2QT

T;0

Prove that the operator Z U.u/ u0 C

t 0

Z

1 @u2 .y; s/ ds dy G.x; y/ C u3 .y; s/ 2 @y1

acts from C.Œ0; TI C .1/ .// into C.Œ0; TI C .1/ .//. We use the fact that G.x; y/ D and

0 x1 .x; y/

.x; y/ C

1 exp.jx yj/ 4 jx yj

2 C. / and represent U .u/ in the form U.u/ D u0 C U1 .u/ C U2 .u/;

(3.194)

where

@u 3 ds dy .x; y/ u Cu ; U1 .u/ D @y1 0 Z Z t @u 1 exp.jx yj/ 3 U2 .u/ D u ds dy Cu : 4 0 jx yj @y1 Z

t

Z

By the fact that .x; y/ 2 C .1/ . /, the operator U1 .u/ acts from C.Œ0; TI into C.Œ0; TI C .1/ .//.

C .1/ .//

307


Now we consider the integral U2 .u/. In the same way as above, using the theorem on the continuous differentiability of potential-type integrals, we conclude that the operator U2 .u/ also acts from C.Œ0; TI C .1/ .// into C.Œ0; TI C .1/ .//. In the Banach space C.Œ0; TI C .1/ .// with the norm defined in (3.193), we introduce the following closed, bounded, convex subset: B1R ¹v 2 C.Œ0; TI C .1/ .// W kvk1;T Rº: Prove that the operator U.u/ acts from B1R into B1R and is a contraction on B1R for sufficiently small T > 0 and sufficiently large R > 0. Indeed, by (3.194) we have kU .u/k1;T ku0 k1;T C CTkuk21;T C CTkuk31;T ku0 k1;T C CTR2 C CTR3 R under the conditions ku0 k1;T R=2 and T 21 .CR2 C CR3 /. Now we prove that the operator U W B1R ! B1R is a contraction. Indeed, kU.u1 / U.u2 /k1;T CRTku1 u2 k1;T C CR2 Tku1 u2 k1;T 1 ku1 u2 k1;T 2 under the condition T 21 .CR C CR2 /1 . Therefore, the operator U W B1R ! B1R is a contraction. Applying the standard extension-in-time algorithm to solutions of integral equation (3.187), we obtain that a unique solution of Eq. (3.187) of the class C.Œ0; T /I C .1/ ./ \ C0 .// exists. Moreover, either T0 D C1 or T0 < C1 and then the limit relation (3.186) holds. Therefore, a unique solution of the integral equation (3.187) of the class C.Œ0; TC .1/ ./ \ C0 .// exists. From the form of the integral equation (3.187) we easily obtain that, in fact, the solution mentioned belongs to the class C .1/ .Œ0; TI C .1/ ./ \ C0 .//. Thus, we have proved the existence and uniqueness of a weakened solution of problem (3.141) in the sense of Definition 3.8.5. Note that the norm introduced above on the Banach space C.Œ0; TI C .1/ ./ \ C0 .// is equivalent for bounded domains with smooth boundaries to the following norm: kvk0T max jrv.x; t /j; .x;t/2QT

QT .0; T/ :

308


Since any weakened solution of problem (3.141) is a strong generalized solution, by virtue of Theorem 3.8.4 under the auxiliary condition ku0 k44 > kru0 k22 C ku0 k22 ; there exists a time moment 0 < T0 T2 such that lim kruk22 D C1;

t"T0

where

kru0 k22 C ku0 k22 : D ln 1 T2 ku0 k44

This means that for the time moment T0 , the inequalities 0 < T0 T0 T2 ; hold, i.e., for the finite time T0 , a weakened solution of problem (3.141) breaks down: lim kuk0T D C1:

T"T 0

Note that the breakdown of solutions of Whitham-type wave equations was studied in the works [161–167, 197, 307–309].

3.9

On an initial-boundary-value problem for a strongly nonlinear equation of the type (3.1) (generalized Boussinesq equation)

In this section, we prove the local and global solvability of the following initialboundary-value problem: @ .u jujq1 u/ C u C jujq2 u D 0; @t u.x; 0/ D u0 .x/; uj@ D 0;

(3.195) (3.196)

where q1 1, q2 0, and RN is a bounded domain with smooth boundary of the class C .2;ı/ , ı 2 .0; 1, N 1. Depending on q1 , q2 , and N , we obtain results on the global-on-time solvability, on the solvability in a finite cylinder Œ0; T (0 < T < C1 is fixed), and on the blow-up of solutions for a finite time.

309

Section 3.9 Blow-up in the generalized Boussinesq equation

3.9.1 Unique solvability of the problem in the weak sense Introduce definitions and assumptions necessary for further consideration and also obtain some auxiliary results. Let the Hilbert space ¹H01 ./; k kº and its dual space ¹H1 ./; k k º be defined. Denote the duality bracket between the Hilbert spaces H01 ./ and H1 ./ by h; i (see, e.g., [347]). In what follows, we use the following notation: k kC.Œ0;T IV / k kT ; k kC.Œ0;T IV / k kT : Note that the operator A0 C .q1 C 1/jujq1 I acts from H01 ./ into H1 ./ and is radially continuous (see, e.g., [168]). First, we prove that the operator B0 .u/ .q1 C 1/jujq1 I acts from H01 ./ into H1 ./. For this, we prove that 1

B0 W Lq1 C2 ./ ! L.Lq1 C2 ./; L.q1 C2/.q1 C1/ .//; where hB0 .u/h1 ; h2 i D .q1 C 1/hjujq1 h1 ; h2 i 8u; h1 ; h2 2 Lq1 C2 ./; kB0 .u/hkp1 D .q1 C 1/kjujq1 hkp1 c1 kjujq1 kp2 khkp3 ; p21 C p31 D p11 ; p1 D .q1 C 2/.q1 C 1/1 ; p3 D q1 C 2; q1 ; p2 q1 D q1 C 2 8h 2 Lq1 C2 ./: p21 D q1 C 2 Hence we obtain q kB0 .u/hk.q1 C2/.q1 C1/1 c1 kukq11 C2 khkq1 C2 :

In addition, we require that q1 1 for N D 2 and 1 q1 4=.N 2/ for N 3; then, by virtue of the Sobolev embedding theorem, H01 ./ Lq1 C2 ./ and, 1 therefore, L.q1 C2/.q1 C1/ ./ H1 ./. Therefore, the following inequalities hold: q kB0 .u/hk c2 kB0 .u/hk.q1 C2/=.q1 C1/ c2 c1 kukq11 C2 khkq1 C2 c3 kukq1 khk:

Therefore,

B0 W H01 ./ ! L.H01 ./; H1 .//:

Thus, A0 2 L.H01 ./; H1 .// and, obviously, it is radially continuous. By the monotonicity of the operator B0 .u/ W Lq1 C2 ./ ! L.q1 C2/=.q1 C1/ ./ and by the fact that hB0 .u/h1 B0 .u/h2 ; h1 h2 i D .q1 C 1/hjujq1 .h1 h2 /; h1 h2 i

310


for u; h1 ; h2 2 H01 ./, we have hB0 .u/h1 B0 .u/h2 ; h1 h2 i D .q1 C 1/hjujq1 .h1 h2 /; h1 h2 i D .q1 C 1/.jujq1 .h1 h2 /; h1 h2 / 0; where the duality bracket h; i between H01 ./ and H1 ./ is the duality bracket between the Banach spaces Lq1 C2 ./ and L.q1 C2/=.q1 C1/ ./ for elements from Lq1 C2 ./ and L.q1 C2/=.q1 C1/ ./ and, therefore, they exactly coincides with the inner product .; / in L2 ./. Hence we obtain hA0 h1 A0 h2 ; h1 h2 i hh1 h2 ; h1 h2 i D kh1 h2 k2 ;

(3.197)

i.e., the operator A0 W H01 ./ ! H1 ./ is strongly monotonic. By [168, Consequence 2.3, p.97] and the Browder–Minty theorem, (3.197) implies that the inverse operator A0 1 W H1 ./ ! H01 ./ is Lipschitz-continuous: kA0

1

x A0

1

yk kx yk

8x; y 2 H1 ./:

(3.198)

Now we consider the operators Fi u jujqi u;

i D 1; 2:

Prove that the operator F W H01 ./ ! H1 ./ is boundedly Lipschitz-continuous under the condition 0 < qi 4=.N 2/ for N D 3 and 0 < qi for N D 1; 2. For this, we note that kFi u Fi vk kjujqi u jvjqi vk : The following inequality holds: kjujqi u jvjqi vk.qi C2/=.qi C1/ .qi C 1/kf .u v/k.qi C2/=.qi C1/ ; where f D max.jujqi ; jvjqi /, kf .u v/kpi kf kr1i pi ku vkr2i pi ; pi D

qi C 2 ; qi C 1

r2i D qi C 1;

r1i D

r2i pi D qi C 2; qi C 1 ; qi

r1i pi D

qi C 2 ; qi

kjujqi u jvjqi vk.qi C2/=.qi C1/ .qi C 1/kf k.qi C2/=qi ku vkqi C2 ; kf k.qi C2/=qi 2.max¹kukqi C2 ; kvkqi C2 º/qi :


311

Thus, kjujqi u jvjqi vk i .R/ku vk; i .R/ D c1 Rqi ;

R D max¹kuk; kvkº;

and, therefore, kFi u Fi vk i .R/ku vk:

(3.199)

Now consider the operator A.u/ D u C jujq1 u W H01 ./ ! H1 ./: It is easy to prove that this operator has a Lipschitz-continuous inverse operator with Lipschitz constant equal to 1. Finally, the operator A0 u D u W H01 ./ ! H1 ./ has a Lipschitz-continuous inverse operator with Lipschitz constant equal to 1. The following theorem holds. Theorem 3.9.1. For any u0 2 H01 ./ (0 < qi 4=.N 2/ for N 3 or 0 < qi for N D 1; 2, i D 1; 2), there exists Tu0 > 0 such that for any 0 < T < Tu0 , problem (3.195), (3.196) has a unique solution in the class C .1/ .Œ0; TI H01 .//. Proof. In the class v 2 C .1/ .Œ0; TI H1 .//, problem (3.195), (3.196) is equivalent to the problem dv D F1 .A1 v/ C F2 .A1 v/; dt

v.0/ D v0 D A.u0 / 2 H1 ./:

In its turn, problem (3.200) is equivalent to the integral equation Z t ds ŒF1 .A1 v/ C F2 .A1 v/.s/: v.t / D H.v/ v0 C

(3.200)

(3.201)

0

Using properties (3.199) of the operators meant in the definition, (3.201) of the operator H.v/, and the method of contraction mappings, we can prove the local solvability in the class v 2 C .1/ .Œ0; T0 /I H1 .//, and either T0 D C1 or T0 < C1; in the latter case, the limit relation lim sup kvk1 D C1 t"T0

holds. Now consider the equation A.u/ D u C jujq1 u D v 2 C .1/ .Œ0; T0 I H1 .//;

(3.202)

312


which, by the properties of the operator A and the Browder–Minty theorem, is equivalent to the following: u D A1 .v/:

(3.203)

Let vn ! v strongly in H1 ./; then (3.203) implies ku un kC1 Ckv vn k1 ! C0 as n ! C1. Note that, by virtue of v 2 C.Œ0; TI H1 .//, for any T 2 .0; T0 / we have kv.t / v.t0 /k1 ! C0 as t ! t0 , t; t0 2 .0; T/. Therefore, (3.203) implies u 2 C.Œ0; T0 I H01 .//. Now in the class u 2 C .1/ .Œ0; T0 I H01 .// from (3.202) we obtain that .A0 C B0 .u//u0 D v 0 2 C.Œ0; T0 I H1 .//;

u D A1 .v/:

(3.204)

We apply the operator A1 0 to both sides of Eq. (3.204); then we obtain the equivalent equation 0 0 1 0 1 ŒI C A1 0 B .u/u D A0 v 2 C.Œ0; T0 I H0 .//:

(3.205)

Prove that the operator 0 O I C A1 C 0 B .u/

(3.206)

O it is invertible for fixed u 2 H01 ./. By virtue of the linearity of the operator C, suffices to prove that the operator equation O D0 Cw

(3.207)

has only trivial solution. Indeed, we apply the operator A0 to both sides of Eq. (3.207); then we obtain the equation A0 .u/w D ŒA0 C B0 .u/w D 0; which, by virtue of (3.198), implies the invertibility of the operator A0 .u/ for fixed u 2 H01 ./. Hence we conclude that w D 0 in the class H01 ./. By the linearity O defined by expression (3.206) and by the inverseand the continuity of the operator C O 1 is linear and bounded. mapping Banach theorem, we conclude that the operator C Then from (3.205) we obtain the following equivalent equation: 0 O 1 .u/A1 u0 D C 0 v :

(3.208)

313


Now we must prove only that 0 1 O 1 A1 u0 D C 0 v 2 C.Œ0; T0 /I H0 .//

for fixed u 2 C.Œ0; T0 /I H01 .//. Indeed, by (3.208), the following inequalities hold: 0 0 O 1 .t0 /A1 ku0 .t / u0 .t0 /kC1 kC 0 Œv .t / v .t0 /kC1 0 O 1 .t0 / C O 1 .t //A1 C k.C 0 v .t0 /kC1

(3.209)

O 1 .t0 / C O 1 .t /k 1 Ckv 0 .t / v 0 .t0 /k1 C CkC H !H1 : 0

0

O is continuous and, Note that the linear (for fixed u 2 C.Œ0; T0 /I H01 .//) operator C O 1 is linear and therefore, by the inverse-mapping Banach theorem, the operator C continuous and, therefore, bounded owing to the linearity. Thus, we can use spectral O 1 W H1 ./ ! H1 ./. representation for the linear bounded operator C 0 0 O First, introduce the resolvent of the operator C: O D .I C/ O 1 : R.; C/ Let be a circle jj D r with sufficiently large radius, greater than sup t2Œt0 ";t0 C"

O 1 kCk H ./!H1 ./ : 0

0

The introduced variable is well defined since, for t 2 Œt0 "; t0 C " Œ0; T0 /, the inequality kukC1 < C1 sup t2Œt0 ";t0 C"

O 1 .t / and holds. Now we can use the spectral representation for the operators C 1 O .t0 / with the same contour introduced above: C Z Z 1 1 1 1 1 O O O 0 //: O d R.; C.t //; C .t0 / D d 1 R.; C.t C .t / D 2 i 2 i Obviously, we have O 1 .t0 / D 1 O 1 .t / C C 2 i

Z

O // R.; C.t O 0 //: d 1 ŒR.; C.t

Now we use the well-known representation for operator resolvents: O // R.; C.t O 0 // D R.; C.t O 0 // R.; C.t

C1 X nD1

O / C.t O 0 //R.; C.t O 0 //n Œ.C.t

314


under the condition O 0 / C.t O /k 1 O kC.t H ./!H1 ./ kR.; C.t0 //kH1 ./!H1 ./ ı < 1: 0

0

0

0

The following inequality holds: O // R.; C.t O 0 //k 1 kR.; C.t H ./!H1 ./ 0

0

O 0 //k 1 kR.; C.t H ./!H1 ./ 0

C1 X nD1

0

O 0 //kn 1 O // C.t O 0 //kn 1 kR.; C.t kC.t : H ./!H1 ./ H ./!H1 ./ 0

0

0

0

Now we note that 0 0 O / C.t O 0 / D A1 C.t 0 ŒBu .u.t // Bu .u.t0 //:

Prove the continuity of the Fréchet derivatives Bu0 .u/. Let un ! u strongly in H01 ./ ,! L2 ./. Therefore, un ! u strongly in Lq1 C2 ./ and then kBu0 .u/ Bu0 .un /kH1 !H1 D 0

sup krhk2 D1

C

kBu0 .u/h Z

sup krhk2 D1

sup krhk2 D1

kBu0 .u/h Bu0 .un /hk1

Bu0 .un /hk.q1 C2/=.q1 C1/


.q1 C1/=.q1 C2/ :

Consider the integral Z Z ˇ q ˇ q1 C2 ˇ ˇ q1 C2 q1 C2 q1 ˇ q1 C1 1 ˇ dx Œjuj jun j h dx ˇjujq1 jun jq1 ˇ q1 C1 jhj q1 C1 Jn D

Z

ˇ.q C2/=q1 ˇ dx ˇjujq1 jun jq1 ˇ 1 .q C2/=.q1 C1/

Z

Ckrhk2 1 Z C

q1 =.q1 C1/ Z

dx jhj



q1 C2

1=.q1 C1/

q1 =.q1 C1/

q1 =.q1 C1/

1 C2/q1 =.q1 C1/ 1 C2/q1 =.q1 C1/ C max¹kruk.q ; krun k.q º C: 2 2

Consider the operator f .x; u/ D jujq1 W Lq1 C2 ./ ! L.qC2/=q1 ./, for which sufficient conditions of its continuity hold: Z ˇ q1 C2 ˇ dx ˇjujq1 jun jq1 ˇ q1 ! C0 Jn C

as n ! C1, since the sequence un converges strongly to u in Lq1 C2 ./.

315


Therefore, according to the Krasnoselskii theorem, we have Jn ! C0 as n ! C1. Therefore, 0 0 O / C.t O 0 /k 1 kC.t H ./!H1 ./ kBu .u.t // Bu .u.t0 //kH1 !H1 ./ ! C0; 0

0

0

O // R.; C.t O 0 //k 1 kR.; C.t H ./!H1 ./ ! 0; 0

O / kC.t

1

O 0/ C.t

1

0

kH1 ./!H1 ./ ! C0 0

0

as t ! t0 . Therefore, by virtue of (3.209) we have u 2 C .1/ .Œ0; T0 /I V0 /. Note that if q1 1, then the operator C.u/ is boundedly Lipschitz-continuous and, therefore, the operator C 1 .u/ is also boundedly Lipschitz-continuous and hence Eq. (3.208) has a local solution in the class u.t / 2 C .1/ .Œ0; T00 /I H01 .//. It is easy to prove that T0 D T00 . Theorem 3.9.1 is proved.

3.9.2 Blow-up of solutions and the global solvability of the problem Let q C2 E0 E.0/; E.t / 21 kruk22 C .q1 C 1/=.q1 C 2/kukq11 C2 ; .q2 C2/.q1 C2/1 q2 C 2 q2 C2 q1 C 2 B C1 ; q1 C 1 q1 C 2

under the conditions 0 < qi 4=.N 2/ for N 3 or qi > 0 for N D 1; 2. Moreover, let kukq2 C2 C1 kukq1 C2 for q1 q2 > 0. The following theorem holds. Theorem 3.9.2. Let the conditions of Theorem 3.9.1 hold. Then, under the condition q1 q2 > 0, problem (3.195), (3.196) is uniquely solvable in any finite cylinder in the class u.x; t / 2 C .1/ .Œ0; TI H01 .// and the following upper estimates hold: E.t / E0 exp.Bt /

under the condition q1 D q2 ;

E.t / ŒE0 C .1 /Bt 1=.1/

under the condition q1 > q2 :

Finally, if 0 < q1 < q2 and q C2

q2 C 2 kru0 k22 ; 2 p 2q2 .q1 C 2/1=2 1 q1 C 1 q C2 2 kru0 k22 C ku0 kq11 C2 ; > kru0 k2 C q2 q1 2 q1 C 2

ku0 kq22 C2 > q C2

ku0 kq22 C2

316


then there exists maximal T0 such that for u0 .x/ 2 H01 ./, a unique strong generalized solution of the class u.x; t / 2 C .1/ .Œ0; TI H01 .// exists for any T 2 .0; T0 /, T0 < C1, and the limit relation lim E.t / D C1

t"T0

holds together with the following upper estimate for the blow-up time: T0 T1 D

1 E1˛ 0 ; A

1 .E00 /2 ˇ1 .˛1 1/E0 A Œ.1 ˛1 /2 E2˛ 0

.1˛1 /2 1=2

;

where q C2 E00 D ku0 kq22 C2 kru0 k22 ;

ˇ1

q22 ; q2 q1

˛1 D

4 C q1 C q2 : 2.q1 C 2/

Proof. Multiplying Eq. (3.195) by u.x; t / 2 C .1/ .Œ0; TI H01 .// and integrating by parts over the domain we obtain 1 d q1 C 1 d q C2 q C2 kruk22 C kukq11 C2 C kruk22 D kukq22 C2 : 2 dt q1 C 2 dt

(3.210)

Multiplying Eq. (3.195) by u t .x; t / 2 C.Œ0; TI H01 .// and integrating over the domain , we obtain Z 1 1d d q C2 2 kru t k2 C .q1 C 1/ dx jujq1 .u t /2 D kukq22 C2 kruk22 : q C 2 dt 2 dt 2 (3.211) From (3.210), under the condition q1 q2 > 0, we obtain the inequalities Z t ds kukqq22 C2 E.t / E0 C C2 .s/ 0

E0 C Cq12 C2

Z

t

ds 0

q1 C 2 q1 C 1

.q2 C2/.q1 C2/1

E.q2 C2/.q1 C2/ .s/:

This integral inequality implies, by the Gronwall–Bellman and Bihari theorems (see [112]), the inequalities E.t / E0 exp.Bt / 1

E.t / ŒE0

C .1 /Bt 1=.1/

where B

under the condition q1 D q2 ;

q C2 C12

q1 C 2 q1 C 1

under the condition q1 > q2 ;

.q2 C2/.q1 C2/1 ;

q2 C 2 : q1 C 2

317


Prove the last part of the theorem. According to the condition q C2 ku0 kq22 C2 >

q2 C 2 kru0 k22 ; 2

the energy equalities (3.210) and (3.211) imply 2 kukqq22 C2 C2 kruk2 > 0:

Hence from (3.210) we obtain that dE q C2 D kukq22 C2 kruk22 > 0: dt The following inequalities hold: ˇZ ˇ2 ˇ ˇ ˇ dx hru; ru t iˇ kruk2 kru t k2 ; 2 2 ˇ ˇ ˇ2 ˇZ Z ˇ ˇ ˇ dx jujq1 uu t ˇ kukq1 C2 dx jujq1 .u t /2 : q1 C2 ˇ ˇ

(3.212)

We also have ˇZ ˇ2 ˇ ˇ q 1 ˇ u.u C juj u/ t dx ˇ ˇ ˇ

j.ru; ru t /j2 C .q1 C 1/2 j.jujq1 u; u t /j2 C 2.q1 C 1/j.ru; ru t /jj.jujq1 u; u t /j q C2

kruk22 kru t k22 C .q1 C 1/2 .jujq1 ; u2t /kukq11 C2 q 1 C2/=2 C 2.q1 C 1/kruk2 kru t k2 .jujq1 ; u2t /kuk.q q1 C2 Z kru t k22 C .q1 C 1/ dx jujq1 u2t Œkruk22 C .q1 C 1/kukqq11 C2 C2

.q1 C 2/

kru t k22

Z

C .q1 C 1/

q1

dx juj .u t /

2

1 q1 C2 2 kruk2 C .q1 C 1/=.q1 C 2/kukq1 C2 : 2

(3.213)

From (3.210)–(3.213) we obtain 0 2

.E / .q1 C 2/E.t /

kru0 k22

Z C .q1 C 1/

q1

0 2

dx juj .u /

:

318


The following inequalities hold: Z 0 2 kru k2 C .q1 C 1/ dx jujq1 .u0 /2 D

d 1 1d q C2 kukq22 C2 kruk22 q2 C 2 dt 2 dt

d 1 q2 E00 kruk22 q2 C 2 2.q2 C 2/ dt 1 q2 1 " 00 0 2 2 E C kru k2 C kruk2 q2 C 2 q2 C 2 2 2" Z q2 1 1 "q2 00 0 2 q1 0 2 kru E C E: k2 C .q1 C 1/ dx juj .u / C q2 C 2 2.q2 C 2/ q 2C2" Hence from (3.210)–(3.213) we obtain the following second-order ordinary differential inequality (see [210]): EE00 ˛1 .E0 /2 C ˇ1 E2 0; where ˛1 D

q2 q2 C 2 1" ; q1 C 2 2.q2 C 2/

ˇ1 D

q2 ; "

(3.214)

q2 > q1 :

From (3.214) we obtain

1 .E1˛1 /00 C ˇ1 E1˛1 0: ˛1 1

(3.215)

Introduce the notation Z.t / E1˛1 .t /. Then by virtue of the fact that E0 0 we have Z0 .t / D .1 ˛1 /E˛1 E0 0: From (3.215) we obtain .Z0 /2 .Z00 /2 1 Z20 C 1 Z2 ;

1 D ˇ1 .˛1 1/:

(3.216)

Now we require the validity of the condition .Z00 /2 1 Z20 > 0; which is equivalent to the condition 1 /2 1 .E00 /2 > ˇ1 .˛1 1/E.1˛ : .1 ˛1 /2 E2˛ 0 0

(3.217)

From (3.217), owing to (3.210), we obtain ˛1 1 0 2 .E0 / > E20 ; ˇ1

(3.218)

319


with ˛1 > 1 under the condition " 2 .0; 2.q2 q1 /=q2 /. Introduce the notation " q2 q 1 q2 " ˛1 1 D : (3.219) f ."/ D ˇ1 q2 q1 C 2 2.q1 C 2/ The maximum of function (3.219) is reached at the point q2 q 1 q2

"0 D

(3.220)

and is equal to f ."0 / D

.q2 q1 /2 : 2q22 .q1 C 2/

At the point (3.220), inequality (3.218) takes the optimal form p 2q2 .q1 C 2/1=2 1 q1 C 1 q2 C2 q1 C2 2 2 ku0 kq2 C2 > kru0 k2 C k C k kru0 2 ku0 q1 C2 : q2 q 1 2 q1 C 2 Now consider inequality (3.216) at the point (3.220). It takes the form .Z0 /2 A2 C 1 Z2 ; where

(3.221)

A2 .Z00 /2 1 Z20 :

From (3.221) we obtain the inequalities Z0 A;

Z Z0 At:

Hence we directly obtain the last statement of the theorem. Theorem 3.9.2 is proved.

Remark 3.9.3. For problem (3.195), (3.196), which is meant in the strong generalized sense, in the case of sufficiently small initial data, we can prove the global-ontime solvability in the class C .1/ .Œ0; C1I H01 .// under the conditions 0 qi

4 N 2

for N 3

and

0 qi

for N D 1; 2:

Indeed, in the strong generalized sense, we obtain the following equation: dw C w D F.A1 .w//; dt where w D A.u/ D u C jujq1 u;

F.u/ D jujq1 u C jujq2 u:

320


In the class w 2 C .1/ .Œ0; TI H1 .//, we obtain the integral equation w D H.w/ w0 e

t

Z C

0

t

ds e .ts/ F.A1 .w//:

Using successive approximations in the Banach space L1 .0; C1I H1 .// under the condition kw0 k1 ", where " > 0 is sufficiently small, we can prove the existence of a global-on-time solution of the integral equation. Finally, from the properties of smoothing on time of the operator H.v/ we obtain H.v/ W L1 .0; C1I H1 .// ! AC.Œ0; C1I H1 .//; H.v/ W AC.Œ0; C1I H1 .// ! C .1/ .Œ0; C1I H1 .//: Therefore, we obtain the following nonlinear operator equation with the known righthand side: A.u/ D w 2 C .1/ .Œ0; C1I H1 .//: Further, in the same way as above, we can prove that u 2 C .1/ .Œ0; C1I H01 .//:

3.10

Blow-up of solutions of a class of quasilinear wave dissipative pseudoparabolic equations with sources

In this section, we consider a certain class of quasilinear wave pseudoparabolic equations with linear dissipation, i.e., we take into account both nonlinearities of convective nature and linear dissipation. Specifically, we consider the following abstract Cauchy problem: .A0 C L/

du C Lu C DP .u/ D F.u/; dt u.0/ D u0 :

(3.222) (3.223)

3.10.1 Unique local solvability of the problem in the strong sense and blow-up of its solutions Let the operators L, A0 , D, P , and F, the Banach spaces V0 , W1 , W0 , W2 , W3 , and W4 , and the Hilbert space H be defined as in Section 3.2 and the conditions (A0 ), (L), (DP), and (F) hold.

Section 3.10 Blow-up in some classes of quasilinear wave equations

321

Further we assume, that the following conditions are valid.

Conditions (V1). (1) Let the Banach spaces V0 and W1 be continuously embedded in H. (2) The following continuous embeddings hold: ds

ds

ds

ds

V V0 H V0 V ; ds

ds

ds

ds

V Wj H Wj V ;

j D 0; 1;

i.e., V is dense in V0 , V0 is dense in H, V is dense in Wj , and Wj is dense in H. (3) The space V is infinite-dimensional and separable. Here the symbol V denotes the space dual to V . Remark 3.10.1. The condition (V1) implies the fact that the intersection V V0 \ W1 equipped with the norm k k D k k0 C j j1 can be made a Banach space. In the further items of the condition (V), by V we mean just this Banach space. Denote by h; i the duality bracket between the Banach spaces V and V . By the conditions (V1.2), the following equalities hold.

Conditions (V2). hA0 v; wi D hA0 v; wi0

for all v; w 2 V ;

hF.v/; wi D .F.v/; w/0

for all v; w 2 V ;

hLv; wi D .Lv; w/0

for all v; w 2 V :

We give a definition of a strong generalized solution of problem (3.222), (3.223). Definition 3.10.2. A solution of the class C .1/ .Œ0; TI V / satisfying the conditions

du du L ; w C A0 ; w C .Lu; w/1 C hDP .u/; wi D .F.u/; w/0 dt dt (3.224) 1 0 8w 2 V ; 8t 2 Œ0; T; u.0/ D u0 2 V ; where the time derivative is meant in the classical sense, is called a strong generalized solution of problem (3.222), (3.223).

322


In the considered class, the inclusions A0 u 2 C .1/ .Œ0; TI V0 /; DP .u/ 2 C.Œ0; TI V /;

Lu 2 C .1/ .Œ0; TI W1 /; F.u/ 2 C.Œ0; TI W0 /

hold and, therefore, A0 u and Lu belong to the class C .1/ .Œ0; TI V / and DP .u/ and F .u/ belong to the class C.Œ0; TI V /. Thus, problem (3.224) by virtue of the conditions (V) and the conditions of Theorem 3.10.3 below is equivalent to the following problem: hLu0 ; wi C hA0 u0 ; wi C hLu; wi C hDP .u/; wi D hF.u/; wi 8w 2 V ;

8t 2 Œ0; T;

u.0/ D u0 2 V ;

where h; i denotes the duality bracket between the Banach spaces V and V . Using the methods developed in the previous sections, we can prove the following result. Theorem 3.10.3. Let the conditions (A0 ), (L), (DP), and (F) hold. Assume that V W0 W2 and W4 V . Then for any u0 2 V , there exists maximal T0 D T0 .u0 / > 0 such that a unique strong generalized solution of problem (3.222), (3.223) of the class u.t / 2 C .1/ .Œ0; T0 /I V0 \ W1 / exists and either T0 D C1 or T0 < C1 and in this case the limit relation lim kukV0 \W1 D C1

t"T0

(3.225)

holds. Proof. Rewrite Eq. (3.222) meant in the strong generalized sense in the following form: du C u D DP .u/ C F.u/ C A0 u; A dt where A A0 C L. In the class u 2 C .1/ .Œ0; TI V / we have: u; u0 2 C.Œ0; TI V /. Moreover, DP .u/, F .u/, A0 u 2 C.Œ0; TI V /. From the conditions (A0 1) and (L1) we obtain hAu1 Au2 ; u1 u2 i m0 ku1 u2 k20 C d1 ju1 u2 j21 ;

A A0 C L:

Note that, according to the condition (V2), the Banach space V is equipped with the following norm: kvk D kvk0 C jvj1 ; which, by the conditions (A0 ) and (L), is equivalent to the norm kvk D .hA0 v; vi0 C .Lv; v/1 /1=2 :

323

Section 3.10 Blow-up in some classes of quasilinear wave equations

Then we directly obtain that the operator A acts from V into V and has a Lipschitz continuous inverse operator A1 W V ! V with Lipschitz constant equal to 21 min¹m0 ; d1 º. Indeed, hAu1 Au2 ; u1 u2 i 21 min¹m0 ; d1 ºku1 u2 k2 ; kAu1 Au2 k .21 min¹m0 ; d1 º/1=2 ku1 u2 k; kA1 z1 A1 z2 k .21 min¹m0 ; d1 º/1=2 kz1 z2 k : Now consider the operators DP .u/ and F.u/. The following estimates bounds hold: kDP .u1 / DP .u2 /k CjDP .u1 / DP .u2 /j4 CjP .u1 / P .u2 /j3 C2 .R/ju1 u2 j2 ;

kDP .u1 / DP .u2 /k C2 .R/ku1 u2 k;

R D max¹ku1 k; ku2 kº:

Thus, the operator DP .u/ W V ! V is boundedly Lipschitz-continuous. Now consider the operator F.u/. The following estimates hold: kF .u1 / F.u2 /k CjF.u1 / F.u2 /j0 C1 .R/ku1 u2 k: Thus, the operator F.u/ W V ! V is boundedly Lipschitz-continuous. Owing to the properties of the operators in Eq. (3.222) meant in the strong generalized sense, we obtain the equation u.t / D H.u/ D u0 e

t

Z C

0

t

ds e .ts/ ŒA1 A0 u C A1 F.u/ A1 DP .u/:

Consider the equality u D H.u/: Introduce the Banach space L1 .0; TI V / and its closed, convex, bounded subset BR ¹u 2 L1 .0; TI V / W kukT D ess sup kuk Rº: t2.0;T/

Now we prove that the operator H.v/ acts from BR into BR and is a contraction on BR . Indeed, qC1

kH.u/kT ku0 k C CT¹kukT C kukT

We require that R > 0 should be so large that the inequality ku0 k

R 2

1Cq=2

C kukT

º:

324


holds and T > 0 should be sufficiently small for T

1 1 : 2C 1 C Rq C Rq=2

Then kH.u/k R. Now we prove that the operator H.u/ on BR is a contraction. Indeed, kH.u1 / H.u2 /kT CTŒ1 C 1 .R/ C 2 .R/ku1 u2 kT ; where R D max¹ku1 kT ; ku2 kT º. Let the condition T

1 1 2C 1 C 1 .R/ C 2 .R/

hold. Hence 1 kH.u1 / H.u2 /kT ku1 u2 kT : 2 Therefore, a unique solution of the integral equation of the class L1 .0; TI V / exists. Note that H.u/ W L1 .0; TI V / ! AC.Œ0; TI V /;

H.u/ W AC.Œ0; TI V / ! C .1/ .Œ0; TI V /:

Using the standard algorithm of extension of solutions of integral equations with variable upper limit, we obtain that there exists maximal T0 > 0 such that either T0 D C1 or T0 < C1 and in the latter case, the limit relation (3.225) holds. Thus, a unique strong generalized solution of the problem of the class C .1/ .Œ0; T0 /I V / exists. Theorem 3.10.3 is proved. Now we prove the main result of this section. Theorem 3.10.4. Let .F.v/; v/0 c > 0 for all v 2 W0 , jvj0 D 1, and for u0 2 V , let the condition .Lu0 ; u0 /1 C hA0 u0 ; u0 i0
0, by virtue of (3.230) and (3.228) we obtain that, under the condition ˛1 0 ˆ0 < (3.238) ˆ .0/; ˇ the estimate T0 T1 holds, where for the value T1 , an equality of the form (3.226) is valid. Finally, using the arbitrariness of the choice of "0 2 .0; q.q C 1/1 /, we can obtain the optimal condition (3.238). As a result, we obtain Theorem 3.10.4.

3.10.2 Examples We propose certain examples of equations that satisfy the conditions introduced in Theorems 3.10.3 and 3.10.4. In this subsection, we use the simplest Sobolev embedding theorems; the appropriate results can be found, for example, in [108–111]. In all examples, H D L2 ./. Example 3.10.5. @jujq=2C1 @ .u u/ C u C C jujq u D 0; @t @x1 uj@ D 0;

u.x; 0/ D u0 .x/ 2 H01 ./;

where R3 is a bounded domain with smooth boundary @ 2 C .2;ı/ , ı 2 .0; 1. In this case, the operator coefficients satisfy the conditions A0 u Iu W V0 L2 ./ ! L2 ./; Lu u W W1 H01 ./ ! W1 H1 ./; F .u/ jujq u W W0 LqC2 ./ ! L.qC2/=.qC1/ ./ W0 ; P .u/ juj1Cq=2 W W2 LqC2 ./ ! L2 ./ W3 ; Du

@u W L2 ./ W3 ! H1 ./ W4 : @x1

Next, we have V V0 \ W1 D H01 ./;

V D H01 ./ W2 LqC2 ./; N D 3;

W4 H1 ./;

q 2 .0; 4;

V H1 ./:

Example 3.10.6. @jujq=2C1 @ div.jrujq ru/ D 0; .2 u C u/ C u C @t @x1 ˇ @u ˇˇ uj@ D D 0; u.x; 0/ D u0 .x/ 2 H02 ./; @n ˇ@ where R3 is a bounded domain with smooth boundary @ 2 C .4;ı/ , ı 2 .0; 1.

328


The operator coefficients satisfy the conditions A0 u 2 u W V0 H02 ./ ! H2 ./ V0 ; Lu u W W1 H01 ./ ! W1 H1 ./; 1;qC2 ./ ! W 1;.qC2/=.qC1/ ./ W0 ; F .u/ div.jrujq ru/ W W0 W0

P .u/ u1Cq=2 W W2 LqC2 ./ ! L2 ./ W3 ; Du

@u W L2 ./ W3 ! H1 ./ W4 : @x1

Moreover, V V0 \ W1 D H02 ./;

V D H02 ./ W2 LqC2 ./

for N D 3 and W4 H1 ./ H2 ./ V ;

V H02 ./ W0 W01;qC2 ./

for N D 3 and q 2 .0; 4. Example 3.10.7. @jujq=2C1 @ C jujq u D 0; .2 u C u/ C u C @t @x1 ˇ @u ˇˇ uj@ D D 0; u.x; 0/ D u0 .x/ 2 H02 ./; @n ˇ@ where R3 is a bounded domain with smooth boundary @ 2 C .4;ı/ , ı 2 .0; 1. The operator coefficients satisfy the conditions A0 u 2 u W V0 H02 ./ ! H2 ./ V0 ; Lu u W W1 H01 ./ ! W1 H1 ./; F .u/ jujq u W W0 LqC2 ./ ! L.qC2/=.qC1/ ./ W0 ; P .u/ u1Cq=2 W W2 LqC2 ./ ! L2 ./ W3 ; Du

@u W L2 ./ W3 ! H1 ./ W4 ; @x1 V D H02 ./ W0 W01;4 ./;

V D H02 ./ LqC2 ./ W2 ;

N D 3;

W4 H1 ./ H2 ./ D V :

329

Section 3.11 Blow-up in the OBBMB equation

Example 3.10.8.

@ @u @u @u @u C 2 @x2 @x3 @x2 @x3 @x1 @ @u @u div.jruj2 ru/ D 0; C3 @x3 @x1 @x2

@ @ .2 u C u/ C u C 1 @t @x1

uj@

ˇ @u ˇˇ D D 0; @n ˇ@

u.x; 0/ D u0 .x/ 2 H02 ./;

where R3 is a bounded domain with smooth boundary @ 2 C .4;ı/ , ı 2 .0; 1, 1 C 2 C 3 D 0, j1 j C j2 j C j3 j > 0. The operator coefficients satisfy the conditions A0 u 2 u W V0 H02 ./ ! H2 ./ V0 ; Lu u W W1 H01 ./ ! W1 H1 ./; 1;4=3 ./ W0 ; F .u/ div.jruj2 ru/ W W0 W01;4 ./ ! W0

P .u/ 1

@u @u @u @u @u @u e1 C 2 e2 C 3 e3 W @x2 @x3 @x3 @x1 @x1 @x2 W2 W01;4 ./ ! L2 ./ L2 ./ L2 ./ W3 ;

Dv div.v/ W L2 ./ L2 ./ L2 ./ W3 ! H1 ./ W4 : In this case, we have V D H02 ./ W0 W01;4 ./; V D H02 ./ W01;4 W2 ;

3.11

N D 3;

W4 H1 ./ H2 ./ D V ;

N D 3:

Blow–up of solutions of the Oskolkov–Benjamin–Bona–Mahony–Burgers equation with a cubic source

In this section, we obtain sufficient conditions of the global-on-time solvability and the blow-up for a finite time of solutions of the first initial-boundary-value problem for the three-dimensional Oskolkov–Benjamin–Bona–Mahony–Burgers (OBBMB) equation in a bounded domain with smooth boundary: @ .u u/ C u C uux1 C u3 D 0; @t u.x; 0/ D u0 .x/; u.x; t /j@ D 0;

(3.239) (3.240) (3.241)

330


where x D .x1 ; x2 ; x3 / 2 R3 , is a bounded domain with smooth boundary @ 2 C .2;ı/ , ı 2 .0; 1. This problem appears in the study of nonstationary processes in semiconductors in the presence of sources and an external constant homogeneous electric field. The one-dimensional model OBBMB equation was obtained in [37, 69, 312, 313]. There are a wide literature devoted to the OBBMB equation. As is known to the authors, To all appearance, the problem on the blow-up of solution of the OBBMB equation with a cubic source has not been considered. We note the classical work of Levine [260], the ideas of which have been developed by the authors of the present monograph to prove the blow-up of solutions of problem (3.239)–(3.241) under certain conditions for the initial function. We also note that the method of [260] cannot be applied in the case of Eq. (3.239) since Eq. (3.239) includes the convective nonlinearity term uux1 .

3.11.1 Unique local solvability of the problem First, we give a definition of a strong generalized solution. Definition 3.11.1. A solution of the class C .1/ .Œ0; TI H01 .// satisfying the condition hu0 u0 C u C uux1 C u3 ; wi D 0 u.0/ D u0 2

8w 2 H01 ./;

8t 2 Œ0; T;

H01 ./;

(3.242) (3.243)

where z 0 is the classical time derivative of the function z.t / and h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./ (see [347]), is called a strong generalized solution of problem (3.239)–(3.241). Here the boundary condition (3.241) is meant in the sense of solution of problem (3.242)–(3.243) belonging to the class C .1/ .Œ0; TI H01 .//. Introduce the operators Au u C u;

F1 .u/

1 @u2 ; 2 @x1

F2 .u/ u3 :

We denote by k kC1 the norm in the space H01 ./, by k k1 the norm in H1 ./, and by k kp the norm in Lp ./, p 2 Œ1; C1. Subject to the introduced operators in the strong generalized sense, we obtain the following abstract Cauchy problem for the operator differential equation: A

du C Au D u C F1 .u/ C F2 .u/; dt

u.0/ D u0 2 H01 ./:

(3.244)

331


It is easy to verify that the operator A W H01 ./ ! H1 ./ has a Lipschitz-continuous inverse operator: kA1 z1 A1 z2 k1 kz1 z2 kC1

8zi 2 H01 ./:

Therefore, in the strong generalized sense, problem (3.244) is equivalent to the abstract integral equation Z t ds e .ts/ A1 .u C F1 .u/ C F2 .u// : (3.245) u.t / D u0 e t C 0

Using the method of contraction mappings and the smoothing properties of the operator in the right-hand side of the integral equation (3.245), we prove the following theorem. Theorem 3.11.2. For any u0 2 H01 ./, there exists T0 D T0 .u0 / > 0 such that a unique strong generalized solution u.t / of problem (3.239)–(3.241) from C .1/ .Œ0; T0 /I H01 .// exists and either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim kukC1 D C1 t"T0

holds. Proof. Introduce the operator H.u/ D u0 e

t

C

Z

t 0

ds e .ts/ A1 .u C F1 .u/ C F2 .u// :

The following estimates hold: kF1 .u1 / F1 .u2 /k1 Cku21 u22 k2 Z 1=2 C dx max¹ju1 j2 ; ju22 jºju1 u2 j2

C max¹ku1 k4 ; ku2 k4 ºku1 u2 k4 1 .R/ku1 u2 kC1 ; kF2 .u1 / F2 .u2 /k1 Cku31 u32 k4=3 3=4 Z 8=3 8=3 4=3 C dx max¹ju1 j ; ju2 j ºju1 u2 j

C max¹ku1 k24 ; ku2 k24 ºku1 u2 k4 2 .R/ku1 u2 kC1 ; where R D C max¹ku1 k2C1 ; ku2 k2C1 º. In the Banach space L1 .0; TI H01 .//, we introduce the closed, bounded, convex subset ® ¯ BR u 2 L1 .0; TI H01 .// W kukT D ess sup kukC1 R : t2.0;T/

332


We prove that the operator H.u/ acts from BR into BR . Indeed, kH.u/kT ku0 kC1 C CTR¹1 C R C R2 º: Now we require that R > 0 should be sufficiently large for the validity of the inequality R ku0 kC1 ; 2 and T > 0 should be sufficiently small for the validity of the inequality T

1 1 : 2C 1 C R C R2

Now prove that the operator H.u/ is a contraction on BR . Indeed, kH.u1 / H.u2 /kT CRŒ1 C R C R2 ku1 u2 kT : If T

1 1 ; 2C 1 C R C R2

then we have 1 kH.u1 / H.u2 /kT ku1 u2 kT : 2 Therefore, a unique solution of the integral equation (3.245) of the class L1 .0; TI H01 .// exists. Using the standard algorithm of extension of solutions of integral equations with variable upper limit, we obtain that there exists maximal T0 > 0 such that either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim kukC1 D C1

t"T0

holds. Note that H.u/ W L1 .0; TI H01 .// ! AC.Œ0; TI H01 .//; H.u/ W AC.Œ0; TI H01 .// ! C .1/ .Œ0; TI H01 .//: Therefore, u.x; t / 2 C .1/ .Œ0; C1/I H01 .//: Theorem 3.11.2 is proved.

333


3.11.2 Global solvability and the blow-up of solutions Introduce the notation ‰.t / kruk22 C kuk22 ;

‰0 D ‰.0/;

C1 is the best constant of the embedding H01 ./ L2 ./: kzk2 C1 kzkC , C2 is the best constant of the embedding H01 ./ L4 ./: kzk4 C2 kzkC for any z 2 H01 ./. The following theorem holds. Theorem 3.11.3. Let u0 2 H01 ./. Then the following assertions hold: (1) if p kru0 k22 C ku0 k22 < . 2 1/2 .ku0 k22 C ku0 k44 /; kru0 k22 C ku0 k22 >

C42 Œ1

1 ; C C21

then T0 2 ŒT1 ; T2 and the limit relation lim kukC1 D C1

t"T0

holds, where 1 1 1 C C21 1 ln 1 ; 2 1 C C21 C42 ‰0 ku0 k22 C kru0 k22 1 1 ; T2 D ln 1 p ˇ . 2 1/2 ku0 k22 C ku0 k44 p p 2 2 I ˛ D 3 2; ˇ D p 21

T1 D

(2) if ‰0 then T0 D C1 and ‰.t /

1 ; Œ1 C C21 C42 1 : Œ1 C C21 C42

Proof. We pass to the new function v D ue t in problem (3.242), (3.243). Then from Eq. (3.242) we obtain hv 0 v 0 C e t vvx1 C e 2t v 3 C v; wi D 0 8w 2 H01 ./:

(3.246)

334


As w, we take the function v; then after integrating by parts we obtain the first energy equality 1d Œkrvk22 C kvk22 D kvk22 C e 2t kvk44 : 2 dt

(3.247)

Now we take w D v 0 ; then after integrating by parts we obtain the second energy equality Z 1d 1 2t d 1 t 0 2 0 2 2 4 dx v 2 vx0 1 t : (3.248) kvk2 C e kvk4 e krv k2 C kv k2 D 2 dt 4 dt 2 Introduce the energy functional depending on time t 2 Œ0; T0 /: ˆ.t / krvk22 .t / C kvk22 .t /:

(3.249)

By Theorem 3.3.2 the solution v.x; t / of Eq. (3.246) belongs to the class C .1/ .Œ0; T0 /I H01 .//. The energy equalities (3.247) and (3.248) imply that ˆ.t / 2 C.2/ .Œ0; T0 //. Using the Cauchy–Schwarz inequality (see [293]) for the function ˆ.t / we obtain the following inequality: 1 dˆ 2 ˆ Œkrv 0 k22 C kv 0 k22 : (3.250) 4 dt From the second energy equality (3.248), we obtain an upper estimate for the value of krv 0 k22 C kv 0 k22 . Using the Hölder inequalities, we obtain ˇ ˇZ ˇ ˇ 2 0 ˇ dx v v ˇ 1 ".t /krv 0 k2 C 1 kvk4 ; (3.251) x1 t ˇ 2 4 ˇ 2 2".t / where ".t / > 0 is a certain function continuous on R1C . Moreover, we require the validity of the following inequality: ˇ ˇ ˇd ˇ ˇ kvk2 ˇ 1 kvk2 C "0 kv 0 k2 ; "0 > 0: (3.252) 2 2 2 ˇ dt ˇ " 0 From inequality (3.248), owing to (3.247), (3.249), (3.251), and (3.252), we obtain the following inequalities: d 1 2t d 1 0 2 0 2 2t dˆ e e 2t .e 2t kvk22 / krv k2 C kv k2 e 8 dt dt 4 dt e t ".t /e t 1 d kvk22 C kvk44 C krv 0 k22 2 dt 4".t / 4 1d 1 2t d et 2t dˆ e C e kvk22 C ˆ0 8 dt dt 4 dt 8".t / C

C

e t ".t /Œkrv 0 k22 C kv 0 k22 : 4

(3.253)

335


From (3.253), owing to inequality (3.252), where ".t / D "0 e t , "0 2 .0; 1/, we obtain the inequality h 1 "0 i 1 1 0 2 0 2 ˆ0 C ˆ00 : 1 1C (3.254) krv k2 C kv k2 2 4 "0 8 From (3.250) and (3.254) we obtain the inequality (see [210]) ˆˆ00 ˛.ˆ0 /2 C ˇˆˆ0 0;

(3.255)

where

2 ; "0 2 .0; 1/: "0 Note that, by virtue of the condition "0 2 .0; 1/, the inequalities ˛ > 1 and ˇ > 0 are valid. Moreover, by (3.247) and the condition ˆ0 > 0, we have that ˆ0 .t / > 0 for t 2 Œ0; T0 /. Subject to this inequality, (3.255) can be easily integrated and we obtain the inequality ˛ 2 "0 ;

ˇ 2C

ˆ1˛ ˆ1˛ 0 ƒD

2 2ˆ˛ 0 Œku0 k2

C

˛1 ƒŒ1 e ˇ t ; ˇ ku0 k44 ;

(3.256)

ˆ0 D ˆ.0/:

Now we require the validity of the inequality ˆ0
ab 1 D

C42 Œ1

1 ; C C21

then from (3.264) we obtain the lower estimate of the blow-up time under condition (3.259): 1 1 1 C C21 1 ln 1 : T1 D 2 1 C C21 C42 ‰0 Theorem 3.11.3 is proved.

337

Section 3.12 Blow-up in generalized BBMB equation with pseudo-Laplacian

3.11.3 Physical interpretation of the obtained results The obtained results imply the fact that under sufficiently small initial energy reserved in the semiconductor, the energy stays finite for all future time moments. On the other hand, under sufficiently large initial energy, there exists a time moment such that the system energy becomes infinite, i.e., the disruption of the semiconductor occurs. The obtained qualitative picture of nonstationary phenomena in bounded semiconductors coincides with the results of experiment (see [54, 55]).

3.12

On generalized Benjamin–Bona–Mahony–Burgers equation with pseudo-Laplacian

In this section, we obtain a sufficient condition of blow-up of solutions of the first initial-boundary-value problem for the three-dimensional Oskolkov–Benjamin–Bona– Mahony–Burgers equation with pseudo-Laplacian p , where p D 4, in a bounded domain with smooth boundary: @ .u u/ C u C ux1 C uux1 div.jruj2 ru/ D 0; @t u.x; 0/ D u0 .x/; u.x; t /j@ D 0;

(3.265) (3.266) (3.267)

where x D .x1 ; x2 ; x3 / 2 R3 , is a bounded domain with smooth boundary @ 2 C .2;ı/ , ı 2 .0; 1. This problem appears in the study of nonstationary processes in semiconductors in the presence of sources and an external constant homogeneous electric field. As is known to the authors, the blow-up of solution of BBMB equation with cubic sources and 4-Laplacian has not been studied. We emphasize the classical work of Levine [260], the ideas of which have been developed by the authors of the present monograph to prove the blow-up of solution of problem (3.265)–(3.267) under certain conditions for the initial function. We note that the method of [260] cannot be applied in the case of Eq. (3.265) since Eq. (3.265) includes the convective nonlinearity term uux1 . On the other hand, Eq. (3.265) does not satisfy the general conditions introduced in our work [238], as far as in Eq. (3.265) linear dissipation u and convective nonlinearity uux1 are taken into account together. Therefore, obtaining of sufficient conditions of the blow-up of solutions of problem (3.265)–(3.267) is of undoubted interest.

3.12.1 Blow-up of strong generalized solutions We give definitions of a strong generalized solution and a weakened solution of the considered problem.

338


Definition 3.12.1. A solution of the class C .1/ .Œ0; TI W01;4 .// satisfying the condition hu0 u0 C u C uux1 C ux1 div.jruj2 ru/; wi D 0; u.0/ D u0 2 W01;4 ./ 8w 2 W01;4 ./;

8t 2 Œ0; T;

(3.268) (3.269)

where z 0 is the classical time derivative and h; i is the duality bracket between the Banach spaces W01;4 ./ and W 1;4=3 ./, is called a strong generalized solution of problem (3.265)–(3.267) (see [347]). Here the boundary condition (3.267) is meant in the sense that problem (3.268), (3.269) belongs to the class C .1/ .Œ0; TI W01;4 .//. Introduce the notation ˆ.t / kruk22 C kuk22 ;

ˆ0 D ˆ.0/:

The following theorem holds. 1;p Theorem 3.12.2. Let u.x; t / 2 C .1/ .Œ0; T00 /I W0 .// be a strong generalized solution and let the inequality p kru0 k22 C ku0 k22 < . 5 2/2 .ku0 k22 C kru0 k44 /

hold. Then T00 < C1 and the upper estimate kru0 k22 C ku0 k22 1 1 ; T00 T1 ln 1 p ˇ . 5 2/2 ku0 k22 C kru0 k44 p 2 5 ; ˇDp 52 is valid. Proof. We pass to the new function v D ue t in problem (3.268), (3.269). Then from Eq. (3.268) we obtain hv 0 v 0 C vx1 C e t vvx1 C v e 2t div.jrvj3 rv/; wi D 0

(3.270)

for all w 2 W01;4 ./. We take w D v in Eq. (3.270); then after integrating by parts we obtain the first energy equality 1d Œkrvk22 C kvk22 D kvk22 C e 2t krvk44 : 2 dt

(3.271)

Now let w D v 0 ; then after integrating (3.270) by parts we obtain the second energy equality Z Z 1 d e t e 2t d 0 2 0 2 2 0 krv k2 C kv k2 D kvk2 dx vx1 t v krvk44 : dx v 2 vx0 1 t C 2 dt 2 4 dt (3.272)

339

Section 3.12 Blow-up in generalized BBMB equation with pseudo-Laplacian

From (3.272), owing to (3.271), we obtain the inequality Z Z 1d e t 0 2 0 2 2 0 vx1 t v dx dx v 2 vx0 1 t kvk2 krv k2 C kv k2 D 2 dt 2 e 2t d 2t e 2t d 2t dˆ e .e kvk22 / C 8 dt dt 4 dt Z Z e t 1d 2 0 dx vx1 t v dx v 2 vx0 1 t kvk2 4 dt 2 2t dˆ d e e 2t : (3.273) C 8 dt dt In what follows, we need the auxiliary estimates Z 1 d dx vv 0 kvk22 C "0 kv 0 k22 ; kvk22 D 2 dt " 0 Z " 1 0 dx vx0 1 t v krv 0 k22 C kvk22 ; 2 2"0 Z ".t / 1 krv 0 k22 C kvk44 dx v 2 vx0 1 t 2 2".t / ".t / 1 krv 0 k22 C B4 krvk44 ; 2 2".t /

(3.274) (3.275)

(3.276)

where B is a constant of embedding W01;4 ./ L4 ./: kwk4 Bkrwk4 for any w 2 W01;4 ./. For simplicity, we assume that B D 1. The interested reader can consider the general case according to the scheme proposed below. Owing to (3.271) and (3.274)–(3.276), we obtain from (3.273) the inequalities krv 0 k22 C kv 0 k22

1 dˆ "0 "0 C Œkrv 0 k22 C kv 0 k22 C Œkrv 0 k22 C kv 0 k22 8"0 dt 4 2 1 dˆ e t C ".t /Œkrv 0 k22 C kv 0 k22 4"0 dt 4 1 t dˆ e 2t d 2t dˆ e C e : C 8".t / dt 8 dt dt

C

Now we take "0 e t as ".t /; then from (3.277) we obtain 1 d 2ˆ 1 1 dˆ 0 2 0 2 C C : .1 "0 /Œkrv k2 C kv k2 8 dt 2 4 2"0 dt

(3.277)

(3.278)

By virtue of the Cauchy–Schwarz inequality (see [293]), for the value ˆ.t / 2 C 2 Œ0; T00 / the inequality 1 dˆ 2 ˆŒkrv 0 k22 C kv 0 k22 4 dt

340


holds. Hence, owing to (3.278), we obtain (see [210]) ˆˆ00 ˛.ˆ0 /2 C ˇˆˆ0 0;

(3.279)

where

4 ; "0 2 .0; 1=2/: "0 Note that, by virtue of the condition "0 2 .0; 1=2/, we have ˛ > 1 and ˇ > 0. Moreover, by (3.271) and the definition of the function ˆ.t /, we obtain that ˆ0 .t / > 0 for t 2 Œ0; T00 /. Due to this fact, inequality (3.279) can be easily integrated, and for t 2 Œ0; T00 / we have the inequality ˛ D 2 .1 "0 / ;

ˇ D2C

ˆ1˛ ˆ1˛ 0

˛1 ƒŒ1 e ˇ t ; ˇ

2 4 ƒ D 2ˆ˛ 0 Œku0 k2 C kru0 k4 ;

(3.280)

ˆ0 D ˆ.0/:

Now we require the validity of the inequality ˆ0
p2 , then T0 D C1 and the upper estimate C .1 ˛/C2 t 1=.1˛/ ; ˆ.t / Œˆ1˛ 0

˛D

p2 ; p1

is valid; (3) if p1 < p2 and p2 kru0 k22 ; 2 h i p1 1 > 1C kru0 kpp11 ; kru0 k22 C 2 p1

kru0 kpp22 > kru0 kpp22

then T0 2 .0; T1 and the limit relation (3.287) is valid, where p p2 =p1 p1 .p2 2/ 2p1 p ; C2 D C1 2 ; D p1 1 .p2 p1 / T1 D

1 ˆ1C˛ 0

.˛1 1/2 Œ.ˆ0 .0//2

ˇ 2 ˛1 1 ˆ.0/

;

ˆ0 .0/ D kru0 kpp22 kru0 k22 ; 1 p1 1 kru0 kpp11 ; ˆ0 D ˆ.0/ D kru0 k22 C 2 p1 ˛1 D

p2 C p1 ; 2p1

ˇD

.p2 2/2 ; p2 p1

C1 is the best constant of the embedding W01;p1 ./ W01;p2 ./ under the condition p1 p2 .

Section 3.13 Blow-up of solutions of one problem with pseudo-Laplacian

343

Proof. In Eq. (3.285), we take as w the function 1;p

u.x; t / 2 C .1/ .Œ0; T0 /I W0

.//I

then after integrating by parts we obtain the first energy equality dˆ C kruk22 D krukpp22 : dt

(3.288)

If we take as w the function u0 , then after integrating by parts we obtain the second energy equality kru0 k22 C .p1 1/

Z

dx jrujp1 2 jru0 j2 D

1 d 1d krukpp22 kruk22 : p2 dt 2 dt (3.289)

Let the condition 1 1 kru0 kpp22 kru0 k22 < 2 p2

(3.290)

hold; then from (3.289) 1 1 krukpp22 : kruk22 < 2 p2 Hence in its turn, we have kruk22 < krukpp22 : Therefore, the first energy equality (3.288) and (3.290) yield ˆ0 .t / > 0;

t 2 Œ0; T0 /: 1;p

Finally, in the class u.x; t / 2 C .1/ .Œ0; T0 /I W0 .2/ C Œ0; T0 /. Now consider the value .ˆ0 /2 :

.//, (3.288) implies ˆ.t / 2

Z .ˆ0 /2 p ku0 k22 C kru0 k22 C .p 2/ dx jrujp4 j.ru0 ; ru/j2 Z C

dx jruj

p2

0 2

jru j

(3.291)

(3.292)

1 p1 1 2 2 p kuk2 C kruk2 C krukp : 2 2 p

The following estimates are valid: .ˆ0 /2 pˆJ;

(3.293)

344


where JD

ku0 k22

C

kru0 k22

Z

C .p 2/

Z C

dx jrujp2 jru0 j2

dx jrujp4 j.ru0 ; ru/j2 :

(3.294)

Owing to (3.289), from (3.294) we obtain JD

p2 2 d 1 d Œkrukpp22 kruk22 kruk22 : p2 dt 2p2 dt

(3.295)

Now we use the inequality 1d " 1 kruk22 kru0 k22 C kruk22 I 2 dt 2 2" then from (3.289), (3.294), (3.295), and the definition of the function ˆ.t / we obtain 1 00 p2 2 p2 2 1 J 1" ˆ .t / C ˆ.t /: (3.296) 2p2 p2 p2 " From (3.11) and (3.296) we obtain the second-order ordinary differential inequality (see [210]) ˆ00 ˆ ˛1 .ˆ0 /2 C ˇˆ2 0; where

(3.297)

p2 p2 2 1" ; ˛1 D p1 2p2 p2 2 p2 p1 ˇD ; " 2 0; 2 ; p1 < p2 : " p2 2

Note that for p2 > p1 we have ˛ > 1 and ˇ > 0. Introduce the new function Z D ˆ1˛1 , Z1 D .1 ˛1 /ˆ˛1 ˆ0 0, by (3.291). Equation (3.297) written for the function Z has the form Z00 .˛1 1/ˇZ 0:

(3.298)

Multiplying both sides of inequality (3.298) by the function Z0 , owing to the fact that this function is of constant sign, after integrating we obtain .Z0 /2 .˛1 1/ˇZ2 A2 ; where

1 1 1=2 .ˆ0 .0//2 .˛1 1/ˇˆ22˛ >0 A D Œ.˛1 1/2 ˆ2˛ 0 0

(3.299)

Section 3.14 Blow-up in higher-order generalized Boussinesq equation

under the condition ˆ0 .0/ >

ˇ ˛1 1

345

1=2 ˆ.0/:

Then from (3.299) we conclude that Z0 A;

Z.t / Z0 At:

(3.300)

From (3.300) we have that the function ˆ.t / for the finite time T0 2 .0; T1 , T1 D A1 Z0 , becomes equal to infinity. Thus, the third result of the theorem is proved. Now let p1 p2 . Consider the first energy equality (3.288). The following inequalities hold: p2 =p1 p 2 dˆ p1 p2 p p2 krukp2 C1 krukp1 C1 2 ˆ D C2 ˆ ; (3.301) dt p1 1 where D p2 =p1 . From (3.301) we easily obtain the following estimates: ˆ.t / ˆ0 e C2 t ; p2 D p1 ; 1 p pp p1 p2 1 2 p1 p2 p1 C C2 t ; p1 > p2 : ˆ.t / ˆ0 p1 From these estimates and Hypothesis 3.13.2 we have that the statements (1) and (2) of the theorem are valid.

3.13.2 Physical interpretation of the obtained results As we have already mentioned, problem (3.283), (3.284) is a model that describes nonstationary processes in a semiconductor with negative differential conductivity and nonlinear dependence of the dielectric permittivity coefficient. The qualitative picture obtained in Theorem 3.13.3 is confirmed experimentally as the phenomenon of electric disruption watched in semiconductors (see [54, 55]).

3.14

Sufficient, close to necessary, conditions of the blow-up of solutions of strongly nonlinear generalized Boussinesq equation

In this section, we consider sufficient, close to necessary, conditions of the blow-up of solutions of the first initial-boundary-value problem in a bounded domain for the strongly nonlinear generalized Boussinesq equation: @ .u u jujq u/ C u C u.u C ˛/.u ˇ/ D 0; ˛; ˇ > 0; @t uj@ D 0; u.x; 0/ D u0 .x/ 2 H01 ./;

(3.302) (3.303)

where is a bounded domain in R3 with smooth boundary @ 2 C 2;ı , ı 2 .0; 1.

346


Definition 3.14.1. A solution of the class C .1/ .Œ0; TI H01 .// satisfying the conditions

@ q .u u juj u/ C u C u.u C ˛/.u ˇ/; v D 0; @t u.x; 0/ D u0 .x/ 2 H01 ./ for all v 2 H01 ./ and all t 2 Œ0; T, with the time derivative meant in the classical sense, is called a strong generalized solution of problem (3.302), (3.303). The following theorem holds. Theorem 3.14.2. For any u0 2 H01 ./ under the condition q 2 Œ1; 4, there exists 0 < T0 such that a unique strong generalized solution of problem (3.302), (3.303) from C .1/ .Œ0; T0 /I H01 .// exists and either T0 D C1 or T0 < C1, and in the latter case we have lim sup kruk2 D C1:

(3.304)

t"T0

Proof. Consider the operators A0 u u C u W H01 ./ ! H1 ./; Lu u W H01 ./ ! H1 ./; A1 .u/ D jujq u W H01 ./ LqC2 ./ ! L.qC2/=.qC1/ ./ H1 ./; Fm .u/ D um W LmC1 ./ ! L.mC1/=m ./;

m D 1; 3:

Analyze the properties of the operators Fm .u/: kF3 .u1 / F3 .u2 /k1 CkF3 .u1 / F3 .u2 /k4=3 3 .R/ku1 u2 k4 ; 2

3 .R/ D CR ;

R D max¹ku1 k4 ; ku2 k4 º:

Therefore, kF3 .u1 / F3 .u2 /k1 3 .R/ku1 u2 kC1 ;

R D max¹ku1 kC1 ; ku2 kC1 º:

Similarly we can prove that kF2 .u1 / F2 .u2 /k1 2 .R/ku1 u2 kC1 ; kF1 .u1 / F1 .u2 /k1 Cku1 u2 kC1 ;

347


where R D max¹ku1 kC1 ; ku2 kC1 º. Now consider the operator A D A0 C A1 : hA.u1 / A.u2 /; u1 u2 i D ku1 u2 k22 C kru1 ru2 k22 Z C dx .ju1 jq u1 ju2 jq u2 /.u1 u2 /

ku1 u2 k2C1 : Therefore, by the Browder–Minty theorem, there exists a Lipschitz-continuous inverse operator A1 W H1 ./ ! H01 ./ with Lipschitz constant equal to 1. Similarly we can prove that for the operator L W H01 ./ ! H1 ./, there exists a Lipschitz-continuous inverse operator with Lipschitz constant equal to 1. We perform the substitution w D A.v/; then, in the strong generalized sense, owing to the properties of the operator coefficients in the class w 2 C .1/ .Œ0; TI H1 .// we obtain dw D LA1 .w/ C F3 .A1 .w// C .˛ ˇ/F2 .A1 .w// ˛ˇA1 .w/: dt In the class w 2 C .1/ .Œ0; TI H1 .//, we obtain the integral equation Z t w D w0 C ds ŒLA1 .w/ C F3 .A1 .w// C .˛ ˇ/F2 .A1 .w// ˛ˇA1 .w/ 0

H.w/: Now in the Banach space L1 .0; TI H1 .// we take the closed, convex, and bounded subset ® ¯ BR w 2 L1 .0; TI H1 .// W jkwkj D ess sup kwk1 R : t2.0;T/

We prove that the operator H.w/ acts from BR into BR and is a contraction on BR . Indeed, jkH.w/kj kw0 k1 C CT¹jkwkj C jkwkj2 C jkwkj3 º

R R C DR 2 2

under the condition kw0 k1

R ; 2

T

1 1 : 2C 1 C R C R2

Now we prove that the operator H.w/ is a contraction on BR . Indeed, jkH.w1 / H.w2 /kj CT.1 C R C R2 /jkw1 w2 kj; R D max¹jkw1 kj; jkw2 kjº;

348


and under T

1 1 2C 1 C R C R2

we have

1 jkH.w1 / H.w2 /kj jkw1 w2 kj: 2 Therefore, the operator H.w/ is a contraction on BR . Thus, there exists a unique solution of the integral equation of the class L1 .0; TI H1 .//. Using the standard algorithm of extension on time of solutions of integral equations with variable upper limit, we obtain that u 2 L1 .0; T0 I H1 .//, and either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim sup kwk1 D C1 t"T0

holds. Using the time smoothness properties of the operator H.w/, we have H.w/ W L1 .0; TI H1 .// ! AC.Œ0; TI H1 .//; H.w/ W AC.Œ0; TI H1 .// ! C .1/ .Œ0; TI H1 .// for all T 2 .0; T0 /. So we obtain the following nonlinear equation with the known right-hand side: A.v/ D w 2 C .1/ .Œ0; TI H1 .//;

A.v/ D v C v C jvjq v:

By the properties of the operators A0 and A1 .v/ proved above we have v D A1 .w/ and, moreover, kv.t / v.t0 /kC1 D kA1 .w/.t / A1 .w/.t0 /kC1 kw.t / w.t0 /k1 ! C0; as t ! t0 . Therefore, v.t / 2 C.Œ0; TI H01 .// for any T 2 .0; T0 / and, moreover, the limit relation (3.304) holds. Consider the Fréchet derivative of the operator A1 .v/ D jvjq v: A01;v .v/h D .q C 1/jvjq h: Now we analyze the properties of the operator A01;v .v/ W L.LqC2 ./I L.qC2/=.qC1/ .// L.H01 ./I H1 .// for fixed v 2 LqC2 ./. Note that the Carathéodory function jzjq generates the Nemytskii operator that acts from LqC2 ./ into L.qC2/=q ./. Therefore, by the properties of Nemytskii operators (see [417]), we obtain that if zn ! z strongly in LqC2 ./, then jzn jq ! jzjq strongly in L.qC2/=q ./, i.e., Z ˇ ˇ.qC2/=q dx ˇjzn .x/jq jz.x/jq ˇ ! C0

349


as n ! C1. We now prove that kA01;v .v/ A01;v .vn /kH1 !H1 ! C0 0

as vn ! v strongly in LqC2 ./. Indeed, kA01;v .v/ A01;v .vn /kH1 !H1 D 0

Z .q C 1/

sup khkqC2 D1

sup krhk2 D1

kA01;v .v/h A01;v .vn /hk1

ˇ.qC2/=q ˇ dx ˇjvjq jvn jq ˇ

q=.qC2/Z

dx jhj

qC2

1=.qC2/

Ckjvjq jvn jq k.qC2/=q ! C0 as vn ! v strongly in LqC2 ./. First, consider the Fréchet derivative of the operator A: A0v .v/ D A0 C A01;v .v/: By the conditions of the theorem we have hA0v .v/v1 A0v .v/v2 ; v1 v2 i kv1 v2 k2C1 for any v; v1 ; v2 2 H01 ./. Therefore, for any fixed v 2 H01 ./, there exists a Lipschitz-continuous inverse operator ŒA0v 1 2 L.H1 ./I H01 .//. Consider the equation A0 v C A1 .v/ D w.x; t / 2 C .1/ .Œ0; T0 /I H01 .//: For further consideration, we must prove that the operator O D I C B; O C

0 BO D A1 0 A1;v .v/

has a bounded inverse operator. Denote by A01;v .v/ the Fréchet derivative on the fixed element v 2 C.Œ0; T0 /I H01 .// of the operator A1 . Indeed, consider the equation O ŒI C Bw D f 2 H01 ./: Prove that this equation has only trivial solution. For this, we apply the operator A0 to both sides of this equation and obtain A0u w D A0 w C A1;v .v/w D A0 f

8v 2 H01 ./:

But we have proved above that for the operator A0v , there exists a Lipschitz-continuous inverse operator ŒA0v 1 W H1 ./ ! H01 ./. Now we apply this operator to both sides of the latter equality. Therefore, a solution of the equation exists for any f 2 H01 ./. It is easy to prove the uniqueness. Therefore, there exists an inverse operator C 1 .

350


In the class v 2 C .1/ .Œ0; T0 /I H01 .//, the equation A.v/ D w is equivalent to the following problem: ŒA0 C A01;v .v/v 0 D w 0 2 C.Œ0; T0 /I H01 .//;

v D A1 .w/:

We apply the operator A1 0 to the latter equation and obtain 0 O 0 D A1 ŒI C Bv 0 w :

Introduce the operator O D I C BO W H1 ./ ! H1 ./; C 0 0 for which, as we have already proved above, there exists a linear, bounded inverse operator. Therefore, we obtain 0 O 1 A1 v0 D C 0 w :

Now we must prove only that 0 1 O 1 A1 v0 D C 0 w 2 C.Œ0; T0 /I H0 .//

for fixed u 2 C.Œ0; T0 /I H01 .//. Indeed, the following inequalities hold: 0 0 O 1 .t0 /A1 kv 0 .t / v 0 .t0 /k0 kC 0 Œw .t / w .t0 /k0 0 O 1 .t0 / C O 1 .t //A1 C k.C 0 w .t0 /k0

O 1 .t0 / C O 1 .t /k 1 Ckw 0 .t / w 0 .t0 /k0 C CkC H !H1 : 0

0

O is continuous and, Note that the linear (for fixed v 2 C.Œ0; T0 /I H01 .//) operator C O 1 is linear and therefore, by the inverse-mapping Banach theorem, the operator C continuous and, therefore, bounded owing to the linearity. Thus, we can use the specO 1 W H1 ./ ! H1 ./. tral representation for the linear bounded operator C 0 0 O First, we introduce the resolvent of the operator C: O D .I C/ O 1 : R.; C/ Let be a circle jj D r with sufficiently large radius, greater than sup t2Œt0 ";t0 C"

O 1 kCk H !H1 : 0

0

351


The introduced value is well defined since, under t 2 Œt0 "; t0 C " Œ0; T0 /, the inequality sup kukC1 < C1 t2Œt0 ";t0 C"

O 1 .t / and holds. Now we can use the spectral representation for the operators C 1 O C .t0 / with the same contour introduced above: Z Z 1 1 1 1 1 O O O O 0 //: C .t / D d R.; C.t //; C .t0 / D d 1 R.; C.t 2 i 2 i Obviously, we have O 1 .t0 / D 1 O 1 .t / C C 2 i

Z

O // R.; C.t O 0 //: d 1 ŒR.; C.t

Now we use the well known representation for the operator resolvents: O 0 // O // R.; C.t O 0 // D R.; C.t R.; C.t

C1 X

O / C.t O 0 //R.; C.t O 0 /n Œ.C.t

nD1

under the condition O 0 / C.t O /k 1 O kC.t H !H1 kR.; C.t0 //kH1 !H1 ı < 1: 0

0

0

0

The following inequality holds: O // R.; C.t O 0 //k 1 kR.; C.t H !H1 0

O 0 //k 1 kR.; C.t H !H1 0

Note that

0

C1 X nD1

0

O 0 //kn 1 O // C.t O 0 //kn 1 kR.; C.t kC.t : H !H1 H !H1 0

0

0

0

0 0 O / C.t O 0 / D A1 C.t 0 ŒA1;u .u.t // A1;u .u.t0 //:

By the continuity of the Fréchet derivatives Aj;u with respect to u 2 H01 ./ and by the fact that u 2 C.Œ0; T0 /I H01 .//, we have O / C.t O 0 /kV !V kA0 .u.t // A0 .u.t0 //k 1 kC.t 1;u 1;u 0 0 H !H1 ! C0; 0

O // R.; C.t O 0 //k 1 kR.; C.t H !H1 ! 0; 0

0

O /1 C.t O 0 /1 k 1 kC.t H !H1 ! C0 0

as t ! t0 . Therefore, u 2 C .1/ .Œ0; T0 /I H01 .//.

0

352


Note that under the condition q 1, the operator C.u/ is boundedly Lipschitzcontinuous and, therefore, the operator C 1 .u/ and hence the equation 0 O 1 A1 v0 D C 0 w

has a local solution in the class u.t / 2 C .1/ .Œ0; T00 /I V0 /. It is easy to prove that T0 D T00 . Theorem 3.14.2 is proved. Introduce the following functional with the sense of energy: 1 qC1 1 qC2 kukqC2 : ˆ.t / D kruk22 C kuk22 C 2 2 qC2

(3.305)

Now we prove the main result of this section. Theorem 3.14.3. Let all the conditions of Theorem 3.14.2 hold. Then the following assertions hold: (1) if q > 2, then T0 D C1 and the inequality

q2 qC2

ˆ.t / ˆ0

q2 C Dt qC2

qC2 q2

is valid; (2) if q D 2, then T0 D C1 and the inequality ˆ.t / ˆ0 exp.Dt / is valid; (3) if 0 < q < 2 and Z 1 ˛ˇ ˛ˇ 1 4 2 2 dx u30 ; ku0 k4 > kru0 k2 C ku0 k2 4 2 2 3 ˇ1 2 1C 4q .ˆ0 .0//2 > ˆ20 C ˆ0 2 ; ˛1 1 2˛1 3 C q=2 then T0 2 ŒT1 ; T2 and the limit relation lim sup kruk2 D C1 t"T0


353

(see (3.304)) holds, where 1 T1 D B1 T2 D ˆ0 1˛1 A1 ; 3 ˆ0 ; 1=2 ˇ1 2 1C 4q 2 2˛1 0 2 2 2 ; ˆ ˆ .ˆ .0// A D .˛1 1/ ˆ0 ˛1 1 0 2˛1 3 C q=2 0 .2 q/2 4 40.1 C ˛ 2 ˇ 2 / 20B2 ˛1 D 1 ; ˇ1 D ; D ; q 2 .0; 2/; 2 qC2 32 .2 q/ .2 q/2 .4q/=2 j˛ ˇj2 B4q j˛ ˇj2 1 2 B2 D ; B3 D 1 C 4B44 ; 2.q C 1/ 2˛ˇ 4 q C 2 qC2 4 j˛ ˇj2 B5 ; DD 1C 2˛ˇ qC1 Z ˆ0 D ˆ.0/; ˆ0 .0/ D ku0 k44 kru0 k22 C .˛ ˇ/ dx u30 ˛ˇku0 k22 ;

B1 is the best constant of the embedding H01 ./ L4q ./ for q 2 .0; 2/, B4 is the best constant of the embedding H01 ./ L4 ./, and B5 is the best constant of the embedding LqC2 ./ L4 ./ for q 2. Proof. Introduce the notation JD

kru0 k22

C

ku0 k22

Z C .q C 1/

dx ju0 j2 jujq :

(3.306)

It is easy to verify that .ˆ0 .t //2 .q C 2/Jˆ.t /:

(3.307)

Multiplying both sides of Eq. (3.302) by u in the sense of the duality bracket h; i between the Hilbert spaces H01 ./ and H1 ./, after integrating by parts we obtain the first energy equality dˆ D kuk44 kruk22 C .˛ ˇ/ dt

Z

dx u3 ˛ˇkuk22 :

(3.308)

Further, multiplying both sides of Eq. (3.302) by u0 in the sense of the duality bracket h; i between the Hilbert spaces H01 ./ and H1 ./, after integrating by parts we obtain the second energy equality JD

1d ˛ˇ d 1d kruk22 C kuk44 C 2 dt 4 dt 3 dt

Z

dx u3

˛ˇ d kuk22 : 2 dt

(3.309)

354


From (3.309) we obtain Z 1d 1d 2 4 3 2 dx u ˛ˇkuk2 JD kruk2 C kuk4 C .˛ ˇ/ 2 dt 4 dt Z ˛ˇ d ˛ˇ d dx u3 (3.310) C kuk22 : 12 dt 4 dt The following auxiliary estimates bounds hold: ˇ ˇ ˇ " ˇ1 d 1 " 1 2ˇ 0 2 2 ˇ ˇ 4 dt kruk2 ˇ 4 kru k2 C 4" kruk2 4 J C 2" ˆ; ˇ ˇ ˇ " 0 2 ˛2 ˇ2 ˇ ˛ˇ d ˛2ˇ2 " 2ˇ 2 ˇ kuk ku kuk J C ˆ; k C 2ˇ 2 2 ˇ 4 dt 4 4" 4 2" ˇ ˇ Z Z ˇ˛ ˇ d ˇ j˛ ˇj 3ˇ ˇ dx u dx ju0 jjuj2 ˇ 12 dt ˇ 4 Z 1=2 Z 1=2 j˛ ˇj 0 2 q 4q dx ju j juj dx juj 4 Z Z " j˛ ˇj2 0 2 q .q C 1/ dx ju j juj C dx juj4q 8 8".q C 1/ .4q/=2 j˛ ˇj2 B4q " 1 2 JC ˆ.4q/=2 ; 8 8".q C 1/

q 2 .0; 2/:

Hence from (3.310) owing to (3.308) and (3.309) we obtain the inequality

.4q/=2 j˛ ˇj2 B4q 5 1 C ˛2ˇ2 1 1 2 1 " J ˆ00 C ˆC ˆ.4q/=2 : 8 4 2" 8".q C 1/

(3.311)

From (3.305)–(3.307) and (3.311) we obtain the second-order ordinary differential inequality ˆˆ00 ˛1 .ˆ0 /2 C ˇ1 ˆ2 C ˆ1C.4q/=2 0;

(3.312)

where ˛1 D

5 4 1 " ; qC2 8

ˇ1 D 2

1 C ˛2 ˇ2 ; "

D

.4q/=2 j˛ ˇj2 B4q 1 2 : 2.q C 1/"

We require that ˛1 > 1; then " 2 .0; .4 2q/=5/, q 2 .0; 2/. Introduce the new function Z.t / D ˆ1˛1 :

(3.313)

355


Then from (3.312) and (3.313) we obtain .4q/=2˛1 1 Z00 C ˇ1 Z C Z 1˛1 0: 1 ˛1

Now we require that

then we obtain

(3.314)

.4 q/=2 ˛1 > 1I 1 ˛1 .2 q/2 " 2 0; 10

for 0 < q < 2:

Note that .2 q/2 =10 < .4 2q/=5 for q 2 .0; 2/. Let the condition 1 1 ˛ˇ ˛ˇ ku0 k44 > kru0 k22 C ku0 k22 4 2 2 3

Z dx u3

holds. Then from (3.308) and (3.309) we have the inequality ˆ0 .t / 0;

t 2 Œ0; T0 /:

Indeed, from (3.309) we have Z 1 ˛ˇ ˛ˇ 1 u3 dx; kruk44 > kruk22 C kuk22 4 3 3 3 Z 3 kruk44 > kruk44 > kruk22 C ˛ˇkuk22 .˛ ˇ/ u3 dx: 4 Therefore,

Z0 D .1 ˛1 /ˆ˛1 ˆ0 0;

t 2 Œ0; T0 /:

Multiplying both sides of (3.314) by Z0 and integrating, we obtain the inequality .Z0 /2 .˛1 1/ˇ1 Z2 C

2.˛1 1/ 1C1 Z C A2 ; 1 C 1

where 1 A D .˛1 1/2 ˆ2˛ .ˆ0 .0//2 0

1 D

. 4q 2 ˛1 / ; 1 ˛1

ˇ1 2 1C 4q ˆ20 ˆ0 2 ˛1 1 2˛1 3 C q=2

(3.315)

1=2 > 0;

and by the condition " 2 .0; .2 q/2 =10, q 2 .0; 2/, we have 1 C 1 > 0. From (3.315) we conclude that Z Z0 At; 1 from which we obtain that T0 T2 D A1 ˆ1˛ . 0

356


By the arbitrariness of " 2 .0; .2 q/2 =10 we obtain the optimal conditions for the initial function. Next, using the same argumentation as earlier, we obtain "D

.2 q/2 ; 20

q 2 .0; 2/:

Now we derive a lower estimate for the blow-up time. The first energy equality (3.308) implies the following inequalities: Z 0 2 4 dx juj3 ˆ C ˛ˇkuk2 kuk4 C j˛ ˇj

kuk44 C

˛ˇ j˛ ˇj2 kuk22 C kuk44 : 2 2˛ˇ

(3.316)

From (3.316) we obtain j˛ ˇj2 4B44 ; B3 D 1 C 2˛ˇ

ˆ 0 B3 ˆ 2 ;

ˆ

ˆ1 0

1 : B3 t

1 Therefore, T1 D B1 3 ˆ0 and T0 T1 . Thus, statement (3) of the theorem is proved. Now we prove statements (1) and (2). Indeed, from the first energy equality (3.308) we obtain Z dx juj3 ˆ0 C ˛ˇkuk22 kuk44 C j˛ ˇj

Z

kuk44

C j˛ ˇj

dx juj

2

1=2 Z dx u

4

1=2

1 j˛ ˇj2 kuk44 C ˛ˇ kuk22 C kuk44 : 2 2˛ˇ

(3.317)

From (3.317), under the condition q 2, we obtain the following inequality: 0

ˆ Dˆ ;

4

D ; qC2

4 j˛ ˇj2 q C 2 qC2 4 B5 ; DD 1C 2˛ˇ qC1

where B5 is the best constant of the embedding LqC2 ./ L4 ./. Hence we complete the proof of Theorem 3.14.3.

Chapter 4

Blow-up of solutions of strongly nonlinear, dissipative wave Sobolev-type equations with sources

In this chapter, we consider an abstract Cauchy problem for a second-order ordinary differential equation with nonlinear operator coefficients. As applications, we give examples of strongly nonlinear, dissipative wave equations of Sobolev type. For this problem, we obtain sufficient, close to necessary, conditions of the global solvability and of the blow-up for a finite time. Moreover, under additional conditions on nonlinear operators, the solvability of the problem in any finite cylinder is proved; the blow-up of solutions for a finite time is proved under certain conditions for the norm of an initial function (sufficiently large initial function). We give examples of problems for Sobolev-type equations that satisfy the conditions introduced (see [238, 267]).

4.1

Introduction. Statement of problem

In this chapter, we obtain optimal results of the type of existence/nonexistence theorems for a class of strongly nonlinear, dissipative wave equations of Sobolev type, in the abstract formulation, for Cauchy problems for equations with operator coefficients in Banach spaces: n X d A0 u C Aj .u/ C Lu C DP .u/ D F.u/; dt

u.0/ D u0 ;

(4.1)

j D1

and obtain lower and upper estimates for the blow-up time of solution of problem (4.1). We give examples of model, multi-dimensional, strongly nonlinear equations of Sobolev type (some of them were derived in Chapter 1): n X @u @ pj 2 u u C div.jruj ru/ u C u C u3 D 0; (4.2) @t @x1 j D1

@jujq2 C1 @ C juj2q2 u D 0; (4.3) .u u jujq1 u/ C u C @t @x1 @ @u .2 u C u C div.jrujp1 2 ru// C u C u div.jruj2 ru/ D 0; (4.4) @t @x1

358

Chapter 4 Blow-up in wave and dissipative equations with sources

@ .2 u C u C div.jrujp1 2 ru// div.jruj2 ru/ C u @t @u @u @u @u @u @u @ @ @ Cˇ1 C ˇ2 C ˇ3 D 0; (4.5) @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2 where ˇ1 C ˇ2 C ˇ3 D 0; p; p1 ; p2 > 2;

4.2

jˇ1 j C jˇ2 j C jˇ3 j > 0; q; q1 ; q2 0:


We assume that all the conditions of Section 2.2 of Chapter 2 and Section 3.2 of Chapter 3 are fulfilled. Definition 4.2.1. A solution satisfying the conditions Z

T 0

N d X .t / hAj .u/; wij .F.u/; w/0 C hDP .u/; wi0 C .Lu; w/1 dt D 0 dt j D0 (4.6) 8w 2 V ; 8 2 L2 .0; T/;

u.0/ D u0 2 V ; is called a weak generalized solution of the abstract Cauchy problem (4.1). We search for a solution of problem (4.6) in the class du 2 L2 .0; TI V0 /; dt Lu 2 L1 .0; TI W1 /; F.u/ 2 L1 .0; TI W0 /;

u.t / 2 L1 .0; TI V /;

DP .u/ 2 L1 .0; TI W4 /; Aj .u/ 2 L1 .0; TI Vj /;

j D 0; N ;

N d X d A.u/ D Aj .u/ 2 L2 .0; TI V / dt dt j D0


N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /;

j D0

i.e., A.u/ is strongly absolutely continuous and weakly differentiable on the segment Œ0; T. Therefore, by the conditions (A), (A0 ), (F), (L), and (DP) and by Theorem 4.2.2


359

below, we obtain that A.u/ D

N X

Aj .u/ 2 H1 .0; TI V /;

F.u/ 2 L2 .0; TI V /;

j D0

A.u/ D

N X

.1/ Aj .u/ 2 Cw .Œ0; TI V /:

j D0

Thus, problem (4.6), by the conditions (V) and the conditions of Theorem 4.2.2 below, is equivalent to the following problem: X Z T N d .t / hAj .u/; wi C hLu; wi C hDP .u/; wi hF.u/; wi dt D 0 dt 0 j D0

8w 2 V ;

8 .t / 2 L2 .0; T/;

u.0/ D u0 2 V ; where h; i is the duality bracket between the Banach spaces V and V . Using the result of of Appendix A.12 we conclude that problem (4.6) is equivalent to the following problem:

Z T d dt A.u/; v C hLu; vi C hDP .u/; vi hF .u/; vi D 0 dt 0 for all v 2 L2 .0; TI V /, u.0/ D u0 2 V . The following theorem holds. Theorem 4.2.2. Let the conditions (A), (A0 ), (F), (L), and (DP) hold. Assume that V0 ,! W0 and V0 ,! W1 , i.e., the embedding operator is a compact operator, W0 W2 . Assume that either p < q C 2 or p q C 2, and in the latter case, Vj W0 , where p D max pj ; pj D p: j D1;N

Let .F.v/; v/0 c > 0 for all v 2 W0 , jvj0 D 1. Then for any u0 2 V , there exists maximal T0 Tu0 > 0 such that the Cauchy problem (4.1) has a unique solution of the class du.t / u.t / 2 L1 .0; TI V /; 2 L2 .0; TI V0 / 8T 2 .0; T0 /; dt .1/ .Œ0; TI V /: A.u/ 2 H1 .0; TI V / \ Cw

Moreover, for the function X pj 1 1 hA0 u; ui0 C hAj .u/; uij ; 2 pj N

ˆ.t /

j D1

360


which is positive definite by virtue of the conditions (A0 2) and (A3) and has the sense of kinetic energy, and for the existence time T0 > 0 of solution, depending on the value of the variable qC2 ; p max pj ; ˛ p j D1;N we have the following estimates: (1) if ˛ D 1, then T0 D C1 and ˆ.t / ˆ0 exp¹C2 t ºI (2) if ˛ 2 .0; 1/, then T0 D C1 and t 1=.1˛/ I ˆ.t / ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 0 (3) if ˛ > 1 and 0

r

ˆ .0/ >

ˇ ˆ.0/ C ˆ.0/; ˛1 1 ˛1 1

then T0 > 0 is such that lim ˆ.t / D C1;

t"T0

T1 T0 T2 ;

where T1 D

2 q=2 1 ˛ 1 1 A ; C2 BCjqC2 ˆ0 B ; T2 D ˆ1˛ 2 ; 0 q 1=p jvj0 Cj hAj v; vij j ;

ˆ0 ˆ.0/; ˛1

pCqC2 ; 2p

ˆ0 .0/ .F.u0 /; u0 /0 .Lu0 ; u0 /1 ; ˇ

.q C 2/.q C 1/ ; qC2p

4.q C 1/2 ; qC2p

q C 2 > p max pj ;

2 2˛1 ˆ0 .0/ A .˛1 1/ ˆ0

j D1;n

2 1=2 ˇ 2 ˆ.0/ ˆ .0/ ; ˛1 1 ˛1 1

B D B2.qC2/=2 CqC2 ; 1 C1 is the constant of the best embedding V ,! W0 , and B is the constant from the condition (F4). Remark 4.2.3. By the fact that the solution belongs to the class u.t / 2 L1 .0; TI V /;

du 2 L2 .0; TI V0 /; dt

361


we have that, after a possible change on a set of zero Lebesgue measure, the mapping u.t / W Œ0; T ! V0 becomes strongly continuous. Therefore, the initial condition u.0/ D u0 makes sense. Proof. Step 1. Galerkin approximations. By the separability of V , there exists a countable, everywhere dense in V , linearly independent system of functions ¹wi ºm i D1 . Prove the solvability of problem (4.6) by the Galerkin method together with the monotonicity and compactness methods [275]. First, consider the following finite-dimensional approximation of problem (4.6): Z T N d d X dt .t / hAj .um /; wk ij hA0 um ; wk i0 C dt dt 0 j D1 (4.7) C .Lum ; wk /1 C hDP .um /; wk i0 .F.um /; wk /0 D 0; k D 1; m, for all .t / 2 L2 .0; T/, um D

m X

cmi .t /wi ;

um0 D

i D1

m X

cmi .0/wi ;

cmi .0/ D ˛mi ;

i D1

um0 ! u0

strongly in V :

From (4.7) we obtain that in the class cmk .t / 2 C .1/ Œ0; Tm , the following system of ordinary differential equations relative to the unknowns cmk .t /, k D 1; m, holds: m X

N X 0 0 hA0 wi ; wk i0 C cmi hAj;u .u /w ; w i m i k j m j D1

iD1

C hDP .um /; wk i0 C .Lum ; wk /1 D .F.um /; wk /0 ;

k D 1; m: (4.8)

Introduce the notation aik hA0 wi ; wk i0 C

N X


j D1

Obviously, m;m X i;kD1;1


N X


j D1

D

m X

i wi ;

i D1

since A0 is a positive definite operator and, therefore, hA0 ; i0 0, and the relation hA0 ; i0 D 0 holds if and only if D 0. On the other hand,

m X i D1

i wi ;

362


and by the linear independence of the system of functions ¹wi ºm i D1 in V we conclude that D 0 if and only if ¹i ºm D 0. Hence (see [335]) we obtain that i D1 > 0. det¹aik ºm;m i;kD1;1 0 .um /wi ; wk ij are continuous with Now prove that the functionals fj hAj;u m respect to the set of the variables cmi , i D 1; m. Indeed, let c m1 ; : : : ; c mm be a certain point of the Euclidean space Rm . Fix arbitrary " > 0. The following inequality holds: jfj .c m1 ; : : : ; c mm / fj .cm1 ; : : : ; cmm /j jfj .c m1 ; : : : ; c mm / fj .cm1 ; c m2 ; : : : ; c mm /j C jfj .cm1 ; c m2 ; : : : ; c mm / fj .cm1 ; cm2 ; c m3 ; : : : ; c mm /j C jfj .cm1 ; cm2 ; c m3 : : : ; c mm / fj .cm1 ; cm2 ; cm3 ; : : : ; c mm /j C C jfj .cm1 ; cm2 ; : : : ; c.m1/m ; c mm / fj .cm1 ; : : : ; cmm /j:

(4.9)

By the condition (A2), there exists ı."/ > 0 such that each term in the right-hand side of inequality (4.9) under the condition m X

jc mk cmk j ı."/

kD1

is less than the value of "=.m C 1/. Now we prove the Lipschitz continuity of the functionals f0k D .F .um /; wk /0 with respect to the set of the variables ¹cmi ºm i D1 . Let uj D

m X i D1

j cmi wj ;

j D 1; 2:

By the condition (F1), the inequalities j.F.u1 / F.u2 /; wk /0 j jF.u1 / F.u2 /j0 jwj0 Bju1 u2 j0 B1

m X

1 2 jcml cml j

lD1

hold. Thus, the function f0 D f0 .cm1 ; : : : ; cmm / is Lipschitz-continuous and, moreover, continuous with respect to the set of variables. Now consider the functionals gk hDP .um /; wk i0 of the variables .cm1 ; cm2 ; : : : ; cmm /. Similarly, by the conditions (DP), we prove the Lipschitz continuity of the functionals gk . Note that the inverse matrix to the matrix aik hA0 wi ; wk i0 C

N X

0 hAj;u .um /wi ; wk ij m

j D1


363

is continuous with respect to cm D .cm1 ; : : : ; cmm / (see Appendix A.18. Therefore, the system of ordinary differential equations (4.8) is a system of Cauchy–Kovalevskaya type and satisfies the conditions that guarantee its solvability on a certain segment Œ0; Tm , Tm > 0, in the class cmk .t / 2 C 1 .Œ0; Tm /, k D 1; m (see, e.g., [319]). Step 2. A priori estimates Lemma 4.2.4. There exists T > 0 independent of m 2 N such that for the sequence ¹um º of Galerkin approximations, the following properties holds uniformly with respect to m 2 N: um


u0m


A 0 um


Aj .um /

is bounded in L1 .0; TI Vj /I

F.um /


Lum


DP .um /

is bounded in L1 .0; TI V0 /:

Proof. Multiply both sides of Eq. (4.8) by cmk .t / and sum over k D 1; m; we obtain hA0 u0m ; um i0 C

N X

0 hAj;u .um /u0m ; um ij C .Lum ; um /1 D .F.um /; um /0 : m j D1 (4.10)

On the other hand, by virtue of the fact that um D

N X


kD1

and by the conditions (A2) and (A4), we obtain the relations d hAj .um /; um ij D pj hAj .um /; u0mt ij ; dt

0 h Aj .um / t ; um ij C hAj .um /; u0mt ij D pj hAj .um /; u0mt ij ;

0 pj 1 d h Aj .um / t ; um ij D .pj 1/hAj .um /; u0mt ij D hAj .um /; um ij : pj dt Prove the equality d hAj .um /; um ij D pj hAj .um /; u0mt ij : dt

364


Indeed, we have

Z

Jmj .t / d Jmj .t / D dt

Z Z

D Z

1 0 1 0 1 0



ds ŒshA0sum .sum /u0m ; um ij C hAj .sum /; u0m ij


1

d ŒshAj .sum /; u0m i D hAj .um /; u0m ij : ds 0 Here we have used the fact that the Fréchet derivatives of the operators Aj are symmetric by the conditions (A). Hence from (4.10) we obtain N X pj 1 d 1 hAj .um /; um ij C .Lum ; um /1 D .F.um /; um /0 : hA0 um ; um i0 C dt 2 pj j D1 (4.11) D

ds

From (4.11), integrating over t 2 .0; Tm /, we obtain X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj N

j D1

X pj 1 1 D hA0 um0 ; um0 i0 C hAj .um0 /; um0 ij 2 pj j D1 Z t C ds Œ.F.um /; um /0 .Lum ; um /1 : N

(4.12)

0

By the condition (A0 3), we can choose in the Banach space V0 a norm equivalent to the initial norm: kvk0 D hA0 v; vi1=2 0 :

(4.13)

On the other hand, V0 ,! W0 ;

qC2


:

From (4.12) by virtue of (4.13) we obtain that X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj j D1 N X pj 1 1 hA0 um0 ; um0 i0 C hAj .um0 /; um0 ij C B2.qC2/=2 C1 qC2 2 pj j D1 .qC2/=2 Z t N X pj 1 1 hA0 um ; um i0 C ds hAj .um /; um ij : (4.14) 2 pj 0 N

j D1

365


0 , sum over Now we obtain the second a priori estimate. Multiply (4.8) by cmk k D 1; m, and integrate over t 2 .0; Tm /; we obtain

Z

t 0

N X ds hA0 u0m ; u0m i0 C h.Aj .um //0 ; u0m ij j D1

1 1 .F.um0 /; um0 /0 .Lum0 ; um0 /1 qC2 2 Z t 1 1 .F.um /; um /0 .Lum ; um /1 D ds hDP .um /; u0m i0 : qC2 2 0 C

(4.15)

Introduce the notation X pj 1 1 hAj .um /; um ij ; ˆm hA0 um ; um i0 C 2 pj N

ˆm0 ˆm .0/:

(4.16)

j D1

From (4.14) we obtain Z ˆm ˆm0 C B

0

t

ds ˆ1Cq=2 .s/; m

where

qC2

B 2.qC2/=2 BC1

(4.17)

;

C1 is the constant of the best embedding V ,! W0 , and B is the constant from the condition (F4). From (4.17) and the Bihari theorem (see, e.g., [112]) we have ˆm

Œ1

ˆm0 : q q=2 2=q 2 ˆm0 Bt

(4.18)

Since um0 ! u0 strongly in V , the inequality ˆm0 C0 holds and the constant C0 is in dependent of m 2 N. For a certain subsequence of the sequence ¹um º, either ˆm0 # ˆ0 or ˆm0 " ˆ0 . First, consider the case where ˆm0 " ˆ0 . The following inequalities hold: q=2 q

1 Bˆm0

2

q=2 q

t 1 Bˆ0

ˆ m C1

8t 2 .0; T/;

2

t

8t 2 .0; T1 /; T 2 .0; T1 /;

T1 D B

T1 D B

1 2

1 2

q

q

q=2

ˆ0

q=2 ˆ0 ;

where C1 is independent of m 2 N. Now consider the case where ˆm0 # ˆ0 . For any m < m, we have ˆm0 > ˆm0 ;

q q q=2 q=2 1 Bˆm0 t 1 B ˆm0 t 2 2

; (4.19)

366


and for any t 2 Œ0; B

1 2 .qC1/ / q ˆm0

from (4.18) we obtain

ˆm

Œ1

C0 : q q=2 2=q 2 ˆm0 Bt

Therefore, for any fixed T from the interval .0; T1 /, there exists m 2 N such that inequality (4.19) holds. By the condition p q C 2 of the theorem, there exists j 2 1; N such that p D pj , the embedding Vj W0 holds, and the following inequality is valid: 1=pj


8v 2 Vj :

(4.20)

From (4.20) we directly obtain that .qC2/=pj pj qC2 qC2 jvj0 Cj pj 1 .qC2/=pj N X pj 1 1 hAj .v/; vij : hA0 v; vi0 C 2 pj

(4.21)

j D1

Owing to (4.12), (4.16), (4.20), and (4.21) we conclude that .qC2/=pj Z t pj qC2 ds ˆ˛m .s/; ˆm ˆm0 C BCj pj 1 0

˛

qC2 : pj

(4.22)

Consider two cases: ˛ < 1 and ˛ D 1. Using the Gronwall–Bellman and Bihari theorems (see [112]), from (4.22) we obtain 1=.1˛/ ˆm Œˆ1˛ ; ˛ 2 .0; 1/; m0 C .1 ˛/C2 t

(4.23)


(4.24)

where

˛ D 1;

.qC2/=pj pj : pj 1 By the fact that um0 ! u0 strongly in V , we have that ˆm0 C0 and the constant C0 is independent of m 2 N. Therefore, we obtain qC2 C2 BCj

ˆm0 C3

8T 0;

8˛ 2 .0; 1:

(4.25)

From the condition (A3) and (4.17), (4.19), and (4.25) we obtain that jF .um /j0 Bjum jqC1 0

BCqC1 2.qC1/=2 1

X pj 1 1 hAj .um /; um ij hA0 um ; um i0 C 2 pj

C3 ˆ.qC1/=2 C30 ; m

N

j D1

.qC1/=2


367

where 0 < C30 < C1 and the constant C30 is independent of m 2 N for any T > 0 in the case ˛j 2 .0; 1 and for any T 2 .0; T1 /, where the constant T1 is defined by formula (4.21), in the case ˛ > 1. Now from (4.15) and the conditions (A2) and (A0 2) we obtain that Z t Z t 0 2 ds kum k0 ds hA0 u0m ; u0m i0 C40 ; m0 0

0

where 0 < C40 < C1 and the constant C40 is independent of m 2 N for any T > 0 in the case ˛j 2 .0; 1 and for any T 2 .0; T1 /, where the constant T1 is defined by formula (4.19), in the case ˛ > 1. Thus, um


u0m


A0 um


Aj .um /


F.um /

is bounded in

Lum

is bounded in

DP .um / is bounded in

(4.26)

L1 .0; TI W0 /; L1 .0; TI W1 /; L1 .0; TI V0 /;

where the inclusions mentioned hold for any T > 0 in the case ˛ 2 .0; 1 and for any T 2 .0; T1 /, where the constant T1 is defined by formula (4.19), in the case ˛ > 1. By (4.26), there exists a subsequence of the sequence ¹um º such that um * u


u0m * u0 weakly in L2 .0; TI V0 /: Note that (4.27)2 implies u0m * u0

in D 0 .0; TI V /:

Therefore, by virtue of the weak convergence L2 .0; TI V0 / we have u0m * : Hence we conclude that .t / D u0 .t / for almost all t 2 .0; T/. By Lemma A.15.1 (see Appendix A.15), where we set W D W0 , we obtain um .s/ ! u.s/


(4.27)

368


On the other hand, by virtue of the condition (F1) we have jF.um / F.u/j0 .R/jum uj0 ! 0

as m ! C1;

(4.28)

where R D max¹juj0 ; jum j0 º. From (4.28) we obtain the strong convergence F.um /.t / ! F.u/.t /

in W0 for almost all t 2 .0; T/:

(4.29)

Note that, since V0 ,! W1 , by virtue of Lemma A.15.1 (see Appendix A.15), where we set W D W1 , we obtain um .s/ ! u.s/


Hence from the condition (L3) we obtain the strong convergence Lum ! Lu in W1 for almost all t 2 .0; T/. It is easy to verify that the conditions (DP) imply the strong convergence DP .um / ! DP .u/

in V0 for almost all t 2 .0; T/:

Step 3. Monotonicity method. Now we apply the monotonicity method. Recall that in Step 2 we have obtained the limit inclusions (4.27) and (4.29) and, moreover, A.um / *


A.u/

N X

Aj .u/:

(4.30)

j D0

P Lemma 4.2.5. Let A.u/ jND0 Aj .u/. Then for certain T > 0, there exists a subsequence of the sequence ¹um º such that A.um / * A.u/

-weakly in L1 .0; TI V /:

Moreover, for a certain subsequence of the sequence ¹um º we have um ! u strongly in Lpj .0; TI Vj /; um ! u strongly in L2 .0; TI V0 /: Proof. Rewrite Eq. (4.7) in the equivalent form hA.um /; wj i D hA.um0 /; wj i Z t ds Œ.F.um /; wj /0 .Lum ; wj /1 hDP .um /; wj i0 ; (4.31) C 0

where h; i is the duality bracket between the Banach spaces V and V .

369


Fix j 2 N and pass in Eq. (4.31) to the limit as m ! C1; we obtain Z D

t

0

ds ŒF.u/ Lu DP .u/ C A.u0 /:

(4.32)

Let Z 0 X Z D Z

T 0

T 0 T 0


dt hA.u /; u i D Z

C

Z

T

0

Z dt hA.u /; vi

0

T

dt hA.v/; u vi;

dt hA.u 0 /; u i

t

dt 0

T

0

T

0

ds Œ.F.u /.s/; u .t //0 .Lu .s/; u .t //1 hDP .u /.s/; u .t /i0 :

Hence we obtain that 0 lim sup X !C1

Z

T

dt 0

Z

T 0

T 0

0

Z

C D

Z

t

ds Œ.F.u/.s/; u.t //0 .Lu.s/; u.t //1 hDP .u/.s/; u.t /i0 Z

dt hA.u0 /; ui

T 0

Z dt h ; vi

0

T

dt hA.v/; u vi

dt h A.v/; u vi:

(4.33)

Now we set v D u w for any v; w 2 Lr .0; TI V /, > 0, u 2 L1 .0; TI V / 0 Lr .0; TI V /, ; A.v/ 2 Lr .0; TI V /, r > 1, r 0 D r=.r 1/. By virtue of (4.33) the following inequality holds: Z

T 0

dt h A.u w/; wi 0;

from which, by the semicontinuity of the operators Aj .v/, j D 0; N , we obtain D A.u/:

370


On the other hand, by (4.32) Z T lim sup dt hAj .u /; u ij

(4.34)

!C1 0

N X

lim inf

!C1

kD0;k¤j

N X

kD0;k¤j

Z

T 0

Z

T

0

Z

T

dt hAk .u /; u ik C lim sup

!C1 0

Z dt hAk .u/; uik C

T

0

dt hA.u /; u i

Z dt hA.u/; ui D

0

T

dt hAj .u/; uij ;

where we have used the fact that each of the operators Aj .v/, j D 0; N , by virtue of the conditions (A0 4) and (A4), generates norms of the uniformly convex Banach spaces L2 .0; TI V0 / and Lpj .0; TI Vj /, and these norms are equivalent to the initial norms by virtue of the conditions (A0 3) and (A3) according to the following formulas: 1=2 1=pj Z T Z T dt hA0 u; ui0 ; dt hAj .u/; uij : 0

0

Finally, by the weak convergence u * u in Lpj .0; TI V /, j D 0; N , and by virtue of the fact that the operators Aj .v/, j D 0; N , generate norms of reflexive Banach spaces according to the above-mentioned rules, we have Z T Z T lim inf dt hAj .um /; um ij dt hAj .u/; uij : (4.35) m!C1 0

0

From (4.34) and (4.35) we have Z Z T dt hAj .um /; um ij D lim m!C1 0

T 0

Now by the conditions (A3) we have the norms Z T 1=pj Z dt hAj .u/; uij ;

T

which are equivalent to the initial norms 1=pj Z T pj dt kukj ;

T

0

0

Z

dt hAj .u/; uij : 1=2

dt hA0 u; ui0

0

;

1=2 dt

0

(4.36)

kuk20

of the Banach spaces Lpj .0; TI Vj /. Therefore, (4.36) implies the existence of a subsequence of the sequence ¹um º such that Z T 1=pj 1=pj Z T dt hAj .um /; um ij ! dt hAj .u/; uij ; j D 1; N ; 0

Z

T 0

Z

1=2 dt hA0 um ; um i0

!

0

T 0

1=2 dt hA0 u; ui0

:

371




j D 0; N I

therefore, there exists a subsequence of the sequence ¹um º such that um ! u strongly in each Lpj .0; TI Vj /, j D 0; N . Lemma 4.2.5 is proved. Step 4. Passage to the limit and the uniqueness. The limit relations (4.27), (4.29), and (4.30) imply the possibility of passage to the limit as m ! C1 in Eq. (4.31). Further, similarly to Step 4 of the proof of Theorem 2.3.2 of Chapter 2, we can prove that u.t / is a unique solution of problem (4.6). Moreover, .1/ .Œ0; TI V /: A.u/ 2 H1 .0; TI V / \ Cw

Step 5. Blow-up of solutions Lemma 4.2.6. Let X pj 1 1 hAj .u/; uij ; hA0 u; ui0 C 2 pj N

ˆ

˛

j D1

qC2 ; p

and .F.v/; v/0 c > 0

8v 2 W0 ;

jvj0 D 1:

Then the following assertions hold: (1) if ˛ 2 .0; 1/, then T0 D C1 and ˆ ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 t 1=.1˛/ I 0 (2) if ˛ D 1, then T0 D C1 and ˆ ˆ0 exp.C2 t /I (3) if ˛ > 1 and ˆ0 .0/ >

r

ˇ ˆ.0/ C ˆ0 ; ˛1 1 ˛1 1

then there exists T0 2 ŒT1 ; T2 such that the limit relation lim ˆ.t / D C1

t"T0

p max pj ; j D1;N

372


holds, where T1 D

2 q=2 1 B ; ˆ q 0

T2 D ˆ1˛ A1 ; 0

qC2

1=pj

jvj0 Cj hAj v; vij ˆ0 ˆ.0/; ˛1

pCqC2 ; 2p

C2 BCj

p p1

˛ ;

;

ˆ0 .0/ .F.u0 /; u0 /0 .Lu0 ; u0 /1 ; ˇ

.q C 2/.q C 1/ ; qC2p

4.q C 1/2 ; qC2p

q C 2 > p max pj ; 2

A .˛1 1/

1 ˆ2˛ 0

j D1;n

ˇ ˆ0 ˆ .0/ ˛1 1 0

2

1=2 ˆ2 .0/ ; ˛1 1

B D B2qC2 CqC2 ; 1 C1 is the constant of the best embedding V ,! W0 , and B is the constant from the condition (F4). Proof. By the conditions (A2), the Schwarz inequality (see [293]) for the Fréchet 0 W Vj ! L.Vj I Vj / of the operators Aj W Vj ! Vj holds: derivatives Aj;u 0 .um /u0m ; um ij j jh.Aj .um //0 ; um ij j D jhAj;u m 0 0 hAj;u .um /u0m ; u0m ij1=2 hAj;u .um /um ; um ij1=2 m m

D h.Aj .um //0 ; u0m ij1=2 .pj 1/1=2 hAj .um /; um ij1=2 ; jhA0 u0m ; um i0 j hA0 u0m ; u0m i0 hA0 um ; um i0 : 1=2

1=2

(4.37) (4.38)

Here we have used the relation 0 Aj;v .v/v D .pj 1/Aj .v/:


0 0 Aj;v .v/ D pj 1 Aj;v .v/;

0 0 Aj;v .v/ D pj 2 Aj;v .v/; j D 0; N ;

Z 1

Z 1 d 0 d d Aj;v .v/v; w 8w 2 Vj ; Aj .v/; w D d 0 0 j j 0 .v/v; wij D 0 hAj .v/ .pj 1/1 Aj;v

Therefore,

0 .pj 1/Aj .v/ D Aj;v .v/v

8w 2 Vj ;

8v 2 Vj ;

j D 0; N ;

j D 0; N :

j D 0; N :

373



(4.39)

j D1

From (4.37)–(4.39) we have ˇ ˇ2 ˇ2 ˇ N X ˇd ˇ ˇ ˇ 0 0 ˇ ˆm ˇ ˇjhA0 u ; um i0 j C jh.Aj .um // ; um ij jˇˇ m ˇ dt ˇ ˇ j D1

N X hA0 u0m ; u0m i0 C h.Aj .um //0 ; u0m ij j D1

N X hA0 um ; um i0 C .pj 1/hAj .um /; um ij j D1

N X 0 0 0 0 p hA0 um ; um i0 C h.Aj .um // ; um ij j D1


j D1

N X p hA0 u0m ; u0m i0 C h.Aj .um //0 ; u0m ij j D1

N X 1 pj 1 hAj .um /; um ij ; hA0 um ; um i0 C 2 pj

(4.40)

j D1

where p D maxj D1;N pj > 2. The following statement holds. Proposition 4.2.7. Let the conditions (L) and (A0 ) hold and moreover, V0 W1 . Then the inequality .Lu; u/1 ChA0 u; ui0 8u 2 V0 holds for certain C > 0. Proof. Indeed, let u 2 V0 W1 ; then the following inequalities hold: .Lu; u/1 D1 juj21 D1 C1 kuk20 D1 C1 m1 0 hA0 u; ui0 D ChA0 u; ui0 : Proposition 4.2.7 is proved.

374


Without loss of generality, we can set the constant C in Proposition 4.2.7 equal to 1. Indeed, introduce the new operators e L C1 L;

e F C1 F

and, moreover, perform the substitution e t Ct . Then for the introduced operators, Proposition 4.2.7 with the constant C D 1 holds. By Proposition 4.2.7 and the Hölder inequality, we have 1=2 1=2 Lum ; u0m 1 .Lum ; um /1 .Lu0m ; u0m /1

"0 1 hA0 um ; um i0 : hA0 u0m ; u0m i0 C 2 2"0

(4.41)

By Proposition 3.2.1 and the Hölder inequality, we have hDP .um /; u0m i0 hA0 u0m ; u0m i0

1=2

.F.um /; um /1=2 0

"0 1 hA0 u0m ; u0m i0 C .F.um /; um /0 : 2 2"0

(4.42)

Introduce the notation Jm hA0 u0m ; u0m i0 C

n X

h.Aj .um //0 ; u0m ij :

j D1

Then from the energy equalities (4.11) and (4.15), owing to inequalities (4.41) and (4.42), for the function Jm we have the inequalities "0 1 .F.um /; um /0 hA0 u0m ; u0m i0 C 2 2"0 1 2 ˆ00m C .Lum ; u0m /1 C qC2 qC2 q "0 1 .F.um /; um /0 Lum ; u0m 1 C hA0 u0m ; u0m i0 C qC2 2 2"0 1 q "0 1 C ˆ00 .Lum ; u0m /1 C Jm C ˆ00 qC2 m qC2 2 qC2 m 1 1 0 1 1 0 C .Lum ; um /1 C ˆm ˆ ˆ00m C 2"0 2"0 qC2 2"0 m 2 qC1 qC1 C (4.43) ˆm C "0 Jm : "0 q C 2 qC2

Jm .Lum ; u0m /1 C

From (4.43) we obtain the inequality 1 1 0 2 qC1 qC1 "0 Jm ˆ00m C ˆm : ˆm C 1 qC2 qC2 2"0 "0 q C 2

(4.44)

375


From (4.40) and (4.44) we obtain the following second-order ordinary differential inequality of (see [210]): ˆm ˆ00m ˛1 .ˆ0 /2 C ˇˆm ˆ0m C ˆ2m 0; where ˛1

qC1 qC2 1 "0 ; p qC2

ˇD

qC2 ; 2"0

D

(4.45) 2q C 2 : "0

Require that "0 2 .0; .q C 2 p/.q C 1/1 /; then ˛1 > 1. Now consider inequality (4.45) in detail. This inequality can be easily reduced to the linear differential inequality Z00m C ˇZ0m ıZm 0;

ı D .˛1 1/;

(4.46)

for the new function 1 Zm D ˆ1˛ : m

(4.47)

Ym Zm e ˇ t

(4.48)

Now we pass to the new function

in Eq. (4.46); then, owing to (4.47), we obtain the inequality 00 0 Ym ˇYm ıYm 0:

(4.49)

From (4.48) we have 0 Ym

ˇt

D e .˛1

1 1/ˆ˛ m

ˆ0m

ˇ C ˆm : ˛1 1

(4.50)

Require the validity of the conditions ˇ ˆ.0/; ˆ .0/ > ˛1 1 0

X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij : 2 pj n

(4.51)

j D1

Then, passing to a subsequence if necessary, we obtain ˆ0m .0/ > Indeed,

ˇ ˆm .0/: ˛1 1

(4.52)

ˆ0m .0/ D .F.um0 /; um0 /0 .Lum0 ; um0 /1

and ¹um0 º strongly converges in the Banach space V , which is continuously embedded in the Banach spaces W0 and W1 . Therefore, the bounded Lipschitz-continuity of

376


the operator F W W0 ! W0 and the continuity of the linear operator L W W1 ! W1 imply the limit relations .F.um0 /; um0 /0 ! .F.u0 /; u0 /0 as m ! C1; .Lum0 ; um0 /1 ! .Lu0 ; u0 /1

as m ! C1;

X pj 1 1 ˆm0 D hA0 um0 ; um0 i0 C hAj .um0 /; um0 ij 2 pj N

j D1

X pj 1 1 ! ˆ0 D hA0 u0 ; u0 i0 C hAj .u0 /; u0 ij 2 pj N

as m ! C1:

j D1

It follows from the conditions (A3), (A4), (A0 3), and (A0 4) that the functionals hA0 u; ui1=2 W V0 ! R1 ; 0

1=pj

hAj .u/; uij

W Vj ! R1

are norms. Therefore, we have ˆ0m .0/ D .F.um0 /; um0 /0 .Lum0 ; um0 /1 ! .F.u0 /; u0 /0 .Lu0 ; u0 /1 D ˆ0 .0/: Indeed, since u.t / 2 C.Œ0; T0 /I V /, we have .F.u.t //; u.t //0 .Lu.t /; u.t //1 D ˆ0 .t / ! ˆ0 .0/ D .F.u0 /; u0 /0 .Lu0 ; u0 /1

as t ! 0:

By the continuous differentiability of the function Ym , from condition (4.52), owing to (4.50), we conclude that there exists an interval Œ0; T3m / belonging to the domain 0 0. Therefore, for t 2 Œ0; T of definition of ˆm .t / such that Ym 3m , we have 0 ˇYm .t / 0. Hence from inequality (4.49) we obtain the differential inequality 00 Ym ıYm 0 for t 2 Œ0; T3m :

Require the validity of the inequality p ˇ 0 ˆ.0/ C ˆ.0/: ˆ .0/ > p ˛1 1 ˛1 1

(4.53)

(4.54)

Then, as above, passing to a subsequence if necessary, we obtain the inequality p ˇ 0 ˆm .0/ C (4.55) ˆm .0/: ˆm .0/ > p ˛ ˛1 1 11 From inequality (4.53) we have 0 2 2 .Ym / ıYm C A2m ;

(4.56)

377


and by virtue of (4.55) we have 2 2 2˛1 Am .1 ˛1 / ˆm .0/ ˆ0m .0/

2 ˇ ı 2 .ˆm .0// > 0: ˆm .0/ ˛1 1 .˛1 1/2

From (4.56) we obtain 0 .t / Am Ym

8t 2 Œ0; T3m :

(4.57)

0 < 0 in the domain of definition of the function From (4.57) we have the fact that Ym ˆm .t /, hence from (4.57) we have

Ym .t / Ym0 Am t: Therefore, there exists a time moment Tm0 2 .0; T2m / such that lim sup ˆm .t / D C1; t"Tm0

where T2m D Ym .0/A1 m . Consider conditions (4.51) and (4.54) separately. Note that we can choose "0 2 .0; .q C 2 p/.q C 1/1 / arbitrarily. Now as "0 , we choose the minimum point of the functions r ˇ f1 D ; f2 D : ˛1 1 ˛1 1 It is easy to verify that the minimum is reached at the point "0 D

qC2p : 2q C 2

From (4.45) we obtain a lower estimate for the function ˆm .t /: ˆm

‰m0 e t ŒT2m t 1=.˛1 1/

;

(4.58)

where 1 1 T2m ˆ1˛ m0 Am ;

1/ ‰m0 D A1=.1˛ ; m

D

ˇ : ˛1 1

Now prove that ˆm .t / ! ˆ.t / for almost all t 2 .0; T/, where X pj 1 1 hAj .u/; uij : ˆ.t / hA0 u; ui0 C 2 pj N

j D1

By Lemma A.15.1 (see Appendix A.15), where we set W D W0 , we obtain um .s/ ! u.s/


378


Similarly, we obtain that, by Lemma A.15.1 (see Appendix A.15), where we set W D W1 , we obtain um .s/ ! u.s/


On the one hand, from (4.41) we obtain Z t ˆm D ˆm0 C ds Œ.F.um /; um /0 .s/ .Lum ; um /1 .s/ : 0

Therefore,

Z ˆ m ! ˆ0 C

t 0

ds Œ.F.u/; u/0 .s/ .Lu; u/1 .s/ :

On the other hand, by virtue of the definition of a weak solution, where we set ´ u.s/; s 2 Œ0; t ; v.s/ D 0; s 2 .t; T; we obtain

Z ˆ D ˆ0 C

t 0

ds Œ.F.u/; u/0 .s/ .Lu; u/1 .s/ :

Hence directly obtain that ˆm .t / ! ˆ.t /

(4.59)

for almost all t 2 .0; T/. Owing to the fact that um0 ! u0 strongly in V , we have as m ! C1 1=.˛ 1/ ‰m0 ! ‰0 A0 1 ;

X pj 1 1 ! ˆ0 hA0 u0 ; u0 i0 C hAj .u0 /; u0 ij : 2 pj N

ˆm0

j D1

From (4.59), passing in inequalities (4.23) and (4.24) to the limit as m ! C1, we obtain that ˆ.t / ˆ0 exp¹C2 t º;

˛ D 1;

(4.60)

t 1=.1˛/ ; ˆ.t / ˆ0 Œ1 C .1 ˛/C2 ˆ˛1 0

˛ < 1;

where ˛

qC2 ; p

qC2 C2 BCj 1=pj


;

p p1

.qC2/=p ;

p D max pj : j D1;N

(4.61)

379


Now consider inequalities (4.18) and (4.58). Note that as m ! C1 ‰m0 ! ‰0 A1=.˛1 1/ ; T1m ! T1 D

1 1 T2m ! T2 ˆ1˛ A ; 0

2 q=2/ 1 B ; ˆ q 0

B B2.qC2/=2 CqC2 ; 1 where C1 is the constant of the best embedding V ,! W0 and B is the constant from the condition (F4). First, consider inequality (4.58). Without loss of generality, by the passage to a subsequence, we can consider the sequence T2m , which is positive uniformly with respect to m since T2 > 0. By the convergence of the sequence T2m ! T2 as m ! C1, we can choose a monotonically converging subsequence (we denote it again by T2m ). We denote the corresponding subsequences of the sequences ¹u0m º and ¹um º also by ¹u0m º and ¹um º. Let T2m " T2 ; then inequality (4.46) holds uniformly with respect to t 2 Œ0; T2m /, m m, for certain fixed m 2 N. Passing in inequality (4.58) to the limit as m ! C1 for such t , we obtain ˆ

‰0 e t ŒT2 t

1=.˛1 1/

;

D

ˇ ˛1 1

(4.62)

for t 2 Œ0; T2m /. Hence by the arbitrariness of m 2 N we directly obtain that (4.62) holds for all t 2 Œ0; T2 / and the blow-up time of solutions of problem (4.6) satisfies the upper estimate T0 T2 . Now let T2m # T2 . Assume that T0 > T2 and M sup t2Œ0;T2 ˆ.t / < C1. Then (4.58) holds uniformly with respect to t 2 Œ0; T2 /. Passing in inequality (4.58) to the limit as m ! C1, we obtain the inequality ˆ

‰0 e t ŒT2 t

1=.˛1/

M < C1 for t 2 Œ0; T2 /;

D

ˇ : ˛1

However, from this inequality we obtain that our assumption does not hold. Hence we directly obtain that T0 T2 . Similarly, from inequality (4.18) we obtain that ˆ.t /

Œ1

ˆ0 q q=2 2=q 2 ˆ0 Bt

Lemma 4.2.6 is proved. Theorem 4.2.2 is completely proved.

8t 2 .0; T1 /;

T0 T1 :

380

4.3



Definition 4.3.1. A solution of the class C .1/ .Œ0; TI V0 / satisfying the condition h.A0 u/0 ; vi0 C

N X

h.Aj .u//0 ; vij C hDP .u/; vi0 C .Lu; v/1 D .F.u/; v/0

j D1

8v 2 V0 ;

t 2 Œ0; T;

u.0/ D u0 2 V0 ; where the time derivative is meant in the classical sense, is called a strong generalized solution of problem (4.1). 0 .u/ W V ! L.V I V / of Under the assumption that the Fréchet derivative Aj;u j j j the operator Aj is strongly continuous with respect to u 2 Vj , by virtue of the conditions of Theorem 4.3.2 below, we obtain that in the considered class, the following inclusions hold:

A0 u 2 C .1/ .Œ0; TI V0 /; Aj .u/ 2 C .1/ .Œ0; TI Vj /; DP .u/ 2 C.Œ0; TI W4 /;

Lu 2 C.Œ0; TI W1 /;

F.u/ 2 C.Œ0; TI W0 /; A0 u and Aj .u/ belong to the class C .1/ .Œ0; TI V0 /, and DP .u/ and Lu, F.u/ belong to the class C.Œ0; TI V0 /. Thus, by virtue of the conditions of Theorem 4.3.2 below, the problem is equivalent to the following problem: h.A0 u/0 ; vi0 C

N X

h.Aj .u//0 ; vi0 C hDP .u/; vi0 C hLu; vi0 D hF.u/; vi0

j D1

8v 2 V0 ;

t 2 Œ0; T;

u.0/ D u0 2 V0 ; where the time derivative is meant in the classical sense and h; i0 is the duality bracket between the Banach spaces V0 and V0 . The following theorem holds. Theorem 4.3.2. Let the conditions (A), (A0 ), (F), (L), and (DP) hold. Assume that V0 Vj , j D 1; N , V0 W0 W2 , and V0 W1 , and let either p < q C 2 or p q C 2, and in the latter case, the embedding Vj W0 holds, where p D max pj ; j D1;N

p D pj :

381


Let the Fréchet derivatives 0 ./ W Vj ! L.Vj ; Vj / Aj;u

of the operators Aj W Vj ! Vj be monotonic in the sense that 0 0 hAj;u .u/u1 Aj;u .u/u2 ; u1 u2 ij 0 8u; u1 ; u2 2 Vj ;

j D 1; N ;

and, moreover, let 0 ./ 2 BC.V0 I L.V0 I V0 //; Aj;u

i.e. these derivatives be continuous and bounded. Let .F .v/; v/0 c > 0

8v 2 W0 ;

jvj0 D 1:

Then for any u0 2 V0 , there exists maximal T0 Tu0 > 0 such that the Cauchy problem (4.1) has a unique solution of the class u.t / 2 C .1/ .Œ0; T0 /I V0 /: Moreover, for various values of ˛ D .q C 2/=p, p D maxj D1;n pj , the following results hold: (1) if ˛ 2 .0; 1/, then T0 D C1 and C .1 ˛/At 1=.1˛/ I ˆ.t / Œˆ1˛ 0 (2) if ˛ D 1, then T0 D C1 and ˆ.t / ˆ0 exp¹At ºI (3) if ˛ > 1 and ˆ0 .0/ >

r

ˇ ˆ.0/ C ˆ0 ; ˛1 1 ˛1 1

then there exists T0 2 ŒT1 ; T2 such that the limit relation lim sup ˆ.t / D C1 t"T0

382


holds, where X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij ; 2 pj N

ˆ0 ˆ.0/;

j D1

2 q=2 1 T1 D ˆ0 B ; q

T2 D ˆ1˛ A1 ; 0

qC2

A Cj M2.qC2/=p ; ˆ0 ˆ.0/; ˛1

pCqC2 ; 2p

B D BCqC2 2.qC2/=2 ; 1 1=pj

jvj0 Cj hAj v; vij

;

0

ˆ .0/ .F.u0 /; u0 /0 .Lu0 ; u0 /1 ; ˇ

.q C 2/.q C 1/ ; qC2p

4.q C 1/2 ; qC2p

q C 2 > p max pj ; j D1;n 1=2 2 ˇ 0 2 1 .0/ .0/ ; ˆ ˆ ˆ A .˛1 1/2 ˆ2˛ 0 0 ˛1 1 ˛1 1 C1 is the constant of the best embedding V ,! W0 , and B is the constant from the condition (F4). Proof. Step 1. Local solvability Lemma 4.3.3. For any u0 2 V0 , there exists maximal T0 > 0 such that a unique solution of problem (4.1) of the class C .1/ .Œ0; TI V0 / exists for any T 2 .0; T0 /. Proof. The proof is similar to that of Lemma 2.4.3 of Chapter 2. Step 2. A priori estimates and blow-up Lemma 4.3.4. Let X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij ; 2 pj N

˛

j D1

qC2 : p

Moreover, let .F.v/; v/0 c > 0

8v 2 W0 ;

jvj0 D 1:

Then for various values of ˛ D .q C 2/=p, p D maxj D1;n pj , the following results hold: (1) if ˛ 2 .0; 1/, then T0 D C1 and C .1 ˛/At 1=.1˛/ I ˆ.t / Œˆ1˛ 0 (2) if ˛ D 1, then T0 D C1 and ˆ.t / ˆ0 exp¹At ºI

383


(3) if ˛ > 1 and 0

r

ˆ .0/ >

ˇ ˆ.0/ C ˆ.0/; ˛1 1 ˛1 1

then there exists T0 2 ŒT1 ; T2 such that the limit relation lim ˆ.t / D C1

t"T0

holds, where X pj 1 1 hAj .u/; uij ; ˆ.t / hA0 u; ui0 C 2 pj N

ˆ0 ˆ.0/;

j D1

2 q=2 1 1 1 B ; T2 D ˆ1˛ A ; B D BCqC2 2.qC2/=2 ; ˆ 1 0 q 0 .qC2/=p p 1=p qC2 A Cj M ; jvj0 Cj hAj v; vij j ; p1

T1 D

ˆ0 ˆ.0/; ˛1

pCqC2 ; 2p

ˆ0 .0/ .F.u0 /; u0 /0 .Lu0 ; u0 /1 ; ˇ

.q C 2/.q C 1/ ; qC2p

4.q C 1/2 ; qC2p

q C 2 > p max pj ; 2 2˛1 ˆ0 .0/ A .˛1 1/ ˆ0

j D1;n

1=2 2 ˇ 2 ; ˆ.0/ ˆ .0/ ˛1 1 ˛1 1

C1 is the best constant of the embedding V ,! W0 , and B is the constant from the condition (F4). Proof. Let u 2 C .1/ .Œ0; T0 /I V0 / be a strong generalized solution of problem (4.1). Multiply both sides of (4.1) by u.t / and u0t .t /; after integrating by parts we obtain the equalities N X pj 1 d 1 hAj .u/; uij D .F.u/; u/0 .Lu; u/1 ; hA0 u; ui0 C dt 2 pj

(4.63)

j D1

hA0 u0 ; u0 i0 C

N X

h.Aj .u//0 ; u0 ij

j D1

D

1 d 1d .F.u/; u/0 .Lu; u/1 hDP .u/; u0 i0 : q C 2 dt 2 dt

(4.64)

384


Integrating Eq. (4.63) over t 2 .0; T/, we obtain Z t ds Œ.F.u/; u/0 .Lu; u/1 ; ˆ.t / D ˆ0 C

(4.65)

0

where

X pj 1 1 ˆ.t / hA0 u; ui0 C hAj .u/; uij ; 2 pj N

ˆ0 D ˆ.0/:

j D1

According to the condition V0 W0 , there exists a constant C1 > 0 such that 1=2 jvj0 C1 hA0 v; vi0 ;

and (4.65) implies the inequalities Z t Z t qC2 ds hA0 v; vi.qC2/=2 ˆ C B ds ˆ.qC2/=2 .s/; ˆ.t / ˆ0 C MC1 0 0 0

0

MCqC2 2.qC2/=2 : 1

B

According to the Bihari theorem (see [112]) we have ˆ.t /

Œ1

ˆ0 q q=2 2=q 2 ˆ0 Bt

8t 2 Œ0; T1 /;

T1

2 q=2 1 B : ˆ q 0

(4.66)

Therefore, in the case p < q C 2 from (4.66) we obtain T0 T1 . Now let p q C 2. We use the fact that there exist j 2 1; N and pj D p, pj q C 2, such that Vj W0 . Let Cj be the constant of the embedding Vj W0 : 1=p

jvj0 Cj hAj .v/; vij j ; Z t .qC2/=pj qC2 ds hAj .v/; vij ˆ.t / ˆ0 C Cj M 0

ˆ0 C

qC2 Cj M

p p1

.qC2/=p Z

t

ds ˆ.qC2/=p .s/:

0

Consider separately the cases ˛ D 1 and ˛ 2 .0; 1/, where ˛

qC2 : p

In the first case, by to the Gronwall–Bellman theorem (see [112]) we have ˆ.t / ˆ0 exp¹At º;

qC2

A Cj M

p p1

.qC2/=p :

(4.67)

385


In the second case, by the Bihari theorem we have ˆ.t / Œˆ1˛ C .1 ˛/At 1=.1˛/ : 0

(4.68)

From (4.67) and (4.68) we obtain that T0 D C1 in the case ˛ 2 .0; 1. Now from relations (4.63) and (4.64) in just the same way as in Step 5 of the proof of Theorem 4.2.2, we derive the following second-order ordinary differential inequality (see [210]): 2 ˆ00 ˆ ˛1 ˆ0 C ˇˆ0 ˆ C ˆ2 0;

(4.69)

where ˛1

pCqC2 ; 2p

4.q C 1/2 ; qC2p

ˇ

.q C 2/.q C 1/ ; qC2p

q C 2 > p max pj : j D1;n

Integrating the differential inequality (4.69) and taking into account (4.63), (4.65), and the inequality r ˇ 0 ˆ.0/ C ˆ0 ; ˆ .0/ > ˛1 1 ˛1 1 we obtain ˆ

e t A1=.1˛1 / ŒT2 t 1=.˛1 1/

8t 2 Œ0; T2 /;

1 1 T2 ˆ1˛ A ; 0

D

ˇ : ˛1 1

Lemma 4.3.4 is proved. Thus, Theorem 4.3.2 is completely proved.

4.4

Examples

In this section, we consider Eqs. (4.2)–(4.5). In these examples, we use standard Sobolev embedding theorems, which can be found, for example, in [111]. In all examples, H D L2 ./. Example 4.4.1. n X @u @ div.jrujpj 2 ru/ u C u C u3 D 0; u u C @t @x1 j D1

uj@ D 0;

u.x; 0/ D u0 .x/;

pj 2;

386


where RN is a bounded domain with smooth boundary of the class C .2;ı/ , ı 2 .0; 1, and the operator coefficients have the form A0 u u C u W H01 ./ ! H1 ./; 1;pj

Aj .u/ div.jrujpj 2 ru/ W W0 F .u/ u3 W L4 ./ ! L4=3 ./; Dv

0

./ ! W 1;pj ./;

P .u/ u2 W L4 ./ ! L2 ./;

@v W L2 ./ ! H1 ./; @x1

pj > 2;

Lu D u W W1 D L2 ./ ! L2 ./ D W1 ; 1;pj

V0 H01 ./;

Vj W0

V0 H1 ./;

Vj W 1;pj ./; W0 L4=3 ./;

./;

W0 L4 ./;

0

W3 L2 ./; W4 H1 ./; \ n 1;p W0 j ./ : V H01 ./ \

W2 L4 ./;

j D1

The embedding V0 H01 ./ W0 L4 ./ holds under the condition N 4. Finally, the embedding V ,! W0 D W2 holds under the conditions n \ j D1

1;pj

W0

./ ,! L4 ./;

which, in its turn, holds under the conditions N p max pj ; j D1;n

4
p:

We require the validity of the following condition for the initial function u0 2 V : r ˇ ˆ0 .0/ > ˆ.0/ C ˆ.0/; ˛1 1 ˛1 1 where X pj 1 1 p krukpjj ; ˆ.0/ D kru0 k22 C 2 pj n

ˆ0 .0/ D ku0 k44 ku0 k22 ;

j D1

˛1 D

pC4 ; 2p

ˇD

12 ; 4p

D

36 4p

under the condition p < 4. One can verify that the operator coefficients satisfy all the conditions of Theorem 4.2.2.

387


Example 4.4.2. @ @u div.jruj2 ru/ D 0; .2 u C u C div.jrujp2 ru// C u C u @t @x1 ˇ @u ˇˇ D 0; u.0; x/ D u0 .x/; p 3; uj@ D @n ˇ@ where RN is a bounded domain with smooth boundary @ of the class C .4;ı/ , ı 2 .0; 1, and the operator coefficients have the form A0 u 2 u u W H02 ./ ! H2 ./; F .u/ div.jruj2 ru/ W W01;4 ./ ! W 1;4=3 ./; 1;p


0

./ ! W 1;p ./;

P .u/ D u2 W H01 ./ L4 ./ ! L2 .//; Dv D

@v W L2 ./ ! H1 ./; @x1

Lu D u W W1 D H01 ./ ! H1 ./ D W1 ; V0 H02 ./; 1;p

V1 W 0

./;

V0 H2 ./;

W0 W01;4 ./;

W2 L4 ./;

W3 L2 ./;

W4 H1 ./;

W0 W 1;4=3 ./;

V1 W 1;p ./;

1;p

V H02 ./ \ W0

0

./:

The embedding V ,! W0 W1 holds under the conditions N 3 and p 2 .2; 6. Also, the embeddings V0 ,! W1 and V0 W0 W2 hold. Now we require the validity of the following condition for the initial function u0 2 H02 ./: r ˇ 0 ˆ.0/ C ˆ.0/; ˆ .0/ > ˛1 1 ˛1 1 where 4Cp 12 36 ; ˇD ; D ; 2p 4p 4p 1 1 p1 ˆ.0/ D ku0 k22 C kru0 k22 C kru0 kpp ; 2 2 p ˛1 D

ˆ0 .0/ D kru0 k44 kru0 k22 under the condition p < 4. Thus, all the conditions of Theorem 4.3.2 hold.

388


Example 4.4.3. @ .2 u C u C div.jrujp2 ru// C u div.jruj2 ru/ @t @u @u @u @u @u @u @ @ @ C ˇ2 C ˇ3 D 0; C ˇ1 @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2 ˇ @u ˇˇ uj@ D D 0; u.x; 0/ D u0 .x/; p 3; @n ˇ @

where ˇ1 C ˇ2 C ˇ3 D 0, jˇ1 j C jˇ2 j C jˇ3 j > 0, RN is a bounded domain with smooth boundary @ of the class C .4;ı/ , ı 2 .0; 1, and the operator coefficients have the form 0

1;p A1 u D div.jrujp2 ru/ W V1 D W0 ./ ! V1 D W 1;p ./;

Lu D u W W1 D H01 ./ ! W1 D H1 ./; A0 u 2 u u W H02 ./ ! H2 ./; F .u/ div.jruj2 ru/ W W01;4 ./ ! W 1;4=3 ./; P .u/ D ˇ1

@u @u @u @u @u @u e1 C ˇ2 e2 C ˇ3 e3 W @x2 @x3 @x3 @x1 @x1 @x2 W01;4 ./ ! L2 ./ L2 ./ L2 ./;

Dv div.v/ W L2 ./ L2 ./ L2 ./ ! H1 ./; V0 H02 ./;

W0 W01;4 ./;

W2 W01;4 ./;

W3 L2 ./ L2 ./ L2 ./; W4 H1 ./;

V0 H2 ./;

W0 W 1;4=3 ./;

V H02 ./ under the conditions N 3 and p 2 .2; 6. Moreover, the embeddings V ,! W0 W1 , V0 W1 , V0 W0 W2 hold. Now we require the validity of the condition for the initial function u0 2 H02 ./: r ˇ ˆ.0/ C ˆ.0/; ˆ0 .0/ > ˛1 1 ˛1 1 where

4Cp 12 36 ; ˇD ; D ; 2p 4p 4p 1 p1 1 kru0 kpp ; ˆ.0/ D ku0 k22 C kru0 k22 C 2 2 p ˛1 D

ˆ0 .0/ D kru0 k44 kru0 k22 under the condition p < 4. Thus, all the conditions of Theorem 4.3.2 hold.

389


Example 4.4.4. @ @jujq2 C1 C juj2q2 u D 0; .u u jujp1 2 u/ C u C @t @x1 uj@ D 0;

u.x; 0/ D u0 .x/;

p1 3;

where RN is a bounded domain with smooth boundary @ of the class C .2;ı/ , ı 2 .0; 1, and the operator coefficients have the form A0 u u C u W H01 ./ ! H1 ./; 0

A1 .u/ jujp1 2 u W Lp1 ./ ! Lp1 ./; 00

F .u/ juj2q2 u W L2q2 C2 ./ ! Lq2 ./; 2q2 C2

P .u/ jujq2 C1 W L0

./ ! L2 ./;

2q2 C 2 ; 2q2 C 1 @v Dv W L2 ./ ! H1 ./; @x1 q200 D

Lu u W W1 D H01 ./ ! W1 D H1 ./; V0 H01 ./;

W0 L2q2 C2 ./;

V1 Lp1 ./;

W2 L2q2 C2 ./; W3 L2 ./; 0

V0 H1 ./; p1 D q1 C 2;

W4 H1 ./; 00

V1 Lp1 ./; W0 Lq2 ./;

q1 > 0;

p10

p1 ; p1 1

q200

2q2 C 2 ; 2q2 C 1

q2 > 0:

The embedding V0 W0 holds if 0 < q2

2 N 2

for N 3

or

0 < q2 < C1 for N D 1; 2:

This means that V H01 ./ \ Lq1 C2 ./ ,! W0 L2q2 C2 ./: Moreover, W0 D W2 . We require that r ˇ 0 ˆ.0/ C ˆ.0/; ˆ .0/ > ˛1 1 ˛1 1 where 2q C2

ˆ0 .0/ D ku0 k2q22 C2 kru0 k22 ; 1 1 1 ˆ.0/ D kru0 k22 C ku0 k22 C ku0 kpp11 ; 2 2 p1 ˛1 D

2q2 C 2 C p1 ; 2p1

ˇD

2.q2 C 1/.2q2 C 1/ ; 2q2 C 2 p1

D

4.2q2 C 1/2 2q2 C 2 p1

390


under the condition p D p1 < 2q2 C 2. Now we verify the monotonicity of the Fréchet derivative of the operator A1 .u/ D jujq1 u in the sense of Theorem 4.3.2. Indeed, the Fréchet derivative has the form 0

A01;u .h/ .q1 C 1/jujq1 h W Lq1 C2 ./ ! Lq1 ./; q1 C 2 q1 C 1

q10

8h 2 Lq1 C2 ./;

hA01;u .u/u1 A01;u .u2 /u2 ; u1 u2 i1 D .q1 C 1/ihjujq1 u1 jujq1 u2 ; u1 u2 i1 Z .q1 C 1/ dx .jujq1 u1 jujq1 u2 /.u1 u2 / 0 8u; u1 ; u2 2 Lq1 C2 ./;

where q1 D p1 2. Now prove that under the condition krun ruk2 ! 0 as n ! C1, the Fréchet derivatives converge in the uniform operator topology L.H01 ./; H1 .//: Indeed, under the conditions 0 < q1 4=.N 2/ for N 3 and q1 > 0 for N D 1; 2, we have kA01;u .u/ A01;u .un /kH1 !H1 D 0

sup krhk2 D1

C

kA01;u .u/h Z

sup krhk2 D1

sup krhk2 D1


A01;u .un /hk.q1 C2/=.q1 C1/


.q1 C1/=.q1 C2/ :

Consider the integral Z ˇ ˇ q1 C2 dx ˇŒjujq1 jun jq1 hˇ q1 C1 Jn D Z ˇ q1 C2 q1 C2 ˇ dx ˇjujq1 jun jq1 ˇ q1 C1 jhj q1 C1

Z

C


1 C2/=.q1 C1/ Ckrhk.q 2

Z C

Z

dx jhjq1 C2


ˇ ˇ.q C2/=q1 dx ˇjujq1 jun jq1 ˇ 1 .q C2/q1 =.q1 C1/

C max¹kruk2 1

q1 =.q1 C1/ Z

q1 =.q1 C1/

q1 =.q1 C1/ .q1 C2/q1 =.q1 C1/

; krun k2

1=.q1 C1/

º C:

391

Section 4.5 Blow-up in a Sobolev-type equation with nonlocal sources

Note that since kru run k2 ! C0 as n ! C1, un ! u strongly in Lq1 C2 ./ for q1 > 0. Consider the function f .x; u/ D jujq1 W Lq1 C2 ./ ! L

q1 C2 q1

./

for any fixed h 2 Lq1 C2 ./. By the definition, the continuity of the operator f W Lq1 C2 ./ ! L.q1 C2/=q1 ./ means that the limit relation “ q1 C2 dx jf .x; u/ f .x; un /j q1 D 0 lim n!C1

holds under the strong convergence un ! u in Lq1 C2 ./. Therefore, according to the Krasnoselskii’s theorem (see Appendix B) we see that Jn ! C0 as n ! C1 uniformly with respect to h, where krhk2 D 1. Thus, all the conditions of Theorem 4.3.2 hold.

4.5

Blow-up of solutions of a Sobolev-type wave equation with nonlocal sources

In this section, we obtain sufficient conditions of the blow-up of solutions of the following initial-boundary-value problem: Z @u @u @ Cu u dx jruj2 D 0; .u u jujq u/ C u C @t @x1 @x1 (4.70) uj@ D 0; u.x; 0/ D u0 .x/; where N D 3, x D .x1 ; x2 ; x3 / 2 R3 , and is a bounded domain with smooth boundary @ 2 C .2;ı/ , ı 2 .0; 1. This problem is a mathematical model of wave processes in semiconductors in external electric field subject to dissipation and nonlocal sources of free electron current.

4.5.1 Unique local solvability of the problem We give the definition of a strong generalized solution of problem (4.70). Definition 4.5.1. A solution of the class C .1/ .Œ0; TI H01 .// satisfying the conditions hD.u/; wi D 0

8w 2 H01 ./;

u.0/ D u0 2

8t 2 Œ0; T;

(4.71)

H01 ./;

where @u @u @ Cu u D.u/ D .u u jujq u/ C u C @t @x1 @x1

Z

dx jruj2 ;

392


where h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./ and the time derivative is meant in the classical sense, is called a strong generalized solution of problem (4.70). Theorem 4.5.2. Let q 2 Œ1; 4 and u0 2 H01 ./. Then there exists T0 > 0 such that a unique strong generalized solution of problem (4.70) exists and either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim sup kruk2 D C1 t"T0

holds. Proof. Denote by k kC1 the norm in H01 ./ and by k k1 the norm in H1 ./. Now consider the following operators: A.u/ D u C u C jujq u; F1 u D ux1 ;

F2 .u/ D uux1 ;

Lu D u;

F3 .u/ D kruk22 .u/:

(4.72)

In the strong generalized sense, owing to notation (4.72), problem (4.71) takes the form d .A.u// C Lu D F1 u C F2 .u/ C F3 .u/; dt

u.0/ D u0 2 H01 ./:

(4.73)

Consider the properties of the operator A: hA.u1 / A.u2 /; u1 u2 i D kru1 ru2 k22 C ku1 u2 k22 Z C dx .ju1 jq u1 ju2 jq u2 /.u1 u2 /

kru1 ru2 k22 : Therefore, the operator A has a Lipschitz-continuous inverse operator with Lipschitz constant equal to 1. The operator L W H01 ./ ! H1 ./ has a Lipschitz-continuous inverse operator with Lipschitz constant equal to 1: F1 u D

@u W H01 ./ ! L2 ./ ,! H1 ./: @x1

Here the operator F1 is Lipschitz-continuous. Now consider the operator F2 .u/ W H01 ./ ! H1 ./: kF2 .u1 / F2 .u2 /k1 Cku21 u22 k2 C1 .R/ku1 u2 kC1 ; 1 .R/ D R;

R D max¹ku1 kC ; ku2 kC º:

(4.74)

393


Now consider the operator F3 .u/ D kruk22 .u/ W H01 ./ ! H1 ./: kF3 .u1 / F3 .u2 /k1 Ckru1 k22 ku1 u2 kC1 C Cjkru1 k22 kru2 k22 jku1 kC C2 .R/ku1 u2 kC1 ;

(4.75)

where R D max¹ku1 kC ; ku2 kC º, 2 .R/ D R2 . Now consider the Fréchet derivative of the operator A: A0u D A0 C B.u/;

(4.76)

where B.u/h D .q C 1/jujq h:

A0 h D h C h; By virtue of (4.76), the inequality

hA0 .u/h1 A0 .u/h2 ; h1 h2 i krh1 rh2 k22 holds. Therefore, by the Browder–Minty theorem, the operator A0 .u/ has a Lipschitzcontinuous inverse operator with Lipschitz constant equal to 1. Now we consider the operator B.u/ D .q C 1/jujq I. Prove that this operator is bounded and continuous with respect to u 2 H01 ./ in the uniform topology of the space L.H01 ./I H1 .//: B./ W BC.H01 ./ W L.H01 ./I H1 .///. Indeed, by the conditions N D 3 and q 2 .0; 4 we have Z kB.u/hk1 CkB.u/hk qC2 C qC1

Z C

qC2

dx juj

q qC2 qC1

dx juj

q Z qC2

dx jhj

jhj

qC2

qC2 qC1

qC1 qC2

1 qC2

(4.77) Ckrukq2 krhk2 :

Now let un ! u strongly in H01 ./; then, obviously, un ! u strongly in LqC2 ./. The following relations hold: kB.un / B.u/kH1 ./!H1 ./ D 0

sup krhk2 D1

kB.un /h B.u/hk1 ;

kB.un /h B.u/hk1 CkB.un /h B.u/hk qC2 qC1

Z C

dx jhj

CkhkqC2

ˇ

.qC2/=.qC1/ ˇ

Z

q

jun j juj

ˇ ˇ qC2 dx ˇjun jq jujq ˇ q

ˇ qC2

q ˇ qC1

q qC2

qC1 qC2

C < C1;

394


where C is independent of n 2 N. Consider the function f .x; u/ D jujq W LqC2 ./ ! L

qC2 q

./:

One can verify that all the conditions of the Krasnoselskii theorem hold. By the definition, the continuity of the operator f W LqC2 ./ ! L.qC2/=q ./ means that the limit relation “ qC2

dx jf .x; u/ f .x; un /j qC1 D 0

lim

n!C1

holds under the strong convergence un ! u in LqC2 ./. From (4.77) we have the strong boundedness of the operator B.u/ D .q C 1/jujq I. By the Krasnoselskii theorem (see Appendix A.5) we conclude that kB.un / B.u/kH1 ./!H1 ./ ! C0 0

as n ! C1. Introduce the notation v D A.u/:

(4.78)

Then in the class v 2 C .1/ .Œ0; TI H1 .//, owing to (4.78) and the Browder–Minty theorem, problem (4.73) is equivalent to the following problem: dv D LA1 .v/ C F1 A1 .v/ C F2 .A1 .v// C F3 .A1 .v//; dt

(4.79)

v.0/ D v0 D A.u0 / 2 H1 ./. Problem (4.79) is equivalent to the integral equation v.t / D H.v/; (4.80) Z t H.v/ D v0 C ds ŒLA1 .v/ C F1 A1 .v/ C F2 .A1 .v// C F3 .A1 .v//.s/: 0

In the Banach space L1 .0; TI H1 .//, consider the following closed, convex, bounded subset: ® ¯ BR v 2 L1 .0; TI H1 .// W kvkT D ess sup kvk1 R : t2.0;T/

It is easy to prove that, by virtue of properties (4.74) and (4.75), there exists T > 0 such that the operator H acts from BR into BR under sufficiently large R. By the theorem on contraction mappings, there exists a unique solution of the integral equation (4.80) of the class L1 .0; TI H1 .//. Using the standard extension-in-time algorithm for solutions of integral equations of the form (4.80), we obtain that there


395

exists T0 > 0 such that either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim sup kvk1 D C1 t"T0

holds. Time smoothing properties imply the fact that v.x; t / 2 C .1/ .Œ0; T0 /I H1 .// is a strong generalized solution of problem (4.79). Now consider Eq. (4.78): A.u/ D u C u C jujq u D v:

(4.81)

From the properties of the operator A we obtain that u D A1 .v/:

(4.82)

Moreover, ku.t / u.t0 /kC1 kA1 .v.t // A1 .v.t0 //kC1 kv.t / v.t0 /k1 ! C0 as t ! t0 . Thus, by (4.82) we have u 2 C.Œ0; T0 /I H01 .//. In the class u 2 C .1/ .Œ0; T0 /I H01 .//, problem (4.81) is equivalent to the following problem: A0u u0 D v 0 ;

u D A1 .v/:

(4.83)

From (4.83) we obtain that ŒA0 C B.u/ u0 D v 0 ;

A0 D C I:

(4.84)

We apply the operator A1 0 to both sides of Eq. (4.84); then we obtain 0 1 0 ŒI C A1 0 B.u/u D A0 v :

(4.85)

O D I C A1 B.u/ C 0

(4.86)

Now prove that the operator

has a bounded inverse operator for fixed u 2 H01 ./. Indeed, consider the equation O D f 2 H01 ./: Cw We apply the operator A0 to both sides of the latter equation; we obtain ŒA0 C B.u/ w D A0 f I

396


hence, by the invertibility of the operator A0 C B.u/, we conclude that in the class H01 ./, the equation is uniquely solvable. Therefore, the inverse operator C 1 is defined. On the other hand, the operators A1 0 and B.u/ are bounded in the corresponding O spaces. Therefore, the operator C W H01 ./ ! H01 ./ defined by the formula (4.86) is linear and bounded. Hence by the inverse-mapping Banach theorem, we conclude O 1 is linear and bounded. Therefore, from (4.85) we obtain that the operator C 0 O 1 A1 u0 D C 0 v :

(4.87)

Now we must prove only the fact that 0 1 O 1 A1 u0 D C 0 v 2 C.Œ0; T0 /I H0 .//


O Ckv .t / v .t0 /k1 C CkC 0

0

1

O .t0 / C

(4.88) 1

.t /kH1 ./!H1 ./ : 0

0

O is continuous and, Note that the linear (for fixed u 2 C.Œ0; T0 /I H01 .//) operator C O 1 is linear and therefore, by the inverse-mapping Banach theorem, the operator C continuous and, therefore, bounded owing to the linearity. Thus, we can use the specO 1 W H1 ./ ! H1 ./. tral representation for the linear bounded operator C 0 0 O First, introduce the resolvent of the operator C: O D .I C/ O 1 : R.; C/ Let be a circle jj D r with sufficiently large radius greater than sup t2Œt0 ";t0 C"

O 1 kCk H ./!H1 ./ : 0

0

The introduced value is well defined since for t 2 Œt0 "; t0 C " Œ0; T0 /, the inequality sup kukC1 < C1 t2Œt0 ";t0 C"

O 1 .t / and holds. Now we can use the spectral representation for the operators C O 1 .t0 / with the same contour introduced above: C Z Z 1 1 1 1 1 O O O 0 //: O d R.; C.t //; C .t0 / D d 1 R.; C.t C .t / D 2 i 2 i

397


Hence we obtain O 1 .t / C O 1 .t0 / D 1 C 2 i

Z

O // R.; C.t O 0 //: d 1 ŒR.; C.t

Now we use the known representation of the operator resolvents: O 0 // O // R.; C.t O 0 // D R.; C.t R.; C.t

C1 X

O / C.t O 0 //R.; C.t O 0 /n Œ.C.t

nD1

under the condition O /k 1 O O 0 / C.t kC.t H ./!H1 ./ kR.; C.t0 //kH1 ./!H1 ./ ı < 1: 0

0

0

0


0

O 0 //k 1 kR.; C.t H ./!H1 ./ 0

C1 X nD1

0


0

0

0

Note that O / C.t O 0 / D A1 ŒBu .u.t // Bu .u.t0 // : C.t 0 By the continuity of the Fréchet derivatives Bu with respect to u 2 H01 ./ and by the fact that u 2 C.Œ0; T0 /I H01 .//, we have O / C.t O 0 /k 1 kC.t H ./!H1 ./ kBu .u.t // Bu .u.t0 //kH1 ./!H1 ./ ! C0; 0

0

0

O // R.; C.t O 0 //k 1 kR.; C.t H ./!H1 ./ ! 0; 0

0

O /1 C.t O 0 /1 k 1 kC.t H ./!H1 ./ ! C0 0

0

as t ! t0 . Therefore, by (4.88) we have: u 2 C .1/ .Œ0; T0 /I H01 .//. Note that under the condition q 1, the operator C.u/ is boundedly Lipschitz-continuous and, therefore, the operator C 1 .u/ is also boundedly Lipschitz-continuous (see Appendix A.17) and hence Eq. (4.87) has a local solution in the class u.t / 2 C .1/ .Œ0; T00 /I H01 .//. It is easy to prove that T0 D T00 . Theorem 4.5.2 is proved.

398


4.5.2 Blow-up of strong generalized solutions In the previous subsection, we proved the local-on-time solvability of problem (4.70) in the strong generalized sense: u 2 C .1/ .Œ0; T0 /I H01 .//. Here we prove that, under certain sufficient conditions, the inequality T0 < C1 holds. Thus, the following limit relation lim sup kruk2 D C1

(4.89)

t"T0

holds. The following theorem is valid. Theorem 4.5.3. Let the conditions of Theorem 4.5.2 and the condition q 2 .0; 2/ and r ˇ 0 ˆ .0/ > ˆ.0/ C ˆ.0/ ˛1 1 ˛1 1 hold. Then T0 2 ŒT1 ; T2 and the limit relation (4.89) holds, where ˛1 D

6Cq ; 2q C 4

ˇD

8B4 ; 2q

D

16 Œ3 C B4 ; 2q

ˆ0 .0/ D kru0 k42 kru0 k22 ; ˆ.0/ D

1 qC1 1 qC2 kru0 k22 C ku0 k22 C ku0 kqC2 ; 2 2 qC2 1 1 T2 D ˆ1˛ A ; 0

A .1 ˛1 /2 ˆ2˛1 .0/ ˆ0 .0/

1 ; 4ˆ0 2 ˆ.0/

T1 D

ˇ ˛1 1

.ˆ.0//2 ˛1

1=2 ;

kvk4 Bkrvk2 : Proof. First, we take w D u.x; t / in Definition 4.5.1. After integrating by parts we obtain the first energy relation dˆ C kruk22 D kruk42 ; dt

(4.90)

where ˆ.t / D

1 1 qC1 qC2 kruk22 C kuk22 C kukqC2 : 2 2 qC2

(4.91)

399


Now let w D u0 2 C.Œ0; T0 /I H01 .//; after integrating by parts we obtain the second energy relation kru0 k22

C

ku0 k22

Z

dx jujq .u0 /2 Z Z 1 d 1d kruk42 kruk22 C D dx u0 ux1 C dx uux1 u0 : (4.92) 4 dt 2 dt C .q C 1/

The following inequalities hold: ˇ ˇZ ˇ ˇ ˇ dx .ru0 ; ru/ˇ kru0 k2 kruk2 ; ˇ ˇ

ˇZ ˇ ˇ ˇ ˇ dx u0 uˇ ku0 k2 kuk2 ; ˇ ˇ ˇ Z ˇZ 1=2 Z 1=2 ˇ ˇ q 0 2 qC2 ˇ dx jujq uu0 ˇ dx juj .u / dx juj : ˇ ˇ

(4.93) (4.94)

By (4.91), (4.93), and (4.94), we have Z 2 Z Z 0 2 0 0 q 0 .ˆ / D dx .ru ; ru/ C dx u u C .q C 1/ dx juj uu

kru0 k22 kruk22

Z

C .q C 1/

C

0 2 ku k2 kuk22

dx jujq .u0 /2 .q C 1/

C 2kru0 k2 kruk2 .q C 1/

Z

Z

0

C 2ku k2 kuk2 .q C 1/

Z

dx jujqC2 C 2kru0 k2 kruk2 ku0 k2 kuk2

dx jujq .u0 /2 0 2

q

dx juj .u /

1=2

1=2

Z .q C 1/ Z

qC2

.q C 1/

Z kru0 k22 C ku0 k22 C .q C 1/ dx jujq .u0 /2

.kruk22

C

kuk22

C .q C

qC2 1/kukqC2 /

.q C 2/ kru0 k22 C ku0 k22 C .q C 1/

Z

dx jujqC2

dx juj

1=2

1=2

dx jujq .u0 /2 ˆ.t /:

Thus, the inequality 0 2

.ˆ / .q C 2/ˆ

kru0 k22

C

ku0 k22

Z C .q C 1/

holds. Introduce the notation J D kru0 k22 C ku0 k22 C .q C 1/

Z

q

0 2

dx juj .u /

dx jujq .u0 /2 :

(4.95)

(4.96)

400


Now we obtain upper estimates for the function J.t /. From (4.90)–(4.92) and (4.96) we obtain ˇZ ˇ ˇ ˇZ ˇ ˇ ˇ 1ˇ ˇ 1 00 ˇ ˇ 1 ˇˇ d 2ˇ 0 2 0 ˇ ˇ ˇ ˇ J ˇ kruk2 ˇ C ˆ C ˇ dx ux1 uˇ C ˇ dx u ux1 ˇ 4 dt 4 2 " 1 1 00 1 " " 1 (4.97) J C ˆ C ˆ C ˆ C J C J C kuk44 ; 4 2" 4 " 2 4 4" where we have used the following inequalities: ˇ ˇZ ˇ ˇ ˇ dx .ru0 ; ru/ˇ " kru0 k2 C 1 kruk2 ; 2 2 ˇ 2 ˇ 2" ˇ ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ dx u0 uˇ " ku0 k2 C 1 kuk2 ; ˇ dx uu0 ˇ " kru0 k2 C 1 kuk2 ; 2 2 x1 ˇ 2 2 ˇ 2 ˇ ˇ 2" 2 2" ˇZ ˇ ˇ ˇ ˇ dx u2 u0 ˇ " kru0 k2 C 1 kuk4 ; " > 0: x 2 4 ˇ 1ˇ 2 "2 As a result, from (4.97) we obtain Œ1 "J

1 B4 0 1 00 Œ3 C B4 ˆ C ˆ C ˆ ; 2" 4" 4

(4.98)

where B is the constant of the best embedding kuk4 Bkruk2 . From (4.95) and (4.98) we obtain the following second-order ordinary differential inequality (see [210]): ˆˆ00 ˛1 .ˆ0 /2 C ˇˆ0 ˆ C ˆ2 0;

(4.99)

where 4 B4 .1 "/ ; ˇ D ; qC2 " 2 2q 4 for 0 < q < 2: D Œ3 C B ; " 2 0; " 4 ˛1 D

Note that under the condition q 2 .0; 2/ we have ˛1 > 1, ˇ > 0, > 0. Inequality (4.99) can be easily reduced to the linear differential inequality Z00 C ˇZ0 ıZ 0;

ı D .˛1 1/;

(4.100)

for the new function Z D ˆ1˛1 :

(4.101)


401

Now we pass to the new function Y Ze ˇ t

(4.102)

in Eq. (4.100); then we obtain the inequality Y 00 ˇY 0 ıY 0:

(4.103)

From (4.101) and (4.102) we have 0

˛1

ˇt

Y D e .˛1 1/ˆ

ˆ0 C

ˇ ˆ : ˛1 1

(4.104)

We require the validity of the condition ˆ0 .0/ >

ˇ ˆ.0/: ˛1 1

(4.105)

By the continuous differentiability of the function Y , from condition (4.105), owing to (4.104) we conclude that there exists an interval Œ0; T3 / belonging to the domain of definition of ˆ.t / such that Y 0 < 0. Therefore, for t 2 Œ0; T3 , we have ˇY 0 .t / > 0. Hence from inequality (4.103) we obtain the differential inequality Y 00 ıY 0

for t 2 Œ0; T3 :

We require the validity of the inequality p ˇ 0 ˆ .0/ > p ˆ.0/ C ˆ.0/: ˛ ˛1 1 11

(4.106)

(4.107)

From inequality (4.106) we have .Y 0 /2 ıY 2 C A2 :

(4.108)

Then (4.108) implies the inequality .Y 0 /2 A2 ;

(4.109)

where by virtue of (4.107) 2

A .1 ˛1 / ˆ

2˛1

.0/ ˆ0 .0/

2 1=2 ˇ ı 2 .ˆ.0// > 0: ˆ.0/ ˛1 1 .˛1 1/2

From (4.109) we obtain Y 0 .t / A

8t 2 Œ0; T3 :

(4.110)

402


From (4.110) we obtain that Y 0 < 0 in the domain of definition of the function ˆ.t /; hence from (4.110) we have Y .t / Y0 At: Thus, there exists a time moment T0 2 .0; T2 / such that lim sup ˆ.t / D C1; t"T0

T2 D Y .0/A1 :

Now we obtain a lower estimate for the blow-up time of solution of problem (4.70). Indeed, the first energy relation (4.90) implies the inequalities 2 dˆ 1 4 2 kruk2 4 kruk2 4ˆ2 : dt 2

(4.111)

From (4.111) we have the inequality ˆ

ˆ0 ; 1 4ˆ0 t

from which we obtain the estimate T1 T0 , where T1 D 41 ˆ1 0 . Theorem 4.5.3 is proved.

4.6

Blow-up of solutions of a strongly nonlinear equation of spin waves

In this section, we obtain sufficient, close to necessary, conditions of the blow-up of solutions of the following initial-boundary-value problem: @ .2 u C u C p u/ C u 4 u @t @u @u @u @u @u @u @ @ @ C ˛2 C ˛3 D 0; C˛1 @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2 ˇ @u ˇˇ uj@ D D 0; u.x; 0/ D u0 .x/ 2 H02 ./; @nx ˇ@

(4.112) (4.113)

where p 3, ˛1 C ˛2 C ˛3 D 0, j˛1 j C j˛2 j C j˛3 j > 0, x D .x1 ; x2 ; x3 / 2 R3 , is a bounded domain with smooth boundary @ 2 C 4;ı , ı 2 .0; 1, p u D div.jrujp2 ru/, 4 u D div.jruj2 ru/. Note that for N D 3, H02 ./ W01;p ./ under the condition p 2 .2; 6 by the Sobolev embedding theorem. Equation (4.112) describes spin waves in magnetics subject to dissipation, strong spatial dispersion, and magnetic domain sources.

403

Section 4.6 Blow-up of spin waves

4.6.1 Unique local solvability in the strong generalized sense We give the definition of a strong generalized solution. Definition 4.6.1. A solution of the class C .1/ .Œ0; TI H02 .// satisfying the conditions hD.u/; wi D 0 D.u/ D

8w 2 H02 ./;

u.0/ D u0 2 H02 ./

8t 2 Œ0; T;

(4.114)

@ .2 u C u C p u/ C u @t @u @u @u @u @u @u @ @ @ C ˛2 C ˛3 4 u; C ˛1 @x1 @x2 @x3 @x2 @x3 @x1 @x3 @x1 @x2

where h; i is the duality bracket between the Hilbert spaces H02 ./ and H2 ./, is called a strong generalized solution of problem (4.112). Theorem 4.6.2. Let p 2 Œ3; 6 and u0 2 H02 ./. Then there exists T0 > 0 such that a unique strong generalized solution of problem (4.112), (4.113) of the class C .1/ .Œ0; T0 /I H02 .// exists, and either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim sup kuk2 D C1 t"T0

holds. Proof. Introduce the notation A0 u 2 u u W H02 ./ ! H2 ./; A1 .u/ div.jruj Lu u W P .u/ ˛1

p2

H01 ./

ru/ W !H

W01;p ./ 1

!W

(4.115) 1;p 0

./;

./;

(4.116) (4.117)

@u @u @u @u @u @u e1 C ˛2 e2 C ˛3 e3 W @x2 @x3 @x3 @x1 @x1 @x2 W01;4 ./ ! L2 ./ L2 ./ L2 ./; (4.118)

Dz div.z/ W L2 ./ L2 ./ L2 ./ ! H1 ./; F .u/ div.jruj2 ru/ W W01;4 ./ ! W 1;4=3 ./:

(4.119) (4.120)

Denote by k kCk , k 2 N, the norm of the Hilbert space H0k ./ and by k kk the norm in Hk ./. Note that for N D 3, the embedding H02 ./ W01;r ./ holds under the condition r 2 Œ2; 6.

404


Owing to the notation (4.115)–(4.120), problem (4.114) takes the following abstract form: d .A0 u C A1 .u// C Lu C DP .u/ D F.u/; dt

u.0/ D u0 2 H02 ./:

(4.121)

Introduce the operator A.u/ A0 u C A1 .u/ W H02 ./ ! H2 ./:

(4.122)

Analyze the properties of this operator. The operator A./ is radially continuous, monotonic, and coercive. Indeed, by the conditions (V2), the following relations hold: hA.u1 / A.u2 /; u1 u2 i0 D hA0 u1 A0 u2 ; u1 u2 i0 C hA1 .u1 / A1 .u2 /; u1 u2 i0 for all u1 ; u2 2 H02 ./. Therefore, by the monotonicity of the operators A0 u and A1 ./, hA.u1 / A.u2 /; u1 u2 i0 0 for all u1 ; u2 2 H02 ./. On the other hand, we have hA.u/; ui0 D hA0 u; ui0 C hA1 .u/; ui0 kuk22 for any u 2 H02 . Hence we directly obtain that lim

kuk0 !C1

hA.u/; ui0 D C1: kuk2

Note that by (4.122), the relation hA.u1 / A.u2 /; u1 u2 i D ku1 u2 k22 C kru1 ru2 k22 Z C dx .jru1 jp2 ru1 jru2 jp2 ru2 ; ru1 ru2 /

ku1 u2 k22 D ku1 u2 k2C2 holds since the operator 2 W H02 ./ ! H2 ./ is an isometric isomorphism. Therefore, for the operator A./, by the Browder–Minty theorem, there exists a Lipschitz-continuous inverse operator with Lipschitz constant equal to 1. Similarly, we can prove that the operator A0 W H02 ./ ! H2 ./ has a Lipschitz-continuous inverse operator with Lipschitz constant equal to 1.

405


Now consider the operator 1;p


0

./ ! W 1;p ./:

The Fréchet derivative of the operator A1 .u/ has the form A01;u .u/h D A11 .u/h C A12 .u/h;

A11 .u/h D div.jrujp2 rh/;

A12 .u/h D .p 2/ div.jrujp4 ru.ru; rh//;

(4.123)

0

where A01;u ./ W W01;p ./ ! L.W01;p ./I W 1;p .//, p 0 D p=.p 1/. Now prove that the Fréchet derivative A01;u ./ is a strongly continuous and bounded mapping H02 ./ ! L.H02 ./I H2 .//. For, by the definition, we have kA11 .u/ A11 .un /kH2 ./!H2 ./ D 0

C

sup khkC2 D1

sup khkC2 D1

kA11 .u/h A11 .un /hk2

kA11 .u/h A11 .un /hkW 1;p0 ./ D

sup khkC2 D1

Jn .h/; (4.124)

where Z Jn .h/ D .p 1/

0

dx jrhjp jjrun jp2 jrujp2 jp

0

1=p0 :

(4.125)

From (4.125) we obtain an upper estimate for the value of Jn .h/: Z Jn .h/ .p 1/

dx jrhj

1=p Z

p

.p2/=p ˇ ˇ p2 p2 ˇp=.p2/ ˇ dx jrun j jruj : (4.126)

Consider the operator f D jrujp2 D .u21 C u22 C u23 /.p2/=2 ;

ui D

@u : @xi

(4.127)

Operator (4.127) is generated by a Carathéodory function, and it is easy to verify that the Nemytskii operator (4.127) from Lp ./ Lp ./ Lp ./ into Lp=.p2/ ./, p > 2, p=.p 2/ > 1. Therefore, according to the Krasnoselskii theorem (see [417]), the operator f W Lp ./ Lp ./ Lp ./ ! Lp=.p2/ ./ is bounded and strongly continuous, i.e., Z ˇ ˇp=.p2/ dx ˇjrun jp2 jrujp2 ˇ ! C0;

406


as run ! ru strongly in Lp ./. From (4.126) we obtain that Jn .h/ ! C0 uniformly with respect to h on the sphere khkC2 D 1. Therefore, by virtue of (4.123)– (4.127) we have kA11 .u/ A11 .un /kH2 ./!H2 ./ ! C0 0

as n ! C1. Now consider operator (4.123) A12 .u/h .p 2/ div.jrujp4 .ru; rh/ru/: Introduce the functions E D jj E p2 i j ; fij ./ E jj E jj

E D .1 ; 2 ; 3 / 2 R3 ;

i; j D 1; 3;

1;p where m D uxm , m 2 1; 3, h 2 H02 ./ W0 ./ (p 2 .2; 6). It is easy to verify that the Nemytskii operators defined by the functions fij act from Lp ./ Lp ./ Lp ./ into Lp=.p2/ ./. Then the inequality

kA12 .u/ A12 .un /kH2 !H2 D 0

C

3 X

sup

ˇ ˇ ux uxj dx ˇˇjrujp2 i jruj jruj

jrun j C

p2

ˇ 0 1=p0 unxi unxj ˇˇp p0 jhxj j jrun j jrun j ˇ

ˇ ˇ ux uxj dx ˇˇjrujp2 i jruj jruj ˇ unxi unxj ˇˇp=.p2/ .p2/=.p1/ jrun jp2 ! C0 jrun j jrun j ˇ

3 Z X i;j D1

kA12 .u/h A12 .un /hkH2

0

Z

i;j D1 khkH02 D1

sup khkH2 D1

holds as run ! ru strongly in Lp ./. The monotonicity of the operator A12 is obvious. Therefore, kA01;u .u/ A01;un .un /kH2 ./!H2 ./ ! C0 0

as un ! u strongly in H02 ./.

(4.128)

407


Now consider the operators DP .u/ and F.u/. For the operator DP .u/, the following inequality holds: @v1 @v1 @v2 @v2 C C @v1 @v1 @v2 @v2 kDP .v1 / DP .v2 /k2 C @x @x @x2 @x3 1 @x3 @x1 @x3 @x1 1 2 3 @v1 @v1 @v2 @v2 D K1 C K2 C K3 : C C @x @x @x2 @x1 1 2 1 Consider, for example, the term K1 ; the terms K2 and K3 can be estimated similarly. Indeed, @v1 @v1 @v2 @v2 K1 D C @x2 @x3 @x2 @x3 1 @v1 @v1 @v2 @v2 @v1 @v2 C C C @x2 @x3 @x3 1 @x3 @x2 @x2 1 ˇ ˇ ˇ ˇ Z Z ˇ @v1 ˇ2 @v1 @u2 2 1=2 ˇ @v2 ˇ2 @v1 @u2 2 1=2 ˇ ˇ ˇ C dx ˇˇ C C dx ˇ @x ˇ @x @x2 ˇ @x3 @x3 @x2 3 2 C

2 X

krvi k4 krv1 rv2 k4 C

2 X

kvi kC2 kv1 v2 kC2

i D1

iD1

CRkv1 v2 kC2 for all vi 2 H02 ./, kvi kC2 R. Thus, we conclude that the following inequality holds: kDP .v1 / DP .v2 /k2 1 .R/kv1 v2 kC2 ;

(4.129)

where vi 2 H02 ./, 1 .R/ D CR, kvi kC2 R. Now consider the operator F.u/. The following inequalities hold: kF .v1 /

F.v2 /k2

Z C C

2 Z X i D1

C kF .v1 /

2 Z X i D1

F.v2 /k2

ˇ ˇ4=3 dx ˇjrv1 j2 rv1 jrv2 j2 rv2 ˇ

dx jrvi j

8=3

dx jrvi j4

1=2

2 .R/krv1 rv2 kC2

for all vi 2 H02 ./, kvi kC2 R, 2 .R/ D CR2 .

jrv1 rv2 j

4=3

3=4

3=4

kv1 v2 k4 ; (4.130)

408


Now consider problem (4.121). Introduce the notation v D A.u/. We search for v in the class C .1/ .Œ0; TI H2 .//. By virtue of the properties of the operator A./ W H02 ./ ! H2 ./ proved above, there exists a Lipschitz-continuous inverse operator A1 ./ W H2 ./ ! H02 ./. Thus, u D A1 .v/ and problem (4.121) takes the following equivalent form: dv C LA1 .v/ C DP .A1 .v// D F.A1 .v//; dt v.0/ D v0 D A.u0 / 2 H2 ./:

(4.131)

In the class v 2 C .1/ .Œ0; TI H2 .//, we obtain the integral equation Z t v.t / D v0 C ds Q.v/.s/; Q.v/ D LA

0 1

.v/ DP .A1 .v// C F.A1 .v//:

(4.132)

We search for solutions of the integral equation (4.132) in the class v 2 L1 .0; TI H2 .// for sufficiently small T > 0. Apply the method of contraction mappings. To this end, introduce a closed, convex, bounded subset ® ¯ (4.133) BR v 2 L1 .0; TI H2 .// W kvkT D ess sup kvk2 R t2.0;T/

of the Banach space L1 .0; TI H2 .//. Prove that the operator Z t H.v/ D v0 C ds Q.v/

(4.134)

0

acts from BR into BR and is a contraction on BR for sufficiently small T > 0 and sufficiently large R > 0. Thus, let kv0 k2 R=2; then kH.v/kT R=2 C TkvkT Œ1 C 1 .R/ C 2 .R/ ; where 1 .R/ D CR, 2 .R/ D CR2 , and R kvkT ; therefore, kH.v/kT R under the condition T .2Œ1 C 1 .R/ C 2 .R//1 . Now we prove that the operator H.v/ is a contraction on BR . Indeed, by (4.129)–(4.134) kH.v1 / H.v2 /kT TkQ.v1 / Q.v2 /kT T Œ1 C 1 .R/ C 2 .R/ kv1 v2 kT ;

409


R D max¹kv1 kT ; kv2 kT º under the condition T < 1=Œ2.1 C 1 .R// C 2 .R/. Therefore, the operator H is a contraction on BR for under sufficiently large R > 0 and sufficiently small T > 0. Therefore, there exists a unique solution of the integral equation (4.132) of the class L1 .0; TI H2 .//. Using the standard extension-intime algorithm for solutions of the integral equation (4.132), we obtain that there exists maximal T0 > 0 such that either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim sup kvk2 D C1

(4.135)

t"T0

holds. Note that the explicit form of the operator H.v/ implies the fact that H.v/ W L1 .0; TI H2 .// ! AC.Œ0; TI H2 .//; H.v/ W AC.Œ0; TI H2 .// ! C .1/ .Œ0; TI H2 .//: Therefore, there exists a unique solution of problem (4.131) of the class C .1/ .Œ0; T0 /I H2 .// and either T0 D C1 or T0 < C1, and in the latter case, the limit relation (4.135) holds. Now consider the equation A.u/ D v 2 C .1/ .Œ0; TI H2 .//:

(4.136)

By the properties of the operator A we have u D A1 .v/:

(4.137)

Here the following inequalities hold: ku.t / u.t0 /kC2 kA1 .v/.t / A1 .v/.t0 /kC2 kv.t / v.t0 /k2 ! C0 as t ! t0 , t; t0 2 Œ0; T0 /. Therefore, u.x; t / 2 C.Œ0; T0 /I H02 .//. Now we prove that, in fact, u from (4.137) belongs to the class C .1/ .Œ0; T0 /I H02 .//. Indeed, by the Fréchet differentiability of the operator A in the class u.x; t / 2 C .1/ .Œ0; T0 /I H02 .//, from (4.136) we obtain that A0u .u/u0 D v 0 2 C.Œ0; T0 /I H2 .//:

(4.138)

Write Eq. (4.138) in more detail: ŒA0 C A01;u .u/u0 D v 0 :

(4.139)

By virtue of the properties of the operator A0 and the Browder–Minty theorem, Eq. (4.139) is equivalent to 0 0 1 0 ŒI C A1 0 A1;u .u/u D A0 v :

(4.140)

410


Now prove that the linear (for fixed u 2 C.Œ0; T0 /I H02 .//) operator 0 O D I C A1 C 0 A1;u .u/

(4.141)

is bounded and invertible. Indeed, the boundedness follows from the properties of the operator A0 , which has a Lipschitz-continuous inverse operator, and the operator A01;u ./ is bounded as an operator H02 ./ ! L.H02 ./I H2 .//. Now we prove that the operator C defined by (4.141) is invertible. To this end, we consider the equation O D f: Cw

(4.142)

We apply the operator A0 to both sides of the latter equation; then we obtain that ŒA0 C A01;u .u/w D A0 f: Prove that the operator E D A0 C A01;u .u/ is invertible. Indeed, we have hEw1 Ew2 ; w1 w2 i D kw1 w2 k22 C krw1 rw2 k22 Z C dx jrujp2 jrw1 rw2 j2 Z C .p 2/ dx jrujp4 j.rw1 rw2 ; ru/j2 kw1

2 w2 kC2 :

Hence by the Browder–Minty theorem we obtain the invertibility of the operator E D A0 C A01;u .u/. Therefore, for any f 2 H02 ./, there exists a solution w 2 H02 ./ of Eq. (4.142). Moreover, we can similarly prove the uniqueness of a solution of Eq. (4.142). Therefore, the inverse operator C 1 exists. Therefore, from (4.140) we obtain 0 O 1 A1 u0 D C 0 v :

(4.143)

Now we must prove only the fact that 0 2 O 1 A1 u0 D C 0 v 2 C.Œ0; T0 /I H0 .//


(4.144)

O 1 .t0 / C O 1 .t /k 2 Ckv 0 .t / v 0 .t0 /k2 C CkC H ./!H2 ./ : 0

0

411


O is continuous and, Note that the linear (for fixed u 2 C.Œ0; T0 /I H02 .//) operator C O 1 is lintherefore, by virtue of the inverse-mapping Banach theorem, the operator C ear and continuous and, therefore, bounded owing to the linearity. Thus, we can use O 1 W H2 ./ ! H2 ./. the spectral representation for the linear bounded operator C 0 0 O First, introduce the resolvent of the operator C: O D .I C/ O 1 : R.; C/ Let be a circle jj D r with sufficiently large radius greater than sup t2Œt0 ";t0 C"

O 2 kCk H ./!H2 ./ : 0

0

The introduced value is well defined since for t 2 Œt0 "; t0 C " Œ0; T0 /, the inequality sup kukC2 < C1 t2Œt0 ";t0 C"

O 1 .t / and holds. Now we can use the spectral representation for the operators C 1 O C .t0 / with the same contour introduced above: Z Z 1 1 1 1 1 O O O O 0 //: C .t / D d R.; C.t //; C .t0 / D d 1 R.; C.t 2 i 2 i Obviously, we have O 1 .t / C O 1 .t0 / D 1 C 2 i

Z

O // R.; C.t O 0 //: d 1 ŒR.; C.t

Now we use the well-known representation for the operator resolvents: O // R.; C.t O 0 // D R.; C.t O 0 // R.; C.t

C1 X

O / C.t O 0 //R.; C.t O 0 /n Œ.C.t

nD1

under the condition O 0 / C.t O /k 2 O kC.t H ./!H2 ./ kR.; C.t0 //kH2 ./!H2 ./ ı < 1: 0

0

0

0


0

O 0 //k 2 kR.; C.t H ./!H2 ./ 0

C1 X nD1

0


0

0

0

412 Note that


O / C.t O 0 / D A1 ŒA0 .u.t // A0 .u.t0 //: C.t 0 1;u 1;u

From (4.128) we obtain the continuity of the Fréchet derivatives A01;u .u/ with respect to u 2 H02 ./. By the continuity of the Fréchet derivatives A01;u .u/ with respect to u 2 H02 ./ and by the fact that u 2 C.Œ0; T0 /I H02 .//, we have 0 0 O / C.t O 0 /k 2 kC.t H ./!H2 ./ kA1;u .u.t // A1;u .u.t0 //kH2 ./!H2 ./ ! C0; 0

0

0

O // R.; C.t O 0 //k 2 kR.; C.t H ./!H2 ./ ! 0; 0

0

O 0 /1 k 2 O /1 C.t kC.t H ./!H2 ./ ! C0 0

0

as t ! t0 . Therefore, by (4.144) we have u 2 C .1/ .Œ0; T0 /I H02 .//: Note that, under the condition p 3, the operator C.u/ is boundedly Lipschitzcontinuous and, therefore, the operator C 1 .u/ is also boundedly Lipschitz-continuous (see Appendix A.18) and hence Eq. (4.143) has a local solution in the class u.t / 2 C .1/ .Œ0; T00 /I H02 .//. It is easy to prove that T0 D T00 . Theorem 4.6.2 is proved.

4.6.2 Blow-up of strong generalized solutions and the global solvability In the previous subsection, we have proved the local-on-time unique solvability of problem (4.112), (4.113) in the strong generalized sense: u 2 C.1/ .Œ0; T0 /I H02 .//. Here we prove that under certain sufficient conditions, the inequality T0 < C1 holds and, therefore, the following limit relation is valid: lim sup kuk2 D C1:

(4.145)

t"T0

Introduce the following functional with the sense of energy: 1 1 p1 ˆ.t / kuk22 C kruk22 C krukpp : 2 2 p

(4.146)

The following theorem holds. Theorem 4.6.3. Let the conditions of Theorem 4.6.2 hold. Then the following assertions are valid: (1) if p 2 .4; 6, then T0 D C1 and ˆ.t / .ˆ1 C .1 /B3 t /1=.1/ ; 0

D

4 I p

413


(2) if p D 4, then T0 D C1 and ˆ.t / ˆ0 e B3 t I (3) if p 2 .2; 4/ and

r ˆ0 .0/ >

ˇ ˆ.0/ C ˆ.0/; ˛1 ˛1

then T0 2 ŒT1 ; T2 and the limit relation (4.145) holds, where 3 3 X 84 X 2 4Cp 28 2 ˛D ˛i ; ˇ D ˛i ; ; D 1C6 2p 4p 4p i D1

p 2 .2; 4/;

i D1

ˆ0 .0/ D kru0 k44 kru0 k22 ; 1 1 p1 ˆ0 D ˆ.0/ D ku0 k22 C kru0 k22 C kru0 kpp ; 2 2 p 1 A1 ; T1 D ; T2 D ˆ1˛ 0 4 4B1 ˆ0 1=2 2 ˇ A .1 ˛/2 ˆ2˛ .0/ ˆ0 .0/ ; ˆ.0/ .ˆ.0//2 ˛1 ˛1 1;p

B1 is the best constant of the embedding H02 ./ W0 4=p p B3 D B42 ; p1

./,

and B2 is the best constant of the embedding W01;p ./ W01;4 ./ for p 4. Proof. In the definition of a strong generalized solution, we take the function u 2 C .1/ .Œ0; T0 /I H02 .// as w; then for any T 2 .0; T0 /, after integrating by parts, we obtain the first energy relation 1 p1 d 1 (4.147) kuk22 C kruk22 C krukpp C kruk22 D kruk44 : dt 2 2 p Now take the function u0 2 C.Œ0; T0 /I H02 .// as w; then after integrating by parts, we obtain the second energy relation Z Z ku0 k22 C kru0 k22 C dx jrujp2 jru0 j2 C .p 2/ dx jrujp4 j.ru0 ; ru/j2 Z @u @u 1d 1d 4 2 0 @ D dx u kruk4 kruk2 C ˛1 4 dt 2 dt @x1 @x2 @x3 Z Z @ @u @ @u @u @u C ˛2 dx u0 dx u0 C ˛3 : (4.148) @x2 @x3 @x1 @x3 @x1 @x2

414


Note that in the class u.x; t / 2 C .1/ .Œ0; T0 /I H02 .//, by virtue of the first energy relation (4.147), the functional ˆ.t / defined by (4.146) belongs to the class C .2/ Œ0; T0 /. Now we obtain an upper estimate for .ˆ0 .t //2 : Z .ˆ0 /2 p ku0 k22 C kru0 k22 C .p 2/ dx jrujp4 j.ru0 ; ru/j2 Z C

dx jrujp2 jru0 j2

Thus, the inequality

1 1 p1 2 2 p kuk2 C kruk2 C krukp : 2 2 p

.ˆ0 /2 pˆJ;

holds, where Z 0 2 C kru k C dx jrujp2 jru0 j2 JD 2 Z C .p 2/ dx jrujp4 j.ru0 ; ru/j2 : ku0 k22

(4.149)

Note that the function J.t / coincides with the right-hand side of the second energy relation (4.148). Now we obtain an upper estimate for J using the first and second energy relations (4.147) and (4.148). The following inequality holds: Z 1d 1 00 @u0 @u @u 2 J dx kruk2 C ˆ ˛1 4 dt 4 @x1 @x2 @x3 Z Z 0 @u @u @u @u0 @u @u ˛2 dx ˛3 dx : (4.150) @x2 @x3 @x1 @x3 @x1 @x2 The following auxiliary inequalities hold: ˇ Z ˇ ˇ @u0 @u @u ˇˇ j˛i j2 " 4 0 2 ˇ˛i dx ˇ ˇ 2" kruk4 C 2 kru k2 ; @x @x @x i j k ˇ ˇ ˇd ˇ 2 ˇ kruk ˇ "kru0 k2 C 1 kruk2 : 2ˇ 2 2 ˇ dt "

(4.151) (4.152)

From (4.147) and (4.150)–(4.152) we obtain the inequalities 1 00 " 1 3" 3 X j˛i j2 kruk44 ˆ C kru0 k22 C kruk22 C kru0 k22 C 4 4 4" 2 2" 3

J

i D1

1 3 1 7" ˆ00 C J C ˆ C 4 4 2" 2"

3 X i D1

j˛i j2 Œ2ˆ C ˆ0 :

(4.153)

415


From (4.153) we obtain 3 3 1 00 3X 7 3 X 1 2 C 1 " J ˆ C j˛i j ˆ C j˛i j2 ˆ0 : 4 4 2" " 2"

i D1

(4.154)

i D1

From (4.149) and (4.154) we obtain the following second-order ordinary differential inequality (see [210]): ˆˆ00 ˛.ˆ0 /2 C ˇˆˆ0 C ˆ2 0;

(4.155)

where ˛D

4 7 1 " ; p 4

2 12 X j˛i j2 ; C " " 3

D

6X j˛i j2 : " 3

ˇD

i D1

i D1

We require that " 2 .0; .4 p/=7/ for p 2 .2; 4/. It is easy to verify that in this case ˛ > 1, ˇ > 0, and > 0. As above, we can obtain the following explicit optimal expression for ": 4p : "D 14 Inequality (4.155) can be easily reduced to the linear differential inequality Z00 C ˇZ0 ıZ 0;

ı D .˛ 1/;

(4.156)

for the new function Z D ˆ1˛ :

(4.157)

Y Ze ˇ t

(4.158)

Now we pass to the new function

in Eq. (4.156); then we obtain the inequality Y 00 ˇY 0 ıY 0:

(4.159)

From (4.157) and (4.158) we have that 0

˛

ˇt

Y D e .˛ 1/ˆ

ˆ0 C

ˇ ˆ : ˛1

(4.160)

We require the validity of the condition ˆ0 .0/ >

ˇ ˆ.0/: ˛1

(4.161)

416


By the continuous differentiability of the function Y , from condition (4.161), owing to (4.160) we conclude that there exists an interval Œ0; T3 / belonging to the domain of definition of ˆ.t / such that Y 0 0. Therefore, for t 2 Œ0; T3 , we have ˇY 0 .t / 0. Hence from inequality (4.159) we obtain the differential inequality Y 00 ıY 0

for t 2 Œ0; T3 :

We require the validity of the inequality p ˇ 0 ˆ.0/: ˆ.0/ C ˆ .0/ > p ˛1 ˛1

(4.162)

(4.163)

Inequality (4.162) implies .Y 0 /2 ıY 2 C A2 ;

(4.164)

where, by (4.163), A .1 ˛/2 ˆ2˛ .0/ ˆ0 .0/

1=2 2 ˇ ı 2 .ˆ.0// > 0: ˆ.0/ ˛1 .1 ˛/2

From (4.164) we obtain Y 0 .t / A

8t 2 Œ0; T3 :

(4.165)

From (4.165) we have that Y 0 < 0 in the domain of definition of the function ˆ.t /, hence from (4.165) we have Y .t / Y0 At: Thus, there exists a time moment T0 2 .0; T2 / such that lim sup ˆ.t / D C1; t"T0

where T2 D Y .0/A1 . Now we obtain a lower estimate for the blow-up time of solution of problem (4.112), (4.113). To this end, we use the first energy relation (4.147). Indeed, from (4.147) we have dˆ kruk44 B41 kuk42 4B41 ˆ2 .t /: dt From (4.166) we obtain ˆ.t / Therefore, T0 T1 D .4B41 ˆ0 /1 .

ˆ0 : 1 4B41 ˆ0 t

(4.166)

417

Section 4.7 Blow-up in a strongly nonlinear dissipative equation

Now we consider the case where p 4. Then we have W01;p ./ W01;4 ./. Let

D 4=p. Then from the first energy relation (4.147) we obtain the inequalities dˆ p 1 4=p ˆ ; kruk44 B42 krukp4 B42 dt p

D

4 : p

(4.167)

From (4.167) we obtain for 2 .0; 1/ the inequality ˆ.t / .ˆ1 C .1 /B3 t /1=.1/ ; 0

B3 D B42

p1 p

4=p :

From (4.167) we obtain for D 1 the inequality ˆ.t / ˆ0 e B3 t ;

B3 D B42

p1 p

4=p :


4.6.3 Physical interpretation of the obtained results From Theorem 4.6.3 we obtain that for a sufficiently large initial function of the magnetostatic energy in a magnetic, the avalanche growth for a finite time appears, i.e., the phenomenon of magnetic disruption occurs. Our qualitative description of disruption is completely confirmed by experimental data (see [54, 55]).

4.7

Blow-up of solutions of an initial-boundary-value problem for a strongly nonlinear, dissipative equation of the form (4.1)

In this section, we obtain sufficient conditions of the blow-up of solutions of the following initial-boundary-value problem for a strongly nonlinear, dissipative, pseudoparabolic equation: ˇ ˇ N X @ ˇˇ @u ˇˇp2 @u @ u C u C ux1 C uux1 C u3 D 0; @t @xi ˇ @xi ˇ @xi

(4.168)

iD1

u.x; t /j@ D 0; u.x; 0/ D u0 .x/;

t 0;

(4.169)

x 2 ;

(4.170)

where p > 2, .x; t / 2 Q D .0; T/, x D .x1 ; x2 ; : : : ; xN / 2 RN .

418


4.7.1 Local unique solvability in the weak generalized sense Let @ 2 C .2;ı/ be the boundary of a bounded domain , ı 2 .0; 1, and a natural number s 2 be such that .s 1/=N 1=2 1=p. Consider the following problem for eigenfunctions and eigenvalues: ˇ @n wk ˇˇ s s .1/ wk k wk D 0; D 0; n D 1; s 1: (4.171) @xin ˇ@ By the definition of s, the following dense and continuous embeddings hold: ds

1;p

H0s ./ W0

ds

ds

./ H01 ./ L2 ./

for p > 2:

Since eigenfunctions of problem (4.171) form a Galerkin basis in H0s ./, by virtue of the embeddings mentioned, eigenfunctions of problem (4.171) form a Galerkin basis 1;p in W0 ./. Definition 4.7.1. A solution of the problem Z T d dt .t / .hu; wi0 C hp u; wi/ dt 0

C .u3 ; w/ .u; w/ C hux1 ; wi0 C huux1 ; wi0 D 0; u.0/ D u0 2 W01;p ./ 8 .t / 2 L2 .0; T/;

8w 2 W01;p ./;

where h; i0 is the duality bracket between the Hilbert spaces H01 ./, H1 ./ and 0 1;p h; i is the duality bracket between the Banach spaces W0 ./ and W 1;p ./, and .; / denotes the inner product in L2 ./, is called a weak generalized solution of problem (4.168)–(4.170). We search for a solution of this problem in the class u 2 L1 .0; TI W0

1;p

.//;

u0 2 L2 .0; TI H01 .//; 0

p u 2 L1 .0; TI W 1;p .//; 0

0

.1/ .Œ0; TI W 1;p .//: u C p u 2 H1 .0; TI W 1;p .// \ Cw

In the considered smoothness class, our problem is equivalent to the following problem:

Z T d dt .t / .u C p u/ u C ux1 C uux1 C u3 ; w D 0 dt 0 8 .t / 2 L2 .0; T/;

1;p

8w 2 W0

./;

419


which, by virtue of results of Appendix A.12, is equivalent to the following problem: Z 0

T

d 3 dt .u C p u/ u C ux1 C uux1 C u ; v D 0 dt 8v 2 L2 .0; TI W01;p .//:

Remark 4.7.2. Since w 2 W01;p ./, one can choose elements wj 2 H0s ./, j D 1; C1, such that their linear combinations are dense in W01;p ./. The following theorem holds. 1;p

Theorem 4.7.3. Let u0 .x/ 2 W0 ./. If N p or if N > p for p 4N=.N C 4/, then there exist T0 D T0 .u0 ; q; p/ > 0 and a unique function u.x; t / W .0; T0 / ! R such that for any T 2 .0; T0 /, the following inclusions hold: u 2 L1 .0; TI W0

1;p

.//;

ˇ ˇ @ ˇˇ @u ˇˇ.p2/=2 @u 2 L2 .Q/; @t ˇ @xi ˇ @xi ˇ ˇ @ ˇˇ @u ˇˇp2 @u 0 2 L2 .0; TI Lp .//; ˇ ˇ @t @xi @xi

u0 2 L2 .0; TI H01 .//; ˇ ˇ ˇ @u ˇ.p2/=2 @u ˇ ˇ 2 L1 .0; TI L2 .//; ˇ @x ˇ @x i i ˇ ˇ ˇ @u ˇp2 @u 0 ˇ ˇ 2 L1 .0; TI Lp .//; ˇ @x ˇ @xi i

0

0

.1/ u C p u 2 H1 .0; TI W 1;p .// \ Cw .Œ0; TI W 1;p .//;

p0 D

p : p1

Moreover, u.0/ D u0 almost everywhere in and @ 0 u C p u u C ux1 C uux1 C u3 D 0 in L2 .0; TI W 1;p .//; @t ˇ ˇ N X @ ˇˇ @u ˇˇp2 @u 1 ; p 1 C p 0 D 1: p u @xj ˇ @xj ˇ @xj j D1

1;p Remark 4.7.4. Since in the conditions of Theorem 4.7.3, u 2 L1 .0; TI W0 .// 0 2 L2 .0; TI H1 .//, by virtue of the result of [275] we have that u.t / W Œ0; T ! and u 0 H01 ./ is a strongly continuous function.

Proof. To prove the local solvability, we use the Galerkin method and the monotonicity and compactness methods (see [275]).

420


We search for an approximate solution of our problem (4.168)–(4.170) in the form Z T d dt .t / .hum ; wi0 C hp um ; wi/ dt 0 3 C .um ; w/ .um ; w/ C humx1 ; wi0 C hum umx1 ; wi0 D 0 (4.172) for all .t / 2 L2 .0; T/, j D 1; m, um D

m X

cmk .t /wk ;

kD1

u0m um .0/ D

m X

(4.173) cmk .0/wk ! u0

strongly in

W01;p ./:

kD1

As above, in the class cmk .t / 2 C .1/ Œ0; Tm0 /, problem (4.172) implies the pointwise relation ˇ N ˇ X ˇ @um ˇp2 @u0m @wj 0 ˇ ˇ .rum ; rwj / C .p 1/ ; ˇ @x ˇ @xi @xi i i D1 1 @wj @wj D .u2m um ; wj /; (4.174) .um ; wj / um ; u2m ; @x1 2 @x1 j D 1; m. From (4.174) we directly obtain ˇ ˇ m X m X N N X X ˇ @um ˇp2 @wk @wj 0 ˇ ˇ 1 C .p 1/ˇ c C ; wk wj cmk @xi ˇ @xi @xi mk kD1 i D1

kD1 iD1

m X kD1

1 2 @wj wk cmk C cmk D .jum jq um ; wj /; @x1 2

j D 1; m:

Now we choose eigenfunctions of problem (4.171) as a Galerkin basis in H01 ./ and 1;p in W0 ./. Introduce the notation ˇ ˇ N X ˇ @um ˇp2 @wk @wj ˇ 1 C .p 1/ ˇˇ : ; akj @xi ˇ @xi @xi i D1

The following inequality holds: m;m X

akj k j D

krk22

ˇ N ˇ X ˇ @um ˇp2 2 ˇ ˇ C .p 1/ ; krk22 ; i ˇ @x ˇ i

i D1

k;j D1;1

i D

m X @wk k ; @xi

kD1

D

m X kD1

k wk :


421

P The norm krk22 vanishes in the class H01 ./ if and only if D m kD1 k wk D 0. P 1;p m Since ¹wk ºC1 is a Galerkin basis in W ./, D w D 0 if and only kD1 k k 0 kD1 m if ¹k ºkD1 D 0; therefore, by virtue of the Sylvester criterion (see, e.g., [335]) we > 0. Note that for the matrix obtain det¹akj ºm;m k;j D1;1 akj

N X i D1

ˇ ˇ ˇ @um ˇp2 @wk @wj ˇ ; 1 C .p 1/ ˇˇ ; @xi ˇ @xi @xi

the inverse matrix is continuous with respect to cm D .cm1 ; : : : ; cmm / (see Appendix A.18). The general results on nonlinear systems of ordinary differential equations (see, e.g., [319]) guarantee the existence of a solution of problem (4.174) on a certain interval Œ0; tm , tm > 0, in the sense ckm 2 C .1/ Œ0; tm . Further, we obtain a priori estimates, which imply tm D T, where T is independent of m. Therefore, 1;p ckm 2 C .1/ Œ0; T, um 2 C .1/ .Œ0; TI W0 .//. Now we obtain a priori estimates. Multiply both parts of (4.174) by cmj and sum over j D 1; m; then we obtain p N p 1 d X @um 1 d 2 2 4 krum k2 C @x C kum k2 D kum k4 : 2 dt p dt i p

(4.175)

i D1

Note that the norms X N @v @x

p 1=p ;

i p

i D1

kvkpp

N X @v C @x i D1

p 1=p

i p

are equivalent on the space v 2 W01;p ./ (see, e.g., [275]). Therefore, we consider 1;p the Banach space W0 ./ with the norm X N @v @x i D1

p 1=p :

i p

From (4.175) we obtain the fact that p N p 1 X @um 1 2 krum k2 C @x 2 p i p i D1

p Z t N 1 p 1 X @u 0m 2 C kru0m k2 C ds kum k44 .s/: (4.176) 2 p @xi p 0 i D1

422


By the conditions imposed to p and N and by virtue of the Sobolev embedding theo1;p rems (see, e.g., [26]), the embedding W0 ./ L4 ./ holds. Now we have X N @um p 1=p ; kum k4 B1 @x i p i D1

p p N N p 1 X 1 p 1 X @um @u0m 1 2 2 krum k2 C @x 2 kru0m k2 C p @x 2 p i p i p i D1

i D1

X N t @um p 4=p ds .s/: @x i p 0

Z C B41

i D1

Thus, we have

4=p 4 p qC2 ˛ D ; B2 B1 ; Em .t / E0m C B2 p p1 0 (4.177) p N 1 p 1 X @um 2 Em .t / krum k2 C @x ; E0m Em .0/: 2 p i p Z

t

ds E˛m .s/;

i D1

Using the Gronwall–Bellman and Bihari theorems (see, e.g., [112]) from (4.177) we obtain 1=.1˛/ Em .t / ŒE1˛ 0m C .1 ˛/B2 t

for 0 < ˛ < 1;

(4.178)

Em .t / E0m exp¹B2 t º

for ˛ D 1;

(4.179)

for ˛ > 1:

(4.180)

.˛1/ 1=.˛1/

Em .t / E0m Œ1 .˛ 1/B2 E0m

t

0 and sum over j D 1; m; then after integrating Now we multiply Eq. (4.174) by cmj over s 2 Œ0; t we obtain ˇ ˇ Z t N Z 4.p 1/ X @ ˇˇ @um ˇˇ.p2/=2 @um 2 0 2 ds krum k2 .s/ C dx (4.181) p2 @t ˇ @xi ˇ @xi 0 i D1 Z t 1 1 2 @u0m 1 1 1 2 4 ds um C um D kum k44 C kum0 k22 : C kum k2 C ku0m k4 C 2 4 2 @x1 4 2 0 The following auxiliary estimates hold: ˇZ ˇ 0 ˇ ˇ ˇ ds um @um ˇ " kru0 k2 C 1 kum k2 ; m 2 2 ˇ @x1 ˇ 2 2" ˇ ˇZ 0 ˇ ˇ ˇ ds u2 @um ˇ " kru0 k2 C 1 kum k4 ; m m 2 4 ˇ @x1 ˇ 2 2" X N @um p 1=p kum k2 B2 kum k4 B3 : @x i p i D1

423


By the conditions on the parameters p and N and by the Sobolev embedding theorems (see, e.g., [26]), from (4.181) we obtain ˇ ˇ Z t N Z 3" 4.p 1/ X @ ˇˇ @um ˇˇ.p2/=2 @um 2 0 2 krum k2 .s/ C ds 1 dx 4 p2 @t ˇ @xi ˇ @xi 0 i D1 X Z t N @um p 4=p 1 1 " " 4 2 2 ku ku B41 C ds k C k m 4 m 2 C ku0m k2 ; 4 @xi p 4 2 2 0 iD1

" 2 .0; 4=3/:

(4.182)

1;p By virtue of (4.173) we obtain that u0m ! u0 strongly in W0 ./ and the inequality E0m A holds, where A is independent of m. Hence from (4.178)–(4.180) we conclude that there exists T0 > 0 such that the inequality Em .t / C , t 2 Œ0; T, holds for all T 2 .0; T0 /, where C is independent of m and t . Thus, from (4.178)–(4.182) we obtain that there exists T0 T0 .u0 ; q; p/ such that for any T 2 .0; T0 /, the following inclusions hold:

is bounded in L1 .0; TI W01;p .//;

um

u0m is bounded in L2 .0; TI H01 .//; ˇ ˇ @ ˇˇ @um ˇˇ.p2/=2 @um is bounded in L2 .0; TI L2 .//; @t ˇ @xi ˇ @xi u3m is bounded in L1 .0; TI L4=3 .//: Finally, it is clear that ˇ ˇ ˇ @um ˇ.p2/=2 @um ˇ ˇ ˇ @x ˇ @xi i

is bounded in L1 .0; TI L2 .//:

(4.183)

(4.184)

By virtue of (4.183) and (4.184) we have tm D T, where T is independent of m 2 N. Therefore, m X 1;p ckm .t /wk 2 C .1/ .Œ0; TI W0 .//: um D kD1

From inclusions (4.183) and (4.184) and the results of [275, p. 24] we conclude that there exists a subsequence of the sequence um (we denote it also by um ) such that um * u u0m * u0 um * u p um * u3m * w @um @u * @x1 @x1

-weakly in L1 .0; TI W01;p .//; weakly in L2 .0; TI H01 .//;

-weakly in L1 .0; TI H1 .//; 0

-weakly in L1 .0; TI W 1;p .//; -weakly in L1 .0; TI L4=3 .//; -weakly in L1 .0; TI Lp .//:

(4.185)

424



in D 0 .0; TI W01;p .//I

therefore, by the weak convergence in L2 .0; TI H01 .// we have u0m * : Therefore, we conclude that .t / D u0 .t / for almost all t 2 .0; T/. Hence weakly in H1 .Q/;

um * u

Q D .0; T/ I

then by virtue of the compact embedding H1 .Q/ b L2 .Q/ we conclude that strongly in L2 .Q/

um ! u

and, therefore, almost everywhere (see, e.g., [168]). Then by virtue of [275, Lemma 1.3] we obtain -weakly in L1 .0; TI L4=3 .//:

u3m * u3

Thus, w D u3 . From (4.183) and (4.184) we obtain that ˇ ˇ ˇ @um ˇ.p2/=2 @um ˇ ˇ 2 H1 .0; TI L2 .//; um 2 H1 .0; TI H01 .//: ˇ @x ˇ @xi i Taking into account the fact that 1;p 0

p um * -weakly in L1 .0; TI W0

.//;

we prove that D p u:

(4.186)

We set in (4.172) .t / D 1; then we obtain Z hum C p um ; wj i C Z C

t

ds 0

t

0

Z t

ds u3m .s/; wj ds um .s/; wj 1 2 2 um /

@.um C @x1

Since um * u

0

; wj D hu0m C p u0m ; wj i: (4.187)


we have hum ; wj i * hu; wj i -weakly in L1 .0; T/;

j D 1; m:

425


Moreover, 0

-weakly in L1 .0; TI W 1;p .//;

p um *

p0 D

p I p1

therefore, -weakly in L1 .0; T/;

hp um ; wj i * h ; wj i Finally, Z t 0

Z ds hu3m ; wj i

D

t

0

Z ds hu3m

t

3

u ; wj i C

0

j D 1; m:

ds hu3 ; wj i;

j D 1; m:

On the other hand, u3m ; u3 2 L1 .0; TI L4=3 .//. Moreover, u3m * u3

-weakly in L1 .0; TI L4=3 .//;

therefore, by virtue of the Lebesgue theorem we obtain ˇ ˇZ t ˇ ˇ 3 3 ˇ ds hum u ; wj iˇˇ ! C0 ˇ 0

as m ! C1 for all t 2 .0; T/. By the condition W01;p ./ L4 ./, we obtain that the right-hand side of the latter inequality tends to zero as m ! C1. 1;p Since u0m ! u0 strongly in W0 ./ H01 ./, we have hu0m ; wj i ! hu0 ; wj i;

hp u0m ; wj i ! hp u0 ; wj i;

j D 1; m:

Therefore, passing in the relation (4.187) to the limit as m ! C1, we obtain Z t Z t Z t ds u ds .ux1 C uux1 / ds u3 .s/: u D u0 C p u0 C 0

0

0

(4.188) Introduce the notation A.v/ D p v. From (4.187) we obtain Z t

3 ds um .s/; um .t / hA.um /; um i D hum ; um i C 0

Z C

t 0

ds .um umx1 Z

D krum k22 C Z C

t 0

0

t

um umx1 /.s/; um .t / hu0m C p u0m ; um i

ds u3m .s/; um .t / hu0m C p u0m ; um i

(4.189)

Z t

ds um .s/; um .t / C ds .umx1 .s/ C um umx1 .s//; um .t / : 0

426


By the fact that um * u weakly in L2 .0; TI W01;p .// L2 .0; TI H01 .// for almost all t 2 .0; T/, and also by a results of [214], we have Z T Z T lim inf dt krum k22 .t / dt kruk22 .t /: m!C1 0

0

Moreover, since um ! u strongly in L2 .Q/, we have Z t

Z t

ds um .s/; um .t / ! ds u.s/; u.t / ; 0

Z

t 0

0

Z t

ds .umx1 .s/ C um umx1 .s//; um .t / ! ds .ux1 .s/ C uux1 .s//; u.t / 0

as m ! C1. Moreover, Z t

Z Z T 3 dt ds um .s/; um .t / D 0

0

0

C

Z

T

t

dt Z

0

ds u3m .s/; um .t /

Z

T

t

ds u3m .s/; u.t /

dt 0

u.t /

0

:

We have proved above that the sequence ¹um º converges weakly in H1 .Q/; therefore, a certain subsequence of the sequence ¹um º converges strongly in L4 .Q/. Moreover, the sequence is bounded in L1 .0; TI L4 .//. Then the following inequality holds: ˇZ T Z t

ˇ ˇ ˇ 3 ˇ ! C0 as m ! C1: ˇ dt ds u .s/; u .t / u.t / m m ˇ ˇ 0

0

Indeed, Z jhu3m .s/; um .t / u.t /ij ˇZ ˇ ˇ ˇ

Z

T

t

dt 0

0

ds u3m .s/; um .t /

Now consider

Z

dx jum .s/j4

ˇ Z ˇ ˇ u.t / ˇ C.T/ Z

T

t

dt 0

3=4 Z

0

T 0

dx jum .t / u.t /j4

dt kum .t /

ds u3m .s/; u.t /

:

Since u.t / 2 L1 .0; TI W01;p .// L1 .0; TI L4 .// and Z t ds jum jq um .s/ 2 L1 .0; TI L4=3 .//; 0

we have that u3m * u3

-weakly in L1 .0; TI L4=3 .//;

1=4 ;

1=4 u.t /k44

:

427


hence we obtain that

Z Z T Z t lim dt ds u3m .s/; u.t / D m!C1 0

0

Z

T

t

dt

0

ds u3 .s/; u.t / :

0

Let Z Xm D

8v 2

T

dt hA.um / A.v/; um vi

0 1;p r L .0; TI W0 .//;

By virtue of (4.189) we obtain Z T Z 2 Xm D dt krum k2 C 0

Z

Z

T 0

Z

t

0

0 T

t

0

dt 0

ds u2m um .s/; um .t /

dt u0m C p u0m ; um dt

0

Z

T

0

T

Z

0

A.v/ 2 Lr .0; TI W 1;p .//; r : r 2 .1; 1/; r 0 D r 1

Z

Z

T

t

dt 0

0

ds um .s/; um .t /

ds .umx1 .s/ C um umx1 .s//; um .t / Z

dt hA.um /; vi

T

0

dt hA.v/; um vi:

(4.190)

Among the terms of expression (4.190), for Xm , we must consider only Z T Z T Z T dt hu0m C p u0m ; um i D dt hu0m ; um i C dt hp u0m ; um i: 0

0

0

(4.191) Consider the first term in (4.191). The functions um are bounded uniformly with 1;p respect to m in L1 .0; TI W0 .// L1 .0; TI H01 .//. On the other hand, u0m ! 1;p u0 strongly in W0 ./ H01 ./ and by the fact that 2 L.H01 .//I H1 ./ (see, e.g., [276]), a set of linear continuous (bounded owing to the linearity of ) operators that act from H01 .// into H1 ./, u0m ! u0 strongly in H1 ./ and the following inequality holds: jhu0m u0 ; um ij C ku0m u0 k1 ! 0 Since

as m ! C1:

um ! u -weakly in L1 .0; TI H01 .//

428


and u0 2 H1 ./, we have Z

T 0

Z dt hu0 ; um i !

T

0

dt hu0 ; ui as m ! C1;

and in this case Z

T

lim

m!C1 0

Z dt hu0m ; um i D

T 0

dt hu0 ; ui:

Consider the second term in (4.191). The functions um are bounded uniformly with 1;p respect to m in L1 .0; TI W0 .//. On the other hand, u0m ! u0 strongly in W01;p ./. By the fact that @=@xi 2 L.W01;p ./I Lp .//, i D 1; N (see, e.g., [276]), @u0m =@xi ! @u0 =@xi strongly in Lp ./ for all i D 1; N . The following inequality holds: kjwjp2 w jvjp2 vkp=.p1/ .p 1/kf .w v/kp=.p1/ ; where f D max.jwjp2 ; jvjp2 /, kf .w v/kp1 kf kq1 p1 kw vkq2 p1 ; q2 D p 1;

q1 D

p1 ; p2

p ; q2 p1 D p; p2 p q1 p1 D ; p2 p1 D

kjwjp2 w jvjp2 vkp=.p1/ .p 1/kf kp=.p2/ kw vkp ; kf kp=.p2/ 2.max¹kwkp ; kvkp º/p2 : We take v D @u0m =@xi and w D @u0 =@xi ; then we directly obtain that ˇ ˇ ˇ ˇ ˇ @u0m ˇp2 @u0m ˇ @u0 ˇp2 @u0 ˇ ˇ ˇ ! ˇˇ ˇ @x ˇ @xi @xi ˇ @xi i 0

0

strongly in Lp ./;

p0 D

p : p1

0

Since @=@xi 2 L.Lp ./; W 1;p .// (see, e.g., [276]), p u0m ! p u0 strongly 0 in W 1;p ./ and the following inequalities hold: jhp u0m p u0 ; um ij C kp u0m p u0 k1;p0 ! 0 Since um ! u

as m ! C1:

-weakly in L1 .0; TI W01 .//

0

and p u0 2 W 1;p ./, we have Z

T 0

Z dt hp u0 ; um i !

T 0

dt hp u0 ; ui as m ! C1:

429


Therefore,

Z

T

lim

m!C1 0

Z dt hp u0m ; um i D

T 0

dt hp u0 ; ui:

Thus, passing in (4.190) to the upper limit as m ! C1, we obtain Z 0 lim sup Xm

Z

T

dt

m!C1

0

Z

T 0

Z

kruk22

Z

t

dt 0

0 T

0

t

2

dt

ds u u.s/; u.t /

0

0

Z

dt hu0 C p u0 ; ui

T

Z

C

Z

T

Z

T

t

dt

ds u.s/; u.t /

0

0

ds .ux1 .s/ C uux1 .s//; u.t / Z

dt h ; vi

T

dt hA.v/; u vi:

0

By (4.188) we conclude that Z

T 0

Z dt h ; ui D

dt 0

Z

Z

Z

T

T 0

kruk22

C

Z

T

dt 0

Z dt

0

0

t

2

ds u u.s/; u.t / 0

dt hu0 C p u0 ; ui

T

t

Z

Z

T

dt 0

t

ds u.s/; u.t / 0

ds .ux1 .s/ C uux1 .s//; u.t / :

Hence we obtain that Z

T 0

1;p

8v 2 Lr .0; TI W0

.//;

dt h A.v/; u vi 0 v D u w;

w 2 Lr .0; TI W01;p .//;

In a standard way (see, e.g., [275]) we conclude that Z

T 0

dt h A.u/; wi 0

8w 2 Lr .0; TI W01;p .//;

0

0

0

; A.u/ 2 L1 .0; TI W 1;p .// Lr .0; TI W 1;p .//: Therefore, relation (4.186) holds: D p u. Now prove that ˇ ˇ ˇ @u ˇ.p2/=2 @u ˇ ˇ 2 H1 .0; TI L2 .//; ˇ @x ˇ @xi i

i D 1; N :

r > 1:

430


Let @v p ; hAi .v/; vi D @xi p

ˇ ˇ @ ˇˇ @v ˇˇp2 @v Ai .v/ ; @xi ˇ @xi ˇ @xi

1;p

By the fact that um * u weakly in Lp .0; TI W0 Z

T

lim sup m!C1 0

dt hAi .um /; um i

.//, we have Z

N X j D1;j ¤i

T

lim sup m!C1 0

Z

T

dt 0

Z

Z

T 0

kruk22

Z

Z dt

Z

T

0 T

0

Z

T

t

t

dt 0

p

dt hA.um /; um i; 2

ds u u.s/; u.t / 0

dt hu0 C p u0 ; ui

T 0

C

@u dt @x

j p

m!C1 0

Z hA.um /; um i

T 0

C lim sup Z

i D 1; N :

Z

Z

T

t

dt 0

ds u.s/; u.t /

0

ds .ux1 .s/ C uux1 .s//; u.t / Z

dt hp u; ui D

T

0

N X @u dt @x j D1

p :

j p

Thus, we obtain Z

T

lim sup m!C1 0

Z T @um p dt dt @xi p 0

@u p @x : i p

(4.192)

On the other hand, @um @xi

is bounded in L1 .0; TI Lp .// Lp .Q/;

Q D .0; T/ I

therefore, by the reflexivity of Lp .Q/ (see, e.g., [436]), there exists a subsequence (we denote it again by um ) such that @u @um * @xi @xi

weakly in Lp .Q/;

Therefore, by a result of [436] we obtain Z T Z T @um p dt dt lim inf m!C1 0 @xi p 0

@u p @x ; i p

i D 1; N :

i D 1; N :

(4.193)

(4.194)

431


From (4.192) and (4.194) we obtain that Z T Z T @um p lim dt D dt m!C1 0 @xi p 0

@u p @x ; i p

i D 1; N :

(4.195)

From a result of [214] and (4.193) and (4.195) we have that @u @um ! @xi @xi

strongly in Lp .Q/;

i D 1; N :

(4.196)

In its turn, this means (see [129]) that @u @um ! @xi @xi

almost everywhere in Q .0; T/ ;

i D 1; N :

(4.197)

Passing to a subsequence, we obtain ˇ ˇ ˇ @um ˇ.p2/=2 @um ˇ ˇ * i ˇ @x ˇ @xi i

-weakly in L1 .0; TI L2 .//;

i D 1; N :

By (4.197) and a the result of [275] we obtain ˇ ˇ ˇ ˇ ˇ @u ˇ.p2/=2 @u ˇ @um ˇ.p2/=2 @um ˇ ˇ ˇ ˇ * ˇ @x ˇ ˇ @x ˇ @xi @xi i i

-weakly in L1 .0; TI L2 .//:

Thus, ˇ ˇ ˇ ˇ @ ˇˇ @um ˇˇ.p2/=2 @um @ ˇˇ @u ˇˇ.p2/=2 @u * @t ˇ @xi ˇ @xi @t ˇ @xi ˇ @xi

in D 0 .0; TI L2 .//:

By (4.183) we can choose a subsequence such that ˇ ˇ @ ˇˇ @um ˇˇ.p2/=2 @um * @t ˇ @xi ˇ @xi

weakly in L2 .Q/;

This means that ˇ ˇ @ ˇˇ @u ˇˇ.p2/=2 @u D 2 L2 .Q/; @t ˇ @xi ˇ @xi Thus, ˇ ˇ @ ˇˇ @u ˇˇ.p2/=2 @u 2 L2 .Q/; @t ˇ @xi ˇ @xi ˇ ˇ ˇ @u ˇ.p2/=2 @u ˇ ˇ ˇ @x ˇ @xi i

Q D .0; T/ :

Q D .0; T/ ;

i D 1; N :

ˇ ˇ ˇ @u ˇ.p2/=2 @u ˇ ˇ 2 L1 .0; TI L2 .//; ˇ @x ˇ @xi i 1

2

2 H .0; TI L .//:

(4.198)

432


Similarly, using (4.197) we can prove that ˇ ˇ ˇ ˇ ˇ @um ˇp2 @um ˇ @u ˇp2 @u ˇ ˇ ˇ ˇ * ˇ @x ˇ ˇ @x ˇ @xi @xi i i

0

-weakly in L1 .0; TI Lp .//:

From (4.198) and embedding theorems we have that ˇ ˇ @ ˇˇ @u ˇˇp2 @u 0 2 L2 .0; TI Lp .//; ˇ ˇ @t @xi @xi ˇp2 ˇ ˇ @u ˇ @u 0 ˇ ˇ 2 L1 .0; TI Lp .//: ˇ @x ˇ @xi i

(4.199)

Prove (4.199). Introduce the notation ˇ ˇ ˇ @u ˇ.p2/=2 @u ˇ ˇ ˇ I @xi ˇ @xi in this case j j

˛p 0

dt 0

i D 1; N :

0

Z

T

p2 ; p

˛D

2 L1 .0; TI Lp .//. Prove that .j j˛ /0t 2 L2 .0; TI Lp .//.

Note that j j˛ Indeed, Z

ˇ ˇ ˇ @u ˇp2 @u ˇ D ˇˇ ; @xi ˇ @xi

˛

dxj j Z

0 p0 tj

j

0

2=p0

Z

T

dt

0

˛p 0 p1

dx j j

2=.p0 p1 / Z

dx j

0 p 0 p2 tj

2=.p 0 p2 / ;

where ˛p 0 p1 D 2;

Z

T

0

dt 0

p11 C p21 D 1;

2 L1 .0; TI L2 .// and

Since Z

p 0 p2 D 2;

dx j j˛p j

0 p0 tj

0 t

2=p0

p1 D

2.p 1/ ; p2

p2 D

2.p 1/ : p

2 L2 .0; TI L2 .//, we have Z sup k k2˛ 2 t2.0;T/

Q

dx dt j

0 2 tj

< C1:

Now we can pass in Eq. (4.172) to the limit as m ! C1. First, we set .t / D 1 in (4.172); then we obtain the integral equality Z t um C p um D um0 C p um0 ds Œumx1 C um umx1 C u3m u; 0

433

Section 4.7 Blow-up in a strongly nonlinear dissipative equation 0

which is meant in the sense of W 1;p ./ for almost all t 2 .0; T/. By the results obtained above, we can pass to the limit as m ! C1 and obtain Z t u C p u D u0 C p u0 ds Œu3 C ux1 C uux1 u 0

in the sense W

1;p 0

./ for almost all t 2 .0; T/. This implies the fact that 0

0

.1/ .Œ0; TI W 1;p .//: u C p u 2 H1 .0; TI W 1;p .// \ Cw

Note that from (4.179) we have the pointwise equality d Œum C p um D u3m umx1 um umx1 C um : dt Since um C p um * u C p u we have

0

weakly in L2 .0; TI W 1;p .//;

d d .um C p um / ! .u C p u/ dt dt 0

in the sense of D 0 .0; TI W 1;p .//. On the other hand, u3m * u3 ;

umx1 * ux1 ;

um umx1 * uux1 ;

um * u

0

weakly in L2 .0; TI W 1;p .// and, therefore, d .um C p um / * dt 0

weakly in L2 .0; TI W 1;p .//. Thus, d d .um C p um / ! .u C p u/ dt dt 0

weakly in L2 .0; TI W 1;p .//. Now we can pass in Eq. (4.175) to the limit and obtain Z T d dt .t / hu C p u; wj i C hu C ux1 C uux1 C u3 ; wj i D 0 dt 0 for all .t / 2 L2 .0; T/, j D 1; C1. This relation is equivalent to the equality

Z T d 1;p 3 Œu C p u C u u C ux1 C uux1 ; v dt 8v 2 L2 .0; TI W0 .//: dt 0

434


Now we prove the uniqueness of a weak generalized solution. Indeed, let ui , i D 1; 2m be two weak generalized solutions of the problem Z T d dt .t / .hu; wi0 C hp u; wi/ dt 0 3 C .u ; w/ .u; w/ C hux1 ; wi0 C huux1 ; wi0 D 0 1;p 1;p for all .t / 2 L2 .0; T/ and all w 2 W0 ./, u.0/ D u0 2 W0 ./, in the classes stated in the conditions of the theorem, with the same initial condition. Note that we have proved above that the solution u satisfies the relation Z t u C p u D u0 C p u0 ds Œux1 C uux1 C u3 u: 0

Now we prove that a solution of our problem in the class of functions u that satisfy this equality is unique. Indeed, hu1 u2 ; u1 u2 i hp u1 p u2 ; u1 u2 i Z Z t ds dx .u31 .s/ u32 .s//.u1 .t / u2 .t // D 0 Z Z t ds dx .u1 .s/ u2 .s//.u1 .t / u2 .t // 0 Z t Z C ds dx .u1x1 .s/ u2x1 .s//.u1 .t / u2 .t // 0 Z Z t 1 ds dx .u21 .s/ u22 .s//.u1x1 .t / u2x1 .t //: C 2 0 By the monotonicity of the operator p and the embedding theorems, it is easy to obtain the inequality kru1 ru2 k22 Z t Z 3 ds dx max¹ju1 j2 .s/; ju2 j2 .s/ºju1 .s/ u2 .s/jju1 .t / u2 .t /j 0 Z Z t ds dx ju1 .s/ u2 .s/jju1 .t / u2 .t /j C 0 Z t Z C ds dx jru1 .s/ ru2 .s/jju1 .t / u2 .t /j 0 Z Z t ds dx max¹ju1 .s/j; ju2 .s/jºju1 .s/ u2 .s/jjru1 .t / ru2 .t /j C 0 Z t ds kru1 ru2 k2 .s/; C.T/kru1 ru2 k2 .t / 0

which implies the fact that u1 D u2 almost everywhere in .0; T/ . Theorem 4.7.3 is proved.

435


4.7.2 Unique solvability of the problem and blow-up of its solution for a finite time Introduce the notation

p N 1 p 1 X @um 2 Em .t / krum k2 C @x ; 2 p i p

E0m Em .0/;

˛D

i D1

1 p1 E.t / kruk22 C 2 p B2

B41

p p1

4=p

N X i D1

@u p @x ;

4 ; p

E0 E.0/;

i p

; 1;p

where B1 is the best constant of the embedding W0 The following theorem holds.

./ L4 ./.

Theorem 4.7.5. Let all the conditions of Theorem 4.7.3 hold. Then the following assertions hold: (1) if p > 4, then T0 D C1 and C .1 ˛/B2 t 1=.1˛/ ; E.t / ŒE1˛ 0

˛D

4 I p

(2) if p D 4, then T0 D C1 and E.t / E0 exp¹B2 t ºI (3) if p 2 .2; 4/ and

0

E .0/ >

˛1 1

1=2

E0 C

ˇ E0 ; ˛1 1

then T0 2 ŒT1 ; T2 , where E0 .0/ D ku0 k44 ku0 k22 ;

E0 D E.0/ D

p N 1 p 1 X @u ; kru0 k22 C 2 p @xi p i D1

4Cp ; 2p

2.B23

C 3/ ; p 2 .2; 4/; 4p 4=p p 1 qC2 1 1˛ 1 1 T1 D ; B2 B1 ; T2 D E1˛ A ; B E 0 ˛1 2 0 p1 2 1=2 ˇ E0 .E.0//2 ; A .1 ˛1 /2 E2˛1 .0/ E0 .0/ ˛1 1 ˛1 1 ˛1 D

ˇD

C 3/ ; 4p

16B23 .B23

D

B3 is the best constant of the embedding H01 ./ L2 ./.

436


Proof. Let um be Galerkin approximations defined in (4.174). Since u0m * u0 1;p 1;p strongly in W0 ./ and um * u weakly in W0 ./ for almost all t 2 .0; T/, then, passing in (4.178) and (4.179) to the limit as m ! C1, owing to the fact that E.t / lim infm!C1 Em .t / (see [436]), we obtain 1

E.t / ŒE1˛ C .1 ˛/B2 t 1˛ 0

for ˛ < 1;

E.t / E0 exp ¹B2 t º

for ˛ D 1:

1;p Consider the case ˛ > 1. Since u0m ! u0 strongly in W0 ./ H01 ./, by a result of [214] we have E0m ! E0 . In this case, we can select from the numerical sequence E0m either a monotonically nondecreasing or a monotonically nonincreasing subsequence. Consider the required subsequence of the sequence E0m ; we denote the corresponding subsequences of the sequences u0m and um by the same symbols. Consider inequality (4.180). Assume that E0m is a monotonically nonincreasing sequence. Denote by ¹E¹nº 0m º the sequence obtained from E0m by removing the first n 2 N terms. Then inequality (4.180) holds uniformly with respect to m for all ¹nº ¹N º 1˛ t 2 Œ0; T1 /, where T1 .˛ 1/1 B1 2 E0n . Since

um * u

-weakly in L1 .0; TI W01;p .//

for all such t , we can pass in (4.180) to the limit as m ! C1 and, owing to a result of [436] we can obtain that t 1=.˛1/ E.t / E0 Œ1 .˛ 1/B2 E˛1 0

for t 2 Œ0; T¹nº 1 /:

(4.200)

¹nº

By the arbitrariness of n 2 N and by the fact that T1 " T1 as n ! C1, we conclude that (4.200) holds for t 2 Œ0; T1 / and, moreover, T0 T1 . Now let E0m be a monotonically nondecreasing sequence. Then, arguing as above, we obtain that (4.200) holds for all t 2 Œ0; T1 / and T0 T1 . In the standard way, using Lemma A.15.1 of Appendix A.15, that for almost all t 2 Œ0; T 8T 2 .0; T0 /, we have Em .t / ! E.t /. 1;p By (4.176) and (4.181) and since um .t / 2 C .1/ .Œ0; TI W0 .//, we have p N p 1 d X @um 1d 2 2 4 krum k2 C @x C kum k2 D kum k4 .s/; 2 dt p dt i p kru0m k22

ˇ ˇ ˇ ˇ ˇ @um ˇp2 ˇ @u0m ˇ2 1 d 2 ˇ ˇ ˇ ˇ C .p 1/ dx ˇ ˇ @x ˇ C 2 dt kum k2 @xi ˇ i i D1 Z Z 1d 1 kum k44 C D dx u0mx1 u C dx u2m u0mx1 : 4 dt 2 N Z X

(4.201)

i D1

(4.202)


437

From (4.201) and (4.202) we obtain that Em .t /; E0m .t / 2 ACŒ0; T. The following inequalities hold: ˇZ ˇ ˇ 0 ˇ2 ˇ dx ru ; rum ˇ kru0 k2 krum k2 ; m m 2 2 ˇ ˇ

ˇZ ˇ ˇp2 ˇ Z ˇ ˇ ˇ @um @u0m ˇˇ ˇ dx ˇ @um ˇ dx ˇ ˇ @x ˇ @xi @xi ˇ i

ˇ ˇ ˇ ˇ ˇ @um ˇp1 ˇ @u0m ˇ ˇ ˇ ˇ ˇ ˇ @x ˇ ˇ @x ˇ i i

(4.203)

(4.204)

ˇ ˇ ˇ ˇ !1=2 Z ˇ @um ˇp2 ˇ @u0m ˇ2 @um p=2 ˇ ˇ ˇ ˇ dx ˇ : ˇ @x ˇ @xi p @xi ˇ i

By the Schwarz inequality (see, e.g., [293]) we have Z N X @um p=2 dx @x i

iD1

p

ˇ ˇ ˇ ˇ ˇ @um ˇp2 ˇ @u0m ˇ2 1=2 ˇ ˇ ˇ ˇ ˇ @x ˇ ˇ @x ˇ i i

N Z X N @um p 1=2 X dx @x i p i D1

i D1

ˇ ˇ ˇ ˇ ˇ @um ˇp2 ˇ @u0m ˇ2 1=2 ˇ ˇ ˇ ˇ : (4.205) ˇ @x ˇ ˇ @x ˇ i i

By (4.201)–(4.205), the following inequalities hold: jhum ; Œum p um 0t ij2

jhru0m ; rum ij2

ˇ ˇX ˇ ˇ Z ˇ @um ˇp2 @um @u0m ˇ2 ˇN ˇ ˇ C .p 1/ ˇ dx ˇˇ @xi ˇ @xi @xi ˇ 2ˇ

i D1

C 2.p

ˇN Z ˇX

1/j.ru0m ; rum /jˇˇ

i D1

kru0m k22 krum k22 C .p 1/2

ˇ ˇ ˇ ˇ @um ˇp2 @um @u0m ˇ ˇ ˇ ˇ dx ˇ @xi ˇ @xi @xi ˇ

ˇ N Z ˇ ˇ ˇ N X ˇ @um ˇp2 ˇ @u0m ˇ2 @um p X ˇ ˇ ˇ ˇ dx ˇ @x ˇ @x ˇ @x ˇ i p i i i D1

i D1

X N @um p 1=2 0 C 2.p 1/krum k2 krum k2 @x i D1

X N Z i D1

i

ˇ ˇ ˇ ˇ ˇ @um ˇp2 ˇ @u0m ˇ2 1=2 ˇ ˇ ˇ ˇ dx ˇ ˇ @x ˇ @xi ˇ i

p

438


krum k22

p

i

i D1

kru0m k22

C .p 1/

N Z X i D1

kru0m k22

N X @um p C .p 1/ @x

C .p 1/

N Z X i D1

p

ˇ ˇ ˇ ˇ ˇ @um ˇp2 ˇ @u0m ˇ2 ˇ ˇ ˇ ˇ dx ˇ ˇ @x ˇ @xi ˇ i

ˇ ˇ ˇ ˇ ˇ @um ˇp2 ˇ @u0m ˇ2 ˇ ˇ ˇ ˇ dx ˇ ˇ @x ˇ @xi ˇ i

p N 1 p 1 X @um 2 krum k2 C @x : 2 p i p i D1

Thus, we obtain the inequality .E0m /2 p Em .t /Jm .t /;

(4.206)

where Jm .t / D kru0m k22 C .p 1/

N Z X i D1

ˇ ˇ ˇ ˇ ˇ @um ˇp2 ˇ @u0m ˇ2 ˇ ˇ ˇ dx ˇˇ ˇ @x ˇ : @x ˇ i

i

Now we use Eqs. (4.201) and (4.202) to prove an upper estimate for Jm . Indeed, the following inequalities hold: Z Z 1 00 1d 1 2 0 Jm D Em C dx umx1 u C dx u0mx1 u2 kum k2 C 4 4 dt 2 1 " 1 " 1 " 1 E00m C B23 Jm C B23 Em C Jm C B23 Em C Jm C kum k44 4 4 2" 2 " 4 4" 2 1 " 2B 1 E00m C .B23 C 3/Jm C 3 Em C E0m : (4.207) 4 4 " 4" From (4.206) and (4.207) we obtain (see [210]) Em E00m ˛1 .E0m /2 C ˇE0m Em C E2m 0; where ˛1 D

4Cp ; 2p

ˇD

2.B23 C 3/ ; 4p

D

16B23 .B23 C 3/ 4p

for p < 4. Then ˛1 > 1 and ˇ; > 0. Arguing further as in the proof of the appropriate part of Theorem 4.5.3, we obtain Theorem 4.7.5

Chapter 5

Special problems for nonlinear equations of Sobolev type

In this chapter, we consider initial-value and initial-boundary-value problems for certain special pseudoparabolic equations, including problems that do not satisfy the conditions formulated in previous chapters; we also consider Cauchy problems in R3 . Results of this chapter were obtained in [232–240] (see also note [267]).

5.1

Nonlinear nonlocal pseudoparabolic equations

In this section, we consider the first initial-boundary-value problem with the homogeneous Dirichlet condition for a nonlinear, nonlocal, pseudoparabolic equation with a nonlocal term of the Kirchhoff type. Here, if we follow the terminology inherited from the studies of the classical wave equation of Kirchhoff type, we consider the degenerate case; moreover, we consider the singular case. For the problem considered, we prove the unique solvability in the strong and weak generalized sense. In the case q 2 .1=2; 0/, the relaxation effect for a finite time is proved and the optimal lower and upper estimates for the relaxation rate and some lower and upper estimates for the relaxation time are obtained. In the case q > 0, we analyze the asymptotic behavior at large time. Now we formulate the problem: q Z @ .u u/ C dx jruj2 u D 0; q > 1=2; (5.1) @t uj@ D 0;

u.x; 0/ D u0 .x/;

(5.2)

where 2 RN , N 1, is a surface-simply-connected bounded domain, @ 2 C 1 . This problem is a mathematical model that describes quasi-stationary processes in semiconductors in the case of a nonlocal relation between the medium conductivity and the intensity E of the electric field.

5.1.1 Global-on-time solvability of the problem We consider the following problem for eigenfunctions and eigenvalues: wj C j wj D 0;

wj 2 H01 ./;

j D 1; C1:

(5.3)

440

Chapter 5 Special problems for nonlinear equations of Sobolev type

The embedding H01 ./ L2 ./ holds. Eigenfunctions of problem (5.3) form an orthogonal basis in H01 ./ relative to the inner product in L2 ./. Moreover, under the smoothness condition of the boundary @ 2 C 1 of the domain , we have wj 2 H01 ./ \ H2 ./, j 2 N. Denote by .x/ an arbitrary function of the class .2/

C0 ./ D ¹u 2 C 2 ./ W uj@ D uxi j@ D uxi xj j@ D 0º;

i; j D 1; NI

we assume that 0. Moreover, denote by h; i the duality bracket between the Hilbert spaces H01 ./ and H1 ./. Theorem 5.1.1. Let q 2 .1=2; C1/, u0 .x/ 2 H01 ./ \ H2 ./. If either N D 1 or q > 0 and N 2, then a unique global-on-time solution of problem (5.1), (5.2) of the class u 2 L1 .0; C1I H01 ./ \ H2 .//;

u0 2 L2 .0; C1I H01 ./ \ H2 .//

exists and if N 2 and q 2 .1=2; 0/, then there exists T0 2 .0; C1/ depending on the initial conditions of the problem such that a unique solution of problem (5.1), (5.2) of the class u 2 L1 .0; C1I H01 .//; u0 2 L2 .0; C1I H01 .//; u 2 L1 .0; T0 I H2 .//;

u0 2 L2 .0; T0 I H2 .//

exists. Here the following lower and upper estimates are valid: (1) if q 2 .1=2; 0/, then .c5 2jqj/1=jqj .T0 t /1=jqj kruk22 .t / C kuk22 .t / .c6 2jqj/1=jqj .T0 t /1=jqj for all t 2 Œ0; T0 , where kru0 k2.qC1/ 2 ; c6 2 Œkru0 k2 C ku0 k22 qC1

c5 ./ c2 ./=.1 C c2 .//;

c2 ./ is the best constant of the embedding H01 ./ L2 ./: c2 ./kvk22 krvk22 , 2jqj

kru0 k2 2jqj

.kru0 k22 C ku0 k22 /jqj kru0 k22 C ku0 k22 T I 0 2jqjc5 ./ kru0 k22

(2) if q > 0, then c7 Œc10 C t 1=q kruk22 .t / C kuk22 .t / c9 Œc8 C t 1=q ;

441

Section 5.1 Nonlinear nonlocal pseudoparabolic equations

where .1Cq/=q .2q/1=q kru0 k2.1Cq/=q ; c7 E0 2 1=q 1=q

1 1 c8 Eq ; 0 c3 q

c9 c3

c10 E0 .2q/1 kru0 k2.1Cq/ ; 2qC1 c2

q

;

E0 D kru0 k22 C ku0 k22 ;

1Cq

c3

:

c21Cq C 1

Finally, if u0 .x/ 2 H01 ./, then a solution of problem (5.1), (1.2) of the class u 2 L1 .0; C1I H01 .//, u0 2 L2 .0; C1I H01 .// is unique. Proof. We prove the global-on-time solvability in the weak generalized sense (see [275]). Definition 5.1.2. A function u.x; t / that coincides almost everywhere with a function of the class C.Œ0; TI H01 .// for certain T > 0 and satisfying the conditions Z

T 0

@ 2q dt Œu u C kruk2 u; w .t / D 0 @t

8 2 L2 .0; T/;

8w 2 H01 ./;

T > 0;

q > 1=2;

for kru0 k2 > 0, u.x; t / D 0 almost everywhere in Q .0; T/ for kru0 k2 D 0, is called a weak generalized solution of problem (5.1), (5.2). Here h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./. Note that the notion of a weak generalized solution is well defined. First, note that for 2q J D kruk2 hu; wi; u; w 2 H01 ./; q > 1=2; the inequality 2q 2qC1 ! C0 jJj kruk2 j.ru; rw/2 j krwk2 kruk2

holds as u ! 0 strongly in H01 ./. Second, take in the definition of a weak generalized solution w D u; then after integrating by parts we obtain Z kruk22 C kuk22 C 2

0

T

dt kruk2qC1 D kru0 k22 C ku0 k22 : 2

Therefore, if u0 D 0 almost everywhere, then u.x; t / D 0 almost everywhere.

442


Remark 5.1.3. Since in the conditions of Theorem 5.1.1 u 2 L1 .0; TI H01 .//;

u0 2 L2 .0; TI H01 .//;

by virtue of the result of [275] we have that u.t / W Œ0; T ! H01 ./ is a strongly continuous function. Therefore, the trace of the function u.x/.t / is defined for t D 0 and the initial condition (5.2) makes sense. Step 1. Galerkin approximations and a priori estimates of first order. To prove the local solvability, we use the Galerkin method together with the monotonicity and compactness methods. We search for an approximate solution of problem (5.1), (5.2) in the form um D

m X

cmk .t /wk ;

u0m um .0/ D

kD1

m X

˛mk wk ! u0 2 H01 ./ \ H2 ./

kD1

Z

H01 ./

˛mk cmk .0/;

T @ Œum um C krum k2q dt 2 um ; wj .t / D 0; @t 0

strongly in

\ H2 ./;

(5.4) (5.5)

j D 1; m, for all 2 L2 .0; T/ and all T 2 .0; Tm0 /, Tm0 D Tm0 .u0m /. Note that by virtue of (5.3) we have wk 2 H3 .G/ for any strictly interior subdomain G of the domain : G with the boundary @G 2 C 1 . We use the form of problem (5.5) in order to reduce it to a rather different form convenient for deriving a priori estimates of second order. Introduce the notation LŒum

@ Œum um C krum k2q 2 um : @t

In this case, problem (5.5) takes the form Z T dt hLŒum ; wj i .t / D 0; j D 1; m; 0

8 .t / 2 L2 .0; T/;

8T 2 .0; Tm0 /:

By virtue of the fact that, according to definition, ¹wj º form a basis in H01 ./, any function of the form hk;i 2 H01 ./, i D 1; N , k D 1; m, can be approximated by linear combinations of wj with any given accuracy. For any m 2 N, there exist natural N.m/ and dkml;i such that the inequality krhk;i rhkm;i k2 C khk;i

1 hkm;i k2 k ; 2

hkm;i D

N.m/ X

dkml;i wl ;

lD1

holds. Now multiply both sides of Eq. (5.5) by dkmj;i and sum over j D 1; N.m/. Next, take the function .t / D c mk .t / 2 L2 .0; T/ (uniformly with respect to m

443


and k) and sum over k D 1; m. We obtain Z

T

dt LŒum ;

0

m X

Z

hk;i c mk D

T

0

kD1

dt LŒum ;

m X

Œhk;i hkm;i c mk Fmi : (5.6)

kD1

Here the following estimates for the functions Fmi hold: m Z X

jFmi j

T

0

kD1

dt jc mk jŒkrhk;i rhkm;i k22 C khk;i hkm;i k22 1=2

Œkru0m k2 C ku0m k2 C krum k2qC1 2 Z T 1=2 m X 2qC1 2 k 0 0 C 2 dt Œkrum k2 C kum k2 C krum k2 0

kD1

Z

sup

0

kD1;m

Z C

T

T

dt 0

1=2 dt jc mk j2

Œkru0m k2

C

ku0m k2

C

krum k2qC1 2 2

1=2

for all T 2 .0; Tm /. Integrating Eqs. (5.5) by parts, we obtain the following equation: Z

Z

T

0

dxŒ.ru0m ; rwj / C u0m wj C krum k2 .rum ; rwj / .t / D 0 2q

dt

for all 2 L2 .0; T/. Hence we obtain that Z

T

dt 0

X m

0 cmk akj

C fmj .t / .t / D 0;

q > 1=2;

kD1

where fmj .t /

m X kD1

cmk .rwk ; rwj / ; q l1 l2 D1;1 cml1 cml2 .rwl1 ; rwl2 /

Œ

Pm

akj D .rwk ; rwj / C .wk ; wj /; q > 1=2, and, using the positive definiteness of the quadratic form formed by the Gram matrix in H01 ./ with entries .rwk ; rwj / and the condition q > 1=2, we can prove the continuity of the functions fmj .t / with respect to the set of variables cmk , k D 1; m, in a neighborhood of zero. Finally, in the class cmk 2 C .1/ Œ0; Tm0 ,

444


by virtue of the principal lemma of calculus of variations, we obtain the following equality: m X

akj

kD1

dcmk C fmj D 0; dt

cmk .0/ D ˛mk ;

j D 1; m;

8t 2 Œ0; Tm0 :

By the choice of the functions wj , the determinant det.akj / does not vanish. The general results on nonlinear systems of ordinary differential equations guarantee the existence of a solution of problem (5.5) on a certain interval Œ0; Tm0 , Tm0 > 0, in the sense cmk 2 C .1/ Œ0; Tm0 . Therefore, cmk 2 C .1/ Œ0; Tm , um 2 C .1/ .Œ0; Tm0 I H01 ./ \ H2 .//: Now we derive a priori estimates. If we take .t / D cmj .t / in Eq. (5.5) and sum over j D 1; m, we obtain the equality Z t ds krum k2.1Cq/ D kru0m k22 C ku0m k22 (5.7) krum k22 C kum k22 C 2 2 0

0 for all t 2 Œ0; Tm0 . If we take the function cmj as .t / and sum over j D 1; m, we obtain the equation Z t 1 1 2.1Cq/ 2.1Cq/ krum k2 kru0m k2 Œkru0m k22 C ku0m k22 ds C D : 2.1 C q/ 2.1 C q/ 0 (5.8)

From Eqs. (5.7) and (5.8), by virtue of the fact that um 2 C .1/ .Œ0; Tm0 I H01 .//; we obtain the following differential equalities: 1d 2.qC1/ D 0; Œkrum k22 C kum k22 C krum k2 2 dt 1 d 2.1Cq/ kru0m k22 C ku0m k22 C D 0: krum k2 2.1 C q/ dt

(5.9) (5.10)

Since, according to the condition um .0/ D

m X

˛mk wk ! u0 2 H01 ./

strongly in H01 ./;

˛mk cmk .0/;

kD1

the right-hand sides of Eqs. (5.7) and (5.8) are bounded by constants independent of m 2 N and t , we obtain that um uniformly with respect to m is bounded in L1 .0; C1I H01 .//; u0m uniformly with respect to m is bounded in L2 .0; C1I H01 .//:

445


Hence we directly obtain that there exists a subsequence of the sequence ¹um º (we denote it also by ¹um º) such that -weakly in L1 .0; C1I H01 .//;

um * u

u0m * u0 weakly in L2 .0; C1I H01 .//:

(5.11)

From (5.11) we obtain that for a certain subsequence of the sequence ¹um º um ! u

weakly in H1 .Q/ ,! L2 .Q/;

Q .0; T/ and strongly in L2 .Q/

8T 2 .0; C1/;

and, therefore, a subsequence converges to u almost everywhere in Q .0; T/ : um ! u almost everywhere in Q .0; T/

8T 2 .0; C1/:

Moreover, from (5.7) and (5.8) by virtue of the choice of the basis wj 2 H02 ./ we have Z t 0 ds jcmk j2 .s/ C < C1; (5.12) jcmk j.t / C < C1; 0

where the constant C is independent of t; k; m. Moreover, for q > 0 the inequality Z t ds jcmk j2 .s/ C < C1 0

holds. Step 2. Lower and upper estimates for Galerkin approximations Lemma 5.1.4. The following upper and lower estimates for rate and time of tending of krum k22 C kum k22 to zero. If q 2 .1=2; 0/, then there exists Tm0 2 .0; C1/ such that .c5 2jqj/1=jqj .Tm0 t /1=jqj krum k22 .t / C kum k22 .t / .c6m 2jqj/1=jqj .Tm0 t /1=jqj ;

(5.13)

where 2.qC1/

c6m

kru0m k2 1; Œkru0m k22 C ku0m k22 qC1

c5 ./ c2 ./=.1 C c2 .//;

446


for all t 2 Œ0; Tm0 , c2 ./ is the best constant of embedding H01 ./ L2 ./: c2 ./kvk22 krvk22 ; 2jqj

kru0m k2 2jqj

.kru0m k22 C ku0m k22 /jqj kru0m k22 C ku0m k22 T : m0 2jqjc5 ./ kru0m k22

If q > 0, then c7m Œc10m C t 1=q krum k22 .t / C kum k22 .t / c9m Œc8m C t 1=q ;

(5.14)

where .1Cq/=q

c7m E0m

2.1Cq/=q

.2q/1=q kru0m k2

1=q 1=q q ; c9m c3

;

q

c8m E0m c31 q 1 ;

c10m E0m .2q/1 kru0m k2.1Cq/ ; 1Cq

E0m D kru0m k22 C ku0m k22 ;

c3

2qC1 c2

c21Cq C 1

:

Proof. Let kru0 k2 > 0; then we can select a subsequence um0 such that krum0 k2 > 0: We take as ¹um º the terminal for the current moment subsequence, which we also denote by ¹um º. By virtue of (5.11), each term um , m 2 N is a global-on-time solution of problem (5.5). Let q 2 .1=2; 0/. By virtue of the conditions for , the embedding H01 ./ 2 ./ holds: c ./k'k2 kr'k2 with a certain constant c ./. Owing to this L 2 2 2 2 embedding, from Eq. (5.9) we obtain the inequality kum k22 1d 0: Œkrum k22 C kum k22 C c2 ./ 2 dt krum k2q

(5.15)

2

Divide both sides of inequality (5.15) by c2 ./ and sum the resulting inequality with Eq. (5.9); we obtain krum k22 C kum k22 d 1 0: Œ1 C c21 ./ Œkrum k22 C kum k22 C 2 dt krum k2q 2

(5.16)

From (5.16) by the negativity of q we have the inequality krum k22 C kum k22 d 1 Œ1 C c21 ./ Œkrum k22 C kum k22 C 0: 2 dt .krum k22 C kum k22 /q

(5.17)

447


Introduce the notation Fm .t / krum k22 C kum k22 ;

c4 ./ D

2c2 ./ : 1 C c2 ./

Then from (5.17) we obtain the following first-order ordinary differential inequality: d Fm C c4 ./Fm .t /1Cq 0; dt

(5.18)

where Fm0 Fm .0/ D krum0 k22 C kum0 k22 > 0;

q 2 .1=2; 0/;

which implies the inequality 1

jqj

0 Fm .t / .Fm0 jqjc4 ./t / jqj : Analyzing this inequality we conclude that since Fm0 D krum0 k22 C kum0 k22 > 0; jqj

there exists a time moment 0 < Tm0 Fm0 jqj1 c41 ./ such that the following equalities hold: krum k22 .Tm0 / D kum k22 .Tm0 / D 0: Therefore, for the finite time Tm0 , the solution um of problem (5.5) of the smoothness class um 2 L1 .0; TI H01 .//; u0m 2 L2 .0; TI H01 .//; T > 0; vanishes for almost all x 2 . This is the essence of the relaxation effect for a finite time. Each solution of problem (5.5) vanishes for almost all x 2 for a certain time Tm0 , generally speaking, different for different solutions. But if for a given solution um , the relaxation time equals Tm0 , then, considering a new problem (5.5) with the initial condition equal to zero for T D Tm0 , we obtain a unique trivial solution in the considered generalized smoothness class by virtue of (5.9). It is easy to verify that the solution glued on time under T D Tm0 belongs to the above-mentioned smoothness class by virtue of (5.11). Indeed, it suffices to prove that the solution um defined as ´ um .x; t /; t 2 Œ0; Tm0 ; um .x; t / D 0; t Tm0 belongs to the class um 2 L1 .Tm0 ; Tm0 C I H01 .//;

u0m 2 L2 .Tm0 ; Tm0 C I H01 .//

448


in a certain neighborhood of the point t 2 .Tm0 I Tm0 C / for all 2 .0; Tm0 =2/. From (5.9) and (5.10) we obtain that Z

krum k22 .t / C kum k22 .t / krum0 k22 C kum0 k22 ; Tm0 C Tm0

ds Œkrums k22 .s/ C kums k22 .s/

1 2.1Cq/ ; krum0 k2 2.1 C q/

t 2 .Tm0 ; Tm0 C /: Now we obtain the optimal upper and lower estimates for the relaxation time and rate of the Galerkin approximations um for a finite time. Introduce the notation 1=2 Mm .t / krum k22 .t / C kum k22 .t / ; M2q m .t /;

vm .s; x/

1 um .s C t ; x/; Mm .t /

t 2 .0; Tm0 /; q Em krvm k22 .s/ C kvm k22 .s/;

where um is an arbitrary solution of problem (5.5) with a fixed relaxation point 0 < jqj Tm0 Fm0 jqj1 c41 ./. Generally speaking, relaxation points are different for different solutions. But all the results below are valid for all the solutions of the initial problem in the generalized sense. It is easy to verify that the introduced function vm .s; x/ satisfies the following initial-boundary-value problem in the sense of L2 .t =; Tm0 = t = I H1 .//: @ 2q .vm vm / C krvm k2 vm D 0; @s um0 t ; q 2 .1; 0/; vm ; x D M.t / t T t vm .s; x/ 2 L1 ; I H01 ./ ; t T t 0 2 1 8T > 0 vm .s; x/ 2 L ; I H0 ./

(5.19)

for all T 2 .0; Tm0 /. Prove that the given solution of problem (5.19) that corresponds to the fixed solution um .x; t / with the relaxation time t0 is such that c6

ˇ d Em .vm /ˇsD0 c5 ; ds

c5 ; c6 > 0:

(5.20)

Note that in the considered smoothness class, each solution of problem (5.19) satisfies the energy equalities (5.9) and (5.10) where um .x; t / is replaced by vm .x; s/ and t by

449


s in the upper limits. In this case we directly obtain that for vm .x; s/, estimate (5.18) holds, which implies the fact that ˇ d Em .vm /ˇsD0 c5 ; ds

c5 ./

c2 ./ ; 1 C c2 ./

(5.21)

where c2 ./ is the the maximal constant of the embedding H01 ./ L2 ./: c2 ./kvk22 krvk22 : On the other hand, by virtue of the mentioned smoothness class and the Cauchy– Bunyakovskii inequality, the following inequality holds: ˇ2 ˇZ ˇ ˇ ˇ dx Œ.rv 0 ; rvm / C .v 0 ; vm /ˇ Œkrvm k2 C kvm k2 Œkrv 0 k2 C kv 0 k2 ; m m 2 2 m 2 m 2 ˇ ˇ

(5.22) together with the equality ˇ ˇ2 ˇZ ˇ2 ˇ ˇ ˇ ˇ ˇ dx Œ.rv 0 ; rvm / C .v 0 ; vm /ˇ D 1 ˇ d Fm ˇ ; m m ˇ ˇ ˇ 4 ds ˇ

(5.23)

where Fm .s/ krvm k22 .s/ C kvm k22 .s/: By (5.22) and (5.23) we obtain ˇ ˇ ˇ d F m ˇ2 d 2 Fm ˇ ˇ 0: Fm .q C 1/ ˇ ds 2 ds ˇ

(5.24)

Dividing both sides of Eq. (5.24) by F2Cq m .s/ > 0, where s 2 Œt =; Tm0 = t =/, after a simple transformation we obtain the inequality d 1 d Fm 0; q 2 Œ1=2; 0/: (5.25) ds FqC1 ds m

Integrating the differential inequality (5.25) over s 2 Œt =; t /, t 2 .t =; Tm0 = t =/, we obtain ˇ ˇ 1 d Fm ˇˇ 1 d Fm ˇˇ qC1 ds ˇsD0 ds ˇsDt = FqC1 Fm m 2.qC1/

D 2

krum0 k2 Mm .t /2.qC1/ Mm .t /2.qC1/ Œkrum0 k22 C kum0 k22 qC1

D 2

krum0 k2 2cm6 : Œkrum0 k22 C kum0 k22 qC1

2.qC1/

(5.26)

450


Since F0m D 2Em E0m , from (5.26) we obtain the inequality ˇ d Em ˇˇ cm6 : ds ˇsD0

(5.27)

From (5.27) and (5.21) we conclude the validity of (5.20). It is easy to verify that, owing to the definition of Em .s/, relation (5.20) is equivalent to the following differential inequalities: cm6

d Mm .t / 1 c5 ; Mm .t /1C2q dt

t 2 .0; Tm0 /;

q 2 .1; 0/:

(5.28)

Integrating relations (5.28) over t 2 .t; Tm0 /, M.Tm0 / D 0, we obtain the inequalities c6m 2jqj.Tm0 t / M2jqj m .t / c5 2jqj.Tm0 t /; which imply the required estimate (5.13). Now let q > 0. Denote by c2 ./ the best constant of the embedding c2 kvk2 kruk22 . From (5.7) we have that 1d qC1 2.qC1/ Œkrum k22 C kum k22 C c2 kum k2 0: 2 dt

(5.29)

Adding (5.29) and (5.9) we obtain qC1 1 c2 C 1 d 2.qC1/ 2.qC1/ C kum k2 0: Œkrum k22 C kum k22 C krum k2 2 c qC1 dt

(5.30)

2

On the other hand, for any q > 0 and any z1 ; z2 0, the obvious inequality z11Cq C z21Cq 2q .z1 C z2 /1Cq holds. Due to the latter inequality and (5.30) we obtain the following first-order differential inequality: d Em 0; C c3 ./E1Cq m dt

1Cq

c3

2qC1 c2

c21Cq C 1

;

which implies the upper estimate (5.14). On the other hand, the second-order differential inequality (5.24) is valid for q > 0, which implies the lower estimate (5.14). Lemma 5.1.4 is proved. Step 3. A priori estimates of second order Remark 5.1.5. For derivation of a priori bounds of second order, it suffices that wk 2 H01 ./ \ H2 ./ \ H3 .G/, where G is an arbitrary subdomain of the domain with the boundary @G 2 C 1 .

451


Consider problem (5.6), where we take as hk;i the function

@ @xi

hk;i

@wk @xi

2 H01 ./;

@wk 2 H02 ./: @xi (5.31)

.2/

0

.x/ 2 C0 ./;

By definition, the function hk;i belongs to the class H01 ./. Now we take in Eq. (5.6) cmk 2 C .1/ .0; Tm0 / from Eq. (5.4) for certain Tm0 > 0 as c mk . The following auxiliary relations hold: Z

T 0

@um @ dt @xi @xi Z Z T @ @um 0 dt dx rum ; r D @xi @xi 0 Z Z T 0 dt dx .rvmi ; r. vmi // D

u0m ;

Z D

0

Z

T

dt

0

1 D 2

Z

Z jrvmi j C

dx Z

Z

T

dt 0

Z Z

Z

T

dt 0

Z D

T 0

dx u0m Z

dt 0

Z

Z

T 0

dt

0

0 dx vmi .rvmi ;r /

dx .rvmi ; r / vmi

0 dx .rvmi ; r / vmi

@ @xi

1 2

Z dx

jrv0mi j2

@um @xi

0 dx vmi vmi

dt jvmi j

1 2

2

dt krum k2q 2 hum ; D

Z

T

dx .rv0mi ; r / v0mi ;

T

1 D 2 Z

rvmi / C

2

Z

0 dx .rvmi ;

Z

dx jv0mi j2 ;

@ . vmi /i @xi

2q

dt krum k2 Œjrvmi j2

C .rvmi ; r /vmi :

452


Thus, by virtue of (5.31) we obtain Z Z Z 1 1 dx .x/jrvm;i j2 C dx .x/jvm;i j2 C dx rvm;i ; r vm;i 2 2 Z Z Z 1 1 2 2 D dx .x/jrvm0;i j C dx .x/jvm0;i j C dx.rvm0;i ; r /vm0;i 2 2 Z Z Z t 0 C ds dx vm;i dx .x/ jrvm;i j2 r ; rvm;i krum k2q 2

0

krum k2q 2

Z

dx vm;i .rvm;i ; r / Fmi ;

(5.32)

where according to the definition vm;i

@um ; @xi

and for the functions Fmi the following estimate holds, which follows from (5.6) by virtue of (5.12): Z jFmi j C

T

dt 0

Œkru0m k2

C

ku0m k2

C

krum k2qC1 2 2

1=2 :

Introduce the notation Z Z 2 I1i D dx .x/jrvm;i j ; I2i D dx .x/jvm;i j2 ; Z Z 2q 0 dx vm;i .r ; rvm;i /; I4i D krum k2 dx .x/jrvm;i j2 ; I3i D Z 2q dx vm;i .rvm;i ; r /; I5i D krum k2 Z dx .rvm;i ; r /vmi : I6i D

Integrating the equalities I5i and I6i by parts, we obtain Z 1 I5i D krum k2q dx jvm;i j2 ; 2 2 Z 1 I6i D dx jvm;i j2 : 2 Now we analyze the integral I3i Z Z @um @ 2 um 0 r ;r I3i D dx vm;i .r ; rvm;i / D dx @xi @t @xi

453


and N X

I3 D

I3i D

iD1

N Z X

dx

i D1

D

D

N Z X

@2 um @xi @t

@um r ;r @xi

0 .rvm;i ; rvm;i /

dx

i D1

N Z 1d X dx 2 dt

jrvm;i j2 C

i D1

C

N Z X

X N Z 1d D dx 2 dt C

i D1

N Z X

i D1

dx

i D1

@u0m @um @xi @xi

@ @u0m um @xi @xi

Z 2

jrvm;i j C

i D1

dx

dx

@2 u0m um @xi2

dx

i D1

N Z X

N Z X

dx

jum j

2

@ @u0m um : @xi @xi

Now we sum (5.32) over i D 1; N ; then owing to the obtained expressions for Ili , l D 1; 5, i D 1; N , we obtain 2

N Z X

.x/jrvm;i j2 C

dx

iD1

Z

C

.x/jum0 j2

dx

D 2

N X

Fmi C 2

iD1

Z C

dx

Z C

t 0

N Z X

dx

i D1

N Z X

i D1

N Z X

dx jvm;i j2

dx

i D1

.x/jum j2

.x/jvm;i j2

.x/jrv0m;i j2 C

N Z X i D1

N Z X i D1

dx

.x/jv0m;i j2

dx jv0m;i j2

Z N Z X @ @u0m 2 ds dx um 2krum k2q dx 2 @xi @xi i D1 Z 2q 2 C krum k2 dx jvm;i j

for all t 2 .0; Tm0 /.

.x/ jrvm;i j2 (5.33)

454


For further consideration, we must obtain the required upper estimate for kum k22 . We use the following equality in problem (5.5):

1 wj D wj 2 L2 ./: j

As .t / 2 L2 .0; T/ we take cmj from (5.4). Then, summing over j D 1; N , we obtain Z t 2q kum k22 C krum k22 C 2 ds krum k2 kum k22 D kum0 k22 C krum0 k22 : 0

(5.34) From (5.6), (5.13), (5.14), and (5.34) we obtain that for all t 2 Œ0; C1/, the inequality kum k22 .t / C krum k22 .t / kum0 k22 C krum0 k22 C.t / < C1

(5.35)

holds for all t 2 .0; C1/, where the constant C.t / is independent of m; this fact is valid by virtue of the strong convergence um0 ! u0 in H2 ./. Introduce the notation N Z X dx .x/jrvm;i j2 : (5.36) Ym .t / i D1

Due to the first-order a priori estimates (5.9) and (5.10) obtained above, estimates (5.6) of the functions of the form Fm;i , and estimates (5.33)–(5.36) we obtain the following integral inequality: Z t Z t 2q ds krum k2 .s/ C A3 C A4 ds krum k2q Ym .t / A1 t C A2 2 .s/Ym .s/ 0

0

(5.37) for all t 2 .0; Tm0 /, An > 0, n D 1; 4, where Tm0 is the relaxation time of the solution .2/ um and the constants An , n D 1; 4, depend, generally speaking, on .x/ 2 C0 ./. First, we consider the case q 2 .1=2; 0/. Owing to the lower estimate (5.13), the integral equation (5.37) takes the form Z t Z t 1 1 C A3 C A4 Ym .s/ Ym .t / A1 t C A2 ds ds T s T m0 m0 s 0 0 for all t 2 .0; Tm0 /. By virtue of the Gronwall–Bellman theorem (see, e.g., [112]), from the latter inequality we obtain ³ ² A2 t A4 t C A3 exp (5.38) 8t 2 .0; Tm0 /: Ym .t / A1 t C Tm0 t Tm0 t


455

From (5.13) we have the fact that the sequence Tm0 satisfies the conditions .2jqj/1 Œkrum0 k22 C kum0 k22 jqj Tm0 .c5 2jqj/1 Œkrum0 k22 C kum0 k22 jqj : Then

lim inf Tm0 D T0 .2jqj/1 Œkrum0 k22 C kum0 k22 jqj > 0:

m!C1

We select from Tm0 a sequence converging to T0 . Here we can select a subsequence converging to T0 monotonically from above or monotonically from below. Let Tm0 # T0 , 0 < T0 Tm0 , Em .2jqj/1=jqj .Tm0 t /1=jqj . Then we can pass to the limit as m ! C1 uniformly with respect to t 2 Œ0; T0 and obtain E lim inf Em .2jqj/1=jqj .T0 t /1=jqj : m!C1

Therefore, in this case kruk2 .T0 / D 0. Now let Tm0 " T0 , Em .2jqj/1=jqj .Tm0 t /1=jqj . Then for any m 2 N, we can select from Tm0 a subsequence and pass to the limit as m ! C1 uniformly with respect to t 2 Œ0; Tm0 : E lim inf Em .2jqj/1=jqj .T0 t /1=jqj : m!C1

By the arbitrariness of m, we obtain that the mentioned upper estimate holds uniformly with respect to t 2 Œ0; T0 . Therefore, from the sequence um , we can select a subsequence such that Tm0 ! T0 W kruk2 .T0 / D 0: Now we pass to the limit as m ! C1 in the right-hand side of inequality (5.38). We consider two cases: Tm0 # T0 and Tm0 " T0 . Consider the first case. Obviously, T0 t Tm0 t , and from (5.38) we obtain ³ ² A2 t A4 t C A3 exp Ym .t / A1 t C T0 t T0 t

(5.39)

for all t 2 .0; T0 /, T0 T0 , where T0 is the relaxation time of the function u 2 C.Œ0; TI H01 .//; here under the condition kru0 k2 > 0 we have T0 > 0. Now we analyze the second case. Prove that Ym is bounded uniformly with respect to m with the constant C.t /: Ym .t / C.t / < C1 8t 2 Œ0; T0 /: 2 .0; T / such that Assume the opposite: let there exist tm 0

lim sup Ym ! C1: t"tm

456


Now we choose a natural number m so that Tm0 > t ; then ³ ² A2 t A4 t 8t 2 .0; Tm0 /: C A3 exp Ym .t / A1 t C Tm0 t Tm0 t Now it suffices to take t 2 .t ; Tm0 /. The contradiction obtained proves our statement. Now consider the case q 2 .0; C1/. From (5.37) we obtain Z t Ym .t / A1 t C A3 C A4 ds Ym .s/: (5.40) 0

From (5.40), according to the Gronwall–Bellman theorem (see [112]) we obtain Ym .t / C.t / < C1

8t > 0;

where C.t / is independent of m 2 N. As a result, we obtain Ym .t / C.t / < C1

(5.41)

in the case t 2 .0; T0 / for q 2 .1=2; 0/ and t 2 .0; C1/ for q > 0, where C.t / is independent of m 2 N. Now we propose a special partition of unit in the domain . First, consider the segment Œ0; 2". We introduce the required partition of unit by smooth functions 1 k .x/ 2 C0 ./ on the segment: 8 ˆ x 2 Œ"; 2"I 1:

Using of the mentioned partition of unit on the segment Œ0; 2", we can construct a partition of unit in a simply-connected bounded domain in the case N D 2. First, choose a sufficiently small " > 0 such that there exists a domain 2" with the boundary @2" distant along normal at any point of the boundary @ for a distance

457


of 2". Next, according to the mentioned rule, we construct the domains 2kC2 " . Fixing an arbitrary point x 2 @ and drawing a ray from this point along the interior normal to @, we find the point of intersection with the boundary @2" of the domain 2" . On the segment obtained, we construct the mentioned partition of unit (5.42). Running all the boundary x 2 @ 2 C 1 , we construct the required partition of unit of the two-dimensional domain R2 . The case of multiply-connected domain can be considered with the use of the mentioned method near each connected component of the boundary @ of the multiply-connected domain . Now consider a way of extending this algorithm over the case of the bounded domain in RN . First, consider an N -dimensional ball, near the boundary of which, as usual, we construct a radially symmetric partition of unit. Next, we mark level lines of each function of the constructed partition of unit. Further, we smooth (in the sense of C 1 ) deform the boundary of the ball. In this case, the mentioned level lines will also be deformed. In any case, the constructed partition of unit is continuous, and this suffices for our argumentation. We assume that the finitely connected domain 2 RN is C 1 diffeomorphic with the N -dimensional ball with some balls removed from inside. In this case, the construction of a partition of unit is similar to the construction in a simply-connected domain. Denote by 'k 2 C0 ./ the functions from the partition of unit near the boundary of the domain 2 RN constructed on the basis of k .x/, x 2 Œ0; 2". Now we obtain required a priori estimates. .2/ Let 'kl 2 C0 ./ be such that k'k 'kl k1 ! 0 as l ! C1. The following equality holds: N Z X dx jrvm;i j2 (5.43) iD1 N Z X

D

iD1

K K K X X X dx 1.x/ 'k .x/ C .'k .x/ 'kl .x// C 'kl .x/ jrvm;i j2 : kD1

kD1

kD1

For any K 2 N, there exists lK 2 N such that the inequality X K 1=4 .' .x/ ' .x// k kl 1

kD1

holds for l lK . Consider the following integral: Z N K X X dx 'k .x/ jrvm;i j2 ; J 1.x/

Z Jm

dx

i D1 N X i D1

1.x/

kD1 K X kD1

'k .x/ fmi :

fmi D

jrvm;i j2 ; krvm;i k22

458


Here for the integral Jm , the conditions of the Lebesgue theorem on the passage to the limit as k ! C1 under the sign of integral. Indeed, ˇX ˇ X K N X ˇN ˇ ˇ ˇ ' .x/ f fmi ; 1.x/ mi k ˇ ˇ i D1

gm D

N X

K X

iD1

kD1

1.x/

kfmi k1 D 1;

i D1

kD1

'k .x/ fmi ! 0 almost everywhere in as K ! C1:

Therefore, by virtue of (5.43), there exists K 2 N such that uniformly by m 2 N Jm 1=4 or, otherwise, Z dx

N X

K X

1.x/

i D1

kD1

Z N X 1 2 'k .x/ jrvm;i j dx jrvm;i j2 : 4 i D1

From (5.43) we obtain N Z X iD1

1X dx jrvm;i j 4 N

Z

2

i D1

1X C 4 N

dx jrvm;i j2 Z 2

i D1

N Z X iD1

dxjrvm;i j2 2

K N X X i D1 kD1

dx jrvm;i j C

i D1 kD1

Z

K Z N X X

dx 'kl .x/jrvm;i j2 ;

dx 'kl .x/jrvm;i j2 C.t / < C1

(5.44)

for l D lK and sufficiently large fixed K 2 N; t 2 .0; T0 / for q 2 .1=2; 0/ and t 2 .0; C1/ for q > 0; C.t / is independent of m 2 N. Now we obtain second-order a priori estimates for the function u0m . For this, we 0 from (5.4) as c mk in Eq. (5.6) and as hk;i , we take take cmk hk;i

@ D @xi

@wk @xi

2 H01 ./;

2 C0.2/ ./:

Integrating by parts, we obtain P1;i C P2;i C P3;i D F3i ;

(5.45)

459


where Z P1;i D

Z

T

dt Z

P3;i D

0

T

2q

Z jF3i j C Z P1;i D

T

0

D

0

dx Z

dt

dx

0

Z

P3;i D Z D

T

dt 0 T

0

C

C

0 jrvm;i j2

1 2

2q

dt 0

Z

0 jrvm;i j2

krum k2q 2

T

dt 0

0 dx jvm;i j2 ;

0 dx vm;i vm;i ;

Z

dx

Z

dt krum k2

Z

Z

T

2qC1 2

T

P2;i D

dt Œkru0m k2 C ku0m k2 C krum k2

Z

T

dt Z

Z

dt krum k2

0

Z

0 0 dx vm;i vm;i ;

krum k2q 2

Z

Z

T

dt 0

;

0 0 dx vm;i .r ; rvm;i /

dt 0

0 dx jvm;i j2 ;

@um @u0m @xi @xi

dx um

Z

Z

T

1=2

@2 u0m @xi2

dx um

@u0m @ : @xi @xi

By virtue of (5.45), the relations obtained imply the equality N Z X

Z

T

dt

iD1 0

dx

D

1 2

Z

T

dt 0 N Z X iD1 0

0 jrvm;i j2 C

N Z X i D1 T

dt

N Z X

Z

T

dt

i D1 0

0 dx jvm;i j2

krum k2q 2

0 jvm;i j2

dx

Z

T

0

Z

dt krum k2q 2

Z

dx um u0m

X @u0m @ F3;i : @xi @xi N

dx um

i D1

Now we obtain an auxiliary estimate for u0m of the form (5.34). To this end, in (5.5) we use the equality wj D j wj 2 L2 ./; 0 as .t /; then integrating by parts and summing over k D 1; m we obtain and take cmk

Z

T

dt 0

ku0m k22

C

kru0m k22

C

2q krum k2

Z

dx u0m um

D 0:

460


Note that the integrand is continuous on the segment Œ0; T for all T 2 .0; Tm0 / and, therefore, we obtain the pointwise equality Z dx u0m um D 0: ku0m k22 C kru0m k22 C krum k2q 2

Owing to the Hölder inequality 1 1 4q ku0m k22 C kru0m k22 ku0m k22 C krum k2 kum k22 ; 2 2 4q ku0m k22 krum k2 kum k22 :

Further, arguing similarly as in the derivation of the estimate (5.41), we obtain N Z X

Z

T

dt

i D1 0

dx

0 jrvm;i j2 C.t / < C1;

(5.46)

where t 2 .0; T0 / for q 2 .1=2; 0/ and t 2 .0; C1/ for q > 0. Further, using the partition of unit obtained above, we obtain from (5.46) N Z X

Z

T

i D1 0

dt

0 dx jrvm;i j2 C.t / < C1;

(5.47)

where t 2 .0; T0 / for q 2 .1=2; 0/ and t 2 .0; C1/ for q > 0. Step 4. Passage to the limit as m ! C1. Now we pass in problem (5.5) to the limit as m ! C1. It follows from a priori estimates (5.44) and (5.47) that we can select a subsequence of the sequence ¹um º (we denote it again by um ) such that um * u

-weakly in L1 .0; C1I H01 .//;

u0m * u0 weakly in L2 .0; C1I H01 .//; um * u


(5.48)

u0m * u0 weakly in L2 .0; TI H2 .// where T 2 .0; C1/ for q > 0 and T 2 .0; T0 / for q 2 .1=2; 0/, T0 is the relaxation time of the limit function u. By the compact embedding H2 ./ ,! H1 ./ and the limit relations (5.11) we obtain um ! u

strongly in H01 ./ for almost all t 2 .0; Tmax /;

(5.49)

where Tmax D C1 for q > 0 and Tmax D T0 for q 2 .1=2; 0/. Owing to relations (5.11) and (5.46)–(5.49), we can pass to the limit as m ! C1 in the problem

Z T @ 2q Œum um C krum k2 um ; wj D 0; j D 1; m: dt .t / (5.50) @t 0

461


Consider the case q 2 .1=2; 0/ (note that the case q > 0 is much simpler). Assume that kru0 k2 > 0; then, passing to a subsequence if necessary, we obtain kru0m k2 > 0 uniformly with respect to m 2 N. By Lemma 5.1.4 we have the inequalities .c5 2jqj/1=jqj .Tm0 t /1=jqj krum k22 C kum k22 .c6m 2jqj/1=jqj .Tm0 t /1=jqj ; i.e., Tm0 is the exact relaxation time of the function um . Here for Tm0 the lower and upper estimates hold: 1 1 .krum0 k22 C kum0 k22 /jqj : .krum0 k22 C kum0 k22 /jqj Tm0 2jqj 2jqjc5

(5.51)

Let T0 D lim inf Tm0 : m!C1

Passing to a subsequence if necessary, we obtain that either Tm0 " T0 or Tm0 # T0 . By the fact that um0 ! u0 strongly in H01 ./, passing in inequalities (5.51) to the limit, we obtain that 1 1 .kru0 k22 C ku0 k22 /jqj T0 .kru0 k22 C ku0 k22 /jqj : 2jqj 2jqjc5 On the other hand, similarly to the derivation of (5.39), we obtain the inequality kruk22 .2jqj/1=jqj .T0 t /1=jqj : Moreover, from (5.11) we obtain that u 2 C.Œ0; TI H01 .//. Then we conclude that for kru0 k2 > 0, the inequality 0 < T0 T0 holds. Here krum k2 > 0 for t 2 Œ0; Tm0 /. Consider two cases: Tm0 " T0 and Tm0 # T0 . In the second case, let T 2 .0; T0 /, where T0 is the relaxation time of the function u from (5.11); therefore, krum k2 > 0 for all t 2 .0; T/. Passing in Eq. (5.50) to the limit, we obtain Z T dt .t /hu0 u0 C kruk2q (5.52) 2 u; wj i D 0 0

for all 2 L2 .0; T/, all j D 1; C1, and all T 2 .0; T0 /. Now consider the first case. We distinguish between two subcases: T0 < T0 and T0 D T0 . In the case where T0 < T0 , removing a finite number of terms of the sequence Tm0 , we obtain Tm0 > T0 for m > m 2 N. Then for all T 2 .0; T0 /, passing in Eq. (5.50) to the limit as m ! C1, we obtain (5.52). Now let T0 D T0 . Fix arbitrary m 2 N. Then for all m > m, we can pass to the limit as m ! C1 for all T 2 .0; T0m /, and by the arbitrariness of m, we conclude that Eq. (5.52) holds.

462


Step 5. Uniqueness. Consider problem (5.52). By definition, functions ¹wj ºjC1 D1 form C1 1 2 a basis in the Hilbert space H0 ./. Let ¹ k .t /ºkD1 be a basis in L .0; T/ for all T 2 .0; T0 /. Linear combinations mX 1 ;m2

cj k wj .x/ k .t /

k;j D1

are dense in the Hilbert space L2 .0; TI H01 .//; therefore, an arbitrary function v 2 L2 .0; TI H01 .// can be approximated by these combinations: Z

T 0

mX 1 ;m2 dt rv cj k rwj .x/ k .t / "; 2

k;j D1

where " > 0 is arbitrarily small. Note that for any u 2 L1 .0; TI H01 .//, u0 2 L2 .0; TI H01 .//, and v 2 L2 .0; TI H01 .//, for arbitrary T 2 .0; T0 / the Lebesgue integral Z T dt hLŒu; vi 0

is defined, where

2q LŒu D u0 u0 C kruk2 u:

Indeed, the following inequalities hold: Z

T

0

dt jhu0 ; vij

Z

dt 0

Z

T 0

T 0

Z

Z

Z

T

dt Z

0

dt jhu ; vij

dt kruk2q 2 jhu; vij

T 0 T

dt Z

0 T 0

ˇ ˇ dx ˇ.ru0 ; rv/ˇ

kru0 k22 ku0 k22

1=2 Z

1=2

T

dt 0

1=2 Z

T

dt 0

dt kruk2.2qC1/ 2

krvk22

I

1=2 kvk22

1=2 Z 0

T

I 1=2

dt krvk22

:

Consider the equality Z 0

T

Z dt hLŒu; vi D

T 0

dt LŒu; v

mX 1 ;m2

cj k wj .x/ k .t / :

(5.53)

k;j D1

Since T 2 .0; T0 /, we have kruk2 > 0 for all t 2 Œ0; T0 / under the condition kru0 k2 > 0. It is easy to prove by integrating by parts that the right-hand side of

463


Eq. (5.53) vanishes as " ! C0. Indeed, the following inequalities hold: ˇ Z T ˇ mX 1 ;m2 ˇ ˇ dt ˇˇhu0 ; v cj k wj .x/ k .t /iˇˇ 0

k;j D1

Z Z

T

dt 0 T

0

T 0

k;j D1

T

dt 0

ku0 k22

1=2 Z

dt 0

Z

ˇ ˇ kruk2q 2 ˇ T

dt 0

T 0

ˇ

T

1=2 Z

2 1=2 mX 1 ;m2 dt rv cj k rwj .x/ k .t / ı1 ."/I

ˇ

ˇ mX 1 ;m2 ˇ 0 ˇ ˇ dt ˇ u ; v cj k wj .x/ k .t / ˇˇ

Z

Z

kru0 k22

u; v

2 1=2 mX 1 ;m2 dt rv cj k rwj .x/ k .t / ı2 ."/I

mX 1 ;m2 k;j D1

kruk2.2qC1/ 2

2

k;j D1

2

k;j D1

ˇ ˇ cj k wj .x/ k .t / ˇˇ

1=2 Z

T 0

2 1=2 mX 1 ;m2 dt rv cj k rwj .x/ k .t / ı3 ."/: k;j D1

2

Here ıi ."; T/ ! C0 as " ! C0 for any fixed T 2 .0; T0 /. Since the left-hand side of Eq. (5.53) is independent of ", (5.53) implies Z T dt hLŒu; vi D 0 8v 2 L2 .0; TI H01 .//; 8T 2 .0; T0 /: (5.54) 0

It is quite obvious that (5.54) implies (5.52). Therefore, problems (5.52) and (5.54) are equivalent. Now take a function u 2 L2 .0; TI H01 .// as v in problem (5.54); integrating by parts we obtain Z t 2.1Cq/ ds kruk2 D kru0 k22 C ku0 k22 (5.55) kruk22 C kuk22 C 2 0

for all t 2 Œ0; T0 . Now, taking the function u0 as v, we obtain the equation Z t 1 1 2.1Cq/ 2.1Cq/ ds Œkru0 k22 C ku0 k22 C D kruk2 kru0 k2 2.1 C q/ 2.1 C q/ 0 (5.56) for all t 2 Œ0; T0 . From Eqs. (5.55) and (5.56) we obtain the following equations: 1d 2.qC1/ D 0; Œkruk22 C kuk22 C kruk2 2 dt d 1 2.1Cq/ kruk2 D 0: kru0 k22 C ku0 k22 C 2.1 C q/ dt

(5.57) (5.58)

464


The following lemma holds. Lemma 5.1.6. The following upper and lower estimates for the rate and time of vanishing of kruk22 C kuk22 under the condition kru0 k2 > 0 hold. (1) If q 2 Œ1=2; 0/, then there exists T0 2 .0; C1/ such that .c5 2jqj/1=jqj .T0 t /1=jqj kruk22 .t / C kuk22 .t / .c6 2jqj/1=jqj .T0 t /1=jqj ;

(5.59)

where 2.qC1/

kru0 k2

c6

kru0 k22

C

qC1 ku0 k22

1;

c5 ./

c2 ./ 1 C c2 ./

8t 2 Œ0; T0 ;

c2 ./ is the best constant of the embedding H01 ./ L2 ./: c2 ./kvk22 krvk22 , 2jqj

kru0 k2 2jqj

.kru0 k22 C ku0 k22 /jqj kru0 k22 C ku0 k22 T0 : 2 2jqjc5 ./ kru0 k2

(2) If q > 0, then c7 Œc10 C t 1=q kruk22 .t / C kuk22 .t / c9 Œc8 C t 1=q ;

(5.60)

where .1Cq/=q .2q/1=q kru0 k2.1Cq/=q ; c 7 E0 2 1=q 1=q

c9 c3 E0 D

q

kru0 k22

C

;

1 1 c8 Eq ; 0 c3 q

c10 E0 .2q/1 kru0 k2.1Cq/ ;

ku0 k22 ;

c3

2qC1 c21Cq 1Cq

c2

C1

:

Proof. The proof is similar to the proof of Lemma 5.1.4 with the substitution um , um0 , Tm0 for u, u0 , T0 , respectively and owing to Eqs. (5.57), (5.58). First, consider the case where q 2 .1=2; 0/. Let ul , l D 1; 2, be two solutions of problem (5.54) that correspond to u0 . Let ´ u1 u2 ; s 2 Œ0; t ; v.s/ D w.s/ D 0; s 2 Œt; TI then from (5.54) we obtain the expression Z t ds hLŒu1 LŒu2 ; wi D 0 0

8t 2 .0; T00 /;

465


where T00 D min¹T01 ; T02 º, T0k is the relaxation time of the function uk , k D 1; 2. Integrating by parts we obtain the equation krwk22 .t / C kwk22 .t / Z Z t 2q ds dx .kru2 k2q D2 2 ru2 kru1 k2 ru1 ; rw/ 8t 2 .0; T00 /: 0

Let T01 T02 . The following inequality holds: ˇ q ˇ ˇr r q ˇ jqj max¹r q1 ; r q1 ºjr1 r2 j: 1 2 1 2 Then the inequality 2q

C jkru1 k22 kru2 k22 j .T01 t /.1Cjqj/=jqj C kru2 k2 krwk2 .T01 t /.1Cjqj/=jqj

2q

jkru1 k2 kru2 k2 j

holds. We have Z krwk22 C kwk22 D

Z

t

ds 0

Z

C

t 0

2q dx Œkru2 k2q 2 kru1 k2 .ru2 ; rw/

2 ds kru1 k2q 2 krwk2 :

We obtain an upper estimate for Z 2q 2q K dx .rw; ru2 /.kru1 k2 kru2 k2 /:

The following inequality holds: 2q 2q 2.q1/ ; kru2 k2.q1/ ºjkru1 k22 kru2 k22 j jkru1 k2 kru2 k2 j C¹kru1 k2 2

C

1 kru1 k2 krwk2 : .T01 t /.1Cjqj/=jqj

Thus, KC Hence we conclude that 2 krwk2 C

1 krwk22 : .T01 t /.1Cjqj/=jqj

1 1 C T01 T .T01 T /.1Cjqj/=jqj

Z

t 0

ds krwk22

466


for all t 2 Œ0; T , where T 2 .0; T01 /. By the Gronwall–Bellman inequality (see [112]) we have krwk22 D 0 8t 2 Œ0; T ;

8T 2 .0; T01 /:

Hence we directly obtain that T01 D T02 D T0 and w D 0 for all t 2 Œ0; C1/. Now consider the case where q > 0. Obviously, this case is simpler, and the uniqueness can be proved as in the previous case, by using estimate (5.51). Theorem 5.1.1 is proved. Theorem 5.1.7. For any u0 .x/ 2 H01 ./, there exists a unique solution of problem (5.1), (5.2) of the class u.x; t / 2 L1 .0; C1I H01 .//, u0 .x; t / 2 L2 .0; C1I H01 .//. Proof. We must prove the fact that problem (5.1), (5.2) is uniquely solvable in the weak generalized sense in the class u 2 L1 .0; C1I H01 .//;

u0 2 L2 .0; C1I H01 .//

under the condition u0 .x/ 2 H01 ./. Let u0 .x/ 2 H01 ./. We take an arbitrary sequence un0 .x/ 2 H01 ./ \ H2 ./ such that un0 ! u0 strongly in H01 ./. Denote by un a unique solution of problem (5.1), (5.2) in the weak generalized sense corresponding to un0 . Definition 5.1.8. A function u.x; t / that coincides with C.Œ0; TI H01 .// almost everywhere for certain T > 0 and satisfies the conditions

Z T @ 2q Œu u C kruk2 u; w .t / D 0 dt (5.61) @t 0 for all 2 L2 .0; T/, all w 2 H01 ./, T > 0, for kru0 k2 > 0, q > 1=2, or u.x; t / D 0 almost everywhere in Q .0; T/ for kru0 k2 D 0, where h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./, and T0 is the relaxation time of the solution u, is called a weak generalized solution of problem (5.1), (5.2). First, take v D un 2 L1 .0; C1I H01 .// as v, where .un /0 2 L2 .0; C1I H01 .//: Integrating by parts we obtain Z krun k22 C kun k22 C 2

0

T

2.1Cq/

dt krun k2

D krun0 k22 C kun0 k22 :

(5.62)


467

Now we set v D .un /0 . Integrating by parts we obtain Z

T 0

dt Œkr.un /0 k22 C k.un /0 k22 C

1 1 2.1Cq/ 2.1Cq/ krun k2 krun0 k2 D : 2.1 C q/ 2.1 C q/ (5.63)

From (5.62) and (5.63) we obtain that un

is bounded in L1 .0; C1I H01 .//;

.un /0

is bounded in L2 .0; C1I H01 .//:

Thus, there exist a subsequence of the sequence ¹un º and, therefore, the corresponding subsequence of the sequence ¹un0 º such that un * u

weakly in L1 .0; C1I H01 .//;

.un /0 * u0

weakly in L2 .0; C1I H01 .//:

For any n 2 N, un satisfies lower and upper estimates of the form (5.59), (5.60), where u is replaced by un , u0 by un0 , and T0 by T0n . Consider the case where q 2 .1=2; 0/. By virtue of (5.59), the relaxation time T0n of the solution un satisfies the inequality .2jqj/1 Œkrun0 k22 C kun0 k22 jqj T0n .c5 2jqj/1 Œkrun0 k22 C kun0 k22 jqj : Then

lim inf T0n D T0 > .2jqj/1 Œkru0 k22 C ku0 k22 jqj > 0;

n!C1

and we can select a subsequence of the sequence ¹T0n º converging either monotonically from above or monotonically from below. Assume that T0n # T0 ;

0 < T0 < T0n ;

En krun k22 C kun k22 .2jqj/1=jqj .T0n t /1=jqj :

Then we can pass to the limit as n ! C1 uniformly with respect to t 2 Œ0; T0 and obtain E lim inf En .2jqj/1=jqj .T0 t /1=jqj : n!C1

Therefore, in this case, kruk2 .T0 / D 0. Now let T0n " T0 . Then for any n 2 N, we can select a subsequence from the sequence ¹T0n º so that we can pass to the limit as n ! C1 uniformly with respect to t 2 Œ0; T0n : E .2jqj/1=jqj .T0 t /1=jqj : By the arbitrariness of n we obtain that the estimate mentioned holds uniformly with respect to t 2 Œ0; T0 . Therefore, in this case kruk22 .T0 / D 0 also.

468


Let n1 and n2 be certain natural numbers. Then from Eq. (5.61) written separately for n D n1 and n D n2 for .t / D 1 and w D un1 un2 we obtain Z

T

d Œkrun1 run2 k22 C kun1 un2 k22 dt Z T n1 n2 n1 dt .krun1 k2q krun2 k2q run2 / D 0; C2 2 ru 2 ru ; ru

dt 0

(5.64)

0 T

ˇ ˇZ ˇ ˇ 2q 2q n n n n n n 1 1 2 2 1 2 dt .kru k2 ru kru k2 ru ; ru ru /ˇˇ ; I ˇˇ 0 ˇZ T ˇ ˇ ˇ 2q 2q n n n n n 1 2 1 1 2 I ˇˇ dt Œkru k2 kru k2 .ru ; ru ru /ˇˇ 0 ˇ ˇZ T ˇ ˇ 2q n n n 2 2 1 2 dt kru k2 kru ru k2 ˇˇ C ˇˇ C

0

1 1 C .1Cjqj/=jqj T0n T .T0n T/

Z

T

dt krun1 run2 k22 ;

0

(5.65)

T0n min¹T0n1 ; T0n2 º. Consider two cases: T0n " T0 and T0n # T0 . As we have obtained above, T0 is the relaxation time of the function u.x; t /. Consider the case where T0n # T0 . Without loss of generality, we assume that n1 < n2 . Then T0n D Tn2 and, moreover, T0n2 T T0 T: From (5.65) we obtain Z T 1 1 IC C dt krun1 run2 k22 : T0 T 0 .T0 T/.1Cjqj/=jqj

(5.66)

Now consider the case where T0n " T0 , T0n D Tn1 : IC

.T0N

1 1 C T0N T T/.1Cjqj/=jqj

Z 0

T

dt krun1 run2 k22 ;

(5.67)

T0n1 T T0N T, n2 > n1 N for all T 2 .0; T0N /, T0N " T0 . Now we consider Eq. (5.64), from which, by virtue of (5.66) and (5.67), we obtain krun1 run2 k22 krun0 1 run0 2 k22 C D.T0 ; T0N ; T/

Z

T 0

dt krun1 run2 k22 .t /

(5.68)

469


for all T 2 .0; T0N /, T0N " T0 . Now we apply the Gronwall–Bellman theorem (see [112]) and obtain from (5.68) krun1 run2 k22 krun0 1 run0 2 k22 2 exp¹D.T0 ; T0N ; T/Tº: Fix T 2 .0; T0 /. Then for any " > 0, there exists N 2 N such that for any n1 ; n2 N we have krun1 run2 k ": Therefore, un is a Cauchy sequence in the strong topology of the space H01 ./ for almost all t 2 .0; T0 /. Then, obviously, this sequence converges to u.x; t / strongly in H01 ./ for almost all t 2 .0; T0 /. Now we can pass in Eq. (5.61) to the limit as n ! C1 and obtain

Z T @ @ 2q dt u u C kruk2 u; v D 0 @t @t 0 for all v 2 L2 .0; TI H01 .//, T 2 .0; T0 /. On the other hand, u 2 L1 .0; TI H01 .//;

u0 2 L2 .0; TI H01 .//:

Thus, we have proved the global-on-time solvability in the case q 2 .1=2; 0/. The case q > 0 is simpler and it can be considered similarly by using estimates (5.60). The proof of the uniqueness is exactly the same as the proof of the uniqueness of a smooth solution in Step 6. Theorem 5.1.7 is proved.

5.1.2 Global-on-time solvability of the problem in the strong generalized sense in the case q 1 In this subsection, we analyze the global-on-time solvability of problem (5.1), (5.2) in the strong sense. Definition 5.1.9. A solution of the class C .1/ .Œ0; TI H01 .// of the problem 2q hu0 u0 C kruk2 u; wi D 0;

u.0/ D u0 2 H01 ./;

8w 2 H01 ./;

where h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./ and the time derivative is meant in the classical sense, is called a strong generalized solution of Problem A. The following theorem holds. Theorem 5.1.10. Let q 1, u0 .x/ 2 H01 ./. Then there exists a unique strong .1/ generalized solution u.x; t / of Problem A of the class Cb .Œ0; C1/I H01 .// exists.

470


Proof. Introduce the notation A

du C B.u/ D 0; dt A I ;

u.0/ D u0 2 H01 ./; 2q

B.u/ kruk2 u:

The following inequality holds: hAz1 Az2 ; z1 z2 i krz1 rz2 k22

8z1 ; z2 2 H01 ./:

Prove the local Lipschitz-continuity of the operator B.u/. We have B.u/ kruk2q 2 u; 2q

2q

2q

kB.u/ B.v/k1 jkruk2 krvk2 jkuk1 C krvk2 ku vk1 C max¹krvk2.q1/ ; kruk2.q1/ ºjkruk22 krvk22 jkruk2 2 2 C Ckrvk2q 2 kru rvk2 2q C max¹krvk2q 2 ; kruk2 ºkru rvk2

C Ckrvk2q 2 kru rvk2 .R/kru rvk2 .R/ku vkC1 ; where .R/ D CR2q and R D max¹kukC1 ; kvkC1 º. Therefore, kB.u1 / B.u2 /k1 .R/ku1 u2 kC1 ; .R/ D CR2q ; kvkC1 krvk2 ; ® ¯ 8u1 ; u2 2 BR v 2 H01 ./ W krvk2 R : Thus, our problem in the strong generalized sense is equivalent to the following problem: du C F.u/.t / D 0; dt

u.0/ D u0 2 H01 ./;

FŒu.t / A1 BŒu.t /;

which, in its turn, is equivalent to the following integral equation: Z t ds FŒu SŒu: u.t / D u0 C 0

Introduce a closed convex bounded subset ® ¯ BR v 2 L1 .0; TI H01 .// W kvkT ess sup kvkC1 R : t2.0;T

471


Prove that S acts from BR into BR and is a contraction on BR . Indeed, 1 kSŒuk T.R/kuk kuk; 2

T
0, and > 1. Then at large time the following asymptotic behavior of a solution of problem (5.1), (5.2) holds: 2 AN K u.; t / ˛N N .x/ ; 1=.2q/ .C .C0 C t / 0 C t/ 2 8 ˆ > 2; ³ ² 1I

therefore, in the worst case jexp¹.k 1=q/f .t /ºj .C1 C t /k 1=q : Therefore, jD.t /j A.t C C0 /1 ;

jf j ln 1 C

A t C C0

Z

t

ds 0

s C C0 s C C0

A:

Therefore, there exists a constant C8 independent of t such that

X

jD.t /j C8

kDN C1

0 2k

1C k ˛k2 C0 k

1 .t C C0 /k

q

C : .t C C0 /

Then from representation (5.75) we obtain 1 jf .t /j C9 t C C0

Z

t

1 (5.76) .s C C0 /.1C˛ /=q 0 8 1 ˆ > 2; Z t 1; otherwise, we obtain a very rough estimate. Now we use estimate (5.76) in order to obtain an estimate of the remainder term of

475

Section 5.2 Blow-up of pseudoparabolic equations with pseudo-Laplacian

the asymptotical expansion at large time: 2 AN u.; t / ˛N N .x/ 1=.2q/ .C0 C t / 2 ² ³ Z t N 2q ˛N N exp ds kruk2 1 C N 0 ² ³2 N 0 Œln.C0 C t / ln.C0 / ˛N N .x/ exp C 1 C N 2 K1 k˛N N exp¹f .t /=2qº 1k22 .C0 C t /1=q K2 K jf .t /j2 C 1=q .t C C0 / .t C C0 /.1C˛ /=q 8 2 ˆ 2; D 2; < 2:


5.2

Blow-up of solutions of nonlinear pseudoparabolic equations with sources of pseudo-Laplacian type

In this section, we consider initial-boundary-value problems for pseudoparabolic equations with the sources of pseudo-Laplacian type. For considered problems, the local and global unique solvability in the weakened sense is proved and sufficient conditions of the blow-up of solutions for a finite time are obtained. For certain problems, we obtain upper estimates for the blow-up time. We obtain sufficient, close to necessary, conditions of the blow-up of solutions of the following first initial-boundary-value problems: @ u D p u; @t

uj@ D 0;

u.x; 0/ D u0 .x/;

p 3;

(5.77)

@ u C u D p u; @t @ .u u/ D p u; @t @ u C uux1 D 4 u; @t

uj@ D 0;

u.x; 0/ D u0 .x/;

p 3;

(5.78)

uj@ D 0;

u.x; 0/ D u0 .x/;

p 3;

(5.79)

uj@ D 0;

u.x; 0/ D u0 .x/;

(5.80)

@ .u u/ C uux1 D 4 u; @t

uj@ D 0;

u.x; 0/ D u0 .x/;

(5.81)

476


where the surface-simply-connected bounded domain 2 RN has a smooth boundary of the class @ 2 C .2;ı/ , ı 2 .0; 1, x 2 , and the nonlinear operator p is defined by the expression p u div.jrujp2 ru/; p > 2: These problems describe quasi-stationary processes in semiconductors under the negative differential conductivity.

5.2.1 Blow-up of weakened solutions of problem (5.77) In this subsection, we prove the unique local-on-time solvability of the problem and the blow-up of its weakened solutions for a finite time. Meanwhile, we obtain an upper estimate of the blow-up time. Introduce some definitions. 1;p

Definition 5.2.1. A solution of the class u.x; t / 2 C .1/ .Œ0; TI W0 .// of problem (5.77) that satisfies the condition

@ 1;p 1;p u p u; w D 0 8w 2 W0 ./; t 2 Œ0; T; u.0/ D u0 2 W0 ./; @t 1;p

0

where h;i is the duality bracket between the Banach spaces W0 ./ and W 1;p ./, is called a strong generalized solution of problem (5.77). Definition 5.2.2. A strong generalized solution of problem (5.77) of the class u.x; t /2 C .1/ .Œ0; TI C0 ./ \ C .1/ .// is called a weakened generalized solution of problem (5.77). The following theorem holds. 1;p Theorem 5.2.3. For any u0 2 W0 ./, a strong generalized solution u.x; t / 2 C .1/ .Œ0; T00 /I W01;p .// blows up for a finite time T00 2 .0; T2 :

lim krukp D C1

t"T00

and T2 D

1 kru0 k22 : p 2 kru0 kpp

Moreover, T0 2 .0; T00 , where T0 is the existence time of a weakened solution of problem (5.77). 1;p

Proof. Let u.x; t / 2 C .1/ .Œ0; TI W0 .// be a local-on-time, strong generalized solution of problem (5.77). Multiplying both sides of Eq. (5.77) in the sense of

477

Section 5.2 Blow-up of pseudoparabolic equations with pseudo-Laplacian 0

the duality bracket between the Banach spaces W01;p ./ and W 1;p ./ (p 0 D p.p 1/1 ) by u.x; t / or by u0 .x; t /, after integrating by parts we obtain 1d kruk22 D krukpp ; 2 dt 1 d kru0 k22 D krukpp : p dt

(5.82) (5.83)

From (5.82), (5.83), and the Cauchy–Schwarz inequality we obtain the ordinary differential inequality for the function ˆ.t / kruk22 .t /: ˆˆ00 ˛.ˆ0 /2 0;

˛D

p ; 2

p 3:

(5.84)

Integrating (5.84), we obtain ˆ.t /

1 ; Œˆ1˛ .˛ 1/E0 t 1=.˛1/ 0

E0

p 2kru0 kp : kru0 kp 2

(5.85)

Since a weakened solution of problem (5.77) is a strong generalized solution, the existence time of a weakened solution is not greater than the existence time of a strong generalized solution. Theorem 5.2.3 is proved.

5.2.2 Blow-up and the global-on-time solvability of problem (5.78) In this subsection, we obtain a sufficient condition of the blow-up of a weakened solution of problem (5.78) and a sufficient condition of the global-on-time solvability. Prove the blow-up of a strong generalized solution of problem (5.78). The following theorem holds. Theorem 5.2.4. Under the condition kru0 k22 < kru0 kpp ;

p 3;

(5.86)

the limit relation lim sup jrx u.x; t /j D C1

t"T0 x2

holds and an upper estimate T0 2 Œ0; T2 is valid, where T0 is the existence time of a strong generalized solution of the problem kru0 k22 1 ln 1 T2 D p : p2 kru0 kp

(5.87)

478


Proof. Prove that any strong generalized solution of problem (5.78) existing locally 1;p on time blows up for a finite time. Let u.x; t / 2 C .1/ .Œ0; TI W0 .// for any T 2 .0; T0 / be a strong generalized solution of problem (5.78). Multiply both sides of Eq. (5.78) by u.x; t / or by u0 .x; t / in the sense of the duality bracket between the 0 1;p Banach spaces W0 ./ and W 1;p ./. After integrating by parts we obtain for the function w.x; t / D exp.t /u.x; t / the following equalities: 1d krwk22 D exp..p 2/t /krwkpp ; 2 dt 1 d krw 0 k22 D exp..p 2/t / krwkpp : p dt

(5.88) (5.89)

Introduce the function ˆ.t / krwk22 . From (5.88), (5.89), and the Cauchy–Schwarz inequality (see [293]) we obtain the second-order ordinary differential inequality 1 0 2 d 1 d exp..p 2/t / ˆ ; .ˆ / ˆ exp..p 2/t / 4 2p dt dt from which, in its turn, we have (see [210]) ˆˆ00

p 0 2 .ˆ / C .p 2/ˆˆ0 0; 2

Introduce the notation ƒ.t /

ˆ0 ; ˆ˛

˛

p 3:

(5.90)

p : 2

Then from (5.90) we obtain ƒ0 C .p 2/ƒ 0;

ƒ.t / ƒ.0/e .p2/t ;

p

ƒ.0/

2kru0 kp

:

(5.91)

p : 2

(5.92)

p

kru0 k2

From (5.91) we have ˆ.t /

1 Œˆ1˛ 0

.˛

1 1/ p2 ƒ.0/Œ1

e .p2/t 1=.˛1/

;

˛D

Now we require the validity of the condition ˆ1˛ < 0

˛1 ƒ.0/I p2

then from (5.92) we obtain that the strong generalized solution blows up for a finite time T0 , for which, by virtue of (5.86), the upper estimate T0 2 .0; T2 holds, where Eq. (5.87) holds for the time T2 . Thus, the blow-up for a finite time of a strong generalized solution of problem (5.78) is proved. Therefore, a weakened solution of problem (5.78) also blows up. Theorem 5.2.4 is proved.

479


5.2.3 Blow-up of solutions of problem (5.79) In this subsection, we prove that there exists a unique weakened solution of problem (5.79) that blows up for a finite time for any initial functions u0 .x/ from a certain class. First, consider the one-dimensional case. The following theorem holds. Theorem 5.2.5. For any u0 .x/ 2 C .1/ .Œ0; 1/ \ C0 .Œ0; 1/, there exists maximal T0 > 0 such that a unique weakened solution of problem (5.79) of the class u.x; t / 2 C .1/ .Œ0; TI C0 .Œ0; 1/ \ C .1/ .Œ0; 1// exists for any T 2 .0; T0 /, T0 < C1, and the following limit relation holds: sup jux .x; t /j D C1:

lim

t"T0 x2.0;1/

(5.93)

For the blow-up time, we have the upper estimate T0 2 .0; T2 ;

T2 D

1 ku0x k22 C ku0 k22 : p p2 ku0x kp

Proof. Just as in Subsec. 5.2.2, we can prove that in the weakened sense, problem (5.79) can be reduced to the following integro-differential equation: @u D @t

Z

where 1 G.x; y/ sinh.1/

1

0

´

dy Gy .x; y/juy jp2 uy ;

(5.94)

sinh.1 y/ sinh.x/; 0 x y; sinh.y/ sinh.1 x/; y x 1:

From (5.94) we obtain the integro-differential equation Z ux;t D K2 .x/jux jp2 ux C where

1 0

dy K1 .x; y/juy jp2 uy ;

(5.95)

´ 1 cosh.y/ cosh.1 x/; 0 y x; K1 .x; y/ sinh.1/ cosh.1 y/ cosh.x/; x y 1; K2 .x/

1 Œcosh.1 x/ sinh.x/ C cosh.x/ sinh.1 x/: sinh.1/

Introduce the notation v ux . Then from (5.95) we obtain the integral equation v D v0 C U1 .v/ C U2 .v/;

(5.96)

480


where Z U1 .v/

t

0

Z ds K2 .x/jvjp2 v;

U2 .v/

Z

t

1

ds 0

0

dy K1 .x; y/jvjp2 v: (5.97)

Introduce the Banach space ® B v.x; t / 2 C.Œ0; T Œ0; 1/ W kvkT

sup

¯ jv.x; t /j ;

.x;t/2QT

where QT .0; T/ .0; 1/. To prove the local-on-time solvability of the integral equation (5.96), we apply the method of contraction mappings. It is easy to prove that from the explicit form of the functions K1 .x; y/ and K2 .x/, we obtain that the operators defined by formula (5.97) act from B into B. In the space B, introduce a closed convex bounded subset BR ¹v 2 B W kvkT Rº : Prove that the operators defined by formula (5.97) act from BR into BR and they are contractions on BR for sufficiently small T > 0 and sufficiently large R > 0. Indeed, p1

kU1 .v/kT TC1 kvkT C1 sup jK2 .x/j; x2.0;1/

p1

kU2 .v/kT TC2 kvkT ; Z C2 sup dy jK1 .x; y/j: ;

x2.0;1/

Hence we obtain that if 0 0, and small T > 0, a unique solution of the integral equation (5.96) of the class v.x; t / 2 C.Œ0; T / exists for any T 2 .0; T0 / and either T0 D C1 or T0 < C1, and in the latter case, lim sup jv.x; t /j D C1:

t"T0 x2

481


It follows from Eq. (5.96) that v.x; t / 2 C .1/ .Œ0; TI C.Œ0; 1// for any T 2 .0; T0 /. Moreover, from the definition v.x; t / D ux .x; t / and the conditions u.0; t / D u.1; t / D 0 we obtain that Z 1 Z x dy v.y; t /; u.x; t / D dy v.y; t / u.x; t / D 0

x

and, therefore, a unique solution of problem (5.79) of the class u.x; t / 2 C .1/ .Œ0; TI C0 .Œ0; 1/ \ C .1/ .Œ0; 1// exists for any T 2 .0; T0 / and either T0 D C1 or T0 < C1, and then the limit relation (5.93) holds. Prove that T0 < C1. Assume that u.x; t / is a strong generalized solution of 1;p problem (5.79) of the class u.x; t / 2 C .1/ .Œ0; TI W0 .0; 1//. Multiply both sides 0 in the sense of the duality bracket between the Banach spaces of (5.79) by u and u 0 1;p W0 .0; 1/ and W 1;p .0; 1/. Then we obtain the following energy equalities: 1d Œkux k22 C kuk22 D kux kpp ; 2 dt 1 d ku0x;t k22 C ku0t k22 D kux kpp : p dt

(5.98) (5.99)

From (5.98), (5.99), and the Cauchy–Schwarz inequality (see [293]) we obtain the second-order ordinary differential inequality for the function ˆ.t / kux k22 C kuk22 : p (5.100) ˆˆ00 .ˆ0 /2 0: 2 Integrating (5.100), we obtain ˆ.t /

Œˆ1˛ 0

1 ; .˛ 1/E0 t 1=.˛1/

p

E0

2kru0 kp p : kru0 k2

(5.101)

From (5.101) we obtain the statement of the theorem. On the other hand, since any weakened solution of problem (5.79) is a strong generalized solution, the existence time of a weakened solution is not greater than the existence time of a strong generalized solution. Theorem 5.2.5 is proved. Now consider the multidimensional case. Theorem 5.2.6. There exists T0 0 such that the following limit relation holds: lim sup jrx u.x; t /j D C1:

(5.102)

t"T0 x2

Moreover, for the blow-up time, the following upper estimate holds: T0 2 Œ0; T2 ;

T2 D

1 kru0 k22 C ku0 k22 ; p p2 kru0 kp

p 3:

(5.103)

482


5.2.4 Blow-up of weakened solutions of problems (5.80) and (5.81) In this subsection, we prove that problems (5.80) and (5.81) are uniquely solvable in the weakened sense and their weakened solutions blow up for a finite time. The following theorem holds. Theorem 5.2.7. Let a weakened solution of problem (5.80) of the class u.x; t / 2 C .1/ .Œ0; TI C0 ./ \ C .1/ .// exists for any T 2 .0; T0 /. If C41 ./kru0 k22 < kru0 k44 ;

(5.104)

lim sup jrx uj D C1

(5.105)

then the limit relation t"T0 x2

holds and we have an upper bound T0 2 .0; T1 , where kru0 k22 1 T1 D 4 ln 1 C41 ; kru0 k44 C1 C1 is the best constant of the embedding W01;4 ./ in L4 ./. Proof. Let u.x; t / be a strong generalized solution of problem (5.80) of the class u.x; t / 2 C .1/ .Œ0; TI W01;4 .//. Multiply Eq. (5.80) by u and u0 in the sense of the duality bracket between the Banach spaces W01;4 ./ and W 1;4=3 ./. Integrating by parts we obtain the following energy equalities: 1 d kruk22 D kruk44 ; 2 dt Z 1 d 1 0 2 4 kru k2 D dy u2 u0x1 t : kruk4 4 dt 2

(5.106) (5.107)

From (5.106) and (5.107) we obtain the following inequalities: 1d kruk44 C 4 dt 1d kruk44 C 4 dt

1d " kru0 k22 kruk44 C 1 4 4 dt kru0 k22

" kru0 k22 C 4 " kru0 k22 C 4 C41 kruk44 : 4"

1 kuk44 4" C41 kruk44 ; 4" (5.108)

From (5.106)–(5.108) and the Cauchy–Schwarz inequality for the function ˆ.t / kruk22 we obtain the following second-order ordinary differential inequality (see [210]): ˆˆ00 ˛1 .ˆ0 /2 C ˇ1 ˆˆ0 0;

(5.109)

483


where

h "i ; ˛1 D 2 1 4

ˇ1 D

C41 : "

Inequality (5.109) has the same form as inequality (5.90). Therefore, similarly integrating, we obtain ˆ.t /

1 1 Œˆ1˛ 0

.˛1

1/ˇ11 ƒ0 Œ1

where ƒ0

exp.ˇ1 t /1=.˛1 1/

;

(5.110)

ˆ0 .0/ : ˆ˛0 1

We require the validity of the condition ˆ0
0 such that a unique classical solution of problem (5.112) exists and T0 2 ŒT1 ; T1 , lim sup ju.x; t /j D C1: t"T0 x2

where

R

exp¹u0 .x/º ; dx dy G.x; y/ exp¹u0 .x/ C u0 .y/º

T1 ’ T1 D

dx

1 C1 ./ supx2 exp.u0 .x//

;

C1 ./ D sup

Z

(5.115) dy G.x; y/;

x2

G.x; y/ is the Green function of the first boundary-value problem for the operator C I.

486


Proof. For classical solutions, problem (5.112) is equivalent to the following double integral equation: Z t Z ds dy G.x; y/ exp¹u.y; t /º; (5.116) u.x; t / D u0 .x/ C 0

where G.x; y/ is the Green function of the first boundary-value problem for the operator C I. Introduce the operator Z t Z ds dy G.x; y/ exp¹u.y; t /º: A.u/ u0 .x/ C 0

By the properties of the Green function, we have that A.u/ W C.Œ0; TI C0 .// ! C .1/ .Œ0; TI C .1/ ./ \ C0 .//:

(5.117)

Introduce the Banach space B C.Œ0; TI C0 .// and its closed, convex, bounded subset BR ¹v.x; t / 2 B W kvkT sup ju.x; t /j Rº: .x;t/2Q

In the standard way, we can prove that the operator A.u/ acts from BR into BR and is a contraction on BR . Therefore, there exists sufficiently small T > 0 such that there exists unique u.x; t / 2 C.Œ0; TI C0 .//. Now, applying the well-known extensionin-time algorithm of solutions to the integral equation (5.116), we obtain that there exists maximal T0 > 0 such that a unique solution of the integral equation (5.116) exists for any T 2 .0; T0 / and either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim sup ju.x; t /j D C1

t"T0 x2

(5.118)

holds. Note that by virtue of (5.116) and (5.117) u.x; t / 2 C .1/ .Œ0; TI C .1/ ./ \ C0 .//: On the other hand, A.u/ W C .1/ .Œ0; TI C .1/ ./ \ C0 .// ! C .1/ .Œ0; TI C .2/ ./ \ C .1/ ./ \ C0 .//: Thus, the first part of the theorem is proved. In the classical sense, the integral equation (5.116) is equivalent to the following Cauchy problem for the integro-differential equation: Z @u D dy G.x; y/ exp.u.y; t //; u.x; 0/ D u0 .x/: (5.119) @t

Section 5.3 Blow-up in pseudoparabolic equations with fast increasing nonlinearities

487

Multiply both sides of Eq. (5.119) by exp.u/ and integrate over the domain ; we obtain “ Z d dx exp.u/ D dx dy G.x; y/ exp.u.y; t / C u.x; t // F.t /: (5.120) dt Now multiply both sides of (5.119) by .exp.u//0 and obtain Z 1d dx exp.u/.u0 /2 D F: 2 dt

(5.121)

The following inequality holds: ˇ2 Z ˇ Z Z ˇ ˇd 0 2 ˇ ˇ dx exp.u/ dx exp.u/.u / dx exp.u/: ˇ ˇ dt Introduce the notation

(5.122)

Z ˆ.t /

dx exp.u/:

(5.123)

Then from (5.120)–(5.123) we obtain the following second-order ordinary differential inequality: 0 2 1 00 ˆ ˆ ˆ; 2

(5.124)

that we can integrate owing to (5.119), (5.120), and (5.124); we obtain ˆ.t / ’

where E0

dx dy

ˆ1 0

1 ; E0 t

G.x; y/ exp¹u0 .x/ C u0 .y/º ˆ20

(5.125)

;

ˆ0 ˆ.0/:

From (5.125) we obtain the existence of T0 T1 , R dx exp¹u0 .x/º ; T1 ’ dx dy G.x; y/ exp¹u0 .x/ C u0 .y/º such that lim ˆ.t / D C1:

t"T 0

(5.126)

From the explicit form of the function ˆ.t / defined by Eq. (5.123) and from (5.126) we obtain that before the time moment T0 , there exists T0 , positive by virtue of the

488


first part of the theorem, such that at a certain point x 2 , the module of the solution ju.x ; t /j becomes infinite. Now multiply Eqs. (5.119) by exp.u/; we obtain Z @v dy G.x; y/v; v exp.u/ Dv @t and hance

Z

w.t / w.0/ C C1 ./

t

ds w 2 .s/;

w.t / D sup j exp.u.x; t //j:

0

x2

Hence we conclude the validity of relations (5.115): w.t / T1 D

w.0/ ; 1 C1 w.0/t

1 ; C1 ./ supx2 exp.u0 .x//

t 2 .0; T1 /; Z dy G.x; y/: C1 ./ D sup x2

From (5.116), (5.123), and (5.125) we obtain that for a finite time 0 T00 < T0 , the solution becomes nonnegative. Thus, the second part of the theorem is proved. Theorem 5.3.2 is proved. Note that for a positive solution of problem (5.112), the Fourier method of unbounded coefficients can be applied (see, e.g., [216]). Lemma 5.3.3. Let u.x; t / be a classical solution of problem (5.112) with an initial function u0 .x/ 2 C .2/ ./ \ C0 ./ \ C .1/ ./. Then 0 < T0 T2 , where Z D dx '1 .y/u0 .y/; T2 .1 C 1/ exp. 0 /; 0

'1 .y/ is the first eigenfunction of the first boundary-value problem for the Laplace operator, 1 if the first eigenvalue, and Z dy '1 .y/ D 1:

Proof. Let u.x; t / be a classical solution of problem (5.112). Since u0 .x/ 0, from the integral equation (5.116) we obtain that u.x; t / 0. Then, multiplying both sides of Eq. (5.112) by '1 , after integrating by parts we obtain Z d dx '1 .x/ exp.u.y; t // exp. /; D .1 C 1/ dt (5.127) Z .t / dx '1 .y/u.y; t /;


489

where we have used the Jensen inequality for convex functions (see, e.g., [445]). Integrating (5.127), we obtain exp. /

1 : exp. 0 / .1 C 1/1 t

(5.128)

From (5.128) we obtain the statement of the lemma. Theorem 5.3.4. For any u0 .x/ 2 C .2/ ./ \ C .1/ ./ \ C0 ./, there exists maximal T0 > 0 such that a unique classical solution of problem (5.113) exists and T0 2 ŒT3 ; T3 , lim sup ju.x; t /j D C1:

t"T 0 x2

where T3

R

dx

’

exp¹u20 .x/º

; G.x; y/u0 .x/u0 .y/ exp¹u20 .x/ C u20 .y/º Z 1 dy G.x; y/; ; C1 ./ D sup T3 D 4C1 ./ supx2 exp.2u20 .x// x2 2

dx dy

(5.129)

(5.130)

G.x; y/ is the Green function of the first boundary-value problem for the operator C I. Proof. We need to prove only the second part of the theorem since the first part is proved similarly to the first part of Theorem 5.3.2. Thus, there exists maximal T0 > 0 such that a unique classical solution of problem (5.113) exists. Now we prove that T0 T3 < C1. Consider the following Cauchy problem for the integro-differential equation equivalent to (5.113): Z @u dy G.x; y/u.y; t / exp.u2 .y; t //; u.x; 0/ D u0 .x/: (5.131) D @t Now multiply both sides of Eq. (5.131) by u exp.u2 / and integrate over the domain ; we obtain Z 1 d dx exp.u2 .x; t // 2 dt “ DF dx dy G.x; y/ exp.u2 .y; t / C u2 .x; t //u.y; t /u.x; t /: (5.132)

0 Multiply both sides of Eq. (5.131) by u exp.u2 / and integrate over the domain ; we obtain Z 1 dF : (5.133) dx Œ.u0 /2 exp.u2 / C 2u2 .u0 /2 exp.u2 / D 2 dt

490


From (5.133) we obtain the inequality Z 1 dF dx .u0 /2 u2 exp.u2 / : 4 dt Note that ˇ2 Z ˇZ Z ˇ ˇ 2 ˇ dy u exp.u2 /u0 ˇ dy exp.u / dy u2 .u0 /2 exp.u2 /: ˇ ˇ


(5.134)

(5.135)

Z ˆ.t /

dx exp.u2 /:

(5.136)

From (5.132) and (5.134)–(5.136) we obtain the second-order ordinary differential inequality ˆˆ00 2.ˆ0 /2 0:

(5.137)

Integrating (5.137), we obtain ˆ

ˆ1 0

1 ; E0 t

where ˆ0 D ˆ.0/;

E0

(5.138)

2F.0/ : ˆ20

From (5.138) we obtain (5.129) and (5.130). Now obtain the lower estimate for the blow-up time of solutions of problem (5.113). Multiply both sides of Eq. (5.131) by u exp.u2 /; we obtain Z 1 @ exp.u2 / 2 D u exp.u / dy G.x; y/u exp.u2 /; 2 @t Z Z t 2 2 2 exp.u / exp.u0 / C 2 ds juj exp u dy G.x; y/juj exp.u2 / 0

Z exp.u20 / C 2

t

Z

ds exp.3u2 =2/

0

dy G.x; y/ exp.3u2 =2/; (5.139)

for u 0. From (5.139) we obtain where we have used the fact that u exp.u the inequality Z t w w0 C 2C1 ./ ds w 3 .s/; w.t / sup j exp.u2 /j; 0 x2 (5.140) Z dy G.x; y/: C1 ./ D sup 2 =2/

x2


491

From (5.140) we obtain the inequality w

w0 : Œ1 4w02 C1 t 1=2

(5.141)

From (5.141) we obtain the lower estimate for the blow-up rate of solutions of problem (5.113). Theorem 5.3.4 is proved. For generality, consider the problem that also includes problem (5.112): @ .u u/ C @t where the function

.u/ D 0;

u.x; 0/ D u0 .x/;

uj@ D 0;

(5.142)

.u/ satisfies the following conditions:

C .1/ .R1 /;

(1)

.s/ 2

(2)

.s/ 0 and is convex;

(3) for the function u0 .x/, we have Z dy w1 .y/u0 .y/ > 0I ˆ0

(5.143)

(4)

Z R1

(5)

ds < C1I .s/

.s/ is an increasing function, where w1 .x/ is the first eigenfunction of the first boundary-value problem for the operator C I.

The following theorem holds. Theorem 5.3.5. For any function u0 .x/ 2 C .2/ ./ \ C .1/ ./ \ C0 ./, there exists maximal T 0 > 0 such that a unique classical solution of problem (5.113) exists, and 2 ŒT ; T 0 4 T4 , lim sup ju.x; t /j D C1;

t"T 0 x2

Z

where T4 .1 C 1 /

R1

(5.144)

ds ; .s/

and T4 > 0 is the existence time of a classical solution of the following Cauchy problem for the ordinary differential equation: Z dZ dy G.x; y/; (5.145) D C1 .Z/; Z.0/ D max ju0 .x/j; C1 sup dt x2 x2 where G.x; y/ is the Green function of the first boundary-value problem for the operator C I.

492


Proof. We need to prove only the second part of the theorem since the uniqueness and existence of a solution of problem (5.142) can be obtained by the method of contraction mappings. In the classical sense, problem (5.142) is equivalent to the following Cauchy problem for the integro-differential equation: Z @u dy G.x; y/ .u/.y; t /; u.x; 0/ D u0 .x/: (5.146) D @t Multiply both sides by the first eigenfunction w1 .x/ 0 of the first boundary-value problem for eigenvalues of the operator C I. Let 1 be the first eigenvalue of the first boundary-value problem for the Laplace operator. Moreover, let Z dxw1 .x/ D 1:

Then we obtain dˆ 1 D dt 1 C 1

Z

dy w1 .y/ .u/

1 .ˆ.t //: 1 C 1

From the conditions (2)–(4) of the theorem we obtain Z 1 t ds : .s/ 1 C 1 R1 Hence we directly obtain the limit relation (5.144). From (5.143), (5.146), and the condition (5) we obtain the following integral inequality: Z v.t / v.0/ C C1

t

ds 0

.v/;

v.t / sup ju.x; t /j; x2

from which, in its turn, we obtain (5.145). Theorem 5.3.5 is proved. Remark 5.3.6. Note that in the case .u/ D exp.u/, the condition (3) is not necessary. Therefore, the result of Theorem 5.3.5 holds also in this case.

5.3.2 Local solvability and blow-up for a finite time of solutions of problem (5.114) In this subsection, we consider the local solvability of problem (5.114) in the so-called weakened sense. We also prove that weakened solutions of problem (5.114) blow up for a finite time. Definition 5.3.7. A strong generalized solution of the class u.x; t / 2 C .1/ .Œ0; TI C .1/ ./ \ C0 .// is called a weakened solution of problem (5.114).


493

Definition 5.3.8. A solution of the class u.x; t / 2 C .1/ .Œ0; TI H01 .// satisfying the condition 2 hu0 u0 .ue u /; wi D 0 8w 2 H01 ./ for all t 2 Œ0; T, u.0/ D u0 2 H01 ./, where the operator is an extension of the Laplace operator to the class H01 ./ corresponding to an isometric isomorphism between the Hilbert spaces H01 ./ and H1 ./, is called a strong generalized solution of problem (5.114). Now we reduce problem (5.114) to an equivalent problem in the weakened sense. Consider problem (5.114). Since for the operator A C I W H01 ./ ! H1 ./, a Lipschitz-continuous inverse operator A1 exists, in the weakened sense, problem (5.114) is equivalent to the following problem: @u 2 2 D ue u A1 .ue u /; @t

u.x; 0/ D u0 .x/ 2 C 1 ./ \ C0 ./:

(5.147)

Indeed, in the class u 2 C .1/ .Œ0; TI C .1/ ./ \ C0 .// from Definition 5.3.8 of a strong generalized solution, we obtain the equality 2

2

A.u0 ue u / C ue u D 0; which is meant in the sense C.Œ0; TI H1 .//, as far as u0 2 C.Œ0; TI C .1/ ./ \ C0 .// C.Œ0; TI H01 .//; 2

ue u 2 C .1/ .Œ0; TI C .1/ ./ \ C0 .// C .1/ .Œ0; TI H01 .// C .1/ .Œ0; TI H1 .//: Since for the operator A W H01 ./ ! H1 ./, a Lipschitz-continuous inverse operator with Lipschitz constant equal to 1 is defined, we obtain Eq. (5.147). Note that the restriction of the operator A1 to the class v.x/ 2 C./ coincides with the operator Z Gv

dy G.x; y/v.y/;

where G.x; y/ is the Green function of the first boundary-value problem for the operator C I. Therefore, in the weakened sense, problem (5.147) coincides with the following problem: Z @u 2 2 D ue u dy G.x; y/ue u ; u.x; 0/ D u0 .x/ 2 C .1/ ./ \ C0 ./: @t (5.148)

494


The following theorem holds. Theorem 5.3.9. For any u0 2 C .1/ ./ \ C0 ./, there exists maximal T0 > 0 such that a weakened solution of problem (5.114) of the class u.x; t / 2 C .1/ .Œ0; TI C .1/ ./ \ C0 .// exists for any T 2 .0; T0 /. Here T0 2 .0; T2 and lim sup jrx u.x; t /j D C1;

t"T0 x2

(5.149)

where

R 2 2 1 dy u0 .y/ exp.2u0 .y// : T2 R ’ 2 2u20 .y/ u20 .x/Cu20 .y/ 2 dy u0 .y/e dx dy G.x; y/u0 .x/u0 .y/e

Proof. First, consider problem (5.148), which, in the weakened sense, is equivalent to the following integral equation: Z t Z t Z 2 2 ds ue u ds dy G.x; y/ue u : (5.150) u.x; t / D u0 .x/ C 0

0

To prove the local-on-time unique solvability of Eq. (5.150) in the class C.Œ0; TI C .1/ ./\C0 .//, we use the method of contraction mappings. Note that Eq. (5.150) is close to Eq. (5.117). Therefore, arguing as in the proof of Theorem 5.3.2, in the method of contraction mappings, we take as the Banach space the following space: B L1 .0; TI C .1/ ./ \ C0 .// with the norm kukT sup jrx u.x; t /j: x2 t2Œ0;T

As a result, we obtain that there exists a unique local-on-time solution of the integral equation (5.150) of the class L1 .0; TI C .1/ ./ \ C./. Further, using the standard extension-in-time algorithm of a solution of problem (5.150), we obtain that there exists maximal T0 > 0 such that a unique solution of problem (5.150) of the class u.x; t / 2 L1 .0; TI C .1/ ./ \ C0 .// exists for any T 2 .0; T0 / and either T0 D C1 or T0 < C1, and in the latter case, the limit relation (5.149) holds. Thus, the first part of the theorem is proved. Now prove the second part. Consider the integro-differential equation (5.150). Assume that u.x; t / 2 C .1/ .Œ0; TI C0 ./ \ C .1/ .// is a locally existing weakened


495

solution of problem (5.150), which really exists by virtue of the first part of the theorem. Now we multiply both sides of Eq. (5.148) by u exp.u2 / and integrate over the domain . As a result we obtain Z Z 1d u2 dy e D dx u2 exp.2u2 / (5.151) 2 dt “ dx dy G.x; y/u.x; t / exp.u2 .x; t //u.y; t / exp.u2 .y; t //: 2

Now multiply both sides of Eq. (5.148) by .ue u /0 and integrate over the domain ; we obtain Z 1d 2 2 dxŒ.u0 /2 e u C 2.u0 /2 u2 e u D F; (5.152) 2 dt where

Z F

dx u2 exp.2u2 /

“ dx dy G.x; y/u.x; t / exp.u2 .x; t //u.y; t / exp.u2 .y; t //:

Consider the condition F.0/ > 0, under which (5.152) implies that F.t / > 0. Introduce the notation v D u exp.u2 /. By definition, a weakened solution belongs to the class C .1/ .Œ0; TI C .1/ ./ \ C0 .//; therefore, v 2 C .1/ .Œ0; TI H01 .//. We can represent v as a series on eigenfunctions of the first boundary-value problem for the Laplace operator in the domain : vD

C1 X

ck .t /wk ;

wk C k wk D 0;

wk j@ D 0;

kD1

kwk k2 D 1;

.wk1 ; wk2 /2 D ık1 k2 :

Substitute this series for v in the condition “ Z F dx u2 exp.2u2 / dx dy G.x; y/u.x/ exp.u2 .x//u.y/ exp.u2 .y// > 0I

after an obvious transformation, we obtain the inequality C1 X kD1

jck .t /j2

C1 X

jck .t /j2

kD1

1 > 0; 1 C k

which is valid except for the trivial case where u0 D 0. The energy equalities (5.151) and (5.152) imply the second-order ordinary differential equality Z 00 0 2 ˆˆ 2.ˆ / 0; ˆ dx exp.u2 .x; t //:

496


After integrating we obtain ˆ

1 ; 1 ˆ0 E0 t

E0

ˆ0 .0/ : ˆ20

(5.153)

It follows from (5.153) that T2 is equal to R 2 2 1 dy u0 .y/ exp.2u0 .y// : T2 R ’ 2 2 2 2 dy u2 .y/e 2u0 .y/ dx dy G.x; y/u0 .x/u0 .y/e u0 .x/Cu0 .y/

0


5.4

Blow-up of solutions of nonhomogeneous nonlinear pseudoparabolic equations

In this section, we consider the first initial-boundary-value problem for nonhomogeneous nonlinear pseudoparabolic equations. Sufficient conditions of the blow-up of solutions of the considered problem are obtained. We prove the local and global solvability of the following initial-boundary-value problem: @ .u u/ C u3 D f .x/; @t u.x; 0/ D u0 .x/; uj@ D 0;

(5.154) (5.155)

where RN is a bounded domain with smooth boundary C.2;ı/ , ı 2 .0; 1, N 1. This problem describes quasi-stationary processes in semiconductors in the presence of a stationary distribution of sources of current of free charges.

5.4.1 Unique local solvability of the problem Introduce the definitions and assumptions necessary for further consideration and obtain some auxiliary results. Consider the Hilbert space ¹H01 ./; k kº and its dual ¹H1 ./; k k º. Denote the duality bracket between the Hilbert spaces H01 ./ and its dual one H1 ./ by h; i (see, e.g., [347]). Further, we use the notation k kC.Œ0;T IV / k kT ;

k kC.Œ0;T IV / k kT :

Note that the operator A C I acts from H01 ./ into H1 ./ and is radially continuous (see, e.g., [168]). Hence we obtain hAh1 Ah2 ; h1 h2 i hh1 h2 ; h1 h2 i D kh1 h2 k2 ; i.e., the operator A W H01 ./ ! H1 ./ is strongly monotonic.

(5.156)

497

Section 5.4 Blow-up of solutions of nonhomogeneous nonlinear equations

It follows from (5.156) that for the operator A, by [168, Corollary 2.3, p. 97], an inverse operator A1 W H1 ./ ! H01 ./ is defined and Lipschitz-continuous: kA1 x A1 yk kx yk

8x; y 2 H1 ./:

(5.157)

Consider the operator F.u/ u3 : Prove that the operator F W H01 ./ ! H1 ./ is boundedly Lipschitz-continuous. Note that kF.u/ F.v/k ku3 v 3 k : The following inequalities hold: ku3 v 3 k4=3 3kg.u v/k4=3 ; kg.u v/kp kgkr1 p ku vkr2 p ;

g D max.juj2 ; jvj2 /; p D 4=3;

r2 p D 3;

3 r1 D ; r1 p D 2; 2 kjuj2 u jvj2 vk4=3 3kgk2 ku vk4 ; r2 D 3;

kgk2 2.max¹kuk4 ; kvk4 º/2 : Thus, under the condition N 4 we have kjuj2 u jvj2 vk .R/ku vk;

.R/ D c1 R2 ;

R D max¹kuk; kvkº:

Therefore, kF.u/ F.v/k .R/ku vk;

.R/ D c1 R2 ;

N 4:

(5.158)

The following theorem holds. Theorem 5.4.1. For any u0 2 H01 ./, N 4, and f 2 L2 ./, there exists Tu0 > 0 such that for any 0 < T < Tu0 , problem (5.154)–(5.155) has a unique solution of the class C .1/ .Œ0; TI H01 .//. Proof. Consider the following problem equivalent to (5.154), (5.155) in the class u.x; t / 2 C .1/ .Œ0; TI H01 .//: @u.t /=@t D G.u/.t /; 8t 2 Œ0; T;

u.x; 0/ D u0 .x/ 2 H01 ./

u 2 C .1/ .Œ0; TI H01 .//;

where G.u/ D A1 .F.u/.t // A1 f by virtue of (5.157).

(5.159)

498


Note that (5.156) and (5.158) imply the inequalities lim kG.u/.t / G.u/.t0 /k lim kF.u/.t / F.u/.t0 /k

t!t0

t!t0

.R/ lim ku.t / u.t0 /k D 0; t!t0

R D max¹ku.t0 /k; ku.t /kº:

Therefore, G W C.Œ0; TI H01 .// ! C.Œ0; TI H01 .//. Introduce the set BR D ¹u.t / 2 C.Œ0; TI H01 .// W kukT Rº: Prove that the operator

Z

U.u/ u0 C

0

t

ds G.u/.s/;

t 2 Œ0; T;

acts from BR into BR and is a contraction on BR . Indeed, since G W C.Œ0; TI H01 .// ! C.Œ0; TI H01 .//; obviously, U W C.Œ0; TI H01 .// ! C .1/ .Œ0; TI H01 .//: On the other hand,

Z

kU.u/kT ku0 k C

T 0

dskG.u/k.s/ ku0 k C T sup kG.u/k.t /; t2Œ0;T

kG.u/k.t / c1 kuk3 C kf k: Therefore, kUkT ku0 k C Tc1 R3 C Tkf k: Let ku0 k R=3 and R > 0 be selected so large that kf k R=3 and T < 1=.3c1 R2 /; then for sufficiently large R > 0 and sufficiently small T > 0 we have kU kT R: Thus, U W BR ! BR . Prove that the operator U is a contraction operator on BR . Let u; v 2 BR ; then by virtue of (5.157) and (5.158) we have Z T Z T kU.u/ U.v/kT dt kG.u/ G.v/k dt kF.u/ F.v/k .t / 0

0

2

Tc1 R ku vkT : From the latter inequality for T < .2c1 R2 /1 we obtain kU.u/ U.v/kT < 1=2ku vkT : Thus, the operator U is a contraction on BR . Therefore, by virtue of the theorem on contraction mappings, for any u0 2 H01 ./, there exists Tu0 > 0 such that a unique solution of problem (5.159) exists. Theorem 5.4.1 is proved.


499

5.4.2 Blow-up of strong generalized solutions of problem (5.154)–(5.155) Introduce the notation E.t / kruk22 C kuk22 ;

E0 E.0/:

The following theorem holds. Theorem 5.4.2. Let the conditions of Theorem 5.4.1 hold. Moreover. assume that the following conditions are valid: 1 ku0 k44 > 4

Z

dx f .x/u0 ;

Z

1 dx f .x/u0 .x/ C p kf k2 Œkru0 k22 C ku0 k22 1=2 : 3

ku0 k44 >

Then Tu0 2 .0; T1 and the limit relation lim E.t / D C1

t"Tu0

holds, where T1 Z0 A1 ; A Œ.Z00 /2

1 Z0 E1˛ ; 0

21 1 0 Z0 1C˛2 1=2 ; Z00 D .1 ˛1 /E˛ 0 E0 ; ˛2 C 1 Z E00 ku0 k44 > dx f .x/u0 ;

˛1 D

5 ; 4

ˇ1 D 3kf k22 ;

1 D

3 kf k22 ; 4

˛2 D 5:

Proof. Multiply Eq. (5.154) by u or by u0t in the sense of the duality bracket h; i between the Hilbert spaces H01 ./ and H1 ./. Then owing to the result of Theorem 5.4.1 and condition (5.155), integrating by parts, we obtain 1d Œkruk22 C kuk22 D kuk44 2 dt kru0 k22 C ku0 k22 D

Z dx f .x/u;

1d kuk44 4 dt

Z

dx f .x/u0 :

(5.160) (5.161)

500


Consider Eq. (5.161). Due to Eq. (5.160), the following inequalities hold: Z Z 1 00 1 0 2 0 2 0 kru k2 C ku k2 E C dx f .x/u dx f .x/u0 8 4 1 3 " 0 2 1 E00 C ku k2 C kf k22 8 4 2 2" 1 3" 3 E00 C Œkru0 k22 C ku0 k22 C kf k22 : 8 8 8" Since, by definition, 1 ku0 k44 > 4 from (5.161) we obtain 1 kuk44 > 4

(5.162)

Z

dx f .x/u0 ;

Z dx f .x/u;

and from Eq. (5.160) we conclude that E0 .t / 0:

(5.163)

By the Cauchy–Schwarz inequality we obtain the following inequality: 1 0 2 .E / EŒkru0 k22 C ku0 k22 : 4

(5.164)

From (5.162) and (5.164) we obtain the second-order ordinary differential inequality 1 00 3 1 3" 0 2 2 1 .E / E E C kf k2 ; 4 8 8 8" which we rewrite for convenience in the following form: EE00 ˛1 .E0 /2 C ˇ1 E 0; where

3" ; ˛1 2 1 8

3 ˇ1 kf k22 ; "

(5.165)

˛1 > 1;

where " 2 .0; 4=3/. From (5.165) we obtain the inequality 1 .E1˛1 /00 C ˇ1 E˛1 0: 1 ˛1

(5.166)

Z.t / E1˛1 :

(5.167)


501


Owing to notation (5.167) from the inequality (5.166) we obtain Z00 C 1 Z˛2 0; where ˛2

˛1 ; ˛1 1

(5.168)

1 .˛1 1/ˇ1 :

Note that Z0 D .1 ˛1 /E˛1 E0 0 by virtue of (5.163). Now multiply both sides of inequality (5.168) by Z0 ; we obtain the inequality .Z0 /2 A2 C

21 1C˛2 ; Z ˛2 C 1

A2 .Z00 /2

21 Z0 1C˛2 > 0: ˛2 C 1

(5.169)

Now we obtain the optimal condition for A2 > 0. From (5.169), (5.167), and (5.160) we obtain 2 Z 2.˛1 1/2 ˇ1 2˛1 C1 4 1 k dx f .x/u > : E0 ku .1 ˛1 /2 4E2˛ 0 4 0 0 2˛1 1 Hence 2 Z 4 ku0 k44 dx f .x/u0 >

2ˇ1 E0 : 2˛1 1

(5.170)

Consider the function Q."/ D

2kf k22 2ˇ1 D : 2˛1 1 3".1 "=2/

The minimum of this function is reached at the point " D 1. From (5.170) we obtain Z ku0 k44 >

1 dx f .x/u0 .x/ C p kf k2 Œkru0 k22 C ku0 k22 1=2 : 3

(5.171)

From inequality (5.169) with "0 D 1 we obtain the following inequalities: Z0 A;

Z.t / Z0 At:

From (5.171) and (5.172) we obtain the statement of the theorem.

(5.172)

502


5.4.3 Blow-up of classical solutions of problem (5.154)–(5.155) In this subsection we prove that problem (5.154), (5.155) is uniquely solvable in the classical sense under certain conditions for the initial function u0 .x/ and the function f .x/ and the solution exists locally on time. The following theorem holds. Theorem 5.4.3. Let u0 .x/ 2 C0 ./ \ C .1/ ./ \ C .2/ ./, f .x/ 2 C0 ./ \ C .1/ ./. Then there exists maximal T00 such that a unique classical solution of problem (5.154), (5.155) of the class u.x; t / 2 C .1/ .Œ0; TI C0 ./ \ C .2/ ./ \ C .1/ .// exists for any T 2 .0; T00 / and T00 2 ŒT2 ; T1 in the case where the conditions for the initial function Z 1 dx f .x/u0 ; ku0 k44 > 4 Z 1 dx f .x/u0 .x/ C p kf k2 Œkru0 k22 C ku0 k22 1=2 ; ku0 k44 > 3 where T1 is defined in Theorem 5.4.2, and T2 is the first (in the increasing order) root of the cubic equation 2 T2 2C1 sup ju0 j C T2 C1 sup jf .x/j D 1: x2

x2

Here the limit relation lim sup ju.x; t /j D C1

(5.173)

t"T00 x2

holds. Proof. In the classical sense, problem (5.154), (5.155) is equivalent to the integrodifferential equation Z u.x; t / D u0 .x/ t

Z

dy G.x; y/f .y/ C

Z

t

dy G.x; y/u3 :

ds 0

(5.174)

It is easy to prove by the method of contraction mappings that a unique solution of Eq. (5.174) of the class u.x; t / 2 C.Œ0; TI C0 .// exists for any T 2 .0; T00 /. Now prove that T00 < C1 and thus the limit relation (5.173) holds. Indeed, from the integral equation (5.174) we obtain that u.x; t / 2 C .1/ .Œ0; TI C0 ./ \ C .2/ ./ \ C .1/ .//

503

Section 5.5 Blow-up of solutions of a nonlinear nonlocal equation

for any T 2 .0; T00 /. Now assume that T00 D C1, but then u.x; t / is a globalon-time, strong generalized solution; we obtain a contradiction with Theorem 5.4.2. Therefore, T00 < C1 and, moreover, T00 T1 . To obtain a lower estimate of the blow-up time, we consider the integral equation (5.174), which implies the integral inequality Z t ds w 3 .s/; (5.175) w.t / v0 .T/ C C1 0

where

Z

v0 sup ju0 j C TC1 sup jf j; x2

w.t / sup ju.x; t /j;

x2

x2

C1 D sup

dy G.x; y/:

x2

From (5.175) subject to the Gronwall–Bellman and Bihari theorem we obtain the following upper estimate: w.t /

v0 .T/ ; Œ1 2C1 v02 .T/t 1=2

t 2 Œ0; T:

(5.176)

From (5.176) we obtain the lower estimate of the blow-up time. Theorem 5.4.3 is proved.

5.5

Blow-up of solutions of a nonlinear nonlocal pseudoparabolic equation

In this section, we consider the first initial-boundary-value problem for one nonlinear, nonlocal pseudoparabolic equation. The local unique solvability of the problem is proved. Under certain conditions for nonlinearities, the solvability in any finite cylinder is proved. Under other conditions, the blow-up of solutions for a finite time is proved. Two-sided estimates for the blow-up time are obtained together with optimal two-sided estimates for blow-up rate under certain conditions for nonlinearities. We obtain an existence/nonexistence theorem for the following nonlocal problem: @ .u jujq u/ kruk2p 2 u D 0; @t

uj@ D 0;

u.x; 0/ D u0 .x/;

(5.177)

where p > 1, q 1, and 2 RN is a bounded domain with smooth boundary @ 2 C 2;ı , ı 2 .0; 1. Depending on q and p, we obtain results on the global-on-time solvability, solvability in any finite cylinder Œ0; T (0 < T < C1 is fixed), or the blow-up for an finite or infinite time. The problem describes quasi-stationary processes in semiconductors in the presence of a nonlocal relation between the medium conductivity and the intensity E of the electric field.

504


5.5.1 Unique local solvability of the problem Introduce the definitions and assumptions necessary for further consideration and obtain some auxiliary results. Consider the Hilbert space ¹H01 ./; k kº and its dual ¹H1 ./; k k º. Denote the duality brackets between the Hilbert spaces H01 ./ and its dual H1 ./ by h; i (see, e.g., [347]). In what follows, we use the following notation: k kC.Œ0;T IH1 / k kT ; k kC.Œ0;T IH1 / k kT : 0

Note that the operator A C .q C 1/jujq I acts from H01 ./ into H1 ./ under the condition 0 q 4=.N 2/ for N 3 and 0 q for the condition N D 1; 2, and is radially continuous (see, e.g., [168]). First, we prove that the operator B0 .u/ .q C 1/jujq I acts from H01 ./ into H1 ./. For this, we prove that 1

B0 W LqC2 ./ ! L.LqC2 ./; L.qC2/.qC1/ .//; where hB0 .u/h1 ; h2 i D .q C 1/hjujq h1 ; h2 i

8u; h1 ; h2 2 LqC2 ./;

kB0 .u/hkp1 D .q C 1/kjujq hkp1 c1 kjujq kp2 khkp3 ; p21 C p31 D p11 ; q p1 D .q C 2/.q C 1/1 ; p3 D q C 2; p21 D ; p2 q D q C 2; qC2 8h 2 LqC2 ./: Hence we obtain kB0 .u/hk.qC2/.qC1/1 c1 kukqqC2 khkqC2 : In addition, we require that q > 0 for N D 2 and 0 < q 4=.N 2/ for N 3; then, by virtue of the Sobolev embedding theorem, H01 ./ LqC2 ./ and, therefore, 1 L.qC2/.qC1/ ./ H1 ./. Therefore, the following inequalities hold: q kB0 .u/hk c2 kB0 .u/hk.qC2/=.qC1/ c2 c1 kukqC2 khkqC2 c3 kukq khk:

Therefore,

B0 W H01 ./ ! L.H01 ./; H1 .//:

Thus, A 2 L.H01 ./; H1 .// and, obviously, this operator is radially continuous. By the monotonicity of the operator B0 .u/ W LqC2 ./ ! L.qC2/=.qC1/ ./ and by the fact that for u; h1 ; h2 2 H01 ./, the equality hB0 .u/h1 B0 .u/h2 ; h1 h2 i D .q C 1/hjujq .h1 h2 /; h1 h2 i

505


holds, where the duality bracket h; i between H01 ./ and H1 ./ is the duality bracket between the Banach spaces LqC2 ./ and L.qC2/=.qC1/ ./ for elements from LqC2 ./ and L.qC2/=.qC1/ ./, and, therefore, they exactly coincide with the inner product .; / in L2 ./, the following relations hold: hB0 .u/h1 B0 .u/h2 ; h1 h2 i D .q C 1/hjujq .h1 h2 /; h1 h2 i D .q C 1/.jujq .h1 h2 /; h1 h2 / 0: Hence we obtain hAh1 Ah2 ; h1 h2 i hh1 h2 ; h1 h2 i D kh1 h2 k2 ; i.e., the operator A W H01 ./ ! H1 ./ is strongly monotonic. From the mentioned properties of the operator A, by virtue of [168, Corollary 2.3] and by the Browder–Minty theorem we obtain that the inverse operator A1 W H1 ./ ! H01 ./ is Lipschitz-continuous: kA1 x A1 yk kx yk

8x; y 2 H1 ./:

(5.178)

Now consider the operator 2p Fu kruk2 u:

Prove that the operator F W H01 ./ ! H1 ./ is boundedly Lipschitz-continuous. For this, note that 2p 2p kF .u1 / F.u2 /k kru1 k2 ru1 kru2 k2 ru2 2 2p 2p kru1 k2p 2 kru1 ru2 k2 C kru2 k2 jkru1 k2 kru2 k2 j 2p kru1 k2 kru1 ru2 k2 2p C C1 .; p/ max¹kru1 k2p 2 ; kru2 k2 ºkru1 ru2 k2 :

Thus, we conclude that kF.u1 / F.u2 /k C2 R2p ku1 u2 k;

(5.179)

where R D max¹kru1 k2 ; kru2 k2 º. The following theorem holds. Theorem 5.5.1. For any initial function u0 2 H01 ./, 1 q < 4=.N 2/ for N 3, 1 q for N D 2 and 1 < p, there exists Tu0 > 0 such that for any 0 < T < Tu0 , problem (5.177) has a unique solution of the class C .1/ .Œ0; TI H01 .//. Proof. By (5.178) and (5.179) the proof is obvious.

506


5.5.2 Blow-up and global solvability of problem (5.177) Introduce the notation ˆ.t / 21 kruk22 C ˛

qC1 qC2 kukqC2 ; qC2

2.p C 1/ ; qC2

kruk2 B1 kukqC2 ;

ˆ0 ˆ.0/;

ˆ0 .0/ D kru0 k2

2.pC1/

B2 D

B2.pC1/ 1

qC2 qC1

˛ ;

; E0

ˆ0 .0/ : ˆ˛0

The following theorem holds. Theorem 5.5.2. Let the conditions of Theorem 5.5.1 hold. Then the following assertions are valid. (1) for ˛ 2 .0; 1/, the two-sided estimate Œˆ1˛ C .1 ˛/E0 t 1=.1˛/ ˆ.t / Œˆ1˛ C .1 ˛/B2 t 1=.1˛/ 0 0 holds; (2) for ˛ D 1, the two-sided estimate ˆ0 exp¹E0 t º ˆ.t / ˆ0 exp¹B2 t ºI holds; (3) for ˛ > 1, there exists a time moment T0 2 ŒT1 ; T2 such that the limit relation lim ˆ.t / D C1

t"T0

holds, where T1 D

1 p p 2pC1 ˆ0

;

T2 D

(5.180)

1 ˆ0 : ˛ 1 ˆ0 .0/

Moreover, in the case where ˛ 2 .0; 1, problem (5.177) is uniquely solvable in any finite cylinder .0; T/ . Proof. Multiplying Eq. (5.177) by u.x; t / 2 C .1/ .Œ0; TI H01 .// and integrating over the domain , we obtain qC1 d 1 d qC2 2pC2 : kruk22 C kukqC2 D kruk2 2 dt q C 2 dt

(5.181)

Now multiplying (5.177) by u t .x; t / 2 C.0; TI H01 .// and integrating over the domain , we obtain Z d 1 2pC2 kruk2 dx jujq .u0t /2 D : (5.182) kru0 k22 C .q C 1/ 2p C 2 dt

507


The following inequalities hold: ˇZ ˇ2 ˇ ˇ ˇ dx hru; ru t iˇ kruk2 kru t k2 ; 2 2 ˇ ˇ

ˇZ ˇ2 Z ˇ ˇ ˇ dx jujq uu t ˇ kukqC2 dx jujq .u t /2 : qC2 ˇ ˇ

(5.183)

Also, the following inequalities hold: ˇ2 ˇZ ˇ ˇ ˇ dx u.u C jujq u/ t ˇ ˇ ˇ

j.ru; ru t /j2 C .q C 1/2 j.jujq u; u t /j2 C 2.q C 1/j.ru; ru t /jj.jujq u; u t /j qC2

kruk22 kru t k22 C .q C 1/2 .jujq ; u2t /kukqC2 q .qC2/=2 C 2.q C 1/kruk2 kru t k2 .jujq ; u2t /kukqC2 Z qC2 kru t k22 C .q C 1/ dx jujq u2t Œkruk22 C .q C 1/kukqC2

.q C 2/

kru t k22

Z

C .q C 1/

q

dx juj .u t /

2

1 qC1 qC2 2 kruk2 C kukqC2 : 2 qC2

(5.184)

From (5.181)–(5.184) we obtain ˆ00 .t /ˆ.t / ˛.ˆ0 .t //2 0;

˛

2p C 2 : qC2

(5.185)

Consider three cases: 2p > q, 2p < q, and 2p D q. First, we obtain lower estimates. Let the condition ˛ 2 .0; 1/ hold. Integrating (5.185), we obtain ˆ0 ˆ0 .0/ E0 I ˆ˛ ˆ˛0 hence, owing to (5.181) we have the inequality ˆ.t /

Œˆ1˛ 0

1 : E0 .˛ 1/t 1=.˛1/

Hence we directly obtain (5.180) and an upper estimate for the blow-up time of a strong generalized equation of problem (5.177): ˆ0 1 ; 0 ˆ .0/ ˛ 1 1 qC1 2.pC1/ qC2 ku0 kqC2 ; ˆ0 .0/ kru0 k2 : ˆ0 ˆ.0/ D kru0 k22 C 2 qC2 T2

508


Now consider the second case ˛ D 1. Then (5.185) implies ˆ0 .0/ ˆ0 : ˆ ˆ0 Integrating this inequality, we obtain ²

³ ˆ0 .0/ t : ˆ ˆ0 exp ˆ0 Finally, consider the third case ˛ > 1. Then from (5.185) we obtain C .1 ˛/E0 t /1=.1˛/ ; ˆ .ˆ1˛ 0

E0 D

ˆ0 .0/ : ˆ˛0

Now we obtain upper estimates. First, consider the case ˛ > 1; from (5.181) we obtain Z t Z t 2.pC1/ ˆ.t / ˆ0 C ds kruk2 ˆ0 C 2pC1 ds ˆpC1 : 0

0

This implies the following lower estimate for the blow-up time of solutions of problem (5.177): 1 : T T1 D pC1 p2 ˆp 0 In the case ˛ 2 .0; 1, from (5.181) we obtain Z ˆ.t / ˆ0 C

t 0

ds kruk2.pC1/ .s/: 2

According to the condition H01 ./ LqC2 ./, with kruk2 B1 kukqC2 , we obtain ˛ Z t 2.pC1/ q C 2 ˆ.t / ˆ0 C B1 ds ˆ˛ .s/: (5.186) qC1 0 From (5.186) for ˛ D 1 we obtain ˆ.t / ˆ0 exp¹B2 t º;

B2

B2.pC1/ 1

qC2 qC1

Finally, for ˛ 2 .0; 1/ from (5.186) we obtain C .1 ˛/B2 t 1=.1˛/ : ˆ.t / Œˆ1˛ 0

˛ :

509


Now we prove the solvability of problem (5.177) for q 2p in any finite cylinder Œ0; T, T 2 .0; C1/. Assume that Tu0 < C1. Consider the auxiliary integral equation for arbitrary T0 2 .0; Tu0 /: 0

u.x; t / D u.x; T / C

Z

t

ds G.u/.s/:

(5.187)

0

Since the norm kruk22 .T0 / is uniformly bounded with respect T0 2 .0; Tu0 /, we can select T 2 .0; Tu0 / such that for any T0 2 .0; Tu0 /, the integral equation has a unique solution of the class u.x; t / 2 C .1/ .Œ0; T I H01 .//. Let T0 D Tu0 T =2; denote by v.x; t / the corresponding solution of the integral equation (5.11) and denote b u.x; t / on the segment Œ0; Tu0 C T =2: b u.x; t / D ¹u.x; t /; t 2 Œ0; T0 I v.x; t T0 /; t 2 ŒT0 ; Tu0 C T =2º: By construction, b u.x; t / is a solution of problem (5.177) on the segment Œ0; Tu0 C T =2 and, by the local uniqueness, an extension of the function u.x; t /. This contradicts the maximality of the segment Œ0; Tu0 . This contradiction proves that Tu0 D C1. Theorem 5.5.2 is proved.

5.5.3 Blow-up rate for problem (5.177) under the condition q D 0 In this subsection, we obtain optimal two-sided estimates for the blow-up rate of strong generalized solutions of problem (5.177) under the condition q D 0. Introduce the notation q 1 ; (5.188) M.t / kruk22 .t / C kuk22 .t /; D 2p M .t / 2pC2

E0

kru0 k2 : Œkru0 k22 C ku0 k22 pC1

(5.189)

The following theorem holds. Theorem 5.5.3. Let the conditions of Theorem 5.5.2 holds. Then under the condition q D 0, the following two-sided estimates for the blow-up rate of solutions of problem (5.177) hold: Œ2p1=p ŒT0 t 1=p Œkruk22 C kuk22 Œ2pE0 1=p :

(5.190)

Proof. Introduce the auxiliary function v .s; x/ D

1 u.s C t ? ; x/: M.t ? /

(5.191)

510


By a direct calculation, owing to (5.188) and (5.191), we can verify that v .s; x/ satisfies all the conditions of the problem @v 2p krv k2 v D 0; @s v .s; x/j@ D 0; ? t 1 u0 .x/: v ; x D M.t ? /

Œ I

(5.192)

Prove that there exists c1 ; c2 > 0 such that 0 < c2

d Œkrv k22 .s/ C kv k22 .s/1=2 .s D 0/ c1 < C1: ds

(5.193)

First, obtain an upper estimate. Multiply the first equation of system (5.192) by v and integrate over the domain taking into account the boundary condition; we obtain ˇ d 2pC2 .0/ 2: Œkrv k22 .s/ C kv k22 .s/ˇsD0 D 2krv k2 ds Now we prove a lower estimate. Let ' .s/ D krv k22 C kv k22 . Then d' .s/ 2 d 2 ' .s/ ' .s/ .p C 1/ 0; ds 2 ds which implies ˇ '0 .s/ ˇ ˇ D E0 ; ' .s/˛ ' .s/˛ ˇsDt ? = '0 .s/

2pC2

E0 D 2

kru0 k2 : Œkru0 k22 C ku0 k22 pC1

Hence for s D 0 from the equality ' .0/ D 1 we conclude that d Œkrv k22 C kv k22 1=2 .0/ E0 : ds Thus, inequality (5.193) is proved. Inequality (5.193) is equivalent to 0 < E0

M0 .t ? / 1 < C1: M.t ? /2pC1

Hence, integrating by t ? in the limits from t to T0 , lim M.t / D C1;

t"T0

we obtain (5.190). Theorem 5.5.3 is proved.

511

Section 5.6 Solvability of the Laplace equation with nonlinear boundary conditions

5.6

Existence of solutions of the Laplace equation with nonlinear dynamic boundary conditions

In this section, we consider initial-boundary-value problems for the Laplace equation with nonlinear, nonclassical, dynamic boundary condition. Under certain conditions for the given data, the global-on-time solvability is proved. Under certain other conditions, the nonexistence of global-on-time solutions is proved. We also obtain Fujita’s result on the nonexistence of global-on-time positive solution even for the initial functions that are small in some sense. Consider the following system:

u D 0;

x3 > 0;

t > 0; ˇ ˇ @u C C jujq u ˇˇ D 0; @t @x3 @x3 x3 D0 @2 u

v.x 0 ; 0/ D v0 .x 0 /;

x 0 2 R2 ;

(5.194) (5.195) (5.196)

where v.x 0 ; t / u.x3 ; x 0 ; t /jx3 D0 , x 0 .x1 ; x2 / 2 R2 , t > 0. This problem describes quasi-stationary processes on the surface of a semiconductor.

5.6.1 Reduction the problem to the system of the integral equations Definition 5.6.1 (class (K)). Introduce the class (K) of functions u.x3 ; x 0 ; t / such that for certain T > 0 and for any t 2 Œ0; T, u.x3 ; x 0 ; t / 2 C .2/ .R3C / and u.x3 ; x 0 ; t / is bounded in the half-space x3 > 0, x 0 .x1 ; x2 /. For any fixed x 0 2 R2 , u.x3 ; x 0 ; t / 2 1 C .1/ .Œ0; TI C .1/ .R1C /\C.RC //, v.x 0 ; t / u.x3 ; x 0 ; t /jx3 D0 , v.x 0 ; t / 2 C .1/ .Œ0; TI W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //, ˇ > 2. Here the boundary condition (5.195) is meant as the pointwise limit .x 0 ; t / 2 R2 Œ0; T, fixed as x3 # C0. Moreover, we assume that v0 .x 0 / 2 W 2;1 ..1 C jx 0 j2 /˛=2 I R2 /; ˛ > 2: A function u.x3 ; x 0 ; t / of class (K) is called a solution of problem (5.194)–(5.196). Theorem 5.6.2. In the class (K), problem (5.194)–(5.196) is equivalent to the following system of the integral equations: Z 1 x3 v0 .y/e t dy 2 u.x 0 ; x3 ; t / D 2 R2 .x3 C jx 0 yj2 /3=2 Z Z t 1 .jvjq v/.y; s/e .ts/ C ds dy q ; x3 > 0; (5.197) 2 0 R2 x32 C jx 0 yj2 Z t Z 1 .jvjq v/.y; s/e .ts/ ; t 2 Œ0; T: ds dy v.x 0 ; t / D v0 .x 0 /e t C 2 0 jx 0 yj R2 (5.198)

512


Proof. Denote by u./ O D FO Œu./ the Fourier transform with respect to the variable 0 2 x .x1 ; x2 / 2 R in the sense of distributions of slow growth P 0 .R2 /. Then applying the Fourier transform to (5.194) with respect to the variables x 0 , we obtain @2 u.; O x3 ; t / jj2 u.; O x3 ; t / D 0; @x32

t 2 Œ0; T:

x3 > 0;

Hence we obtain O t /e x3 jj : u.; O x3 ; t / D v.;

(5.199)

From (5.195) we obtain

1

1 @ jvjq v.; t /: v.; O t / C v.; O t/ D @t jj Hence we obtain v.; O t / D vO 0 ./e t C

Z

t

ds e .ts/

0

1

1 jvjq v.; s/: jj

(5.200)

From (5.199) and (5.200) we obtain u.; O x3 ; t / D vO 0 ./e tx3 jj C

Z

t 0

ds e .ts/

1

1 x3 jj q jvj v.; s/: e jj

(5.201)

Assume that v0 .x 0 / 2 L1 .R2 /, jvjq v 2 L1 .R2 /. Note that the following auxiliary assertion holds (the proof of a similar assertion can be found in [421]). Lemma 5.6.3. If f 2 Cb .R2 /, g 2 L1 .R2 /, then the Fourier transform of the convolution F Œf g in the sense of P 0 .R2 / exists and is equal to F Œf F Œg. Indeed, for all w 2 P .R2 /, .f g; w/ .f .x/; .g.y/; w.x C y///; .F Œf g; w/ .f g; F Œw/ Z w./e i hxCy;i d // D .f .x/; .g.y/; R2 Z .g; e i h;yi /e i h;xi w./ d / D .f .x/; 2 R Z F Œg./w./e i h;xi d / D .f; R2

D .f; F ŒF Œgw/ D .F Œf ; F Œgw/ D .F Œf F Œg; w/:


513

Thus, by virtue of (5.201), any solution of problem (5.194)–(5.196) can be represented in the form Z Z t 1 .jvjq v/.y; s/ ds e .ts/ dy q ; u.x 0 ; x3 ; t / D px3 .x 0 / v0 .x 0 /e t C 2 0 R2 x32 C jx 0 yj2 (5.202) where t 2 Œ0; T, x3 ı > 0, for certain fixed ı > 0 and px3 .x 0 /

x3 1 : 2 2 .x3 C jx 0 j2 /3=2

Thus, in the class (K), problem (5.194)–(5.196) implies the integral equation (5.202). Now we prove the possibility of the passage to the limit as x3 # 0 in the integral equation (5.202) in the class (K). Here we use the standard technique (see, e.g., [289]). Consider separately each of the terms in the right-hand side of Eq. (5.202). We use the obvious relation px3 .x 0 / v0 .x 0 / v0 .x 0 / Z 1 1 dz Œv0 .x 0 x3 z/ v0 .x 0 / D 2 2 R .1 C jzj2 /3=2 Z Z dz 1 D C Œv .x 0 x3 z/ v0 .x 0 /; 2 /3=2 0 2 R2 n…A .1 C jzj …A where …A ŒA; A ŒA; A. The estimate ˇ ˇZ ˇ 1 ˇˇ dz 0 0 ˇ Œv .x x z/ v .x / 0 3 0 ˇ ˇ 2 R2 n…A .1 C jzj2 /3=2 Z dz 2M < "=2 2 R2 n…A .1 C jzj2 /3=2 holds for sufficiently large A > 0. Let 0 < x3 < ı and ı be selected so small that j.x 0 x3 z/ x 0 j < 2Aı D ı1 ; then by the continuity of the function v0 .x 0 / jv0 .x 0 x3 z/ v0 .x 0 /j < and, finally,

" A2

ˇ ˇZ ˇ " dz 1 ˇˇ 0 0 ˇ Œv .x x z/ v .x / 0 3 0 ˇ < 2: ˇ 2 …A .1 C jzj2 /3=2

Thus, for any " > 0, there exist 0 < A < C1 and ı D ı."; A/ such that jpx3 .x 0 / v0 .x 0 / v0 .x 0 /j < "

for 0 < x3 < ı:

514


Now we prove that Z lim

x3 #0 R2

dy q

Z

.jvjq v/.y; t / x32 C jx 0 yj2

D

R2

dy

.jvjq v/.y; t / : jx 0 yj

(5.203)

Since v.x 0 ; t / 2 C.Œ0; TI L1 .1 C jx 0 j2 /ˇ=.2.qC1// I R2 // we have ˇZ ˇ ˇ ˇ

ˇ ˇ 1 1 ˇ dy .jvj v/.y; t / q 0 ˇ jx yj 2 0 2 R2 x3 C jx yj ˇ ˇ Z ˇ ˇ jzj 1 1Cq 0 ˇ dz .x z; t /ˇ q 1ˇˇ jvj jzj R2 x32 C jzj2 ˇ ˇ Z C1 Z 2 ˇ ˇ 1 ˇ C d d' q 1ˇˇ; ˇ 0 2 2 0 ˇ=2 .1 C jx j C 2jx j cos.'// 0 0 x 2 C 2 q

3

where ˇ > 2. The integrand is bounded from above by the function f .x; ; '/ C .1 C jx 0 j2 C 2 2jx 0 j cos.'//ˇ=2 : It can be proved that Z

Z

C1

2

d 0

0

d' jf .x; ; '/j C < C1:

Using the Lebesgue theorem on the passage to the limit under the sign of the Lebesgue integral (see, e.g., [129]) we see that (5.203) is valid. Now we can pass in the integral equation (5.197) to the limit as x3 # 0; we obtain 0

0

v.x ; t / D v0 .x /e

t

1 C 2

Z

Z

t

ds 0

dy

R2

.jvjq v/.y; s/e .ts/ : jx 0 yj

Thus, in the class (K), system (5.194)–(5.196) implies the system of two integral equations (5.197) and (5.198). Now we prove the contrary statement: any solution of the system of integral equations (5.197), (5.198) in the class (K) is a solution of problem (5.194)–(5.196). Consider the integrals J1 D px3 .x 0 / v0 .x 0 /;

J2 D

1 2

Z R2

dy

.jvjq v/.x 0 y/ : q 2 2 x3 C jyj


515

Let x3 > 0; then

1 D 2

1 @2 v0 .x 0 y/ dy 2 q @x3 2 R2 2 x3 C jyj

Z

@J1 1 D @x3 2 Z

R2

dy q

1 x32 C jyj2

2y v0 .x 0 y/:

Since v0 .x 0 / 2 W 2;1 ..1 C jx 0 j2 /˛=2 I R2 /, the limit relation Z @J1 1 1 D dy 2y v0 .x 0 y/ lim 2 R2 jyj x3 #0 @x3 holds. Consider the integral J2 for x3 > 0. We have Z 1 x3 @J2 D dy 2 .jvjq v/.x 0 y; t /: 2 @x3 2 R .x3 C jyj2 /3=2 Prove that

@J2 D .jvjq v/.x 0 ; t /: x3 #0 @x3 lim

Indeed, since u.x 0 ; t / belongs to the class (K), we have .jvjq v/.x 0 ; t / 2 C.Œ0; TI Cb .R2 // and, moreover, the following inequality holds: ˇ ˇ ˇ ˇ @J2 q 0 ˇ ˇ ˇ @x C .jvj v/.x ; t /ˇ 3 Z Z ˇ q ˇ 1 dz ˇ.jvj v/.x 0 x3 z; t / .jvjq v/.x 0 ; t /ˇ : C 2 3=2 2 R2 n…A …A .1 C jzj / Further, it is easy to prove (see, e.g., [289]) that such for all " > 0, there exist A > 0 and ı.A; "/ > 0 such that for all 0 < x3 < ı, ˇ ˇ ˇ @J2 ˇ q 0 ˇ ˇ ˇ @x C .jvj v/.x ; t /ˇ < ": 3 Note that for x3 > 0, the function u.x 0 ; x3 ; t / defined by (5.197) satisfies the following relations: Z 1 t 2y v0 .x 0 y/ @u 0 .x ; x3 ; t / D dy q e @x3 2 R2 x32 C jyj2 Z Z t 1 x3 .jvjq v/.x 0 y; s/ .ts/ ds e dy 2 0 .x32 C jyj2 /3=2 R2

516


for x3 > 0, @u 0 e t lim .x ; x3 ; t / D 2 x3 #C0 @x3

Z

2y v0 .x 0 y/ dy jyj R2

Z

t

ds e .ts/ .jvjq v/.x 0 ; s/:

0

Therefore, for any x 0 2 R2 and t 0, we obtain @u 0 .x ; x3 ; t / 2 C .1/ .Œ0; TI C.R1C //; @x3

1

u.x 0 ; x3 ; t / 2 C .1/ .Œ0; TI C.RC //:

Moreover, we similarly obtain that @2 u 2 C.Œ0; TI C.R1C // @t @x3 for any fixed x 0 2 R2 and t 2 Œ0; T. For x3 > 0, Z

@u 1 @2 u C D @t @x3 @x3 2

R2

dy

.x32

x3 .jvjq v/.x 0 y/: C jyj2 /3=2

Passing in the latter equality to the limit as x3 # 0, we obtain @u @2 u C C jujq u D 0 for x3 D 0: @t @x3 @x3 Moreover, for x3 > 0 u.x 0 ; x3 ; t / D

1 2

Z R2

1 C 2

dy

Z

.x32 Z

t

ds 0

x3 v0 .y/e t C jx 0 yj2 /3=2

.jvjq v/.y; s/e .ts/ dy q : R2 x32 C jx 0 yj2

Obviously, the function u.x 0 ; x3 ; t / is infinitely differentiable in the domain ¹x3 > 0º R2 Œ0; T and is harmonic with respect to the variables .x 0 ; x3 / for x3 > 0, i.e., it satisfies Eq. (5.194). Thus, in the class (K), problem (5.194)–(5.196) is equivalent to the system of integral equations (5.197), (5.198). Theorem 5.6.2 is proved.

517


5.6.2 Global-on-time solvability and the blow-up of solutions Let

v0 .x 0 / 2 W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 /

where 2 < ˇ and 1 C q 1 < ˇ < 1 C q. Theorem 5.6.4. For any function v0 .x 0 / belonging to the above-mentioned class, for q > 1 and for sufficiently small initial function v0 .x 0 /, sup Œjv0 .x 0 /j C jrv0 .x 0 /j.1 C jx 0 j2 /ˇ=.2.qC1// ı;

x 0 2R2

there exists a unique solution of the integral equation 0

0

v.x ; t / D v0 .x /e

t

1 C 2

Z

Z

t

ds 0

R2

dy

.jvjq v/.y; s/e .ts/ jx 0 yj

(5.204)

belonging to the class v.x 0 ; t / 2 Cb .Œ0; C1I W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //: .1/

Proof. Introduce the notation w.x 0 ; t / v.x 0 ; t /.1 C jx 0 j2 /ˇ=.2.qC1// . From (5.204) we obtain that w.x 0 ; t / satisfies the integral equation Z Z 1 t w.x 0 ; t / D w0 .x 0 /e t C ds dy .jwjq w/.y; s/e .ts/f .x 0 ; y/; (5.205) 2 2 0 R w0 .x 0 / v0 .x 0 /.1 C jx 0 j2 /ˇ=.2.qC1// ;

f .x 0 ; y/

.1 C jx 0 j2 /ˇ=.2.qC1// : jyj.1 C jx 0 yj2 /ˇ=2

To prove the theorem, we use the method of successive approximations. Define recursive approximations v1 .x 0 ; t / v0 .x 0 /e t ; vnC1 .x 0 ; t / v0 .x 0 /e t C A.vn /.x 0 ; t /; Z Z t .jvjq v/.x 0 y; s/ 1 .ts/ ds e dy : A.v/ 2 0 jyj R2 Here the iterated integral in the definition of the operator A is meant in the sense of the Lebesgue integral. Introduce the following norms: kvk0 sup j.1 C jx 0 j2 /ˇ=.2.qC1// v.x 0 ; t /j; x2R2 t0

kvk1 kvk0 C krvk0 :

(5.206)

518


Estimate the operator A./ by the norm k k1 . We have kA.v/k1 kA.v/k0 C krA.v/k0 ; Z Z t 1 0 2 ˇ=.2.qC1// .ts/ ds e dy f .x 0 ; y/jwj1Cq .x 0 y; s/: jA.v/j.1 C jx j / 2 0 R2 Consider the integral Z R2

0

Z

dy f .x ; y/ D

R2

dy

.1 C jx 0 j2 /ˇ=.2.qC1// : jyj.1 C jx 0 yj2 /ˇ=2

We can estimate the latter integral; we obtain Z dy f .x 0 ; y/ C.q; ˇ/ < C1 R2

under the additional conditions ˇ > 2 and 1 C q 1 < ˇ 1 C q. Hence and from (5.206) we obtain 1Cq jA.v/j.1 C jx 0 j2 /ˇ=.2.qC1// C.q; ˇ/kvk0 ;

kA.v/k0 C kvk1Cq : 0

Now we obtain an estimate for krA.v/k0 : jrA.v/j.1 C jx 0 j2 /ˇ=.2.qC1// Z Z 1Cq t .1 C jx 0 j2 /ˇ=.2.qC1// .ts/ ds e dy .jvjq jrvj/.x 0 y/ 2 0 jyj R2 Z 1Cq t ds e .ts/ 2 0 Z dy f .x 0 ; y/.jwjq jrvj/.x 0 y/.1 C jx 0 yj2 /ˇ=.2.qC1//

R2 C kvkq0 krvk0 :

Hence we have

krA.v/k0 C kvk1Cq : 1

Thus, 1Cq

kA.v/k1 C kvk1

:

(5.207)

For the introduced nonlocal operator A.v/, the following auxiliary assertion holds. Lemma 5.6.5. Let M > 0 and q > 1 be defined. For any functions z1 and z2 such that kzi k1 M , i D 1; 2, the following estimate holds: Z c1 .1 C q/2 M q kA.z1 / A.z2 /k1 kz1 z2 k1 ; c1 D sup dy f .x 0 ; y/: 2 2 0 2 x 2R R (5.208)


519

Proof. Introduce the notation z1i

@z1 ; @xi

z2i

@z2 ; @xi

i D 1; 2;

˛D

ˇ : qC1

The following inequalities hold: ˇ ˇ ˇ ˇ ˇjz1 jq z1i jz2 jq z2i ˇ jz2i jˇjz1 jq jz2 jq ˇ C jz1 jq jz1i z2i j 1 ŒqMM q1 kz1 z2 k0 C M q kz1i z2i k0 Œ1 C jx 0 j2 ˛=2 1 qM q Œkz1 z2 k0 C kz1i z2i k0 ; Œ1 C jx 0 j2 ˛=2 ˇ ˇ 1 ˇjz1 jq z1 jz2 jq z2 ˇ .1 C q/M q kz1 z2 k0 ; Œ1 C jx 0 j2 ˛=2 jA.z1 / A.z2 /j.1 C jx 0 j2 /ˇ=.2.1Cq// Z Z t ˇ ˇ 1 .ts/ ds e dy f .x 0 ; y/.1 C jx 0 yj2 /ˇ=2 ˇjz1 jq z1 jz2 jq z2 ˇ 2 0 R2 1Cq c1 M q kz1 z2 k0 ; 2 Z 1Cq dy f .x 0 ; y/: c1 M q kz1 z2 k0 ; c1 D sup kA.z1 / A.z2 /k0 2 2 0 2 R x 2R jrA.z1 / rA.z2 /j.1 C jx 0 j2 /ˇ=.2.1Cq// Z Z 1Cq t .ts/ ds e dy f .x 0 ; y/ 2 0 R2 .1 C jx 0 yj2 /ˇ=2

2 X ˇ ˇ ˇjz1 jq z1i jz2 jq z2i ˇ i D1

q.1 C q/ q M c1 2

2 X

Œkz1 z2 k0 C kz1i z2i k0

i D1

q.1 C q/ c1 M q kz1 z2 k1 ; 2

krA.z1 / rA.z2 /k0

q.1 C q/ c1 M q kz1 z2 k1 : 2

Therefore, c1 .1 C q/2 M q kz1 z2 k1 ; kA.z1 / A.z2 /k1 2 Lemma 5.6.5 is proved.

Z c1 D sup

x 0 2R2

R2

dy f .x 0 ; y/:

520


From the iterative scheme we have kunC1 k1 ı C C kun k1Cq ; 1

ku0 k1 ı:

Hence we have (see, e.g., [275]) that kun k M.ı/ ! C0 for sufficiently small ı > 0. From (5.208) we have kunC1 un k1 D kA.un / A.un1 /k1

c1 .1 C q/2 M.ı/q kun un1 k1 : 2

On the other hand, for sufficiently small ı > 0 we obtain c1 .1 C q/2 M.ı/q < 1=2; 2

kunC1 un k1 1=2kun un1 k1 :

Hence we directly obtain the strong convergence of the recursive sequence: un ! u strongly in L1 .0; C1I W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //: Now we prove the uniqueness of a solution of Eq. (5.204). Let w1 and w2 be two solutions; then owing to (5.205) we have jw1 w2 jŒ1 C jx 0 j2 ˛=.2.1Cq// Z Z t ˇ ˇ 1 ds e .ts/ dy f .x 0 ; y/ˇjw1 jq w1 jw2 jq w2 ˇŒ1 C jx 0 yj2 ˛=2 2 0 R2 Z t Z 1Cq ds e .ts/ dy f .x 0 ; y/ 2 0 R2 max¹jw1 jq ; jw2 jq ºjw1 w2 jŒ1 C jx 0 yj2 ˛=2 Z t 1Cq q ds e .ts/ kw1 w2 kB Œ1 C jx 0 yj2 ˛=.2.1Cq// c1 M 2 0 Z t d ds kw1 w2 kB Œ1 C jx 0 yj2 ˛=2.1Cq/ ; 0

where dD

c1 .1 C q/M q ; 2

Mi D sup j.1 C jx 0 j2 /ˇ=.2.qC1// wi j; t0; x 0 2R2

M D max Mi : i

Thus, Z kw1 w2 kB d

0

t

dskw1 w2 kB ;

kvkB sup Œ1 C jx 0 j2 ˛=.2.1Cq// jv.x/j: x2R3

Therefore, there exists t0 > 0 such that w1 D w2 for t 2 Œ0; t0 . Using the well known extension-in-time algorithm of solutions of Volterra-type integral equations


521

(see, e.g., [275]) we obtain that w1 D w2 for any t 2 Œ0; C1 and for all x 0 2 R2 . Thus, there exists a unique solution of the integral equation (5.204) of the class L1 .0; C1I W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //: Prove that .1/ v.x 0 ; t / 2 Cb .Œ0; C1I W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //:

It suffices to prove that the second term in the right-hand side of the integral equation (5.204) belongs to this class: Z t Z .jvjq v/.y; s/e .ts/ ds dy : .x 0 ; t / jx 0 yj R2 0 Since

v.x 0 ; t / 2 L1 .Œ0; C1I W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //;

obviously, .x 0 ; t / 2 AC.Œ0; C1I W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //: Thus,

v.x 0 ; t / 2 AC.Œ0; C1I W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //;

and the integral by s is a Riemann proper integral. Hence we directly obtain v.x 0 ; t / 2 C .1/ .Œ0; C1I W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //: From the explicit form of the function

.x 0 ; t / we obtain that

v.x 0 ; t / 2 Cb.1/ .Œ0; C1I W 1;1 ..1 C jx 0 j2 /ˇ=.2.qC1// I R2 //: Theorem 5.6.4 is proved. The results obtained also imply the existence of u.x 0 ; x3 ; t / from (5.197) for any t 2 Œ0; C1; by Theorem 5.6.4 they imply the global solvability of problem (5.194)– (5.196). Theorem 5.6.6. The following assertions hold. (1) Let the function v0 .x 0 / 2 L1Cq .R2 / \ L1 .R2 / be such that Z Z 1 .jv0 jq v0 /.x/.jv0 jq v0 /.y/ qC2 0 < kv0 kqC2 < dx dy : 2 R2 R2 jx yj Then there exists 0 T0 < C1 such that the solution of Eq. (5.204) v.x 0 ; t / … L1 .Œ0; T0 /I L1Cq .R2 / \ L1 .R2 //:

522


(2) Let q 2 .0; 1=2/, v0 .x 0 / 2 C0 .R2 /, v0 0, and v0 > 0 on a set of a positive measure. Then any positive solution of the integral equation (5.204) is such that there exists 0 T0 < C1 and v.x 0 ; t / … L1 .Œ0; T0 /I L1Cq .R2 / \ L1 .R2 //: Proof. Let a global solution of the integral equation (5.204) exist under the conditions formulated either in (1) or in (2), meaning a solution of the class L1 .Œ0; C1/I L1Cq .R2 / \ L1 .R2 //: Then by virtue of the fact that the operator in the right-hand side of the integral equation (5.204) is smoothing, the solution belongs to the class C .1/ .Œ0; C1/I L1Cq .R2 / \ L1 .R2 //: In this case, the integral equation (5.204) can be written in the form of the following Cauchy problem for the integro-differential equation: Z 1 .jvjq v/.y; t / v t .x 0 ; t / C v.x 0 ; t / D dy (5.209) ; v.x 0 ; 0/ D v0 .x 0 /: 2 R2 jx 0 yj Introduce the new function w D e t v; then (5.209) takes the form Z 1 qt .jwjq w/.y; t / dy e : w t .x 0 ; t / D 2 jx 0 yj R2

(5.210)

Let the conditions of assertion (1) hold. Multiply both sides of Eq. (5.210) by .jwjq w/.x 0 ; t /. Integrating over x 0 2 R2 , we obtain the equation Z Z 1 qt .jwjq w/.x; t /.jwjq w/.y; t / 1 d qC2 dx dy kwkqC2 D e : (5.211) q C 2 dt 2 jx yj R2 R2 Moreover, we have ˇZ ˇ ˇ ˇ

ˇ2 Z ˇ dx jwj ww t ˇˇ 2

R

Z

q

R2

dx jwjq w 2t

R2

dx jwjqC2 :

(5.212)

Multiply both sides of (5.210) by the function .jwjq w/ t ; after integrating over x 0 2 R2 we obtain Z Z Z 1 e qt d .jwjq w/.x; t /.jwjq w/.y; t / q : dx .jwj w/ t w t D dx dy 2 2 dt R2 R2 jx yj R2 (5.213)


The latter equality implies Z R2

dx jwjq w 2t D

Z

where f .t /

R2

Z R2

dx dy

1 qt df 1 e ; 2.1 C q/ 2 dt

523

(5.214)

.jwjq w/.x; t /.jwjq w/.y; t / : jx yj

qC2 Introduce the notation ' kwkqC2 . Estimates (5.211)–(5.214) lead to the inequalities 2 Z Z d' q C 2 qt d 2 q 2 qC2 qt d' e ': e .q C 2/ dx jwj w t dx jwj dt 2.q C 1/ dt dt R2 R2

Hence we obtain ' 00 .t / .' 0 .t //2 ' 0 .t / ˛ C q 0; '.t /˛ '.t /1C˛ '.t /˛

˛ 1 C ;

q=.2 C q/:

(5.215)

From (5.215) we have ' 0 .t / 0 e qt ˛ 0; ' .t /

' 0 .t / q C 2 qt e f .0/'.0/˛ : '˛ 2

Then we conclude that

1 : 2 This inequality can be reduced to the following form: Z Z 1 .jv0 jq v0 /.x/.jv0 jq v0 /.y/ qC2 dx dy : kv0 kqC2 2 R2 R2 jx yj '.0/ f .0/

The inequality obtained contradicts condition (1) of Theorem 5.6.6. This contradiction proves the existence of 0 T0 < C1 such that v.x 0 ; t / … L1 .Œ0; T0 /I L1Cq .R2 / \ L1 .R2 //: Now we prove the second part of Theorem 5.6.6. Let conditions (2) hold. Consider the integro-differential equation Z 1 1 v 1Cq .y; t / vt C v dy 0 2 2 .R/ jx yj Z 1 1 dy p v 1Cq .y; t /; 2 2 2 .R/ .x3 a/ C jx 0 yj2

524


where 2 .R/ R2 is a circle of radius R > 0 centered at the origin. Continue the previous inequality: Z vt C v 2 dy G.x 0 ; x3 I y; a/v 1Cq .y; t /; (5.216) 2 .R/

where G.x 0 ; x3 I y; a/ is the Green function of the first boundary-value problem for the Laplace equation in the cylinder D Œ0; l 2 .R/ 2 R3C . Let '3 sin

l

x3 c2 J0

r

21 c2 2 R2

1 ;

R

Z

1

1

dz J0 .z/z

0

;

where '3 is the first eigenfunction of the first boundary-value problem for the Laplace operator in the cylinder D Œ0; l 2 .R/ 2 R3C , 1 is the first root of the equation J0 .z/ D 0. Obviously, '3 > 0 for 0 r < R and 0 < x3 < l; therefore, multiplying both sides of inequality (5.216) by '3 and integrating by .x 0 ; x3 / 2 D, we obtain Z Z df 0 0 dx dx3 '3 .x ; x3 / dy G.x 0 ; x3 I y; a/v 1Cq .y; t /; Cf 2 dt D 2 .R/ Z 2l f .t / dx 0 dx3 v.x 0 ; t /'3 .x 0 ; x3 / D f1 .t /;

D Z dx 0 v.x 0 ; t / 2 .x 0 /; f1 .t / 2 .R/

0 2 .x /

21 2 R2

Z

1

1

0

dz J0 .z/z

J0

r R

1 :

It follows that df1

C f1 dt l3

Z dy 2 .R/

2 .y/v

1Cq

.y; t / sin

a l

;

3

2 l

C

2 1

R

We set a D l=2 and use the Jensen inequality (see, e.g., [445]); then we obtain

1Cq df1 f : C f1 dt l3 1 The following relations hold: f2 .t / e t f1 .t /; f1 .0/q

df2

qt 1Cq e f2 ; f2 .0/q f2 .t /q Œ1 e qt ; dt l3 l3

2 1 2 1=q ; 1=q l 1=q C f1 .0/: l3 l R

:

525

Section 5.7 Global solvability of a pseudoparabolic Cauchy problem

We take l D R; then c2 R

1=q

21 2 R2

Z

1 Z

1 0

dz J0 .z/z c2

1=q

Z

2

R

ds 0

0

2

2 1=q

Œ. / C .1 /

dr rJ0

r R

1 v0 .r; s/;

:

Assume that supp v0 .x 0 / 2 .R0 /, R0 2 .0; 1/; then the following relations hold: Z

2 0

Z 2 Z R0 r r 1 v0 .r; s/ D 1 v0 .r; s/ ds dr rJ0 ds dr rJ0 R R 0 0 0 Z 2 Z R0 Z 2 Z R0 r r1 ds dr rJ0 1 v0 .r; s/ D J0 1 dr r v0 .r; s/; R0 R0 0 0 0 0 Z

R

where R > R0 ;

bD

r1 1 < 1 ; R

r1 D r1 .R0 / 2 .0; R0 /:

As a result we obtain R1=q c3 R2 ; this fact for sufficiently large R > R0 > 0 under the condition q 2 .0; 1=2/ leads to a contradiction. This proves the second assertion of Theorem 5.6.6.

5.7

Conditions of the global-on-time solvability of the Cauchy problem for a semilinear pseudoparabolic equation

Here we consider the Cauchy problem in R3 for a pseudoparabolic equation that describes quasi-stationary processes in an unipolar semiconductor in the presence of sources of current of free charges. Depending on the nonlinearity index, we prove the global-on-time solvability, the global insolvability, and the blow-up of solutions. Also, we obtain Fujita’s result on the nonexistence of global-in-time solutions even for functions small in some sense. Consider the Cauchy problem u0t C u C jujq u D 0;

x 2 R3 ;

u.x; 0/ D u0 .x/;

t > 0;

(5.217)

3

(5.218)

x2R :

5.7.1 Reduction of the problem to an integral equation In this subsection, we consider a Cauchy problem for a pseudoparabolic semilinear equation in RN .

526


Definition 5.7.1 (class (K)). A function u.x; t / 2 C .1/ .Œ0; TI W 2;1 .Œ1 C jxj2 ˛=.2.1Cq// I R3 //;

˛ > 2;

which satisfies Eq. (5.217) almost everywhere, is called a strong generalized solution of problem (5.217), (5.218). In this case, we say that the function u.x; t / belongs to the class (K). Naturally, we obtain that u0 .x/ 2 W 2;1 .Œ1 C jxj2 ˛=Œ2.1Cq/ I R3 /: Theorem 5.7.2. In the class (K), problem (5.217), (5.218) is equivalent to the integral equation Z Z t 1 e .ts/ q t ds dy (5.219) juj u.y; s/: u.x; t / D u0 .x/e C 4 0 jx yj R3 Proof. Note that the iterated integral in the right-hand side of the integral equation (5.219) is a thermal potential (see, e.g., [158]). Denote by u./ O D FO Œu./ the Fourier transform with respect to the variable x 2 R3 in the sense of the slow-growth distributions P 0 .R3 / (see, e.g., [421]). Then, since u belongs to the class (K), we have u 2 C .1/ .Œ0; TI L1 .R3 //, jujq u 2 C .1/ .Œ0; TI L1 .R3 / \ L1 .R3 //. In this case, from (5.217) and (5.218) we obtain the following equalities in P 0 .R3 /:

1

1 @u.; O t/ C u.; O t / D 2 jujq u.; t /; @t jj Z t e .ts/ q t ds juj u.; s/: u.; O t / D uO0 ./e C jj2 0

1

(5.220)

Note that if f 2 L1loc .R3 /, g 2 L1 .R3 / \ L1 .R3 /, then FO Œf g D FO Œf FO Œg in the sense P 0 .R3 / (see, e.g., [421]). Since jujq u 2 C.Œ0; TI L1 .R3 / \ L1 .R3 // and jxj1 2 L1loc .R3 /, from (5.220) we obtain u.x; t / D u0 .x/e t C

1 4

Z

Z

t

ds 0

R3

dy

e .ts/ q juj u.y; s/: jx yj

Thus, each solution of problem (5.217), (5.218) in the class (K) is a solution of the integral equation (5.219). From (5.219) and the fact that u.x; t / belongs to the class (K) we obtain the initialvalue problem for the integro-differential equation Z 1 1 @u CuD jujq u.y; t /; u.x; 0/ D u0 .x/: dy (5.221) @t 4 R3 jx yj

527


Let f .x; t / .jujq u/.x; t /, q > 0. By the fact that u.x; t / 2 C .1/ .Œ0; TI W 1;1 .Œ1 C jxj2 ˛=.2.1Cq// I R3 //;

˛ > 2;

it is easy to verify that f .x; t / 2 C .1/ .Œ0; TI W 1;1 .Œ1 C jxj2 ˛=2 I R3 //;

˛ > 2:

Consider the volume potential Z VŒf .x; t / D

R3

dy

1 f .y; t /: jx yj

(5.222)

Let x 2 BR .0/ be a ball of radius R > 0 centered at the origin. Then from (5.222) we have Z Z 1 1 VŒf .x; t / D dy dy f .y; t / C f .y; t / jx yj jx yj BR .0/ R3 nBR .0/ V1 Œf .x; t / C V2 Œf .x; t /: Obviously, V2 Œf .x; t / D 0, and from the results of the work [181] we obtain VŒf .x; t / D 4 f .x; t /. Thus, from (5.221), by the fact that u.x; t / belongs to the class (K), we conclude that for any t 2 Œ0; T for certain T > 0 and almost all x 2 R3 , Eq. (5.217) is valid. Theorem 5.7.2 is proved.

5.7.2 Theorems on the existence/nonexistence of global-on-time solutions of the integral equation (5.219) In this subsection, we obtain sufficient conditions of the global-on-time solvability and sufficient conditions of the blow-up for a finite time. Theorem 5.7.3. For any function u0 .x/ 2 W 2;1 .Œ1 C jxj2 ˛=.2.1Cq// I R3 / under the condition 3 < ˛ 1 C q, q > 2, for a sufficiently small initial function u0 .x/, 2 ˛=Œ2.1Cq /

sup .1 C jxj /

x2R3

ˇ ˇ 3;3 ˇ 2 3 ˇ X X ˇ @u0 ˇ ˇ @ u0 ˇ ˇ ˇ ˇ ˇ ju0 j C ˇ @x ˇ C ˇ @x @x ˇ ı; i i j i D1

i;j D1;1

there exists a unique solution of the integral equation (5.219) of the class .1/

u.x; t / 2 Cb .Œ0; C1I W 2;1 .Œ1 C jxj2 ˛=Œ2.1Cq/ I R3 //:

ı > 0;

528


Proof. We use the method of successive approximations: u1 .x; t / D u0 .x/e t ; unC1 .x; t / D u0 .x/e t C A.un /.x; t /; Z Z t 1 e .ts/ A.un / ds dy jun jq un .x y; s/: 4 0 jyj R3

(5.223)

The iterated integral in the definition of the operator A is a Lebesgue integral. Introduce the norms 3 X @v 2 ˛=Œ2.1Cq/

kvk0 sup Œ1 C jxj jv.x; t /j; kvk1 kvk0 C @x ; i 0 x2R3 ;t0 kvk2 kvk0 C

i D1

3 X i D1

3;3 X @v @2 v C @x @x @x : i 0 i j 0 i;j D1;1

For further consideration, we require some auxiliary results. 1Cq 1. kA.u/k2 C kuk2 .

Indeed, from Lemma 5.1.6 we obtain the inequality ´ Z .1 C jxj2 /˛=.2.1Cq// C1 ."/; dy 2 ˛=2 jyj.1 C jx yj / C2 ."/jxj˛=.1Cq/1 ; R3

0 jxj < "; jxj " > 0;

for 3 < ˛ 1 C q. Therefore, the following inequalities hold: jA.u/j.1 C jxj2 /˛=.2.1Cq// C kuk1Cq ; kA.u/k0 C kuk1Cq ; 0 0 ˇ ˇ ˇ @A.u/ ˇ ˇ ˇ .1 C jxj2 /˛=.2.1Cq// C kukq @u : 0 ˇ @x ˇ @x i i 0 Hence we obtain 3 3 X X @A.u/ @u C kukq 0 @x @x i i 0 0 iD1

i D1

3 3 X X @u q @u 1Cq kuk C kuk0 C C 0 @x @x C kuk1 : i 0 i 0 i D1

i D1

Finally, ˇ 2 ˇ 2 ˇ @ A.u/ ˇ ˇ .1 C jxj2 /˛=.2.1Cq// C kukq @ u C C kukq1 @u @u ˇ 0 0 ˇ @x @x ˇ @xi @xj 0 @xi 0 @xj 0 i j C kuk1Cq ; 2

3;3 2 X @ A.u/ 1Cq @x @x C kuk2 : i j 0

i;j D1;1

529


Thus, 1Cq

kA.u/k2 C kuk2

:

(5.224)

The following lemma is similar to the corresponding lemma of [275]. Lemma 5.7.4. Let M > 0 and q > 2. For any functions z1 .x/ and z2 .x/ such that kzk k2 M , k D 1; 2, the following estimate holds: kA.z1 / A.z2 /k2 CM q kz1 z2 k2 : Proof. Introduce the notation z1i

@z1 ; @xi

z2i

@z2 ; @xi

z1ij

@2 z1 ; @xi @xj

z2ij

@2 z2 : @xi @xj

The following relations hold: .jzjq z/xi D .1 C q/jzjq zxi ; .jzjq z/xi xj D .1 C q/jzjq zxi xj C .1 C q/qjzjq2 zzxi zxj : We have ˇ ˇ ˇjz1 jq z1i jz2 jq z2i ˇ

ˇ ˇ jz1 jq jz1i z2i j C jz2i jˇjz1 jq jz2 jq ˇ

jz1 jq jz1i z2i j C jz2i jq max¹jz1 jq1 ; jz2 jq1 ºjz1 z2 j 1 ŒM q kz1i z2i k0 C qMM q1 kz1 z2 k0 .1 C jxj2 /˛=2 1 M q Œkz1i z2i k0 C qkz1 z2 k0 ; .1 C jxj2 /˛=2 ˇ ˇ ˇjz1 jq z1ij jz2 jq z2ij ˇ

1 M q Œkz1ij z2ij k0 C qkz1 z2 k0 ; .1 C jxj2 /˛=2 ˇ ˇ ˇjz1 jq2 z1 z1i z1j jz2 jq2 z2 z2i z2j ˇ ˇ ˇ ˇ ˇ ˇjz1 jq2 z1 z1i z1j jz2 jq2 z2 z1i z1j ˇ C ˇjz2 jq2 z2 z1j z1i jz2 jq2 z2 z1j z2i ˇ ˇ ˇ C ˇjz2 jq2 z2 z2i z1j jz2 jq2 z2 z2i z2j ˇ

1 ŒM 2 .q 1/M q2 kz1 z2 k0 C M q1 M kz1i z2i k0 .1 C jxj2 /˛=2 C M q1 M kz1j z2j k0

1 M q Œ.q 1/kz1 z2 k0 C kz1i z2i k0 C kz1j z2j k0 ; .1 C jxj2 /˛=2

530


ˇ ˇ ˇjz1 jq z1 jz2 jq z2 ˇ .1 C q/ max¹jz1 jq ; jz2 jq ºjz1 z2 j

1 M q .1 C q/kz1 z2 k0 : .1 C jxj2 /˛=2

Hence by Lemma 5.1.4 we have jA.z1 / A.z2 /j.1 C jxj2 /˛=.2.1Cq// Z Z t ˇ ˇ 1 ds e .ts/ dy F .x; y/.1 C jx yj2 /˛=2 ˇjz1 jq z1 jz2 jq z2 ˇ.x y; s/ 4 0 R3 1Cq c1 M q kz1 z2 k0 ; 4 Z .1 C jxj2 /˛=.2.1Cq// dy F .x; y/; F .x; y/ c1 D sup I .1 C jx yj2 /˛=2 jyj x2R3 R3 ˇ ˇ ˇ @A.z1 / @A.z2 / ˇ ˇ .1 C jxj2 /˛=.2.1Cq// ˇ ˇ ˇ @x @x i i Z Z ˇ ˇ 1Cq t ds e .ts/ dy F .x; y/.1 C jx yj2 /˛=2 ˇjz1 jq z1i jz2 jq z2i ˇ 4 0 R3 1Cq c1 M q Œqkz1 z2 k0 C kz1i z2i k0 ; 4 ˇ 2 ˇ ˇ @ A.z1 / @2 A.z2 / ˇ 2 ˛=.2.1Cq// ˇ ˇ ˇ @x @x @x @x ˇ .1 C jxj / i j i j Z Z 1Cq t .ts/ ds e dy F .x; y/.1 C jx yj2 /˛=2 jjz1 jq z1ij jz2 jq z2ij j 3 4 0 R Z Z 1Cq t C ds e .ts/ dy F .x; y/.1 C jx yj2 /˛=2 3 4 0 R ˇ ˇ ˇjz1 jq2 z1 z1i z1j jz2 jq2 z2 z2i z2j ˇ 1Cq q M Œqkz1 z2 k0 C kz1ij z2ij k0 4 q.1 C q/ q M Œ.q 1/kz1 z2 k0 C kz1i z2i k0 C kz1j z2j k0 : C c1 4

c1

From the obtained estimates we have the inequality kA.z1 / A.z2 /k2 CM q kz1 z2 k2 : Lemma 5.7.4 is proved.

(5.225)


531

Return to the proof of Theorem 5.7.3. From the iterative scheme (5.223) and estimate (5.224) we have kunC1 k2 ı C C kun k1Cq ; 2

ku0 k ı:

Hence it follows (see, e.g., [275]) that kun k M.ı/ ! 0 as ı ! C0. From (5.225) we obtain the relation kunC1 un k2 D kA.un / A.un1 /k2 CM q .ı/kun un1 k2 : On the other hand, for sufficiently small ı > 0 we obtain 1 kunC1 un k2 kun un1 k2 : 2

CM q .ı/ < 1=2;

Hence we obtain the strong convergence of the recursive sequence un ! u in L1 .0; C1I W 2;1 .Œ1 C jxj2 ˛=.2.1Cq// I R3 //. Now prove the uniqueness of a solution of the integral equation (5.219). Let w1 and w2 be two solutions of (5.219); then we have Z t Z 1 1 .ts/ jjw1 jq w1 jw2 jq w2 j jw1 w2 j ds e dy 4 0 jyj R3 Z Z F .x; y/ 1Cq t .ts/ ds e dy 2 ˛=Œ2.1Cq/

3 4 0 Œ1 C jxj R M q jw1 w2 j.1 C jx yj2 /˛=Œ2.1Cq/ ;

(5.226)

.1 C jxj2 /˛=Œ2.1Cq/

: jyj.1 C jx yj2 /˛=2

F .x; y/ From (5.226) we obtain

kw1 w2 k0 .t / C

Z

t 0

ds kw1 w2 k0 .s/;

kwk0 sup .1 C jxj2 /˛=Œ2.1Cq/ jwj: x2R3

By the Gronwall–Bellman lemma (see [112]) we obtain that w1 D w2 for all .x; t / 2 R3 Œ0; C1/. Now we must prove only that .1/

u.x; t / 2 Cb .Œ0; C1I W 2;1 .Œ1 C jxj2 ˛=.2.1Cq// I R3 //: Note that the thermal potential Z VŒu.x; t /

Z

t

ds 0

R3

dy

e .ts/ q juj u.y; s/ jx yj

(5.227)

532


is smoothing by t , and it is easy to verify that the conditions u.x; t / 2 L1 .0; C1I W 2;1 ..1 C jxj2 /˛=Œ2.1Cq/ I R3 // imply the fact that VŒu.x; t / 2 AC.Œ0; C1I W 2;1 ..1 C jxj2 /˛=Œ2.1Cq/ I R3 //; hence from (5.219) we directly obtain the inclusions u.x; t / 2 AC.Œ0; C1I W 2;1 ..1 C jxj2 /˛=Œ2.1Cq/ I R3 //; VŒu.x; t / 2 C .1/ .Œ0; C1I W 2;1 ..1 C jxj2 /˛=Œ2.1Cq/ I R3 //; and from the explicit form of the thermal potential (5.227) we obtain .1/ VŒu.x; t / 2 Cb .Œ0; C1I W 2;1 ..1 C jxj2 /˛=Œ2.1Cq/ I R3 //:

Therefore, .1/ u.x; t / 2 Cb .Œ0; C1I W 2;1 ..1 C jxj2 /˛=Œ2.1Cq/ I R3 //:

Theorem 5.7.3 is proved. (1) Let the function u0 .x/ 2 L1Cq .R3 / L1 .R3 / be such that Z Z 1 .ju0 jq u0 /.x/.ju0 jq u0 /.y/ qC2 : dx dy 0 < ku0 kqC2 < 4 R3 R3 jx yj

Theorem 5.7.5.

Then there exists 0 T0 < C1 such that u.x; t / … L1 .Œ0; T0 /I L1Cq .R3 / \ L1 .R3 //: (2) Let q 2 .0; 2=3/, u0 .x/ 2 C0 .R3 /, u0 0, and u0 > 0 on a set of positive measure. Then for any positive solution of the integral equation (5.219), there exists 0 T0 < C1 such that u.x; t / … L1 .Œ0; T0 /I L1Cq .R3 / \ L1 .R3 //: Proof. Let a global solution of the integral equation (5.219) exist under the conditions formulated either in (1) or (2) and belong to the class L1 .Œ0; C1/I L1Cq .R2 / \ L1 .R2 //: Then, by the fact that the operator in the right-hand side of the integral equation (5.219) is smoothing and the solution belongs to the class C .1/ .Œ0; C1/I L1Cq .R2 / \ L1 .R2 //:


533

In this case, the integral equation (5.219) can be written in the form of the following Cauchy problem for the integro-differential equation: Z 1 .jujq u/.y; t / u t .x; t / C u.x; t / D dy (5.228) ; u.x; 0/ D u0 .x/: 4 R3 jx yj Introduce the new function w D e t u; then (5.228) takes the form Z 1 qt .jwjq w/.y; t / dy e : w t .x; t / D 4 jx yj R3

(5.229)

Let conditions (1) of Theorem 5.7.5 hold; then, multiplying both sides of (5.229) by .jwjq w/.x; t / and integrating over x 2 R3 , we obtain Z Z 1 qt .jwjq w/.x; t /.jwjq w/.y; t / 1 d qC2 dx dy kwkqC2 D e : (5.230) q C 2 dt 4 jx yj R3 R3 On the other hand, ˇ2 Z ˇZ ˇ ˇ q ˇ ˇ dx jwj ww tˇ ˇ 3

Z

R3

R

dx jwjq w 2t

R3

dx jwjqC2 :

(5.231)

Now we multiply both sides of (5.229) by the function .jwjq w/ t ; after integrating over x 2 R3 we obtain Z Z Z 1 e qt d .jwjq w/.x; t /.jwjq w/.y; t / dx .jwjq w/ t w t D dx dy : 4 2 dt R3 R3 jx yj R3 (5.232) It follows from the latter equality that Z dx jwjq w 2t D R3

Z

where f .t /

R3

Z R3

dx dy

1 1 qt df e ; 2.1 C q/ 4 dt

(5.233)

.jwjq w/.x; t /.jwjq w/.y; t / : jx yj

Introduce the notation ' kwkqC2 qC2 . Estimates (5.230)–(5.233) lead to the inequality

d' dt

2

Z .q C 2/2

Z R3

dx jwjq w 2t

R3

dx jwjqC2

d' q C 2 qt d e e qt 'I 2.q C 1/ dt dt

hene obtain (see [210]) .' 0 .t //2 ' 0 .t / ' 00 .t / ˛ Cq 0; ˛ 1C˛ '.t / '.t / '.t /˛

˛ 1 C ;

q=.2 C q/:

(5.234)

534


From (5.234) we obtain e qt

' 0 .t / ' ˛ .t /

0

' 0 .t / q C 2 qt e f .0/'.0/˛ : ˛ ' 4

0;

1 , which can be reduced to the form Hence we obtain the inequality '.0/ f .0/ 4 qC2 ku0 kqC2

1 4

Z R3

Z R3

dx dy

.ju0 jq u0 /.x/.ju0 jq u0 /.y/ : jx yj

This inequality contradicts condition (1) of Theorem 5.7.5. The contradiction obtained proves the existence of 0 T0 < C1 such that u.x; t / … L1 .Œ0; T0 /I L1Cq .R3 / \ L1 .R3 //: Now we prove item (2) of Theorem 5.7.5. From (5.228) in the case of a positive solution u.x; t / 0, we obtain the integro-differential inequality Z 1 1 dy u1Cq .y; t /; ut C u 4 BR .0/ jx yj BR .0/ is a ball of radius R > 0 centered in the origin. Let G.x; y/ be the Green function of the first boundary-value problem for the Laplace equation in the ball BR .0/. We have 3 .x/

p 1 C.R/ p J1=2 . 3 r/; r

3

2 ; R2

C.R/

C ; R5=3

where 3 .x/ is the first eigenfunction of the Laplace operator that corresponds to the first eigenvalue 3 . It is known that 3 .x/ > 0 for x 2 BR .0/ and 3 .x/ D 0 for x 2 @BR .0/. We choose C.R/ such that the condition Z dx 3 .x/ D 1 BR .0/

holds. By the properties of the Green function we have (see, e.g., [421]) Z dy G.x; y/u1Cq .y; t /: ut C u

(5.235)

BR .0/

Multiply both sides of inequality (5.235) by 3 .x/ and integrate by x 2 BR .0/; using the Jensen inequality (see, e.g., [445]) we obtain Z df1 1 1 1Cq C f1 dy 3 .y/u1Cq .y; t / f .t /; dt 3 BR .0/ 3 1


535

Z

where f1 .t / D

dy BR .0/

3 .y/u.y; t /:

Integrating the obtained ordinary differential inequality we obtain the inequality Z 1=q dy 3 .y/u0 .y/ (5.236) 3 f1 .0/ D BR .0/

r

Z

Z

Z

p 1 1 dr r 2 p sin. 3 r/ r 4 3 0 0 0 Z Z 2 Z R

1 1=2 d

d' dr r sin r u0 .r; ; '/: C 5=2 R R R 0 0 0 D CR

5=2

2

2

d

R

d'

Note that for any 2 .0; 1/, there exists sufficiently small x0 ./ > 0 such that sin x=x 1 , x 2 Œ0; x0 ./. Let supp u0 BR .0/; for sufficiently large R > R0 from (5.236) we have Z Z 2 Z R0 d

d' dr r 2 u0 .r; ; '/ D CR3 : R2=q R3 0

0

0

Therefore, R3 CR2=q . If q 2 .0; 2=3/, then for sufficiently large R > R0 > 0 the latter inequality does not hold. Theorem 5.7.5 is proved. Theorem 5.7.6. For any function u0 .x/ 2 L1 ..1 C jxj2 /˛=Œ2.1Cq/ I R3 /; 3 < ˛ 1 C q, q > 2, there exists T1 D T1 .u0 / > 0 such that there exists a unique solution of the integral equation (5.219): u.x; t / 2 C .1/ .Œ0; TI L1 ..1 C jxj2 /˛=Œ2.1Cq/ I R3 //

8T 2 .0; T1 /;

and, if qC2 0 < ku0 kqC2
0, q2 > 0 hold for N D 1; 2. Then there exists maximal T0 > 0 such that a unique strong generalized solution of problem (5.240)–(5.242) of the class C .1/ .Œ0; T0 /I H1 .// exists, and either T0 D C1 or T0 < C1, and in the latter case, the limit relation lim sup kukH1 ./ D C1 t"T0

holds.

(5.243)

539

Section 5.8 Blow-up in generalized Boussinesq equation

Theorem 5.8.3. Let Hypothesis 5.8.2 hold. If q1 < q2 and

q C2

ku0 kq22 C2

1 1 1 q C2 ku0 kq22 C2 > kru0 k22 C q2 C 2 2 q1 C 2 Z 2 > kru0 k2 C dx ju0 jq1 C2 C

Z

dx ju0 jq1 C2 ;

ˇ ˛1

1=2

1 1 q1 C 1 kru0 k22 C ku0 k22 C 2 2 q1 C 2

Z

dx ju0 jq1 C2 ;

then T0 2 ŒT1 ; T2 and the limit relation (5.243) holds, where T2 D A1 ˆ1˛ ; 0

2 ˆq2 =2 ; q2 B2 0

T1 D

jq1 q2 j2 .q1 C 2/ 2 2 ˇD q C ; q2 > q1 ; q2 q1 2 2.q1 C 1/2 Z 1 1 q1 C 1 2 2 dx ju0 jq1 C2 ; ˆ0 D ˆ.0/ D kru0 k2 C ku0 k2 C 2 2 q1 C 2 Z q2 C2 0 2 dx ju0 jq1 C2 ; ˆ .0/ D ku0 kq2 C2 kru0 k2

q 2 C q1 C 4 ; ˛D 2.q1 C 2/

2

2

2˛

A D .˛ 1/ ˆ

0

2

.ˆ .0// .˛

1/ˇˆ22˛ ; 0

B2 D B1q2 C2 2.q2 C2/=2 ;

B1 is the best constant of the embedding H1 ./ Lq2 C2 ./. Proof. Take the function u 2 C .1/ .Œ0; T0 /I H1 .// as ; then we obtain the first energy equality d dt

1 q1 C 1 1 kruk22 C kuk22 C 2 2 q1 C 2 C

Z

q1 C2

dx juj

kruk22

Z C

dx jujq1 C2 D kukqq22 C2 C2 : (5.244)

Now take as the function u0 2 C.Œ0; T0 /I H1 .//. Then we obtain the second energy equality kru0 k22 C ku0 k22 C .q1 C 1/

Z

dx jujq1 .u0 /2

d d 1 1d 1 q C2 kukq22 C2 kruk22 D q2 C 2 dt 2 dt q1 C 2 dt

Z

dx jujq1 C2 : (5.245)

540


Let 1 1 q1 C 1 ˆ.t / kruk22 C kuk22 C 2 2 q1 C 2

Z

dx jujq1 C2 :

(5.246)

It is easy to verify that, since the solution u.x; t / of the problem belongs to the class C .1/ .Œ0; T0 /I H1 .//, and by the conditions of Hypothesis 5.8.2 we have that for any element from H1 ./, the trace from Lq1 C2 ./ is defined and H1 ./ Lq2 C2 ./. In this case, ˆ.t / 2 C .1/ Œ0; T0 /. Finally, by virtue of the first energy equality, we obtain that ˆ.t / 2 C .2/ Œ0; T0 /:

(5.247)

It is easy to verify that .ˆ0 /2 .q1 C 2/ˆJ;

J kru0 k22 C ku0 k22 C .q1 C 1/

Z

dx jujq1 .u0 /2 : (5.248)

Indeed, by virtue of (5.246) and (5.247), the following inequalities hold: Z 2 Z Z 0 2 0 0 q1 0 dx .ru ; ru/ C dx u u C .q1 C 1/ dx juj uu .ˆ / D Z Z kru0 k22 kruk22 C ku0 k22 kuk22 C .q1 C 1/2 dx jujq1 .u0 /2 dx jujq1 C2

C 2kru0 k2 kruk2 ku0 k2 kuk2 1=2 Z 1=2 Z C 2.q1 C 1/ dx jujq1 ju0 j2 kruk2 dx jujq1 C2 kru0 k2

Z C 2.q1 C 1/

q1

0 2

dx juj ju j

1=2

Z kuk2

q1 C2

dx juj

Z kruk22 C kuk22 C .q1 C 1/ dx jujq1 C2

kru0 k22

C

ku0 k22

C .q1 C 1/

Z q1

0 2

dx juj .u /

Z 1 1 q1 C 1 dx jujq1 C2 kruk22 C kuk22 C 2 2 q1 C 2 Z 0 2 0 2 q1 0 2 kru k2 C ku k2 C .q1 C 1/ dx juj .u /

.q1 C 2/

.q1 C 2/ˆJ; i.e., inequality (5.248) holds.

1=2

ku0 k2

Section 5.8 Blow-up in generalized Boussinesq equation

541

From (5.244) and (5.245) we obtain the following inequalities: Z d 1 d d q2 C2 2 q1 C2 J dx juj kukq2 C2 kruk2 q2 C 2 dt dt dt Z d d q2 q1 q2 2 kruk2 C dx jujq1 C2 2.q2 C 2/ dt .q1 C 2/.q2 C 2/ dt ˇ ˇ ˇd ˇ 1 q2 00 2ˇ ˇ ˆ C kruk2 ˇ ˇ q2 C 2 2.q2 C 2/ dt ˇ Z ˇ ˇd ˇ jq1 q2 j q1 C2 ˇ ˇ C dx juj (5.249) ˇ ˇ: .q1 C 2/.q2 C 2/ dt The following auxiliary inequalities hold: ˇ ˇ 2 ˇd ˇ 1 q2 q2 ˇ kruk2 ˇ " kru0 k2 C ˆ; 2ˇ 2 ˇ 2.q2 C 2/ dt 2 q2 C 2 " ˇ Z ˇ ˇd ˇ jq1 q2 j q1 C2 ˇ ˇ dx juj ˇ ˇ .q1 C 2/.q2 C 2/ dt Z " jq1 q2 j2 .q1 C 2/ .q1 C 1/ dx jujq1 .u0 /2 C ˆ: 2 .q1 C 1/2 .q2 C 2/2 2" From inequalities (5.249)–(5.251) we obtain the inequality 2 1 1 q2 jq1 q2 j2 .q1 C 2/ J ˆ00 C "J C ˆC ˆ: q2 C 2 q2 C 2 " .q1 C 1/2 .q2 C 2/2 2"

(5.250)

(5.251)

(5.252)

From (5.248) and (5.252) we obtain the second-order ordinary differential inequality (see [210]) ˆˆ00 ˛.ˆ0 /2 C ˇˆ2 0; where q2 C 2 ˛D Œ1 "; q1 C 2

(5.253)

q22 1 jq1 q2 j2 .q1 C 2/ ˇD C : " q2 C 2 2.q1 C 1/2 .q2 C 2/

Using the arbitrariness of " > 0, we obtain the condition under which ˛ > 1 for q2 > q1 . Indeed, take " 2 .0; .q2 q1 /=.q2 C 2//. Now we require the validity of the following initial condition: Z 1 1 1 q C2 dx ju0 jq1 C2 : (5.254) ku0 kq22 C2 kru0 k22 C q2 C 2 2 q1 C 2 If condition (5.254) holds, then from the second energy equality (5.245) we obtain the condition Z 1 1 1 q C2 kukq22 C2 > kruk22 C dx jujq1 C2 : (5.255) q2 C 2 2 q1 C 2

542


In its turn, from (5.255) and (5.244) we obtain the inequality ˆ0 0:

(5.256)

By virtue of (5.253) we obtain the inequality Z00 Œ˛ 1ˇZ 0;

Z D ˆ1˛ :

(5.257)

Since Z0 D .1 ˛/ˆ˛ ˆ0 , by virtue of (5.256) we conclude that Z0 0. Then from (5.257) we have .Z0 /2 A2 C Œ˛ 1ˇZ2 ; where

(5.258)

>0 A2 D .˛ 1/2 ˆ2˛ .ˆ0 .0//2 .˛ 1/ˇˆ22˛ 0

under the condition

0

ˆ .0/ >

ˇ ˛1

1=2 ˆ.0/:

Then from (5.258) we obtain jZ0 j A;

Z Z0 At:

Therefore, for the time T0 , the upper estimate T0 T2 holds, where T2 D A1 Z0 . Now consider the lower estimate for the blow-up time of a strong generalized solution of Eqs. (5.240)–(5.242). Indeed, from the first energy equality (5.244) we obtain the inequality dˆ q C2 q C2 kukq22 C2 B12 Œkruk22 C kuk22 .q2 C2/=2 B2 ˆ ; dt

D 1 C q2 =2;

where B2 D Bq12 C2 2.q2 C2/=2 , B1 is the constant of the best embedding H1 ./ Lq2 C2 ./: Hence it follows that ˆ.t / Theorem 5.8.3 is proved.

ˆ0 Œ1

q2 21 ˆq02 =2 B2 t 2=q2

:

Chapter 6

Numerical methods of solution of initial-boundary-value problems for Sobolev-type equations

In this chapter, we consider some methods of numerical solution of equations unsolved relative to the time derivative. First, taking a pseudoparabolic equation as an example, we consider the possibility of applying the theory of dynamic potentials for numerical solution of initial-boundary-value problems. In the course of arguing, the problem is reduced to an integral equation, for which an effective numerical algorithm is constructed with the possibility of calculations on condensing grids with precision control. Further, we consider certain nonlinear pseudoparabolic equations analyzed in previous chapter. By using the strict methods of lines, the problem can be reduced either to an implicit system of ordinary differential equations of large dimension or to a differential-algebraic system. To solve these problems, we use the complex Rosenbrock scheme applicable for large stiffness problems. Calculations on condensing grids allow one to determine the blow-up time of solutions to within grid step order.

6.1

Numerical methods for initial-boundary-value problems for linear equations of Sobolev type. Method of dynamic potentials

One of the most fruitful methods of analyzing boundary-value-problems is the method of potentials. Applying this method to analyzing Sobolev-type equations requires the construction of the so-called dynamic potentials, which are analogs of the classical volume potential and potentials of simple and double layers. The dynamic potentials for the Sobolev equation u t t C ! 2 ux3 x3 D 0 were constructed and analyzed in [215, 440]. The theory of dynamic potentials for gravity-gyroscopic waves in the Boussinesq approximation (ˇ D 0) @2 Œ3 u ˇ 2 u C !02 2 u C ˛ 2 Œux3 x3 ˇ 2 u D 0; (6.1) @t 2 was developed in the monographs [164, 165]. Gabov constructed dynamical logarithmic and angular potentials, i.e., analogs of the corresponding potentials for the Laplace equation. Further, the theory of dynamic potentials for Eq. (6.1) was devel-

544

Chapter 6 Numerical methods for Sobolev-type initial-boundary-value problems

oped by Gabov, Pletner, and Korpusov. In 1989, Gabov paid attention to the fact that nonstationary ion-sound waves in unmagnetized plasma are also described by a Sobolev-type equation .u u/ t t C u D 0:

(6.2)

The analysis of initial-boundary-value problems for Eq. (6.2) by the method of dynamic potentials can be found in [9]. In the mathematical modelling of wave processes in media with strong anisotropic dispersion (for example, ferromagnetics, ferrites, dielectrics, plasma in external fields and etc.), vector-valued systems of partial differential equations appear. Recent researches show that, in some cases, such vector-valued systems can be reduced to a scalar partial differential equation of high degree, an equation of Sobolev type. One should suppose that for most of them the theory of dynamic potentials can be successfully constructed. Our goal is to show that dynamic potentials can be used not only for the proof of the solvability of initial-boundary-value problems but also for constructing an effective method of numerical solution.

6.1.1 Dynamic potentials for one equation In Chapter 1, we obtained an equation that describes quasi-stationary processes in semiconductors; in the linear approximation, this equation has the from .u u/ t C ˛u C ˇu D 0;

(6.3)

where ˛ 0, ˇ can have any sign, u has the sense of the electric potential. We construct dynamic potentials of simple and double layer for Eq. (6.3) in the twodimensional case, i.e., @2 u @2 u u D 2 C 2 : @x1 @x2 Let the domain R2 be bounded by a simple, closed, smooth curve of the Lyapunoff type. Consider the potentials Z V Œ .x; t / D .t; s/K0 .jx y.s/j/ ds

C

Z tZ 0

.; s/F .jx y.s/j ; t / ds d

D V1 Œ .x; t / C V2 Œ .x; t /; Z @ .t; s/ K0 .jx y.s/j/ ds W Œ .x; t / D @ns Z tZ @ C .; s/ F .jx y.s/j ; t /ds d @ns 0 D W1 Œ .x; t / C W2 Œ .x; t /;

(6.4)

(6.5)

545

Section 6.1 Numerical solution of problems for linear equations

where

s

³ pˇ (6.6) x K0 .x/ exp.p t/ dp: K0 pC˛ i 1 q pˇ Note that the branch of the root .p/ D pC˛ in formula (6.6) is chosen so that .p/ ! 1 as p ! 1 in the half-plane Re p > 0. We mean by K0 .z/ the main branch of the corresponding multi-valued function with a cut along the negative real p semiaxis, jx yj D .x1 y1 /2 C .x2 y2 /2 , > max.˛; ˇ/, ns is an external normal to at the point y.s/ D .y1 .s/; y2 .s//; we choose as the parameter s the arclength, i.e., use the normal parametrization of the curve . Potential (6.4) is called a dynamic potential of a simple layer, and potential (6.5) is called a dynamic potential of a double layer for Eq. (6.3). Before analyzing the behavior of potentials near the boundary, we obtain a representation for the function F .x; t /. The following equality holds: s ² ³ Z C1 pˇ 1 K0 .ix/ 2 K0 x K0 .x/ D d: pC˛

i 1 C1 2 C pˇ 1 F .x; t / D 2 i

Z

Ci1 ²

pC˛

(6.7) The proof is based on the possibility of closing a contour of integration on the upper half-plane and calculating the residues inside the contour. Substitute (6.7) in (6.6) and change the order of integration: ³ ² Z C1 Z i 1 1 exp.pt / dp d K0 .ix/ F .x; t / D 2 pˇ 2 1 i 1 2 C 1 2 C pC˛ Z C1 Z i 1 .˛ C ˇ/ exp.pt / 1 dp d D 2 K0 .ix/ ˇ ˛ 2 2 1 i 1 .2 C 1/2 .p / 2 C1 Z C1 .˛ C ˇ/ expŒ.ˇ ˛2 /t =2 C 1 1 K0 .ix/ d: D

i 1 .2 C 1/2 Taking into account the relation K0 .ix/ D K0 .ix/ and the fact that the second factor is an odd function, we obtain Z C1 .˛ C ˇ/ 2 2 K0 .ix/ 2 exp t d F .x; t / D Im

. C 1/2 2 C 1 0 Z C1 .˛ C ˇ/ ˇ ˛2 J0 .x/ 2 exp t d D . C 1/2 2 C 1 0 Z C1 .˛ C ˇ/ .˛ C ˇ/t D exp.˛t / J0 .x/ 2 exp d: (6.8) . C 1/2 2 C 1 0

546


Above we have taken into account the relation K0 .iz/ D

i .1/ H .z/; 2 0

H0.1/ .z/ D J0 .z/ C iN0 .z/:

z > 0;

Remark 6.1.1. In formula (6.8), the integral converges uniformly in the domain .x; t / 2 Œ0; C1/ Œ0; T , where T is any fixed positive number; therefore, F .x; t / is a continuous function for x 0 and t 0. We transform the integral in formula (6.8) by representing the exponent as the series of powers of .˛ C ˇ/t =.2 C 1/ and using the fact that in this case we can choose the orders of summation and integration: Z

C1 0

.˛ C ˇ/t .˛ C ˇ/ J0 .x/ 2 exp d . C 1/2 2 C 1 Z

D

D

C1 0

J0 .x/

C1 .˛ C ˇ/ X .˛ C ˇ/n t n d .2 C 1/2 nŠ.1 C 2 /n nD0

C1 X t n .˛

C ˇ/nC1 nŠ

nD0

Z 0

C1

J0 .x/

.2

d: C 1/2Cn

The integral here is a Hankel transform; it is equal to Z

C1 0

J0 .x/

x nC1 d D KnC1 .x/: .2 C 1/2Cn 2nC1 .n C 1/Š

Finally we have

F .x; t / D exp.˛t /

C1 X tn nD0

.˛ C ˇ/nC1 x nC1 KnC1 .x/: nŠ 2nC1 .n C 1/Š

(6.9)

Formula (6.9) is useful in two aspects: first, it allows one to analyze the behavior of the function F .x; t / near x D 0; second, this series converges very fast and can be used for approximate calculation of F .x; t / with high precision. Further we need analogs of formulas (6.8) and (6.9) for the derivative @F .x; t /=@x: @ F .x; t / D exp.˛t / @x

Z 0

C1

.˛ C ˇ/2 .˛ C ˇ/t d: (6.10) J1 .x/ 2 exp . C 1/2 2 C 1


547

Acting according to the scheme used above to obtain formula (6.9), we obtain C1

X t n .˛ C ˇ/nC1 x nC1 @F .x; t / D exp.˛t / Kn .x/: @x nŠ 2nC1 .n C 1/Š

(6.11)

nD0

Remark 6.1.2. Since x nC1 Kn .x/ x2n1 .n 1/Š for n 1, ˇ ˇ ˇ @F .x; t / ˇ ˇ ˇ ˇ @x ˇ exp.˛t /.K0 .x/ C C.t //x:

(6.12)

.1/ In what follows, we write v.t; s/ 2 C0 .Œ0; C1/I C.// if v.t; s/ 2 C .1/ .Œ0; C1/I C.// and v.0; s/ D 0. .1/ Lemma 6.1.3. Let v.t; s/ 2 C0 .Œ0; C1/I C.//. Then for any x … and t 0, the function u.x; t / D V Œ.x; t / satisfies Eq. (6.3) in the classical sense and the zero initial conditions uj tD0 D 0.

Lemma 6.1.3 is proved by a direct verification. Remark 6.1.4. The dynamic potential of a simple layer is a continuous function for any x 2 R2 and t 0. Potential (6.4) contains two terms, V1 Œ.x; t / and V2 Œ.x; t /. The first term is the potential of a simple layer for the Helmholtz equation u u D 0 with the density that continuously depends on t as on a parameter. Under the conditions formulated above, the potential of a simple layer V1 Œ.x; t / is, as is known, a continuous function of spatial coordinates. The second term is also a continuous function, as F .x; t / is a continuous function. .1/

Lemma 6.1.5. Let .t; s/ 2 C0 .Œ0; C1/I C.//. Then for any x … and t 0, the function u.t; x/ D W Œ.t; x/ satisfies Eq. (6.3) in the classical sense and the zero initial conditions uj tD0 D 0. Moreover, the limit relation lim

x!y0 2

W Œ.t; x/ D WN Œ.t; y0 / ˙ .t; y0 /:

(6.13)

holds. Here WN Œ.t; y0 / is the direct value of the potential on the boundary, the sign “” must be taken if x ! y0 from inside and “C” in the opposite case. Proof. The first statement of Lemma 6.1.5 is proved similarly to Lemma 6.1.3, i.e., by substituting the potential (6.5) in Eq. (6.3) and a direct calculation. Prove the limit relation (6.13). The potential (6.5) is the sum of two terms W1 Œ.x; t / and W2 Œ.x; t /. The first term is the potential of a double layer for the Helmholtz equation; it has a discontinuity at passage over the boundary by exactly the same formula as (6.13);

548


therefore, we must prove that the second term is a continuous function of spatial coordinates. Transform W2 Œ.x; t / in formula (6.5): Z tZ @ .; s/ F .jx y.s/j; t / ds d W2 Œ.x; t / D @ns 0 Z tZ @ @jx y.s/j D .; s/ F .jx y.s/j ; t / ds d @x @ns 0 Z tZ @ .; s/ F .jx y.s/j; t / cos ' ds d : (6.14) D @x 0 Here ' is the angle between the inner normal y.s/ and the vector joining y.s/ and x. By estimate (6.12) in expression (6.14) we can assume that the kernel is a continuous function since for x D y.s/ it can be redefined with zero limit value. The continuity of the kernel directly follows from the continuity of W2 Œ.x; t /. Lemma 6.1.5 is proved. Remark 6.1.6. By the linearity of Eq. (6.3), any linear combination of potentials (6.4) and (6.5) will also satisfy the equation and the zero initial conditions.

6.1.2 Solvability of Dirichlet problem N function u.t; x/, which belongs to the Problem S. Find a continuous on Œ0; C1/ .2/ .1/ class C.Œ0; C1/I C .// \ C ..0; C1/I C .2/ .// and satisfies in Œ0; C1/ the equation .u u/ t C ˛u C ˇu D 0; the zero initial condition uj tD0 D 0 and the Dirichlet condition on the boundary: uj D f .s; t /. Analyze the solvability of Problem S by the method developed in [163–165]. Theorem 6.1.7. Let f .s; t / 2 C0.1/ .Œ0; TI C.//, then Problem S is solvable. Proof. We search for a solution of Problem S in the form of the dynamic potential of a double layer (6.5): u.x; t / D W Œ.x; t /: Then by Lemma 6.1.5 u.x; t / satisfies the equation and the initial conditions. The boundary condition leads to the integral equation Z .t; / C C

.t; s/

Z tZ

@ K0 .jx./ y.s/j/ ds @ns

.; s/ 0

@ F .jx./ y.s/j; t / ds d D f .; t /: (6.15) @ns


549

We search for a solution of Eq. (6.15) in the form .t; / D

C1 X

k .t; /;

(6.16)

kD0

where k .; t / are found from the relations Z @ BŒ0 .t; / D 0 .t; / C 0 .t; s/ K0 .jx./ y.s/j/ ds D f .; t /; @ns Z @ k .t; / C k .t; s/ K0 .jx./ y.s/j/ ds @n s Z tZ @ k1 .; s/ F .jx./ y.s/j; t / ds d ; k 1: (6.17) D @ns 0 For convenience, rewrite the latter equation in the form Z t BŒk .t; / D AŒk1 .; / d :

(6.18)

0

Equation (6.17) depends on the variable t as on a parameter. Consider the Dirichlet problem for the Helmholtz equation in the domain : wj D h./:

w w D 0;

If we search for a solution of this problem in the form of the classical potential of a double layer for the Helmholtz equation, Z @ w.x; t / D .s/ K0 .jx y.s/j/ ds; @ns we obtain the integral equation BŒ./ D h./: As follows from the well known results of the theory of elliptic boundary-value problems, under the conditions for formulated above, for any continuous function h./ it is uniquely solvable. Then by the Banach theory on inverse operators, there exists a bounded operator B 1 acting from C./ into C./. Equation (6.17) yields Z t AŒk1 .; / d : (6.19) k .t; / D B 1 0

Let M.T / D kB 1 k sup0 T kAk; then kk .; t /kC./ M.T /

Z

t 0

kk1 .; /kC./ d :

(6.20)

550


Equation (6.17) implies the estimate k0 .; t /kC./ kB 1 k sup kf .; /kC./ q.T /:

(6.21)

0 T

By (6.21) for k D 1 we obtain Z k1 .; t /kC./ M.T /

t 0

q.T / d D M.T /q.T /t:

(6.22)

From (6.22) and estimate (6.20) for k D 2 we obtain Z t t2 M.T /q.T / d D M 2 .T /q.T / : k2 .; t /kC./ M.T / 2 0 For any k we finally obtain kk .; t /kC./ M k .T /q.T /

tk : kŠ

(6.23)

Hence we directly obtain the uniform convergence of series (6.16) since it is majorized by the converging number series C1 X

M k .T /q.T /

kD0

Tk : kŠ

Similar estimates hold for the series of derivatives C1 X kD0

@ k .t; /: @t

.1/ Thus, series (6.16) defines a function .t; s/ 2 C0 .Œ0; T I C.//, which is a solution of the integral equation (6.15) and, therefore, the dynamic potential of a double layer generated by this function satisfies the boundary condition. Note that T can be chosen as large as desired and, therefore, exists for all t 0. Theorem 6.1.7 is proved.

Theorem 6.1.8. Problem S has a unique solution. Proof. Assume that there exist two solutions u1 and u2 ; then their difference w D u1 u2 satisfies the problem .w w/ t C ˛w C ˇw D 0;

wj tD0 D wj D 0:

Rewrite this equation in the form .w w/ t C ˛.w w/ D .˛ C ˇ/w

551


and integrate taking into account the zero initial conditions: Z t w.t; x/ w.t; x/ D expŒ˛.t /.˛ C ˇ/w.; x/ d : 0

As is known, under the conditions stated above for the domain and its boundary , there exists a Green function G.x; y/ for an inner Dirichlet problem for the Helmholtz equation. Then due to the zero boundary conditions we obtain Z Z Z t w.t; x/ D G.x; y/ expŒ˛.t /.˛ C ˇ/w.; y/ d dy: 0

Since

Z jw.t; x/j j˛ C ˇj

t

Z Z sup jw.; y/j d

0 y2

G.x; y/ dy;

Z Z

we have u.x/ D

G.x; y/ dy:

The function x 2 satisfies the boundary problem u u D 1;

uj D 0:

By the maximum principle, 0 u 1; therefore, Z t sup jw.; y/j d : jw.t; x/j j˛ C ˇj 0 y2

Here the right-hand side is independent of x; therefore, Z t sup jw.t; x/j j˛ C ˇj sup jw.; y/j d : 0 y2

x2

The latter inequality is possible only if w 0. Theorem 6.1.8 is proved. Remark 6.1.9. In the case of an external initial-boundary-value problem, the formulation of Problem S must be added with the regularity conditions at infinity: limjxj!1 u.x; t / D 0. Instead of Eq. (6.15), the following equation must be considered: Z

.t; / C

.t; s/

C

Z tZ

@ K0 .jx./ y.s/j/ ds @ns

.; s/ 0

@ F .jx./ y.s/j; t / ds d D f .; t /: (6.24) @ns

Then for external problems, analogs of Theorems 6.1.7 and 6.1.8 hold.

552


The integral equation (6.15) contains integration by time from 0 to t , i.e., an integral operator of Volterra type. This fact allows the construction of a numerical method for the solution of this equation in such a way that values of the unknown function .t; s/ lie sequentially on each time layer. On each of these layers, a system of linear algebraic equations must be solved; the matrices of these systems are the same for all layer, but the right-hand sides are different. Describe the numerical algorithm. The spatial grid on the boundary should be chosen to be convenient, the main goal is to make it uniform or quasi-uniform. Thus, in the case of star-shaped domains, it is convenient to define the curve in the polar coordinates: r D r.'/. In this case, the uniform grid by ' (i.e., 'i D 2 i =N ) generates a quasi-uniform grid on the curve: x1i D r.'i / cos 'i , x2i D r.'i / sin 'i . We take a uniform grid by time with step t D const; then tj D jt . Further, we fit a convenient quadrature formula that approximates the integral equation (6.15) on the grid uniform by time and quasi-uniform by space: j i C

N X

Kik j k C

kD1

j X N X

Fj l;i;k lk D fj i :

(6.25)

lD1 kD1

Here j i is the required value of the potential density at the ith spatial point and on the j th time layer and Kik and Fj l;i;k are found by the chosen quadrature formula. We have used the trapezium formula on the uniform grid by time and on the quasi-uniform grid to approximate the integral by the curve . Assume that for l D 1; : : : ; j 1 and k D 1; : : : ; N , all lk are already found; then from the formula (6.25) we obtain j i C

N X kD1

Kik j k C

N X kD1

F0;i;k j k D fj i

jX 1 X N

Fj l;i;k lk :

(6.26)

lD1 kD1

In other words, for finding j i on the j th time layer, we must solve a system of linear equations with a matrix that is independent of j ; only the right-hand side depends on j . This allows the construction of an algorithm to find j i effectively: the essence of the algorithm is to multiply the inverse matrix once calculated by a column of the right-hand side of the system of algebraic equations (6.26), and only this column must be calculated on each time layer again. Emphasize the importance of formula (6.11) obtained above, which allows one to calculate Fm;i;k to within accuracy 1015 –1016 . We performed calculations for different values of ˛ and ˇ, for different domains with different boundary conditions. Tests on condensing grids were also performed. In the case where the domain is a circle, the explicit analytical solution can be constructed, which is convenient to be used for testing the numerical method. For finding an effective order of accuracy of the proposed method the tests on condensing grids have been fulfilled. Calculations were made for the problem in-


553

side a unit circle for Eq. (6.3) with ˛ D 1, ˇ D 0, and the boundary condition ujrD1 D sin t exp.sin '/. The measure of inaccuracy of a numerical solution was estimated as a result of calculations on two neighbor grids with the use of Richardson method. First, under a fixed number N D 64 of nodes of the spatial grid, calculations were performed with different numbers of steps by time: J D 20; 40; : : : ; 1280. Results of the test are represented in Figure 6.1: the numerical solution inaccuracy decrease in the grid norm C under doubling the number of time-grid nodes. The decreasing character corresponds to the accuracy of O. 2 /: the tangent of the inclination angle in the double logarithmic scale is equal to 2. Such order of accuracy of O. 2 / completely agrees with the approximation order of the integral by time with the use of quadrature trapezium formula. If for a fixed number of time grid steps J D 80, we condense the spatial grid, N D 32; 64; : : : ; 1024, the accuracy decrease character relative to the nodes number growth makes a pleasant surprise: the inaccuracy diagram inclination angle tangent in double logarithmic scale is equal to 3. Thus, in spite of the fact that for approximating the spatial integral quadrature trapezium formula is also used, the accuracy order of the constructed method is equal to O.N 3 /. This is related with the fact that under integrating periodic functions periodically quadrature formula of trapezium has a higher accuracy order than in the general case. On the basis of the tests fulfilled we can make the conclusion that in the case of a sufficiently smooth boundary and sufficiently smooth boundary conditions, the proposed method has the accuracy O.N 3 C 2 /. One of the main advantages of the method is that it works equally well both for simple canonical domains and for domains with more complicated boundaries. In Figure 6.2 we show isolines of the potential of the inner and external Dirichlet prob-

Figure 6.1. Inaccuracy decreases as the number of grid nodes increases; this confirms the accuracy O.Nx3 C 2 / of the method.

554


Figure 6.2. Isolines of solutions of inner and external Dirichlet problems.

lems with the same boundary conditions defined on the curve that is described in polar coordinates by the equation D 2 C cos 3'. Let us resume for some results. Applying dynamic potentials for numerical solution of initial-boundary-value problems is very effective by virtue of several causes: the dimension of the problem becomes lower (we have reduced the two-dimensional problem to an integral equation on a curve), the method becomes applicable for a wider spectrum of domains, the algorithm can be simply implemented. However, there are several restrictions: the boundary must be smooth, for calculation of the kernel we must use the representation in the form of a series (6.9), which allows to achieve virtually the computer accuracy of the kernel calculation, if this representation cannot be obtained, one must invert numerically the Laplace transform (6.6).

6.2

Numerical method of solving initial-boundary-value problems for nonlinear pseudoparabolic equations by the Rosenbrock schemes

6.2.1 Stiff method of lines For numerical solution of initial-boundary-value problems for partial differential equations, the following approach is often used: differential operators on spatial variables are replaced by their difference analogs on the grid chosen, and thus the problem ie reduced to a system of ordinary differential equations of large order. This method has been known since the 1940s, but the appearing systems of differential equations were solved by means of explicit methods, for the recent years the method has gained new possibilities due to then fact that ODE systems could be solved by stiff methods

555

Section 6.2 The Rosenbrock schemes

(see [194]), which allows one to avoid splitting into processes, to avoid iterations in nonlinear problems, etc.

6.2.2 Stiff systems of ODE and methods of solving them In numerical solution of systems of ordinary differential equations du D F .u/; dt there often arise problems for which implicit methods yield substantially better results than explicit methods; it is conventional to call such problems stiff. Such factors as the dimension of the system, the smoothness of solution, the eigenvalues of the Jacobi matrix @F =@u, etc. influence the stiffness of the system. In numerical solution of such problems, one imposes increased requirements on the stability.

6.2.3 Stiff stability Starting from the 1950s, for stiff problems, special implicit methods have been created; at that time, new formulations of a number of additional properties which the desired schemes must satisfy were given. Consider the Dahlquist problem du D u: dt When using any linear scheme, the passage to the next temporal layer has the form uO D R. /u, where R./ is called the growth function or the stability function. Definition 6.2.1 (Dahlquist). A scheme is said to be A-stable if jR./j 1 for Re 0. It is desirable that as Re ! 1, the stability function also tends to 0; therefore, one introduces the concept of Lp -stability. Definition 6.2.2 (N. N. Kalitkin). A scheme is said to be Lp -stable if it is A-stable and R./ D O. p / as jj ! 1.

6.2.4 Schemes of Rosenbrock type The implicit Runge–Kutta methods and the (semiimplicit) Rosenbrock methods are the most commonly used methods for solution of stiff problems. In the implicit Runge–Kutta methods, at each temporal level, one needs to solve systems of nonlinear equations by using iterative methods. The principal advantage of the Rosenbrock methods is that no nonlinear systems arise in them, even if the initial problem is nonlinear.

556


Consider the following autonomous system of equations: du D F .u/; dt where u is a column vector composed of functions u1 .t /; u2 .t /; : : : ; un .t /. In what follows, we use a one-parameter family of one-stage Rosenbrock schemes [353]: uO D u C Re k;

.E ˛Fu /k D F .u/I

(6.27)

here, Fu @F =@u is the Jacobi matrix, E is the identity matrix, is the step in time, u is the solution at the current temporal level, uO is the solution at a new temporal level, and ˛ is a numerical parameter defining families of the one-stage Rosenbrock scheme. Therefore, the vector k is found from the system of linear equations (6.27) with the matrix .E ˛Fu /. The solution of the linear system must be performed by using direct methods (for example, by the Gauss method with the choice of the principal element). In this case, the passage from the initial layer to a new layer is performed via a finite number of operations known in advance as in implicit schemes. At the same time, the finding of k by solving the linear system introduces an implicitness into the scheme. Therefore, such schemes are said to be semiimplicit or semiexplicit. The Rosenbrock method is easily generalized for implicit differential equations du D F .u/; dt where M is a constant nonsingular matrix. In this case, the CROS scheme has the following form: M

uO D u C Re kI

.M ˛Fu /k D F .u/:

(6.28)

(6.29)

For different parameters ˛, the scheme has different properties. For ˛ D 0, it is an implicit scheme with accuracy O. /. This variant of the scheme is practically inapplicable for computing stiff problems. For ˛ D 0:5, one obtains the known “half-sum” scheme. Its accuracy is O. 2 /, and it is unconditionally stable; therefore, it is often used in computations. However, it is only A-stable according to the classification of stiff problems. For ˛ D 1, one obtains the so-called (purely implicit) inverse Euler scheme. It is L1 -stable, which ensures the unconditional stability and a good qualitative behavior of the numerical solution. But its accuracy O. / is not high, which prevents its application. The schemes described above are real. However, there exists one complex scheme of this family with ˛ D .1 C i /=2. It has unique properties: its accuracy is O. 2 /, it is L2 -stable, and, respectively, it is unconditionally stable. This scheme has a high reliability and is applicable for computing problems with strong stiffness. Precisely this scheme is used in the computations described below. In the literature, it is conventional to call this scheme the CROS scheme.

557


6.2.5 "-embedding method Consider the following differential-algebraic system: dy D f .y; z/; dt 0 D g.y; z/; y.0/ D y0 I

(6.30)

z.0/ D z0 :

Assume that the matrix gz .y; z/ is invertible in a neighborhood of its solution. In order to construct a method of the second order of accuracy, we assume that the functions f and g are sufficiently smooth. To construct a numerical solution of such systems, one applies the following approach. Consider the system y 0 D f .y; z/;

"

dz D g.y; z/; dt

y.0/ D y0 ;

z.0/ D z0

with compatible initial conditions g.y0 ; z0 / D 0. For this system, one applies some numerical method, and then " D 0 is taken in the obtained formulas. The justification of this approach for real Runge–Kutta and Rosenbrock schemes can be found, e.g., in [194]. Moreover, this approach extends to systems in which the algebraic equations are not separated from the differential equations, i.e., to systems of the form M

du D F .u/ dt

(6.31)

where the matrix M is nonsingular. In this case, the CROS scheme has the same form as in (6.29). In numerical computations, it is important not only to obtain the result but to estimate its accuracy with guarantee. The computation method on condensing grids [10, 286] allows one to do this. It is known that when using grid methods on uniform or quasi-uniform grids, the error of the numerical solution is expanded in inverse powers of node numbers N : X Ck N k ; (6.32) .N / D kDp

here p is the theoretical order of the scheme accuracy. The number of summands in the sum is determined by the number of the bounded derivatives of the exact solution. The main term of the inaccuracy is equal to Cp N P . We make calculations on two neighbor grids with the numbers of nodes N and rN , respectively. If r is an integer, all nodes of the more rough grid coincides with nodes of the more detailed one. At these points, the inaccuracy .rN / is calculated by the Richardson formula .rN / .t / D

u.rN / .t / u.N / .t / C o.N p /: rp 1

(6.33)

558


Using the results of calculations on three neighbor grids with the numbers of nodes N , rN , and r 2 N , one can calculate the effective order of accuracy p eff .t / D logr

u.rN / .t / u.N / .t / u.r 2 N / .t / u.rN / .t /

:

(6.34)

If p eff .t / ! p, then the exact solution is sufficiently smooth and the estimate of inaccuracy (6.33) is asymptotically exact as N ! 1. The effective order of accuracy for certain specific problems can prove to be higher than the theoretical one, for example, when periodic functions are integrated. This shows that Cp D 0 in the decomposition (6.32). Then for obtaining asymptotically exact estimate of the inaccuracy, one must use the real p eff : .rN / .t / D

u.rN / .t / u.N / .t / C o.N p /: r p eff 1

(6.35)

The condensing grid method is laid as a base in the computation technique with guaranteed accuracy: the grid is condensed up to the moment when the error of the numerical solution (6.33), (6.35) becomes less than the required accuracy level. If the exact solution of the problem has a large number of bounded derivatives, then the asymptotically exact estimate of the inaccuracy (6.33), (6.35) can be taken into account as a correction: uQ .rN / .t / D u.rN / .t / C .rN / .t /: This is equivalent to the elimination of the main member of the inaccuracy expansion (6.32). Such an extrapolation revision requires only several arithmetic operations, and the accuracy order becomes higher for 1 or even 2 (for symmetrically written difference schemes): uQ .rN / .t / D u.t / C O.N p /;

D 1–2:

The use of the extrapolation correction substantially accelerates the process of attaining the given accuracy level. If the effective order of the accuracy p eff as N ! 1 tends to the value less than the theoretical order of accuracy of the scheme p, then this means insufficient smoothness of the solution. For example, if we use the method of second order of accuracy for the problem with exact solution with the discontinuous first derivative, the effective order of accuracy will prove to be less for 1 than the theoretical one: p eff D 1. The natural continuation of the ideas of calculations with accuracy control is the singularity diagnostics technique. An effective algorithm for singularity diagnosis of the exact solution of ODEs in computations on condensing grids was proposed in [10]. Assume that at a point t , the exact solution of the problem has a discontinuity of the

559


second kind u.t / D 1 or its first derivative becomes infinite: u0 .t / D 1, and for t > t , the exact solution does not exist. Such problems are said to be ill-posed. Testing the most frequently used methods of numerical integration of ODEs on such problems show that almost all of them led to overflow at calculations. The overflow time moment, generally speaking, is not related with the place of singularity t in any way. Also, it was shown that CROS scheme primarily proposed for stiff systems has also a lot of advantages for solving ill-posed problems, i.e., calculation does not give overflow, after the passage of the singularity t > t the CROS scheme numerical solution is stabilized at a certain equilibrium place u .N / that depends on the grid condensation. By calculating on condensing grids according to the CROS scheme with the control of the value p eff one can track the singularity moment and diagnose its type. At the exact solution smoothness points p eff ! p. Under power singularity of the type u.t / .t t /ˇ (discontinuity of the second kind ˇ > 0 or a root singularity under 1 < ˇ < 0), the effective accuracy order of CROS scheme at all the points t > t tends to the value p eff ! ˇ as N ! 1 (see [10]). If limN !C1 p eff D 1, the solution u.t / grows exponentially: u.t / exp¹.t t /1 º as t ! t . If limN !C1 p eff D 0, the solution grows logarithmically: u.t / ln.t t /. We generalize the singularity diagnostics method for systems of partial differential equations that are, generally speaking, nonlinear. Assume that it is required to find a solution u.x; t / of the initial-boundary-value problem for a partial differential equation, which is nonlinear in general. The dimension of the problem in the spatial variable x is arbitrary, but we illustrate the method by a problem, which is one-dimensional by space. Introduce a uniform or quasi-uniform grid by the spatial variable ¹xl º, 0 l L, with the number of L and use the method of lines, i.e., approximate the spatial variables by finite differences O.L2 /. In the general case, this leads to a differential-algebraic system of great dimension: M

dy D f .y/ ; dt

(6.36)

where y.t / is the values of the unknown function u.x; t / at the nodes of the spatial grid ¹xl º, 0 l L. The methods for reducing concrete initial-boundary-value problems for linear, nonlinear, and even nonlocal partial differential equations are illustrated below. For numerical integration of (6.36), we use the CROS scheme with the accuracy O. 2 / D O.N 2 /, where N is the number of nodes of the time grid; thus, the constructed numerical method will have the accuracy O.N 2 C L2 /. Let us perform the computation on the start grid ¹xl ; tn º, 0 l L, 0 n N . Since the theoretical accuracy order in the time and spatial variables is the same and is equal to 2, we perform a sequential condensation of the spatial and time grids by the same integer number of times r (the most convenient is r D 2). In this case, the nodes of the start grid are those of all subsequent grids; at these points, we control the

560


error .rN;rL/ .x; t / D

u.rN;rL/ .x; t / u.N;L/ .x; t / C o.N 2 C L2 / r2 1

(6.37)

and the effective accuracy order p eff .x; t / D logr

u.rN;rL/ .x; t / u.N;L/ .x; t / u.r 2 N;r 2 L/ .x; t / u.rN;rL/ .x; t /

(6.38)

at these points. At points .x; t / where lim

N;L!1

p eff .x; t / D 2;

(6.39)

the exact solution of the problem has bounded second derivatives with respect to the spatial and time variables and the estimate of the error is asymptotically exact as N; L ! 1. The violation of (6.39) testifies the loss of smoothness of the exact solution. In particular, in the case of power singularity u.x; t / .t t /ˇ for any t > t , the effective accuracy order limN;L!1 p eff .x; t / D ˇ allows one to find the degree ˇ uniquely. If limN;L!1 p eff .x; t / D 1 for t > t , then we can assert that the exact solution grows exponentially, u.x; t / D 1; if limN;L!1 p eff .x; t / D 0 for t > t , then the solution grows logarithmically. The singularity point t can be found with accuracy of the order of the control grid. If the smoothness of the solution vanishes on the whole domain of the spatial variable simultaneously, then the substantial deviation of p eff .x; t / from 2 appear at all nodes of the grid ¹xl º on the first temporal layer t > t . If the singularity of the solution first appears at one point x , then the method described allows one to watch over its propagation. This diagnostics is possible owing to the fact that the scheme CROS does not lead to the overflow even in the case where the exact solution of the problem becomes infinite.

6.3

Results of blow-up numerical simulation

From the viewpoint of application of numerical methods, the equations considered below can be divided into the following three types: (1) equations with a linear operator by the derivative in time; (2) equations with a nonlinear operator by the derivative in time; (3) equations with nonlocal terms (the coefficients of the equation considered depend on the norm of the function). Also, a combination of the second and third types is possible.

561

Section 6.3 Results of blow-up numerical simulation

6.3.1 Blow-up of pseudoparabolic equations with a linear operator by the time derivative We first consider the following equation with a rapidly growing nonlinearity (Chapter 5) in the case of one spatial variable: @ .u u/ C e u D 0; @t

uj tD0 D u0 .x/;

ujxD0 D ujlD0 D 0:

(6.40)

In Chapter 5 it was proved that there exists a unique classical solution up to a certain instant of time T0 , and, moreover, lim sup ju.x; t /j D C1:

t"T0 x2Œ0;l

Also, the following two-sided estimates for the blow-up time T0 2 ŒT1 ; T1 of the solution expressed through the initial data were obtained. To numerically solve problem (6.40), we apply the method of lines, and to diagnose the blow-up time, we apply the condensing grid method. Approximating the differential operator d 2 =dx 2 1 on the uniform spatial grid xk D hk, h D l=K, we arrive at the following system of ordinary differential equations: dU D F .U / : M dt Here U.t / D .u1 .t /; u2 .t /; : : : ; uN 1 .t //T is a column vector consisting of the components of the difference solution at the points x1 ; x2 ; : : : ; xN 1 : 1 0 1 0 1 0 0 2 1 0 0 B 1 2 1 0 0 C B0 1 0 0C B B :: C :: :: C C B: : :C 1 B : C B :: C B 0 1 2 1 (6.41) M D 2 B :: : : C B C : : : : : : B B C : h :C : 0 C B: C B : @ 0 0 1 2 1 A @ ::: 1 0A 0 0 1 2 0 0 0 1 The column F .U / for this problem has the form .e u1 ; e u2 ; : : : ; e uK1 /T . Note that the matrix M is nonsingular. To solve this problem, we use the CROS scheme in the form (6.29). It is convenient to illustrate the methodology for finding the blow-up time described above by using Table 6.1, in which the values of the effective accuracy order p eff computed by formula (6.34) at distinct instant of time and at distinct points of the closed interval considered are presented. As was noted above, the effective accuracy order tends to the theoretical accuracy order as the number of nodes of the grid tends to infinity; therefore, a small deviation of p eff from 2 does not yet indicate the blow-up of the solution; however, at the instant of time t D 0:34, there is a jump-like variation

562


t | x 0.26 0.52 0.79 1.05 0.02 2.00 2.00 2.00 2.01 0.04 2.00 2.00 2.00 2.01 0.06 2.00 2.00 1.99 2.01 0.08 2.00 2.00 1.99 2.01 0.10 2.00 2.00 1.99 2.02 0.12 2.00 2.00 1.99 2.03 0.14 2.00 2.00 1.99 2.06 0.16 2.00 2.00 1.99 1.97 0.18 2.00 2.00 1.99 1.95 0.20 2.00 2.00 1.99 1.97 0.22 2.00 1.99 1.99 1.98 0.24 1.99 1.99 1.99 1.98 0.26 1.99 1.99 1.99 1.98 0.28 1.99 1.99 1.98 1.98 0.30 1.98 1.98 1.98 1.98 0.32 1.95 1.95 1.95 1.95 0.34 1.36 1.36 1.37 1.41 0.36 0.06 0.06 0.09 0.27 0.38 0.32 0.32 0.25 0.22 0.40 0.40 0.38 0.20 0.92

1.31 1.57 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.01 2.00 2.01 2.01 2.01 2.01 2.01 2.01 2.02 2.01 2.04 2.02 1.92 2.07 1.96 1.91 1.97 1.97 1.98 1.97 1.97 1.97 1.95 1.95 1.56 1.71 1.01 2.17 2.51 1.43 1.37 0.57

1.83 2.00 2.00 2.00 2.00 2.01 2.01 2.01 2.01 2.02 2.04 1.92 1.96 1.97 1.98 1.97 1.95 1.56 1.01 2.51 1.37

2.09 2.36 2.62 2.88 2.01 2.00 2.00 2.00 2.01 2.00 2.00 2.01 2.01 1.99 2.00 2.00 2.01 1.99 2.00 2.00 2.02 1.99 2.00 2.00 2.03 1.99 2.00 2.00 2.06 1.99 2.00 2.00 1.97 1.99 2.00 2.00 1.95 1.99 2.00 2.00 1.97 1.99 2.00 2.00 1.98 1.99 1.99 2.00 1.98 1.99 1.99 1.99 1.98 1.99 1.99 1.99 1.98 1.98 1.99 1.99 1.98 1.98 1.98 1.98 1.95 1.95 1.95 1.95 1.41 1.37 1.36 1.36 0.27 0.09 0.06 0.06 0.22 0.25 0.32 0.32 0.92 0.20 0.38 0.40

Table 6.1. Effective accuracy order of the method allows one to diagnose the blow-up time of the solution.

of p eff at all observation points simultaneously (in Table 6.1, this is marked by bold type). We can find the blow-up time of the solution with accuracy up to the step length in time. For Problem 1, the picture of evolution of the solution is presented in Figure 6.3. It is interesting to show how the two-sided estimates obtained for the blow-up time of the solution are in accord with the approximate value T0 obtained by using the condensing grid method. We carried out the computations for problem (6.40) with the initial condition u0 .x/ D A sin x:

(6.42)

In Figure 6.4, we show the two-sided estimates (smooth lines) for the blow-up time depending on the quantity A calculated by formula (5.115) (see Chapter 5). By stars we mark the values of T0 diagnosed in the numerical experiment by using the condensing grid method. If we slightly increase the amplitude of the initial data, u .x; 0/ D 1:2 sin x, then the maximum of the solution also moves to the right, but its amplitude increases; this


563

Figure 6.3. Evolution of the solution at distinct instants of time.

Figure 6.4. Two-sided theoretical estimates for the blow-up time of the solution T1 < T1 and the blow-up time T0 diagnosed in the numerical experiment.

increasing is especially rapid after t 2:5. The profiles of the numerical solutions for this case are shown in Figure 6.5. These profiles are smooth, and we can assume that the classical solution of the initial-boundary-value problem (6.39) also exists after the moment t 2:5. Let us consider one more initial-boundary-value problem for a pseudoparabolic equation with a linear operator by the time derivative, precisely, for the Oskolkov–

564


Figure 6.5. Profiles of the numerical solution of problem (6.43) for the initial data u .x; 0/ D 1:2 sin x; the dotted line corresponds to the time moment after the blow-up of the exact solution.

Benjamin–Bona–Mahony–Burgers equation with cubic source: @ .uxx u/ C uxx C uux C u3 D 0; @t u .0; t / D u . ; t/ D 0;

(6.43)

u .x; 0/ D '.x/: The three-dimensional analog of this problem is considered in Chapter 3. In particular, in this paper, the theorem on the solvability of the problem in the strong generalized sense is proved and sufficient conditions for the global solvability of the problem and those for the blow-up of the strong generalized solution at a finite time are obtained. For simplicity, choose a uniform spatial grid xn D nh, h D =N , 0 n N , and apply the method of lines. For this purpose, it is necessary to replace all spatial derivatives by divided differences. To preserve the general second order of accuracy, we use the symmetric divided difference for the approximation of the first spatial derivative: unC1 2un C unC1 d unC1 2un C unC1 u n C 2 dt h h2 unC1 un1 (6.44) Cun C u3n D 0; 1 n N; 2h u0 D uN D 0; un .0/ D ' .xn / : In numerical computation, the division by a number whose module is small leads to large round-off errors; for difference schemes, this can result in the loss of stability.


565

To avoid this, we multiply each of the equations in system (6.43) by h2 and reduce the result to a form admitting the application of the CROS scheme: d.unC1 2un C un1 h2 un / D .unC1 2un C un1 / dt 0:5huu .unC1 un1 / u3n ; u0 D uN D 0;

1 n N;

un .0/ D ' .xn / :

In this case, the matrix of the linear algebraic system (6.29) is tridiagonal. The inversion of the matrix in this case is very economical performed by using the sweep method, and the number of arithmetical operations needed for the passage to a new temporal level is of order N , where N is the number of nodes of the spatial grid. The constructed numerical method has the same accuracy order in the time and spatial variables (the accuracy O.h2 C 2 /), and, therefore, in testing on condensing grids, we need to refine the time and spatial grids by the same times. The picture of the behavior of the solution to the initial-boundary-value problem (6.43) substantially depends on the initial data. For u.x; 0/ D sin x, the amplitude of the solution damps, and owing to the existence of the transfer term, its module displaces to the right. The profiles of this numerical solution for these initial data are presented in Figure 6.6 for the instants of time t D 0; 1; 2; 3; 4; 5, and 6. The testing on condensing grids justifies the smoothness of the obtained solution during the whole computation. However, the testing on condensing grids allows us to detect the blow-up of the solution: after the instant of time t 2:5, the effective accuracy order peff for t > 2:5

Figure 6.6. Profiles of the numerical solution of problem (6.43) for the initial data u.x; 0/ D sin x.

566 t |x 0.5 1.0 1.5 2.0 2.4 2.5 2.6 2.7 2.8 2.9 3.0

Chapter 6 Numerical methods for Sobolev-type initial-boundary-value problems 0.314 1.993 1.998 2.001 2.006 2.020 2.023 1.555 0.881 1.528 1.824 2.226

0.628 1.997 1.999 2.002 2.007 2.021 2.024 1.559 0.846 1.266 0.618 1.417

0.942 2.000 2.001 2.002 2.008 2.022 2.026 1.564 0.723 0.3 0.279 4.007

1.257 2.001 2.001 2.003 2.008 2.022 2.026 1.667 0.479 1.4 3.99 3.729

1.571 2.001 2.001 2.003 2.008 2.022 2.026 1.666 0.15 1.569 2.664 2.310

1.885 2.001 2.001 2.002 2.008 2.022 2.026 1.564 0.169 3.298 2.071 0.006

2.199 1.999 2.000 2.002 2.007 2.022 2.026 1.562 0.391 5.497 1.951 0.230

2.513 1.989 1.998 2.002 2.007 2.022 2.026 1.561 0.499 7.356 1.941 0.594

Table 6.2. At smoothness points of the solution, the effective accuracy order is close to the theoretical value p D 2; after the blow-up time t 2:6, the deviation of the effective accuracy order from the theoretical accuracy order indicates the blow-up of the solution.

is clearly different from the theoretical value p D 2, which allows us to diagnose the blow-up of the solution. For this computation, the effective accuracy order at a control point is presented in Table 6.2.

6.3.2 Blow-up of strongly nonlinear pseudoparabolic equations Let us consider some problems with a nonlinear operator by the time derivative. If we directly apply the method of lines to such an equation, then the problem reduces to an implicit system of ordinary differential equations of the form M .U /

dU D F .U /; dt

(6.45)

where the matrix approximating the nonlinear operator depends on U ; therefore, it is impossible to apply the Rosenbrock method to this system. However, denoting the whole nonlinear operator by a new variable, we can reduce the problem to a differential-algebraic system with a constant but singular matrix. Let us present concrete examples: @ .uxx u u3 / C uxx C u.u C 1/.u 2/ D 0: @t

(6.46)

Replace this equation by the system @ W C uxx C u.u C 1/.u 2/ D 0; @t

0 D W .uxx u u3 /:

(6.47)

567


After the approximation on a uniform grid, the problem reduces to a differentialalgebraic system of the form @ W C uxx C u.u C 1/.u 2/ D 0; @t

0 D W .uxx u u3 /;

where the matrix M has a very simple structure: E O M D : O O

(6.48)

(6.49)

For numerical computations, we take the problem on the collapse of a “step.” As a result, for various initial data, we observe two possible scenarios: either the solution tends to a stationary limit (Figure 6.8) and the limit profile is determined by the solution of the boundary-value problem for the equation uxx C u.u C 1/.u 2/ D 0;

(6.50)

or the solution rapidly grows and blows-up at a finite time (Figure 6.7). Let us consider one more example of the numerical solution of an equation with a nonlinear operator by the time derivative. In Chapter 4, Section 9, the authors considered an initial-boundary-value problem, which in the one-dimensional spatial case has the form @ .uxx jujq1 u/ C uxx C jujq2 u D 0; @t

ujxD0 D ujxDl D 0;

uj tD0 D u0 .x/: (6.51)

Figure 6.7. Blow-up of the “step.”

568


Figure 6.8. Tending to a stationary limit.

As was shown in Chapter 4, the behavior of the solution substantially depends on the relation between its parameters. Thus, if q1 q2 > 0, then the solution exists for an infinitely long time. The most interesting is the case where 0 < q1 < q2 ; then the solution blows-up for initial data satisfying the conditions q2 C 2 kru0 k22 ; 2 p (6.52) 2 2q2 .q1 C 2/1=2 1 q1 C 1 2 2 2 > kru0 k2 C kru0 k2 C ku0 k2 : q2 q1 2 q2 C 2

q C2 ku0 kq22 C2 > q C2

ku0 kq22 C2

In Chapter 4, the study of the behavior of the solution is based on the analysis of the energy function E.t /

1 q1 C 1 q C2 kruk22 C kukq11 C2 : 2 q1 C 2

(6.53)

Let us proceed in the same way as in the previous example. Introduce the following new variable: w D uxx jujq1 u; we arrive at the system @ W C uxx C jujq2 u D 0; @t

0 D w .uxx jujq1 u/:

Applying the method of lines, we obtain a differential-algebraic system with the same matrix (6.51). To solve the differential-algebraic system, we use the CROS scheme. Below, we present the computations for the case l D , q1 D 2, q2 D 4 (q1 < q2 ). If we take the initial conditions in the form u0 .x/ D A sin mx, then


569

conditions (6.51) are easily calculated analytically, and it is convenient to illustrate them by using the following graph. To satisfy conditions (6.51), the point .m; A/ needs to lie above both curves on the graph (see Figure 6.9). Below, we present the computations of the evolution of the solution for the initial data u0 .x/ D 0:5 sin x (Figure 6.10) and u0 .x/ D 3 sin x (Figure 6.11); in the latter case, conditions (6.51) hold.

Figure 6.9. The boundary of fulfillment of conditions (6.53) for the initial data u0 .x/ D A sin mx.

Figure 6.10. Evolution of the solution for the initial data u0 .x/ D 0:5 sin x.

570


Figure 6.11. Evolution of the solution for the initial data u0 .x/ D 3 sin x.

If the exact solution goes to infinity at a finite time, then in a neighborhood of the singular point, the numerical solution can remain bounded, and looking at the results of computations carried out on a single grid, the investigator can draw a wrong conclusion about the behavior of the solution. In this connection, we must stress the importance of performing the computations on condensing grids with accuracy control, which allows one to find the singularity and its type. In Figure 6.11, by smooth lines we depict the evolution of the numerical solution up to the moment of a singularity, and by dotted lines we do so after that moment. Namely, the tests on condensing grids allow us to conclude up to which moment we can consider our computations as true ones. In Chapter 4, we obtained that describes a semiconductor with nonlinear and anisotropic dependence of the dielectric constant tensor on the field with free charge density current depending on the potential of the self-consistent field according to a power law: ˇ ˇ N X @ ˇˇ @u ˇˇp2 @u @ C jujq u D 0: u C @t @xi ˇ @xi ˇ @xi i D1

Moreover, for the initial-boundary-value problem with zero boundary conditions, results on the local solvability and uniqueness in the weak generalized sense were obtained; also global solvability conditions and conditions for the blow-up at a finite time were obtained, and, moreover, two-sided estimates for the blow-up time were obtained.

571


Consider the spatially one-dimensional case: @ .uxx C .jux jp2 ux /x / C jujq u D 0; @t

ujxD0 D ujxDl D 0;

uj tD0 D u0 .x/: (6.54)

Since the nonlinear operator stands by the time derivative, it follows that to apply the method of lines combined with the Rosenbrock method, we need to pass to a differential-algebraic system introducing the new function w D uxx C .jux jp2 ux /x W @ w D jujq u; 0 D w .uxx C .jux jp2 ux /x /; @t ujxD0 D ujxDl D 0; uj tD0 D u0 .x/: The application of the method of lines leads to a system of differential-algebraic equations with a singular but constant matrix. Under the approximation of the derivatives, in the denominator a small number h arises (the step of the spatial grid); to a certain extent, under the condensation of grids, this can lead to the loss of stability. Therefore, it is appropriate to multiply each of the equations by hp avoiding small numbers in the denominators. After that, we apply the Rosenbrock scheme, namely, the CROS scheme (6.29), to the obtained differential-algebraic system. In Chapter 2 it was shown that for p > q C 2, the solution exists for an infinitely long time, and also two-sided estimates for the power energy growth were obtained; in the one-dimensional case, the energy has the form E.t / D

p1 1 kux k22 C kux kpp : 2 p

(6.55)

For p D q C 2, the solution also exists for an infinitely long time, but the growth of the energy is exponential. For p < q C 2, the energy goes to infinity at a finite time. In the computation carried out, we choose q D 2 and p equal to 3, 4, and 5; thus, all three cases from theorem 6 have been considered. For p D 4, the evolution of the solution with the initial data u0 .x/ D sin x is shown in Figure 6.12. Moreover, in the process of computations, we compute a numerical analog of energy (6.54); in the case p D 4, the energy growth is exponential. In Figure 6.13, we show the energy growth in the logarithmic scale; the graph is close to a straight line, which justifies the exponential dependence. We carried out tests on condensing grids; these tests showed that at all control points in the spatial and time coordinates, the effective accuracy order of the method is equal to 2. In Figure 6.14, the evolution of the solution and its energy for the case p D 3 and the initial data u0 .x/ D sin x are presented. By the dotted line we denote the profile after the blow-up moment diagnosed by the condensing grid method.

572


Figure 6.12. Profiles of the solution of problem (6.54) for the case q D 2, p D 4.

Figure 6.13. Energy growth in problem (6.54) for the case q D 2, p D 4.

In the case p > 4, the behavior of the solution looks like that in the case p D 4, but the growth of the solution is slower; therefore, we present only the graph of the energy growth. It is important to show that the energy growth is power. In the case p D 5, the energy grows as E.t / .A C Bt /5 . If we change the sign of the nonlocal term, then the behavior of the solution substantially changes: .uxx u/ t C kux kp 2 uxx ˇuux D 0; ujxD0 D ujxDl D 0; uj tD0 D u0 .x/:

(6.56)

573


Figure 6.14. Profiles of the solution of problem (6.54) for the case q D 2, p D 3. t |x 1 2 3 4

0.314 2.054 2.033 2.023 2.022

0.628 2.033 2.025 2.019 2.019

0.942 2.003 2.011 2.011 2.014

1.257 2.062 2.088 2.097 2.005

1.571 1.863 1.868 1.984 2.047

1.885 2.062 2.088 2.097 2.005

2.199 2.003 2.011 2.011 2.014

2.513 2.033 2.025 2.019 2.019

2.827 2.054 2.033 2.023 2.022

Table 6.3. Effective accuracy order is close to the theoretical value p D 2, which indicates the smoothness of the solution.

Figure 6.15. Energy growth in problem (6.54) for the case q D 2, p D 3.

574


Figure 6.16. Power energy growth in problem (6.54) for the case q D 2, p D 5 justifies the theoretical estimates obtained in Chapter 4.

t |x 0.35 0.875 0.91 0.98 1.015 1.05 1.085 1.12 1.155 1.19 1.26 1.295 1.365

0.314 2.048 2.098 2.096 2.075 2.024 1.963 2.042 1.915 0.137 0.989 0.997 1.001 1.042

0.628 2.038 2.094 2.093 2.073 2.023 1.929 2.055 1.916 0.137 0.989 0.997 1.001 1.039

0.942 2.031 2.089 2.088 2.07 2.023 1.977 2.067 1.916 0.137 0.989 0.996 1.000 1.027

1.257 2.026 2.083 2.083 2.067 2.024 1.937 2.090 1.917 0.138 0.989 0.996 0.997 0.998

1.571 1.905 1.929 1.935 1.930 1.890 1.955 2.109 1.935 0.139 0.990 0.996 0.994 0.967

1.885 2.026 2.083 2.083 2.067 2.024 1.937 2.039 1.917 0.138 0.989 0.996 0.997 0.998

2.199 2.031 2.089 2.088 2.070 2.023 1.977 2.067 1.916 0.137 0.989 0.996 1.000 1.027

2.513 2.038 2.094 2.093 2.073 2.023 1.929 2.055 1.916 0.137 0.989 0.997 1.001 1.039

Table 6.4. Tests on condensing grids showed the blow-up of the solution at T0 1:155. After the singularity, the limit of the effective accuracy order 1 indicates that the amplitude growth is proportional to .T0 t /1 .

575


6.3.3 Blow-up of equations with nonlocal terms (coefficients of the equation depend on the norm of the function) Let us consider the third type of equations in which their coefficients depend on the norm of the function: .uxx u/ t kux kp 2 uxx ˇuux D 0; ujxD0 D ujxDl D 0;

uj tD0 D u0 .x/I

(6.57)

here kk2 is the norm in the space L2 .0; l/. Since Eq. (6.57) contains a linear operator by the time derivative, the application of the method of lines leads to an implicit system of ordinary differential equations with constant nonsingular matrix having the same form as in formula (6.41): M

dU D F .U /: dt

Each row of this system corresponds to the difference approximation of Eq. (6.57) at a certain spatial point. As was noted above, in this case, it is possible to apply the Rosenbrock method in the form (6.29). However, the existence of the nonlocal term in the equation means that the right-hand side at each point depends on the values of the function at all points. This leads to the Jacobi matrix in formula (6.29) being densely filled in. We can reduce the problem to a differential-algebraic system by introducing the new variable w D kux k2 : .uxx u/ t w p uxx ˇuux D 0;

0 D w kux k2 :

The matrix M for this system is obtained from matrix (6.41) by adding the zero last row and zero last column, and the Jacobi matrix in formula (6.29) has the following simple structure: 0 1 0 0 0 0 B 0 0 0 C B C B0 0 0 C B C B 0 0 0 C; B C B0 0 0 C B C @0 0 0 0 A i.e., it contains 5N nonzero entries. If, in the matrix of the system of linear algebraic equations, there exist dense blocks of zero entries known in advance, then their going around in the direct run of the Gauss method leads to a large acceleration of the computation. In this case, the number of arithmetic operations needed for solving the linear system (6.29) by using the Gauss method is proportional to the number of

576


equations. Recall that in the case of a densely filled-in matrix this number is equal to 2N 3 =3. Therefore, in contrast to equations of the second type, the application of the method of lines and the Rosenbrock method to equations of the third type is possible without reduction to differential-algebraic systems; nevertheless, this method allows us to substantially accelerate the computational process. For numerical computations in problem (6.57), we took the initial data in the form of a solitary “hump”: x 2 Œ0; , q D 2, ˇ D 0:5, u.x; 0/ D 10e5x sin x. The behavior of the solution is stipulated by the interaction of two nonlinear terms in Eq. (6.57). At the first instants of time (Figure 6.17), we see the growth of the solution with a small shift to the right, and a nonlinear transfer dominates; it rapidly shifts the wave to the right, and after the wave, a “deflection” appears; after the passage through zero, this deflection becomes the source of a new wave (Figure 6.18), which now travels to the left, and so on. In conclusion, we consider an equation containing a nonlinear operator by the time derivative, as well as nonlocal coefficients. In Chapter 5, we considered the initialboundary-value problem (5.1) arising in semiconductors when there exists a nonlocal connection between the conductivity of the medium and the electric field strength. The local solvability and the uniqueness of solutions were proved and the blow-up conditions and the global solvability conditions for the problem were also obtained. Here, we consider the one-dimensional analog of this problem: @ 2p .uxx jujq u/ kux k2 uxx D 0; @t ujxD0 D ujxDl D 0; uj tD0 D u0 .x/:

Figure 6.17. Evolution of the solution of problem (6.57), t 0:8.

(6.58)


577

Figure 6.18. Evolution of the solution of problem (6.57) t 0:8.

The equation contains a nonlinear operator by the time derivative, and, therefore, to apply the Rosenbrock method, we need to introduce the new function w D uxx jujq2 u. Furthermore, applying the method of lines, we obtain a differential-algebraic system with constant singular matrix. Strictly speaking, the existence of the nonlocal term in Eq. (6.58) does not require the introduction of the new function v D kux k2 . However, this approach is preferable since the Jacobi matrix in formula (6.29) has a simple structure and contains many zero entries. The numerical computations of problem (6.58) were carried out for the case p D 1, q D 1. In Figure 6.19, we present the profiles of the numerical solution for the initial data u0 .x/ D sin x, 0 x . The finding of the hole moment in the semiconductor was carried out by tests on condensing grids. The constructed difference scheme has the accuracy O.h2 C 2 /, and therefore, the time and spatial grids must be condensed by two times. According to the results of computations on three neighboring grids, we have computed the effective accuracy order p eff . The results of this test are presented in the table for the instants of time corresponding to the profiles of the solution in the figure. In the smoothness region of the solution, we have limN !1 p eff D p D 2; the violation of this convergence indicates the singularity of the solution. In this case, the blow-up of the solution is observed between the control points t D 0:98 and t D 1:00. In Chapter 5 two-sided estimates for the blow-up time were obtained. A number of computations were carried out with different initial data u0 .x/ D A sin x, 0 x , and the dependence of the blow-up time on the amplitude of the initial data was studied. In Figure 6.20, we show the two-sided theoretical estimates for the blow-up time

578


Figure 6.19. Evolution of the solution of problem (6.56); the dotted line shows the numerical solution after the blow-up moment.

Figure 6.20. Two-sided theoretical estimates of the blow-up time Tmin < Tmax and the singularity moment Treal diagnosed on condensing grids.

Tmin ; Tmax of the solution and for the hole moment Treal of the semiconductor defined by tests on condensing grids. We see that the computational experiment completely justifies the theoretical estimates.

579

Section 6.3 Results of blow-up numerical simulation t |x 0.20 0.40 0.60 0.80 0.90 0.94 0.96 0.98 1.00 1.02

0.314 2.004 2.005 2.007 2.006 1.988 1.957 1.919 1.817 1.354 0.700

0.628 2.002 2.002 2.002 1.997 1.983 1.96 1.932 1.855 1.484 0.629

0.942 2.001 2.000 1.998 1.994 1.986 1.973 1.958 1.919 1.803 0.990

1.257 2.000 1.998 1.997 1.994 1.993 1.996 2.005 2.087 1.652 1.564

1.571 1.999 1.998 1.996 1.995 1.998 2.011 2.045 2.398 0.727 1.213

1.885 2.000 1.998 1.997 1.994 1.993 1.996 2.005 2.087 1.652 1.564

Table 6.5. Control the effective accuracy order allows one to diagnose the blow-up moment of the exact solution.

Conclusions The application of the method of lines combined with the complex Rosenbrock scheme to solving initial-boundary-value problems for pseudoparabolic equations allows one to avoid a computational overflow even if the exact solution goes to infinity. Applying the accuracy control techniques in computations on condensing grids by using the proposed method, one can diagnose the blow-up moment of the exact solution. At the smoothness points of the exact solution, the proposed algorithm allows one to obtain the result with an asymptotically sharp error estimate of the computation. All of this makes the proposed method very attractive for applications.

Appendix A

Some facts of functional analysis

A.1

s;p

Sobolev spaces W s;p ./, W0 ./, and W s;p ./

Results of this section can be found in [26]. Consider Sobolev spaces that correspond to an arbitrary bounded domain RN with a smooth boundary 2 C 2;ı , ı 2 .0; 1. Definition A.1.1. If s 2 N and p 2 Œ1; C1, then we denote by W s;p ./ the vector space ¹u 2 D 0 ./ W D ˛ u 2 Lp ./; j˛j sº with the norm kukW s;p ./ D

X

kD

˛

j˛js

ukp Lp ./

1=p :

If s 2 RC nN and p 2 .1; C1/, then we denote by W s;p ./ the vector space ¹u 2 D 0 ./ W u 2 W Œs ;p ./; d;˛ .u/ 2 Lp . /; j˛j D Œsº with the norm kukW s;p ./ D where d;˛ D

p kukW jsj;p ./

C

X j˛jDŒs

jD ˛ u.x/ D ˛ u.y/j ; jx yj.N=p/C˛

p kd;˛ .u/kLp ./

1=p ;

s D Œs C :

k;p Definition A.1.2. If k 2 N and p 2 Œ1; C1, then we denote by W0 ./ the k;p closure of D./ with respect to the norm of the space W ./ with the topology induced by the latter space. If s 2 RC nN and p 2 .1; C1/, then we denote by s;p W0 ./ the closure of D./ with respect to the norm W s;p ./ with the topology induced by the latter space.

Definition A.1.3. If s 2 R n0 and p 2 .1; C1/, then we denote by W s;p ./ the s;p 0 space dual to W0 ./, where p 0 D p=.p 1/.

582

Appendix A Some facts of functional analysis

Theorem A.1.4. The spaces W k;p ./, k 2 N, p 2 Œ1; C1, are Banach spaces. If p 2 .1; C1/, then they are reflexive; if p D 2, then they are Hilbert spaces with the inner product X .u; v/Hk ./ D .D ˛ u; D ˛ v/L2 ./ : j˛jk

The spaces W s;p ./, s 2 RC nN, p 2 .0; C1/, are reflexive Banach spaces; in the case where p D 2, they are Hilbert spaces with the inner product .u; v/Hs ./ D .u; v/HŒs ./ C

X

.d;˛ .u/; d;˛ .v//L2 ./ :

j˛jDŒs

k;p

Theorem A.1.5. The spaces W0 ./, k 2 N, p 2 Œ1; C1, are Banach spaces. If p 2 .1; C1/, then they are reflexive; in the case where p D 2, they are Hilbert s;p spaces, which we denote by H0s ./. Similarly, the spaces W0 ./, s 2 RC nN, p 2 .1; C1/, are reflexive; in the case where p D 2, they are Hilbert spaces denoted by H0s ./. 0

k;p Theorem A.1.6. A generalized function T belongs to .W0 .//0 D W k;p ./, k 2 N, p 2 .1; C1/, if and only if T can be represented in the form

TD

X j˛jk

D ˛ g˛ ;

0

g˛ 2 Lp ./;

p0 D

p : p1

Theorem A.1.7. If RN is a bounded open set of the class C 2;ı , ı 2 .0; 1, k 1 is integer, and p 2 Œ1; C1/, then the following embeddings hold. (1) If N > kp, then W k;p ./ ,! Lq ./ for q Np=.N kp/ and W k;p ./ ,!,! Lq ./ for q < Np=.N kp/. In a more general case, if m 2 N, m k, and N > .k m/p, then W k;p ./ ,! W m;q ./ for q Np=.N .k m/p/ and W k;p ./ ,!,! W m;q ./ for q < Np=.N .k m/p/. (2) If N D kp, then W k;p ./ ,!,! Lq ./ for any q 2 Œ1; C1/. (3) If N < kp, then W k;p ./ ,!,! C 0; ./ with the following values of : D k .N=p/ for k .N=p/ < 1; is arbitrary and < 1 for k .N=p/ D 1; D 1 for k .N=p/ > 1. In a more general case, if m 2 N, m k, and N < .k m/p, then W k;p ./ ,!,! C m; ./ for: D k m .N=p/ if k m .N=p/ < 1; arbitrary < 1 if k m .N=p/ D 1; D 1 if k m .N=p/ > 1. Theorem A.1.8. Let RN be a bounded domain with Lipschitz-continuous boundary , k 1 be an integer, and p 2 Œ1; C1/. Then the following assertions hold.

Section A.2 Weak and -weak convergence

583

(1) If kp < N and 1 q .N 1/p=.N kp/, then there exists a unique operator 0 2 L.W k;p ./I Lq .// such that if u 2 D./, then 0 u D uj , and if u 2 W k;p ./, then 0 u is called the trace of order 0 of a function u on I if p > 1, then 0 is a compact operator. (2) If kp D N , then (1) holds for any q 1. (3) If kp > N , then the trace 0 u of a function u 2 W k;p ./ C./ is the classical restriction of u to . Now we consider the space H1 ./. Let f 2 H1 ./. Then there exist f0 ; f1 ; : : : ; fn in L2 ./ such that

Z hf; vi D

dx f0 v C

n X

fi vxi

8v 2 H01 ./:

i D1

Moreover, kf kH1

A.2

²Z X 1=2 ³ n 2 2 D inf jfi j dx W f0 ; : : : ; fn 2 L ./ : i D0

Weak and -weak convergence

Results of this section can be found in [436]. Definition A.2.1. A sequence xn * x is said to be weakly converging in a normed space X if the numerical sequence hun ; hi converges for any h 2 X , where h; i is the duality bracket between the normed spaces X and X . Definition A.2.2. A sequence hn 2 Xs is said to be -weakly converging if for any u 2 X the numerical sequence hu; hn i converges. Theorem A.2.3. Let ¹xn º be a bounded by norm sequence of elements of a reflexive Banach space X. Then ¹xn º contains a subsequence ¹xn0 º, which weakly converges to some point of the space X. Theorem A.2.4. An arbitrary weakly converging sequence ¹xn º is strongly bounded. In particular, if xn * x, then the sequence ¹kxn kº is bounded and lim inf kxn k kxk:

n!C1

584


Theorem A.2.5. Let B be a separable Banach space and T W B ! L1 .R1 /. Then there exists a measurable function g on R1 with values in B such that sup kgk D kT k

x2R1

Z

and, moreover, T .e/ D

hg.x/; ei dx:

Theorem A.2.6. Let Q be a bounded domain in RN x R t , g and g be functions from Lq .Q/, 1 < q < C1, such that kg kLq .Q/ C, g ! g a.e. in Q. Then g * g weakly in Lq .Q/.

A.3

Weak and strong measurability. Bochner integral

Results of this section can be found in [436]. Definition A.3.1. Let .S; B; m/ be a space with measure and x.s/ be a mapping defined on S with values in B-space X. The mapping x.s/ is said to be weakly Bmeasurable if for any element h 2 X , the number-valued function hx.s/; hi of the variable s is measurable. The mapping x.s/ is said to be simple if it takes constant nonzero values on each of sets Bj that form a finite system of S nonintersecting Bmeasurable sets and m.Bj / 1, and x.s/ D 0 for s 2 S j Bj . The mapping x.s/ is said to be strongly B-measurable if there exists a sequence ¹xn .s/º of simple mappings, which strongly converges to x.s/ m-a.e. on S. Definition A.3.2. A function x.s/ is said to be separable-valued if the set of its values ¹x.s/I s 2 Sº is separable. A function x.s/ is said to be m-almost separablevalued if there exists a B-measurable set B0 with zero m-measure such that the set x.s/I s 2 S B0 is separable. Definition A.3.3. A function x.s/ defined on a space with measure .S; B; m/ and taking its values in a B-space X is said to be Bochner m-integrable if there exists a sequence of simple functions ¹xn .s/º, which strongly converges to x.s/ m-a.e. in S and, moreover, Z lim kx.s/ xn .s/k m.ds/ D 0: n!C1 S

For any set B 2 B, the Bochner m-integral of the function x.s/ over the set B is defined as follows: Z Z x.s/ m.ds/ s lim B .s/xn .s/ m.ds/; B

n!C1 S

where B .s/ is the characteristic function of the domain .

585

Section A.4 Spaces of integrable functions and distributions

Theorem A.3.4. A strongly B-measurable function is Bochner m-integrable if and only if the norm kx.s/k is m-integrable. Theorem A.3.5. Let x./ 2 L.0; TI B/. Then for almost all t; t C h; t h 2 .0; T/, the following relation holds: 1 lim h!0 h

Z

tCh th

ds kx./ x.t /kB d D 0:

Remark A.3.6. A strongly absolutely continuous function can have derivative nowhere. The corresponding example can be found in [198]. Theorem A.3.7. If y.t / is a function with bounded strong variation defined on .0; T/, taking its values in B, and almost everywhere weakly differentiable, then its derivative y 0 .t / belongs to L.0; TI B/. If, moreover, y.t / is a weakly absolutely differentiable function, then it can be represented as the indefinite integral of y 0 .t /.

A.4

Spaces of integrable functions and distributions

Results of this section can be found in [168]. Definition A.4.1. We define by Lp .S I X/, p 2 Œ1; C1/, the set of all Bochnermeasurable functions u 2 .S ! X/ for which the following Lebesgue integral is finite: Z S

ds ku.s/kp < C1:

Theorem A.4.2. The set Lp .S I X/, p 2 Œ1; C1/, which is a vector space with natural linear operations, becomes a Banach space with respect to the norm Z kukLp .SIX/ D

S

ds ku.s/kp

1=p :

Definition A.4.3. A function u 2 .S ! X/ is said to be essentially bounded if it is equivalent to some bounded function, i.e., if there exists a number M < C1 such that ku.s/k M for almost all s 2 S. The infimum of all such numbers M is denoted by vraimax ku.s/k: s2S

The set of all Bochner-measurable, essentially bounded functions from .S ! X/ is denoted by L1 .S I X/.

586


Theorem A.4.4. The set L1 .S I X/ is a nonreflexive Banach space with respect to the norm kukL1 .SIX/ D vraimax ku.s/k: s2S

Theorem A.4.5. If u 2 Lp .S I X/, p 2 Œ1; C1, and v 2 Lq .S I X /, p 1 C q 1 D 1, then hu./; v./i 2 L1 ./ and Z hu.s/; v.s/i ds kukLp .SIX/ kvkLq .SIX / : S

Theorem A.4.6. ¹H; .; /º is a Hilbert space, then the space L2 .S I H/ with the inner product Z .u; v/S D

.u.s/; v.s// ds S

is also a Hilbert space. Theorem A.4.7. For any finite T > 0, the set of polynomials from Œ0; T ! B, i.e., the set ³ ² m X j aj t ; aj 2 B; j D 0; m ; pjp 2 Œ0; T ! B; p.t / D j D0

is dense in C.Œ0; TI B/. Definition A.4.8. We denote the space L.D.S/; Xw / of all continuous linear mappings of a space D.S/ to a locally convex space Xw by D .S I X/. Elements of this space are called distributions on S with values in X. .1/

Definition A.4.9. Cw .Œ0; TI B/ is the set of all demicontinuous functions on the segment Œ0; T with values in a reflexive Banach space B that have demicontinuous weak derivative.

A.5

Nemytskii operator. Krasnoselskii theorem

Results of this section can be found in [122]. Definition A.5.1. Let be a domain in RN and f .xI / be a function defined for almost all x 2 and all 2 Rm . The function f is said to possess the Carathéodory property if (1) for any 2 Rm , the function f .x/ D f .xI / is measurable on as a function of the variable ; (2) for almost all x 2 , the function fx ./ D f .xI / is continuous in Rm with respect to the variable .

587

Section A.5 Nemytskii operator. Krasnoselskii theorem

Definition A.5.2. Assume that a function f .xI / is defined for x 2 and 2 Rm and possesses the Carathéodory property. The operator K defined on all ordered sets of m measurable functions ui D ui .x/ (x 2 , i D 1; m) by the formula K.u1 ; : : : ; um / D f .xI u1 .x/; : : : ; um .x//;

x 2 ;

is called the Nemytskii operator. Theorem A.5.3. If the Nemytskii operator f determined by a function f .xI u.x// acts from Lp1 to Lp2 , where p1 ; p2 1, then it is continuous and bounded. Theorem A.5.4. Let a function f .x; u/, x 2 G, 1 < u < C1, be a continuous in u and satisfy the inequality jf .x; u/j

n X

Ti .x/jujˇi C bjujˇ0 ;

i D1

where p1 p2

Ti 2 L p1 p2 ˇi ;

0 p2 ˇi < p1 ;

i D 1; n;

ˇ0 D

p1 ; p2

b 0:

Then the operator f acts from the space Lp1 to the space Lp2 and it is continuous and bounded. Theorem A.5.5. If the Nemytskii operator f D f .xI u1 ; : : : ; um / acts from Lp1 Lp2 Lpm to Lr , where p1 ; : : : ; pm ; r 1, then it is continuous and bounded. Example A.5.6. Consider the function qC2

f .x; u/ D jhjjujq W LqC2 ./ ! L qC1 ./ for any fixed h 2 LqC2 ./, q > 0. By definition, the continuity of the operator f W LqC2 ./ ! L.qC2/=.qC1/ ./ means that the limit relation “ qC2 dx jf .x; u/ f .x; un /j qC1 D 0 lim n!C1

holds under the strong convergence un ! u in LqC2 ./. Example A.5.7. Consider the function 0

f .x; u/ D jrhjjrujp2 W Lp ./ ! Lp ./ 1;p for p 0 D p=.p 1/ and any fixed h 2 W0 ./, p > 2. By definition, the continuity 0 of the operator f W Lp ./ ! Lp ./ means that the limit relation “ 0 p dx jf .x; u/ f .x; un /jp D 0; p 0 D ; lim n!C1 p 1

holds under the strong convergence run ! ru in Lp ./.

588


Example A.5.8. Consider the function f .x; u/ D jujq W LqC2 ./ ! L

qC2 q

./

for q > 0. By definition, the continuity of the operator f W LqC2 ./ ! L.qC2/=q ./ means that the limit relation “ qC2 lim dx jf .x; u/ f .x; un /j q D 0 n!C1

holds under the strong convergence un ! u in LqC2 ./. Example A.5.9. Consider the function f .x; u/ D jrujp2 W Lp ./ ! Lp=.p2/ ./ 0

1;p for p > 2. By definition, the continuity of the operator f W W0 ./ ! W 1;p ./ means that the limit relation “ lim dx jf .x; u/ f .x; un /jp=.p2/ D 0 n!C1

holds under the strong convergence run ! ru in Lp ./.

A.6

Inequalities

Results of this section can be found in [436]. Theorem A.6.1. Let f W R ! R be a concave function and U 2 RN be a bounded open set. Let u W U ! R be a summable function. Then Z Z 1 1 u dx f .u/ dx; f jU j U jU j U where jU j is the measure of U . " 1 Theorem A.6.2 (Cauchy inequality with "). ab a2 C b 2 , " > 0, a; b > 0. 2 2" Theorem A.6.3 (Cauchy–Schwarz inequality). Let A W V ! V be a symmetric, linear, nonnegative operator with respect to the duality bracket h; i between reflexive Banach spaces V and V . Then the following inequality holds: hAu; vi hAu; ui1=2 hAv; vi1=2 :

589

Section A.7 Operator calculus

A.7

Operator calculus

Results of this section can be found in [359]. Consider a bounded linear operator T 2 L.X; X/, where X is a complex Banach space. We define a function f .T / of the operator T by the following formula similar to the Cauchy integral formula: Z 1 f ./R.I T / d : f .T / D 2 i For this, we denote by F .T / the set of all complex-valued functions f ./ that are holomorphic in neighborhoods of the spectrum .T / of the operator T . These neighborhoods can be nonconnected and can depend on f ./. Let f 2 F .T / and let an open set U .T / of the complex plane be contained in the domain where the function f is holomorphic. We also assume that the boundary of this set consists of a finite number of rectifiable, positively oriented Jordan curves. Then the bounded operator f .T / is defined by the formula Z 1 f ./R.I T / d : f .T / D 2 i

A.8

Fixed-point theorems

Results of this section can be found in [359]. Theorem A.8.1. Let a set K X be compact and convex and X be a real Banach space. Assume that a mapping AWK!K is continuous. Then A has a fixed point. Theorem A.8.2. Let B be a closed bounded subset of a Banach space X. Let an operator A act from B to B and is a contraction operator on B. Then this operator possesses a unique fixed point x 2 B.

A.9

Weakened solutions of the Poisson equation

In this section, we introduce the notion of a weakened solution and study its relations with the well-known notion of the weak solution of the Poisson equation. Consider the following problem: u D f;

uj@ D 0:

(A.1)

590


Definition A.9.1. A weak generalized solution of problem (A.1) is a solution of the class u 2 H01 ./ satisfying the relation .ru; r/2 D hf; i

8 2 H01 ./

(A.2)

for f 2 H1 ./, where .; /2 is the inner product in L2 ./, in a bounded domain with smooth boundary @ 2 C 2;ı , ı 2 .0; 1, and h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./. Definition A.9.2. A weakened solution of problem (A.1) is a weak solution of the class C .1/ ./ \ C0 ./. Consider Eq. (A.1) in the weak sense. Then the operator is an isometric isomorphism between H01 ./ and H1 ./. In this case, if f 2 H1 ./, we obtain an equivalent equation u D ./1 f:

(A.3)

Note that on the set f 2 C .0;h/ ./, h 2 .0; 1, the operator ./1 has an explicit representation (see [421]) Z dy G.x; y/f .y/; (A.4) ./1 f D

where G.x; y/ is the Green function of the first boundary-value problem for the Laplace equation in the bounded domain with smooth boundary @ 2 C 2;ı , ı 2 .0; 1. Taking into account (A.3), we obtain from (A.4) Z dy G.x; y/f .y/: (A.5) uD

Using well-known properties of the Green function G.x; y/, we obtain that for f D f" 2 C .0;h/./, the formula yields a weak solution of problem (A.1) in the sense (A.2), where the space is a Banach space with respect to the norm jf .x/ f .y/j ; jx yjh x;y2

kf kh D sup jf .x/j C sup x2

h 2 .0; 1:

Substituting (A.5) with f D f" in Eq. (A.2) we obtain Z dy G.x; y/f" .y/; r D hf" ; i 8 2 H01 ./: r

2

Now we assume that 2 C./ is an arbitrary fixed function and f" ! strongly in C./ with respect to the standard norm kf k D sup jf .x/j; x2

591

Section A.10 Intersections and sums of Banach spaces

where f" 2 C .0;h/ ./. Note that for , the potential-type integral Z dy G.x; y/ .y/

is well defined. Then the following relation holds: Z Z dy rx G.x; y/ .y/; r C dy rx G.x; y/Œf" .y/ .y/; r

2

2

D h ; i C hf" ; i: (A.6) Using the properties of the Green function, from (A.6) we obtain Z dy rx G.x; y/ .y/; r D h ; i 8 2 H01 ./

2

as " ! C0. Therefore, the function Z u.x/ D dy G.x; y/ .y/;

.x/ 2 C./;

(A.7)

is a weak solution of the problem and belongs to the class C .1/ ./ \ C0 ./, i.e., is a weakened solution of problem (A.1). Thus, we have proved that the restriction of the operator ./1 to the class C./ has the explicit representation (A.7). Moreover, we have obtained the explicit representation (A.7) for the weakened solution of problem (A.1).

A.10

Intersections and sums of Banach spaces

Results of this section can be found in [168]. Definition A.10.1. A seminorm on a vector space X is a functional p.x/, x 2 X, satisfying the following conditions: (1) p.x C y/ p.x/ C p.y/; (2) p.tx/ D jt jp.x/. Definition A.10.2. A vector topological space is called a locally convex space if there exists a family of seminorms on X such that the following conditions hold: (1) if p.x/ D 0 for any p 2 , then x D 0; (2) the set of (convex) sets of the form ¹x j x 2 X; pi .x x0 / < "i ; i D 1; nº forms a basis of topology in X (here x0 is an arbitrary point of X, p1 ; : : : ; pn is an arbitrary finite system of seminorms from , and "1 ; : : : ; "n is an arbitrary finite system of positive real numbers).

592


Theorem A.10.3. Assume that Banach spaces X and Y are continuously embedded in a locally convex space V . Then their intersection X \ Y is a Banach space with respect to the norm kzk D kzkX C kzkY : Theorem A.10.4. Let X and Y be Banach spaces continuously embedded in a locally convex space V . Let the intersection X \ Y equipped with the norm kzkV D kzkX C kzkY is dense in X and Y . Then X C Y D .X \ Y / and .X C Y / D X \ Y in the sense of the equality of sets and in the sense of equality of norms.

A.11

Classical, weakened, strong generalized, and weak generalized solutions of evolutionary problems

In this monograph, we use different notions of solutions. In this section, we discuss the relation between these notions. Namely, the following embeddings hold: classical solutions weakened solutions strong generalized solutions weak generalized solutions : For any specific problem, the notions of these classes are introduced in different ways but the relationship between them is the same. Consider a simple example of the definition of these classes for the following linear problem: @ u u D 0; @t

uj@ D 0;

u.x; 0/ D u0 .x/;

(A.8)

where R3 is a bounded domain with the smooth boundary @ 2 C .2;ı/ , ı 2 .0; 1, .x1 ; x2 ; x3 / 2 . Definition A.11.1. A classical solution of problem (A.8) is a solution of the class C .1/ .Œ0; TI C .2/ ./ \ C .1/ ./ \ C0 .//. Note that in the class considered, all partial derivatives are defined in the classical sense.

Section A.11 Different classes of solutions of evolutionary problems

593

Definition A.11.2. A weakened solution of problem (A.8) is a strong generalized solution of the class C .1/ .Œ0; TI C .1/ ./ \ C0 .//. Definition A.11.3. A strong generalized solution of problem (A.8) is a solution of the class C .1/ .Œ0; TI H01 .// of the following problem:

@ u u; w D 0 8 w 2 H01 ./; 8 t 2 Œ0; T; @t u.x; 0/ D u0 .x/ 2 H01 ./; where h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./ and the derivative with respect to time is treated in the classical sense. Definition A.11.4. A weak generalized solution of problem (A.8) is a solution of the class AC.Œ0; TI H01 .// of the problem

Z T @ dt u u; w .t / D 0 8 w 2 H01 ./; 8 .t / 2 L2 .0; T/; @t 0 u.x; 0/ D u0 .x/ 2 H01 ./; where h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./ and the derivative with respect to time is treated in the Lebesgue sense. Note that one can state a more weak definition of a weak generalized solution; this depends on a specific problem. For example, for problem 1 we can state the following definition. Definition A.11.5. A weak generalized solution of problem (A.8) is a solution of the class u 2 C.0; TI H01 .// of the problem d hu; wi hu; wi D 0 8w 2 H01 ./; dt u.x; 0/ D u0 .x/ 2 H01 ./; where h; i is the duality bracket between the Hilbert spaces H01 ./ and H1 ./ and the derivative with respect to time is treated in the sense of D 0 .0; T/. Obviously, the chain of embeddings presented above is valid for these types of solutions. For stationary solutions, a simpler chain holds: classical solutions weakened solutions weak generalized solutions : This is related to the fact that in Definitions A.11.3–A.11.5 of strong and weak generalized solutions, one uses one or another type of smoothness of solutions in time, but in stationary problems, these properties are absent.

594


A.12

Two equivalent formulations of weak solutions in L2 .0; TI B/

Let B be a reflexive, separable Banach space, B be its dual space, and h; iB be the duality bracket. We denote by k kB the norm in B, and by k kB the norm in B . Let LŒu be a partial differential operator such that for any T 2 .0; T0 /, the following inequality holds: Z

T 0

dt kLŒuk2 B .t / C.T/ < C1

8T 2 .0; T0 /:

(A.9)

We consider the following two formulations of a weak generalized solution of the operator LŒu: Z

T 0

dt hLŒu; wiB .t / D 0

8w 2 B;

8 .t / 2 L2 .0; T/;

8T 2 .0; T0 /; (A.10)

and Z

T 0

dt hLŒu; viB D 0 8v 2 L2 .0; TI B/;

8T 2 .0; T0 /;

(A.11)

where h; iB is the duality bracket between the Banach spaces B and B . By virtue of (A.9), the left-hand sides of (A.10) and (A.11) are defined. We prove that an arbitrary fixed function from L2 .0; TI B/ can be approximated in the sense of L2 .0; TI B/ by functions from C.Œ0; TI B/ with an arbitrary accuracy. Indeed, introduce the following compactly supported in R1 , infinitely differentiable function ´ C exp. t 211 /; jt j < 1; .t / D (A.12) 0; jt j 1; where the constant C > 0 is defined by the condition Z dt .t / D 1: R1

We fix an arbitrary function v 2 L2 .0; TI B/ and denote by v the zero extension of the function v outside the interval .0; T/. Using function (A.12), we introduce the shear of the function v: Z 1 t s v " .t / D ds v.s/; (A.13) " R1 "

Section A.12 Two equivalent formulations of weak solutions in L2 .0; TI B/

595

where the integral is treated in the Bochner sense. Note that by virtue of [198, Theorem 3.8.4], (A.13) implies that v " 2 C.Œ0; TI B/. The following relation holds: Z 1 t s " Œv.s/ v.t / : ds v .t / v.t / D " R1 " This implies the inequality Z 1 t s kv.s/ v.t /kB kv " .t / v.t /kB ds " R1 " and then the inequality 1 kv .t / v.t /kB " "

Z

tC" t"

8t 2 .0; T/

(A.14)

ds kv.s/ v.t /kB :

(A.15)

By [198, Theorem 3.8.5] we obtain that for almost all t 2 .0; T/, the following limit relation holds: Z 1 tC" ds kv.s/ v.t /kB D 0: lim "!C0 " t" Therefore, for almost all t 2 .0; T/, by (A.15), the limit relation lim kv.t / v " .t /kB D 0

"!C0

holds. By the Lebesgue theorem, we obtain that Z T dt kv.t / v " .t /k2B D 0: lim "!C0 0

Thus, we have proved that any function from L2 .0; TI B/ can be approximated by functions from C.Œ0; TI B/ with arbitrary accuracy. Consider the Banach space C.Œ0; TI B/. Since the Banach space B is separable, there exists a countable, everywhere dense system of elements ¹wk ºC1 of this space. kD1 By [168, Chapter 4, Theorem 1.3], the system of functions ¹wk1 t k2 ºC1 , t 2 k1 ;k2 D1 Œ0; T, is everywhere dense in C.Œ0; TI B/. be a basis in L2 .0; T/. The following inequality holds: Let ¹ k .t /ºC1 kD1 2 Z T mX 1 ;m3 dt cm1 m3 k1 k3 k3 .t /wk1 v 0

B

k1 ;k3 D1

Z T Z T 2 dt kv v" kB C dt v" 0

0

mX 1 ;m2 k1 ;k2 D1

2 k2

dm1 m2 k1 k2 wk1 t

B

2 Z T mX m3 X 1 ;m2 k2 C dt dm1 m2 k1 k2 wk1 t k2 k3 k3 .t / 0

k1 ;k2 D1

D I 1 C I2 C I3 ;

k3 D1

B

(A.16)

596


where cm1 m3 k1 k3 D

m2 X

dm1 m2 k1 k2 k2 k3 :

k2 D1

Therefore, for any " > 0, there exists a function v" such that I1 "=3, there exist m1 ; m2 2 N and dm1 m2 k1 k2 such that I2 "=3, and there exist k2 k3 such that I3 "=3. Hence (A.16) implies the inequality Z 0

T

dt v

mX 1 ;m3 k1 ;k3 D1

2 cm1 m3 k1 k3 k3 .t /wk1 ":

(A.17)

B

Consider problems (A.10) and (A.11). Obviously, (A.11) implies (A.10). Now we prove that (A.10) implies (A.11). Indeed, the following equality holds: Z

T 0

Z dt hLŒu; viB D

0

T

dt LŒu; v

mX 1 ;m3

cm1 m3 k1 k3 k3 .t /wk1

k1 ;k3 D1

(A.18) B

for all v 2 L2 .0; TI B/. From (A.9) and (A.17) we see that the right-hand side of (A.18) can be made less than ı."/ ! C0 as " ! C0 for any " > 0. Since the lefthand side is independent of " > 0, we obtain (A.11). Therefore, problems (A.10) and (A.11) are equivalent.

A.13

Gâteaux and Fréchet derivatives of nonlinear operators

Results of this section can be found in [122]. Now we recall the definition of the Gâteaux and Fréchet derivatives of nonlinear mappings and some their properties. Let B be a reflexive Banach space with string dual space B0 . Assume that an operator F .u/ W B ! B0 is defined on the Banach space B. Definition A.13.1. If the limit lim kt 1 ŒF.u C t h/ F.u/ V F .u; h/kB0 D 0;

t!C0

h 2 B;

exists at a point u 2 B, then the operator V F .u; h/ is called the Gâteaux differential of the operator F.u/ at the point x 2 B. Definition A.13.2. In the case where the Gâteaux differential V F .u; h/ W B B ! B0 of the operator F.u/ W B ! B0 is linear with respect to h 2 B, it is called the Gâteaux derivative.

597

Section A.13 Gâteaux and Fréchet derivatives of nonlinear operators

Definition A.13.3. If the relation F.u C h/ F.u/ D d F.u; h/ C !.u; h/ holds at a point u 2 B, where !.u; h/ is a linear operator of u 2 B and k!.u; h/kB0 D 0; khkB khkB D1 lim

then d F .u; / W B ! L.B; B0 / is called the Fréchet derivative of the operator F.u/ W B ! B0 . Now we state sufficient condition under which the existence of the Gâteaux derivative implies the existence of the Fréchet derivative and these derivatives coincide. Theorem A.13.4. If the Gâteaux derivative V F .u; h/ W B B ! B0 of an operator F .u/ W B ! B0 exists in some neighborhood U.u/ of a point u 2 B and is continuous at this point, then d F.u; h/ D V F .u; h/ exists. In other words, the continuous Gâteaux derivative is the Fréchet derivative. Now we prove that the pseudo-Laplacian 1;p

p u div.jrujp2 ru/ W W0

0

./ ! W 1;p ./; 1;p

has the Fréchet derivative defined on W0

p > 2;

p0 D

p ; p1

./:

d F .u; h/ D div.jrujp2 rh/ C .p 2/ div.jrujp4 .ru; rh/ru/ W 1;p

W0 1;p

Indeed, let u; h 2 W0

0

./ W01;p ./ ! W 1;p ./:

./. Introduce the notation D ru;

D rh:

To find the Fréchet derivative of the operator A1 .u/ div.jrujp2 ru/, we must find the Fréchet derivative of the operator jjp2 . First, we find the Gâteaux derivative of this function for any vector-valued function 2 Lp ./ Lp ./ Lp ./, i.e., prove the limit relation lim k.j C t jp2 . C t / jjp2 /t 1

t!C0

jjp2 .p 2/jjp4 .; /kp 0 D 0:

598


For any fixed , 2 Lp ./ Lp ./ Lp ./ the sets E0ı D ¹x 2 W jj < ıº

Eı D ¹x 2 W jj ı > 0º ;

for any fixed ı > 0 are measurable subsets of the set ; moreover, it is obvious that D Eı [ E0ı [ E0 , meas E0 D 0. We must consider the following two cases: p 2 .2; 4/ and p 2 Œ4; C1/. The second case is simpler, so we discuss the first. Let p 2 .2; 4/. We consider the vector-valued function .t / D j C t jp2 . C t /;

p > 2;

for which the Taylor expansion with the remainder term in the Lagrange form can be written: .t / D .0/ C 0 . /t; .0/ D jj

p2

2 .0; t /;

;

0 . / D j C jp2 C .p 2/. jj2 C .; //j C jp2 . C / D j. C /jp2 C .p 2/j C jp2 Therefore,

1 j C j2

. C ; / C : j C j j C j

j 0 . /j .p 1/jjj C jp2 :

In addition, Z Z 0 p0 p0 dx j . /j .p 1/ E0ı

Z p=.p1/

.p 1/

0

E0ı

p

E0ı

0

dx jjp j C jp .p2/

dx jj

1=.p1/ Z

p

E0ı

dx j C j

.p2/=.p1/

Ckkpp=.p1/ Œkkp C kkp p.p2/=.p1/ C1 ı C C2 "=6 for sufficiently small ı > 0 and small t > 0, 2 .0; t /. Therefore, the following estimate holds: k.j C t jp2 . C t / jjp2 /t 1 jjp2 .p 2/jjp4 .; /kp 0 ;E0ı k 0 . /kp 0 ;E0ı C kjjp2 kp0 ;E0ı C .p 2/kjjp4 .; /kp 0 ;E0ı ; where

Z k kp 0 ;E0ı

E0ı

dx j j

p0

1=p0 :

599


The following inequalities hold: Z p2

kjj

kp 0 ;E0ı Z

E0ı

E0ı

dx jj

.p2/p 0

dx jj

.p2/p 0 r1

jj

p0

1=p0

1=.p0 r1 / Z

p 0 r2

E0ı

dx jj

1=.p 0 r2 /

Y:

We take p 0 r2 D p; then r2 D p 1, r1 D .p 1/.p 2/1 , .p 2/p 0 r1 D p, p 0 r1 D p=.p 2/. Therefore, Z Y

E0ı

dx jj

p

.p2/=p Z E0ı

p

dx jj

1=p

Cı

" 6

Similarly, we can prove that .p 2/kjjp4 .; /kp 0 ;E0ı

" 6

for sufficiently small ı > 0. Therefore, k.j C t jp2 . C t / jjp2 /t 1 jjp2 .p 2/jjp4 .; /kp0 ;E0ı

" 2

for sufficiently small t > 0 and ı > 0. Now we consider the vector-valued function .t / D j C t jp2 . C t /;

p > 2;

on the set Eı . The function .t / can be represented in the following equivalent form: ˇ.p2/=2 ˇ . C t / .t / D ˇjj2 C t 2 jj2 C 2 .; /ˇ D jjp2 j1 C zj.p2/=2 . C t /; where zD

t 2 jj2 C 2t .; / : jj2

Let t > 0 be so small that z 2 .0; 1/. Then, applying the Lagrange formula and the fact p 2 .2; 4/, we obtain .1 C z/.p2/=2 D 1 C

z p2 ; 2 .1 C /.4p/=2

2 .0; z/:

600


This implies .t / D .0/ C t jjp2 C t .p 2/ C

jjp4 .; / .1 C /.4p/=2

h 1 p 2 2 p4 2 jj t jj .p 2/t 2 .; / jjp4 C .4p/=2 2 .1 C / p 2 3 p4 2 i jj : t jj C 2

The explicit form of z implies that for fixed ; 2 Lp ./ Lp ./ Lp ./, there exists 2 .0; t / such that the representation D

2 jj2 C 2 .; / jj2

holds. Note that jj >

j2j.; /j jj2 j; jj2

jj > jj for sufficiently small 2 .0; t / on the set Eı . Moreover, for such 2 .0; t /, the estimate ˇ ˇ ˇ ˇ 1 ˇ ˇ ˇ .1 C /.4p/=2 1ˇ Cjj holds. Note that for jj > jj, the inequality jj 3 is valid. Moreover, the following inequalities hold: J kŒ.t / .0/t 1 jjp2 .p 2/jjp4 .; /kp 0 ;Eı k.1 .1 C /.p4/=2 /.p 2/jjp4 .; /kp0 ;Eı C k.1 C /.p4/=2 Œ.p 2/t .; /jjp4 C .p 2/21 jjp4 jj2 t C .p 2/21 jjp4 jj2 t 2 kp 0 ;Eı D J1 C J2 I for J1 , the estimate J1 Ckjjp2 jjkEı ;p 0 C.ı; kkEı ;p0 / < C1 holds uniformly in t > 0. On the other hand, by the explicit form of , we see that ! C0 for t > > 0 tending to zero for almost all x 2 Eı . Therefore, .1 .1 C /.p4/=2 /.p 2/ Cjj ! C0

601


as t > > 0 tends to zero. Hence, by the Lebesgue theorem, we obtain for sufficiently small t > 0 the estimate " J1 : 4 Using the fact that for sufficiently small t > 0, the inequality jjt < jj holds, we obtain the relations ˇ ˇp4 j jjp2 jj; t ˇ.; /j ˇ ˇ ˇ p4 jj ˇ jjp2 jj; t ˇjj ˇ ˇ t 2 ˇjjp4 jj2 ˇ jjp2 jj: This and the explicit expression for J2 imply, on one hand, that for sufficiently small t > 0, J2 C; and, on the other hand, that t .; /jjp4 ! 0;

t jjp4 jj ! 0;

t 2 jjp4 jj2 ! 0

as t ! C0 for almost all x 2 Eı . Applying the Lebesgue theorem, we see that for sufficiently small t > 0, " J2 : 4 Therefore, for sufficiently small t > 0 we have " J : 2 Hence, we obtain that k.j C t jp2 . C t / jjp2 /t 1 jjp2 .p 2/jjp4 .; /kp0 " for sufficiently small t > 0. Therefore, the vector-valued function jjp2 ;

p 2 .2; 4/;

is Gâteaux differentiable. This means that the operator div.jrujp2 ru/ is Gâteaux 1;p differentiable for any u 2 W0 ./ and its Gâteaux derivative has the form 1;p

div.jrujp2 rh/ C .p 2/ div.jrujp4 .ru; rh/ru/ 8h 2 W0

./:

For simplicity, we rewrite the Gâteaux derivative of the operator p u in the form A01;u .u/h D A11 .u/h C A12 .u/h; A11 .u/h D div.jrujp2 rh/; A12 .u/h D .p 2/ div.jrujp4 ru.ru; rh//;

602

Appendix A Some facts of functional analysis 0

where A01;u ./ W W01;p ./ ! L.W01;p ./I W 1;p .//, p 0 D p=.p 1/. Now we prove that the Gâteaux derivative A11 .u/ is a strongly continuous and bounded 0 1;p 1;p mapping W0 ./ ! L.W0 ./I W 1;p .//. By definition, we have kA11 .u/ A11 .un /kW 1;p ./!W 1;p0 ./ 0

D

sup khk

1;p D1 W0

where

kA11 .u/h A11 .un /hkW 1;p0 ./ C

Z Jn .h/ D

p0

p2

dx jrhj jjrun j

p2 p 0

jruj

j

sup khk

1;p D1 W0

Jn .h/;

1=p 0 :

This implies the upper estimate for Jn .h/. Indeed, the following inequality holds: Z Jn .h/

p 0 r1

dx jrhj

1=.r1 p0 / Z

p2

dx jjruj

jrun j

p2 p 0 r2

jj

1=.r2 p 0 / ;

where r11 C r21 D 1, r1 p 0 D p, r1 D p 1, r2 D .p 1/=.p 2/, and p 0 D p=.p 1/. Hence, the inequality Z Jn .h/

dx jrhj

p

1=p Z

ˇp=.p2/ ˇ dx ˇjrun jp2 jrujp2 ˇ

.p2/=p

holds. We consider the operator f D jrujp2 D .u21 C u22 C u23 /.p2/=2 ;

ui D

@u : @xi

This operator is generated by a Carathéodory function and, as is easy to prove, this Nemytskii operator acts from Lp ./ Lp ./ Lp ./ to Lp=.p2/ ./, p > 2, p=.p 2/ > 1. Therefore, by the Nemytskii theorem, f W Lp ./ Lp ./ Lp ./ ! Lp=.p2/ ./ is bounded and strongly continuous, i.e., Z ˇ ˇp=.p2/ dx ˇjrun jp2 jrujp2 ˇ ! C0;

as run ! ru strongly Lp ./. We obtain that Jn .h/ ! C0 uniformly in h on the sphere khkC2 D 1. Now we consider the operator A12 .h/ .p 2/ div.jrujp4 .ru; rh/ru/:

603


Introduce the functions fij ./ D jjp2

i j ; jj jj

D .1 ; 2 ; 3 / 2 R3 ;

i; j D 1; 3;

1;p where m D uxm , m 2 N, h 2 W0 ./. It is easy to verify that the Nemytskii operators determined by the functions fij act from Lp ./ Lp ./ Lp ./ to Lp=.p2/ ./. Then the following inequalities hold:

kA12 .u/ A12 .un /kW 1;p !W 1;p0 0 p4 p4 D .p 2/ sup div.jruj .ru; rh/ru jrun j .run ; rh/run /

1;p 0

khk

1;p D1 W0

C

C

C

C

sup khk

1;p D1 W0

sup khk

1;p D1 W0

sup khk

1;p D1 W0

3 X

X 3 p4 p4 .ru; rh/uxi jrun j .run ; rh/unxi jruj

p0

i D1

3 X jrujp4 .ru; rh/ux jrun jp4 .run ; rh/unx i i

p0

iD1

3 X X 3 p2 uxj uxi p2 unxj unxi jru h jruj j n x j jruj jruj jru j jru j n

iD1 j D1

ˇ ˇ ux uxj dx ˇˇjrujp2 i jruj jruj

Z

sup

i;j D1 khkW 1;p D1 0

jrun jp2 C

3 Z X i;j D1

p0

n

ˇ ˇ ux uxj dx ˇˇjrujp2 i jruj jruj jrun j

p2

ˇ 0 1=p0 unxi unxj ˇˇp p0 jh j xj jrun j jrun j ˇ

ˇ unxi unxj ˇˇp=.p2/ .p2/=.p1/ Sn : jrun j jrun j ˇ

These inequalities imply that Sn ! C0 as run ! ru strongly in Lp ./. Therefore, kA01;u .u/ A01;un .un /kW 1;p ./!W 1;p0 ./ ! C0 0

W01;p ./.

as un ! u strongly in Thus, we obtain that the Fréchet derivative of the operator p u W W01;p ./ ! 0 1;p W 1;p ./ is defined on W0 ./ and has the following explicit form: div.jrujp2 rh/ C .p 2/ div.jrujp4 .ru; rh/ru/ W 1;p

W0

1;p

./ ! L.W0

0

./I W 1;p .//:

604


It is easy to prove that the Fréchet derivative of the nonlinear operator jujq u W ! L.qC2/=.qC1/ ./, q > 0, is defined on LqC2 ./ and has the following explicit form:

LqC2 ./

.q C 1/jujq h W LqC2 ./ ! L.LqC2 ./I L.qC2/=.qC1/ .//:

A.14

On the gradient of a functional

Results of this section can be found in [417]. Let X be a reflexive Banach space and X be its dual space. Denote by h; i the duality bracket between X and X . Assume that f is a Gâteaux-differentiable functional on X with Gâteaux derivative Df .v; h/ W X X ! C 1 for all h; v 2 X. Therefore, there exists F 2 X such that Df .v; h/ D hF; hi;

F D F.v/ W X ! X :

Definition A.14.1. The operator F D F.v/ defined by the formula lim t 1 Œf .v C t h/ f .v/ D hF.v/; hi;

t!C0

is called the gradient of the functional f .v/: F.v/ D grad f .v/: Consider a norm-type functional 1=p Z Z p p jrvj dx C jvj dx ; f .v/ D

where p 2, v 2 W01;p ./, is a bounded domain with a sufficiently smooth boundary. We prove that the gradient of this functional has the following explicit representation: grad f .v/ D .kvkpp C krvkpp /1=p1 .p v C jvjp2 v/; where

Z kukp D

jujp dx

1=p ;

p 2:

Introduce the notation D rv and D rh. Consider the function j C t jp ; by the Lagrange formula, we have j C t jp D jjp C p j C jp2 ¹.; / C jj2 ºt; D .; / 2 .0; t /:

(A.19) (A.20)

605

Section A.14 On the gradient of a functional

Expression (A.19) can be rewritten in the following equivalent form: j C t jp D jjp C pjjp2 .; /t C 1 .; ; ; t /;

(A.21)

1 D pŒj C jp2 jjp2 Œ.; / C jj2 t C p t jj2 jjp2 : (A.22) Similarly, we can obtain jv C t hjp D jvjp C p jvjp2 vht C 2 .v; h; ; t /;

D .v; h/ 2 .0; t /; (A.23)

2 D pŒjv C hjp2 jvjp2 Œvh C h2 t C p t jhj2 jvjp2 :

(A.24)

By virtue of (A.19)–(A.24), the following relations hold: 1=p Z Z p p jrv C t rhj dx C jv C t hj dx Z Z Z p p D jrvjp2 rv; rh dx jrvj dx C jvj dx C tp

Z

C tp

jvj

p2

Z

vh dx C

Œ 1 C 2 dx

1=p

D J;

Z J D kvk 1 C tpkvkp .jrvjp2 rv; rh/ dx C tpkvk

p

p2

jvj

Z kvk D

Z

1=p vh dx C #3 .t / ;

Z

jrvjp dx C

jvjp dx

1=p ;

NN /. For J, the following asymptotic expansion holds as t ! C0: where #3 .t / D o.t Z Z 1p p2 p2 N /; .jrvj rv; rh/ dx C jvj vh dx t C o.t J D kvk C kvk

t 1 .J kvk/ D kvk1p

Z

.jrvjp2 rv; rh/ dx C

Therefore, we have lim t 1 .J kvk/ D kvk1p

t!C0

Z

NN jvjp2 vh dx C o.1/:

Z

Z .jrvjp2 rv; rh/ dx C

jvjp2 vh dx

D kvk1p Œhp v; hi C hjvjp2 v; hi; 0

1;p where h;i is the duality bracket between the Banach spaces W0 ./ and W 1;p ./, p 0 D p .p 1/1 . Therefore, 0

grad f .v/ D .kvkpp C krvkpp /1=p1 .p v C jvjp2 v/ W W01;p ./ ! W 1;p ./:

606

A.15


Lions compactness lemma

Results of this section can be found in [275]. Lemma A.15.1. Let a sequence ¹um ºC1 mD1 satisfy the following conditions uniformly in m 2 N: Z T dt ku0m k2V0 C 8t 2 .0; T/; (A.25) kum kV0 C; 0

where 0 < T < C1 and the constant C depends on T. In addition, let the completely continuous embedding V0 ,! W holds. Then there exists a function u.s/ 2 W for almost all s 2 Œ0; T and the limit relation umk .s/ ! u.s/

strongly in W for almost all s 2 Œ0; T

holds. Proof. Consider the reflexive Banach space R D ¹uju 2 L2 .0; TI V0 /; u0 2 L2 .0; TI V0 /º:

(A.26)

Note that the function um .t /, perhaps after the change on a set of zero Lebesgue measure, on .0; T/ belongs to the class um .t / 2 C.Œ0; TI V0 / C.Œ0; TI W /: By virtue of (A.25), the sequence ¹um ºC1 mD1 is bounded in R. Since R is a reflexive Banach space, the sequence ¹um ºC1 mD1 contains a weakly converging subsequence ¹umk ºC1 . In particular, mk D1 umk * u

weakly in L2 .0; TI V0 /:

(A.27)

We prove that umk .s/ ! u.s/

strongly in W for almost all s 2 Œ0; T:

(A.28)

Without loss of generality, we can assume that u D 0. Moreover, s does not play any special role, and we must only prove that umk .0/ ! 0

strongly in W :

(A.29)

Now we define vk by the relation vk .t / D umk .t /;

> 0 is fixed,

(A.30)

607

Section A.16 Browder–Minty theorem

where umk .0/ D vk .0/;

kvk kL2 .0;TIV0 / c1 1=2 ;

kvk0 kL2 .0;TIW / c2 1=2 : (A.31)

If a function ' belongs to the class C 1 Œ0; T and '.0/ D 1 and '.T/ D 0, then Z T vk .0/ D .'.t /vk /0 dt D ˇk C k ; Z ˇk D

0

0

T

'vk0

Z dt;

k D

T 0

' 0 vk dt:

Therefore, kumk .0/kW kˇk kW C kk kW c3 1=2 C kk kW : For arbitrary " > 0, we choose > 0 so that c3 1=2 "=2. Since vk * 0 weakly in L2 .0; TI V0 /, we see that k * 0 weakly in V0 . But the embedding of V0 in W is completely continuous and hence the strong convergence k ! 0 in W also occurs. Therefore, for any ", there exists N such that for all mk > N we have kumk .0/kW ": Hence (A.29) and the lemma are proved.

A.16

Browder–Minty theorem

Results of this section can be found in [168]. Lemma A.16.1. Let A W X ! X be a monotone operator. Then the following assertions are equivalent: (1) the operator A is radially continuous; (2) the inequality hf Av; u vi 0 for all v 2 X implies Au D f ; (3) the limit relations un * u in X, Aun * f in X , and lim supn!C1 hAun ; un i hf; ui imply Au D f ; (4) the operator A is demicontinuous; (5) if K is a dense subset in X, then hf Av; u vi 0 for all v 2 K implies Au D f . Lemma A.16.2. Let B W Rn ! Rn be a continuous mapping satisfying for some R > 0 the condition .Ba; a/ 0 for jaj D R: Then there exists a 2 Rn such that jaj R and Ba D 0.

608


Theorem A.16.3. Let A W X ! X be a radially continuous, monotone, coercive operator. Then for any f 2 X , the set of solutions of the equation Au D f

(A.32)

is nonempty, weakly closed, and convex. Theorem A.16.4. Let an operator A W X ! X be radially continuous, strongly monotone, and coercive. Then there exists the inverse operator A1 W X ! X, which is strongly monotone, bounded, and demicontinuous.

A.17

Sufficient conditions of the independence of the interval, on which a solution of a system of differential equations exists, of the order of this system

We consider the following typical situation, which appears when the Galerkin method is used for the proof of the local (in time) solvability. Indeed, on some step we prove that hA0 um ; um i0 c.T/ < C1 for all t 2 Œ0; T;

(A.33)

where T > 0 is independent of m 2 N and, moreover, c is a positive constant independent of m 2 N. The linear operator A0 W V0 ! V0 is a norm-type operator on the Banach space V0 with respect to the norm kk0 possessing the strong dual space V0 and the duality bracket h; i0 between V0 and V0 ; moreover, um D

m X

cmk .t /wk ;

(A.34)

kD1

V0 is a linearly independent and dense in and the system of the functions ¹wk ºC1 kD1 V0 family. In addition, the operator A0 satisfies the condition 0 kvk20 hA0 v; vi0

for all v 2 V0

(A.35)

and some 0 > 0, which is independent of v 2 V0 . The following relation holds: hA0 um ; um i0 D

m;m X k;j D1;1

akj cmk .t /cmj .t /;

(A.36)

Section A.17 Independence of the solution domain of the order of system

where

609

˛ ˝ akj A0 wk ; wj :

By (A.35), the determinant of the matrix kakl km;m k;j D1;1 is positive for all m 2 N. Therefore, for this matrix, there exists a matrix Q W Rm ! Rm independent of time such that cm D Qdm ;

cm D .cm1 ; : : : ; cmm /t ;

dm D .dm1 ; : : : ; dmm /t

(A.37)

and m;m X

akj cmk .t /cmj .t / D

k;j D1;1

m X

l .dml /2 ;

(A.38)

lD1

where l > 0 for all l D 1; m. Then inequality (A.33) implies that jdml j .t / c.m; T/ < C1

for all t 2 Œ0; T:

(A.39)

Now recall that any proper matrix Q W Rm ! Rm has a bounded norm: kQkm˝m D sup jQjm < C1; jjm D1

jjm .j1 j2 C C jm j2 /1=2 :

Therefore, inequality (A.39) and equality (A.37) imply jcm jm kQkm˝m jdm jm c.m; T/

for all t 2 Œ0; T;

where jjm .j1 j2 C C jm j2 /1=2 : Therefore, we obtain the estimate jcmk .t /j c.m; T/ < C1

for all t 2 Œ0; T; k D 1; m:

This estimate implies that any solution cm D .cm1 .t /; : : : ; cmm .t //t of the first-order system of differential equations appearing in the Galerkin method can be extended to a segment Œ0; T, where T > 0 is independent of m 2 N, in the same class C .1/ .Œ0; Tm /.

610


A.18

On the continuity of some inverse matrices

Consider the matrix A D kaik km;m ; i;kD1;1

(A.40)

where aik D hA0 wi ; wk i0 C

N X j D1

hAj0 um .um /wi ; wk ij ;

um D

m X

cmk wk ;

(A.41)

kD1

and ¹wk º V is a Galerkin basis in V . Consider the matrix cik

D

N X j D1

hAj0 um .um /wi ; wk ij :

(A.42)

First, we prove the boundedness of this matrix. Indeed, we have kCkm˝m D sup jCj;

(A.43)

jjm D1

where jjm D .21 C C 2m /1=2 : This implies kCkm˝m c.m/

m;m N X X

kAj0 um .um /kVj !Vj kwi kj kwk kj

j D1 i;kD1;1

c.m/

m;m N X X

j .Rm /kum kj kwi kj kwk kj ;

(A.44)

j D1 i;kD1;1

where j .Rm / is nondecreasing and bounded on any compact function and Rm D kum kj . Estimate (A.44) implies that the matrix C is bounded under fixed cm D .cm1 ; : : : ; cmm /. Introduce the resolvent matrices R.; A/ D .I A/1

(A.45)

for sufficiently large jj D rm > 0. Then for the matrix A1 , the integral presentation Z 1 A1 D 1 R.; A/d (A.46) 2 i jjDrm

611

Section A.18 On the continuity of some inverse matrices

holds. Then A1 .c1m / A1 .c2m / D

1 2 i

Z jjDrm

1 ŒR.; A.c1m // R.; A.c2m // d: (A.47)

Now we will use the well-known equality for the resolvent R.; A.c2m // R.; A.c1m // D R.; A.c1m //

C1 X

Œ.A.c2m / A.c1m //R.; A.c1m //n

nD1

(A.48) under the condition k.A.c2m / A.c1m //R.; A.c1m //km˝m kA.c2m / A.c1m /km˝m kR.; A.c1m //km˝m ı < 1; which holds for sufficiently large jj D r > 0. Note that ; A.c2m / A.c1m / D C.c2m / C.c1m / D kki km;m k;iD1;1

(A.49)

where ki

N X

hŒAj0 um .u1m / Aj0 um .u2m /wk ; wi i:

(A.50)

j D1

But then jki j

N X

kAj0 um .u1m / Aj0 um .u2m /kVj !Vj kwi kj kwk kj

j D1

N X

kwi kj kwk kj j .Rm /ku1m u2m kj ;

(A.51)

j D1

where Rm D max¹ku1m kj ; ku2m kj º: From (A.51) we have jki j

N X j D1

kwi kj kwk kj j .Rm /

m X kD1

1 2 kwk kj jcmk cmk j:

(A.52)

612


Thus, (A.47)–(A.52) imply that for all " > 0, there exist ı."/ > 0 such that for jc1m c2m j ı."/ we have jki j ": A1 .cm /

is continuous concerning a column cm 2 Rm . This means that the matrix Then in the following equality it is possible to turn this matrix and to obtain that the right-hand side of the obtained equality is continuous function with respect to a column cm D .cm1 ; : : : ; cmm / and it is possible to apply the Peano theorem, for example, to the system m X iD1

0 cmi

N X

0 hA0 wi ; wk i0 C hAj;u .um /wi ; wk ij m j D1

D .F.um /; wk /0 :

Remark A.18.1. Note that we can prove the obtained result under following condition: the operator Aj0 u .u/ is strongly continuous in the norm of the Banach space L.Vj I Vj /.

Appendix B

To Chapter 6

B.1

Convergence of the "-embedding method with the CROS scheme

We consider an autonomous differential-algebraic problem dy D f .y; z/ ; dt

0 D g .y; z/ ;

y .0/ D y0 ;

z .0/ D z0 :

(B.1)

Let the functions f .y; z/ and g .y; z/ in the right-hand sides of the equations are twice continuously differentiable with respect to both variables and the initial conditions y0 and z0 are consistent. i.e., g .y0 ; z0 / D 0. Introduce the following notation for the numerical solution of system (B.1): v y;

w z:

(B.2)

The formulas for the transition to the next time layer for the numeric solution of problem (B.1) by the "-embedding method with the CROS scheme can be written in the following form, which is convenient to theoretic analysis: v1 D v0 C Re k;

w1 D w0 C Re l;

(B.3)

where .v0 ; w0 / is the numeric solution on the given time layer t and .v1 ; w1 / is the solution on the next time layer t C . The increments k and l in (B.3) are determined from the linear algebraic system k fy fz k f .v0 ; w0 / : (B.4) C ˛ D l gy gz g .v0 ; w0 / 0 For CROS ˛ D .1 C i/=2. Let y .t / ; z .t / be the exact solution of problem (B.1). We fix t and apply the method making one step by . We denote the local errors of the scheme (B.3), (B.4) by ıv .x/ D v1 y .t C / ;

ıw .x/ D w1 z .t C / ;

(B.5)

where .v1 ; w1 / is the numeric solution obtained by formulas (B.3) with the initial data v0 D y.t / and w0 D z.t /.

614

Appendix B To Chapter 6

Theorem B.1.1. For the "-embedding method with the CROS scheme ıv .t / D O. 3 /;

ıw .t / D O. 2 /:

Proof. We expand the exact solution y .t C /, z .t C / of problem (B.1) and the numeric solution v1 , w1 obtained by using the scheme (B.3), (B.4), in the Taylor series in taking a necessary number of terms. For this, we must calculate the derivatives y 0 , y 00 , z 0 , v 0 , v 00 , and w 0 with respect to at the point D 0 and verify that these derivatives for the exact and numeric solutions coincide. For the exact solution we have y 0 D f .y; z/ ;

z 0 D .gz /1 gy f;

y 00 D fy y 0 C fz z 0 D fy f fz .gz /1 gy f:

(B.6)

For the numeric solution, by (B.3) we have v 0 D Re k 0 ;

v 00 D Re k 00 ;

w 0 D Re l 0 :

(B.7)

0 D g C ˛gy k C ˛gz l:

(B.8)

We rewrite (B.3) in the component form: k D f C ˛fy k C ˛fz l; The first equation (B.8) implies k 0 j D0 D f I we have assumes that k; lj D0 D 0. The second equation (B.8) implies l 0 D .gz /1 gy k 0 D .gz /1 gy f: Differentiating the first equation (B.8) two times, we obtain k 00 j D0 D 2˛fy k 0 C 2˛fz l 0 D 2˛fy f 2˛fz .gz /1 gy f: Taking into account that for CROS Re 2˛ D 1, we see that (B.1) and (B.7) coincide for D 0. Therefore, all terms in (B.5) up to ıv .t / D O. 3 /;

ıw .t / D O. 2 /

vanish. Assume that we solve problem (B.1) on a grid with N nodes and the interval of time for the problem is N D const. An estimate of the global error during the whole numeric simulation is given by the following theorem.

Section B.1 Convergence of the "-embedding method with the CROS scheme

615

Theorem B.1.2. Let the matrix gz be regular in some neighborhood of the exact solution y .t /, z .t / of problem (B.1), the functions f .y; z/ and g .y; z/ be twice continuously differentiable with respect to both variables, and the initial conditions y0 ; z0 be consistent, i.e., g .y0 ; z0 / D 0. Then the "-embedding method with the CROS scheme converges with the second order: y .t C N / vN D O. 2 /; z .t C N / wN D O. 2 /; where y .t C N /, z .t C N / and vN ; wN are the exact and numeric solutions, respectively, at the moment tN D N . We apply use the scheme used in [194] for the proof of the convergence of the "-embedding method with schemes of the Rosenbrock–Wanner type with real coefficients. Since the matrix gz is regular in some neighborhood of the solution, we have kgz1 .y; z/ g .y; z/k ı;

(B.9)

where ı is independent of and can be made arbitrarily small by the decreasing the neighborhood. 1 1 , @v1 , @w1 , and @w . For this, we calculate Using expression (B.3), we find @v @v0 @w0 @v0 @w0 @k , @k , @l , @v0 @w0 @v0

and

@l . @w0

We estimate k and l using (B.8):

k D O . / ;

1 l D gz1 g C O . / D O . C ı/ : ˛

Differentiating (B.8), we obtain 1 @ 1 @l @k @k D D O. /; D O. /; gz g C O. /; @v0 @v ˛ @v0 @w0 @l 1 1 D C gz1 gzz gz1 g C O. / D C O . C ı/ : @w0 ˛ ˛ Differentiating (B.3) and using (B.10), we obtain @w1 @v1 @v1 D 1 C O. /; D O.1/; D O. /; @v0 @v0 @w0 @w1 1 D 1 Re CO. C ı/ D O. C ı/I @w0 „ ƒ‚ ˛… D0

we have used the fact that ˛ D .1 C i /=2.

(B.10)

616


We consider two pairs of initial data .v0 ; w0 / and .vQ 0 ; wQ 0 /; these initial data not necessarily consistent but they lie in the neighborhood (B.9) described above. Using the estimates for the derivatives (B.10) and the Lagrange formula we obtain kv1 vQ 1 k .1 C L/ kv0 vQ 0 k C P kw0 wQ 0 k ; kw1 wQ 1 k Q kv0 vQ 0 k C q kw0 wQ 0 k ;

q < 1:

(B.11)

The constants Q, P , L, and q are independent of the initial data and positive; decreasing the neighborhood (B.9), we can obtain q < 1. We estimate the accumulation of the error for N steps. Lemma B.1.3. Let ¹vn º, ¹wn º, n D 0; 1; 2; : : : , be sequences of nonnegative numbers satisfying the inequalities vnC1 m11 vn C m12 wn ; wnC1 m21 vn C m22 wn 11 m12 and all entries of the matrix M D m m21 m22 are positive. Then vn and wn can be estimated through v0 and w0 : O 11 v0 C m O 12 w0 ; vn m

wn m O 21 v0 C m O 22 w0 ;

where m O ij are the entries of the matrix M n . The lemma can be easily proved by induction. Lemma B.1.4. Assume that, starting from the initial data .v0 ; w0 / and .vQ 0 ; wQ 0 /, we have made N steps .N const/ and on each of steps, .vn ; wn / and .vQ n ; wQ n /, 1 n N , lie in the neighborhood (B.9) described above. Then kvN vQ N k C .kv0 vQ 0 k C kw0 wQ 0 k/ ; kwN wQ N k C.kv0 vQ 0 k C . C q N / kw0 wQ 0 k/;

q < 1:

P We use formula (B.11) and apply Lemma B.1.3 with the matrix M D 1C L Q q : O 11 kv0 vQ 0 k C m O 12 kw0 wQ 0 k kvn vQ n k m O 21 kv0 vQ 0 k C m O 22 kw0 wQ 0 k ; kwn wQ n k m

P n . where m O ij are the entries of the matrix M n D 1C L Q q The eigenvalues of the matrix M are the roots of the equation

(B.12)

.1 C L / .q / PQ D 0I we have 1 D 1 C L

PQ C O 2 ; q L 1

2 D q C

PQ C O 2 : q L 1

617


In the sequel, we need the following estimates. There exists constants A > 0 and B > 0 such that n1 A;

n2 Bq n :

(B.13)

Indeed, if N const, since 1 D 1 C O . / ; then 1 1 C C h. Applying a wellknown estimate (see, e.g., [206]), we have

n1

const 1CC N

n

const n 1CC exp .C const/ D A: n

The second estimate is proved similarly. The matrix M n can be represented as follows: n 1 0 1 n ; T M T D 0 n2 where the columns of the matrix T are the eigenvectors of the matrix M : !

P 1 1q C o. / 1 O./ T D D : Q O.1/ 1 1 1q C O . / Therefore, T 1 D

1CO. / O.1/

O. / . 1CO. /

For the matrix M n we obtain

n 1 C O. / O . / n1 0 1 O./ 1 0 1 T D M DT O.1/ 1 C O. / 0 n2 0 n2 O.1/ 1 n n1 O. / 1 O./ 1 .1 C O. // D n n O.1/ 1 2 .1 C O. // 2 O.1/ .1 C O. // n1 C n2 O . / n1 .1 C O. // O. / C n2 O . / .1 C O. // : D n1 O.1/O. / C n2 .1 C O. // n1 O.1/.1 C O. // C n2 O.1/

n

Applying estimates (B.13), we obtain that there exist positive constants C1 , C2 , C3 , C4 , and C5 such that m O 11 D .1 C O. // n1 C n2 O . / C1 C; m O 12 D n1 .1 C O. // O. / C n2 O . / .1 C O. // C2 C ; m O 21 D n1 O.1/.1 C O. // C n2 O.1/ C4 C C3 q n C; m O 22 D

n1 O.1/O. /

C

n2 .1

n

(B.14) n

C O. // C5 C q C C q :

We obtain the lemma if in (B.14) we set C D max ¹C1 ; C2 ; C4 C C3 q n ; C5 º and apply estimate (B.12).

618


Proof. To prove the theorem, we must estimate the accumulation of errors for N steps .N const/. We use the following method. We will hypothetically apply a numerical method with the initial data v0 D y .t C n/ and w0 D z .t C n/ taken from the exact solution. We denote the numeric solution obtained after N n steps by vN n , wN n . Then the norm of the difference of the numeric solution vN ; wN with the initial data v0 D y .t /, w0 D z .t / and the exact solution y .t C N /, z .t C N / after N steps can be estimated as follows: kvN y.t C N /k D kvN vN 1 C vN 1 vN 2 C C v1 y.t C N /k kvN vN 1 k C kvN 1 vN 2 k C C kv1 y.t C N /k;

(B.15)

kwN z.t C N /k D kwN wN 1 C wN 1 wN 2 C C w1 z.t C N /k (B.16) kwN wN 1 k C kwN 1 wN 2 k C C kw1 z.t C N /k: We consider vN n vN .n1/ and wN n wN .n1/ separately. To obtain vN n and wN n we must perform N n steps and to obtain vN nC1 and wN nC1 we must perform N n C 1 steps. We perform one step of our numeric method (B.3) with the initial data v0 D y .t C .n 1/ / and w0 D z .t C .n 1/ / and then perform synchronously N n steps by scheme (B.3). On the first step, an error occurs, which, in fact, is the local error of the method and, by Theorem B.1.1, equals ıv .x/ D O. 3 /, ıw .x/ D O. 2 /. The further accumulation of the error for N n steps is described by Lemma B.1.4, where kv0 vQ 0 k D O. 3 /, kw0 wQ 0 k D O. 2 /, and the number of steps is N n. Thus, kvN n vN .n1/ k C.O. 3 / C O. 2 //; kwN n wN .n1/ k C.O. 3 / C . C q N n /O. 2 //;

Figure B.1. Accumulation of the global error.

(B.17) q < 1:


619

Substituting (B.15) and (B.16) and taking into account the fact that N P (B.17)Nto n < 1 , we obtain the following estimate of the global error: const and N q nD1 1q kvN y .t C N /k const 2 ;

kwN z .t C N /k const 2 :

(B.18)

To complete the proof, we must also prove that the numeric solution lies in neighborhood (B.9) of the exact solution. This can be easily proved by induction in the number of steps. Estimate (B.18) proved the convergence of the "-embedding method for the CROS scheme with the second order of accuracy.

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Index

A-stable scheme, 555 Lp -stable scheme, 555 Bochner integral, 584 Equation Barenblatt–Zheltov–Kochina equation, 3 Benjamin–Bona–Mahony (BBM) equation, 2 Benjamin–Bona–Mahony–Burgers (BBMB) equation, 2 Benney–Luke equation, 7, 9 Camassa–Holm equation, 1 Coleman–Duffin–Mizel equation, 3 Degasperis–Procesi equation, 1 dissipative nonlinear ion-acoustic wave equation with nonlinear damping, 9 dissipative system of ion-acoustic wave equations, 10 Fornberg–Whitham equation, 1 generalized Benjamin–Bona– Mahony–Burgers equation, 5 generalized Boussinesq nonlinear equation, 4 generalized hyperelastic-rod wave equation, 2 generalized porous-media equation, 2 generalized Rosenau–Burgers equation, 5 generalized von Karman system, 10 gravity-gyroscopic wave equation, 8 higher-order improved Boussinesq equation, 7 Higher-order nonlinear ion-acoustic wave equations with nonlinear damping, 9 Hoff equation, 3

improved Boussinesq equations, 7 improved Boussinesq–Schrödinger system, 10 ion-acoustic wave equation, 8 ion-acoustic wave equation in plasma in external magnetic field, 8 Kadomtsev–Petviashvilli equation, 5 Korpusov–Pletner–Sveshnikov equation, 3 Kuramoto–Sivashinsky equation, 12 linear Rossby wave equation, 5 Longern wave equation, 6 modified Benjamin–Bona–Mahony (MBBM) equation, 2 multidimensional Benjamin–Bona– Mahony–Burgers equation, 4 nonlinear ion-acoustic wave equations with nonlinear damping, 9 nonlinear Rossby-type equation, 28 nonlinear telegraph equation with nonlinear damping, 6 nonlocal higher order wave equation, 10 nonlocal pseudoparabolic equation, 5 one-dimensional Boussinesq equation, 3 one-dimensional Camassa–Holm equation, 28 one-dimensional Oskolkov equation, 3 Oskolkov system of equations with sources, 10 Oskolkov–Benjamin–Bona–Mahony equation, 29 Pochhammer–Chree equation, 6 Rabinowitz wave equations with nonlinear damping, 6 Rosenau–Burgers equation, 2

648

Index S. L. Sobolev equation, 7 semiconductor equation, 4 Showalter equation, 4 Showalter inclusion, 4 spin-wave equation, 6 spin-wave equation in magnetics in an external magnetic field, 8 system of two higher-order improved Boussinesq equations, 7 system of two improved Boussinesq equations, 7 three-dimensional Camassa–Holm equation, 5 two-dimensional Petviashvili equation, 28 two-dimensional Rossby waves equation, 27

Lemma Lions compactness lemma, 606 Operator boundedly Lipschitz continuous, 71 completely continuous, 71 Fréchet derivative, 597 Gâteaux differential, 596 locally uniform Fréchet differential, 72

monotonic mapping, 70 Nemytskii operator, 586 semicontinuous, 71 symmetric, 72 the Fréchet differential, 71 weakly lower semicontinuous, 71 Sequence -weakly converging, 583 weakly converging, 583 Sobolev spaces, 581 Solution classical solution, 485, 592 strong generalized solution, 102, 127, 197, 203, 244, 273, 322, 330, 338, 341, 346, 380, 392, 403, 469, 476, 493, 538, 592 weak generalized solution, 78, 112, 143, 166, 219, 254, 358, 418, 441, 466, 592 weakened solution, 198, 302, 476, 492, 589, 592 Theorem Browder–Minty theorem, 607 Krasnoselskii theorem, 586

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