Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems

270 Pages·1995·7.502 MB·English

by  Songmu Zheng

Pitman Monographs and
Surveys in Pure and Applied Mathematics 76
NONLINEAR PARABOLIC
EQUATIONS AND
HYPERBOLIC-PARABOLIC
COUPLED SYSTEMS
SONGMU ZHENG
mmm
HIUI
Nonlinear parabolic equations and hyperbolic-parabolic
coupled systems
Main Editors
H. Brezis, Université de Paris
R.G. Douglas, State University of New York at Stony Brook
A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)
Editorial Board
H, Amann, University of Zürich
R. Aris, University of Minnesota
G.I. Barenblatt, University of Cambridge
A. Bensoussan, INRIA, France
P. Bullen, University of British Columbia
S. Donaldson, University of Oxford
RJ. Elliott, University of Alberta
R. P. Gilbert, University of Delaware
D. Jerison, Massachusetts Institute of Technology
K. Kirchgassner, Universität Stuttgart
B. Lawson, State University of New York at Stony Brook
B. Moodie, University of Alberta
S. Mori, Kyoto University
L. E. Payne, Cornell University
G.F. Roach, University of Strathclyde
W.A. Strauss, Brown University
SJ. Taylor, University of Virginia
It Pitman Monographs and
Surveys in Pure and Applied Mathematics 76
Nonlinear parabolic
equations and
hyperbolic-parabolic
coupled systems
Songmu Zheng
Fudan University
mmm
LONGMAN
Copublished in the United States with
John Wiley & Sons Inc., New York.
Longman Group Limited
Longman House, Burnt Mill, Harlow
Essex CM20 2JE, England
and Associated companies throughout die world.
Copublished in the United States vdth
John Wiley & Sons Inc., 605 Third Avenue, New York, NY 10158
© Longman Group Limited 1995
All rights reserved; no part of this publication may be rq>roduced,
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issued by the Copyright Licensing Agency Ltd, 90 Tottenham
Court Road, London, WIP 9HE
First published 1995
AMS Subject Classifications: (Main) 35K, 35L, 35Q20
(Subsidiary) 82A25, 73B30, 80-xx
ISSN 0269-3666
ISBN 0 582 24488 9
British Library Cataloguing in Publication Data
A catalogue record for this book is
available from the British Library
Library of Congress Cataloging-in-Publication Data
Zheng, S. (Songmu)
Nonlinear parabolic equations and hyperbolic-parabolic coupled
systems / S. Zheng.
p. cm. ~ (Pitman monographs and surveys in pure and applied
mathematics, ISSN 0269-3666;)
1. Differential equations. Parabolic. 2. Differential equations.
Hyperbolic. 3. Differential equations. Nonlinear. I. Tide.
II. Series.
QA377.Z44 1995
515’.353~dc20 94-31194
CIP
Printed and bound by Bookcraft (Bath) Ltd
То Weixi and Leijun

Contents
Preface ..............................................................................................................................xi
1. Preliminaries...................................................................................................................1
1.1. Basic Facts on Sobolev Spaces.........................................................................1
1.1.1. Sobolev Spaces 1
1.1.2. The Gagliardo-Nirenberg and Poincare Inequalities .....................3
1.1.3. Abstract Functions Valued in Banach Spaces .................................4
1.2. Linear Elliptic Equations ...............................................................................5
1.2.1. Boundary Value Problems .................................................................5
1.2.2. Interpolation Spaces ............................................................................8
1.3. Linear Evolution Equations..........................................................................10
1.3.1. Variational Evolution Equations .....................................................10
1.3.2. Setting of Semigroup Theory ...........................................................14
1.3.3. Hyperbolic and Parabolic Systems .................................................16
1.4. Nonlinear Evolution Equations ...................................................................20
1.4.1. Local Existence and Uniqueness......................................................21
1.4.2. Global Existence ................................................................................23
1.5. Asymptotic Behaviour..................................................................................24
1.5.1. Global Attractor.................................................................................24
1.5.2. Convergence to Equilibrium ............................................................26
vii
viii Contents
1.5.3. Inertial Manifolds and Inertial Sets..............................................27
1.6. Notation and Some Useful Inequalities.....................................................31
2. Decay of Solutions to Linear Evolution Equations................................................33
2.1. Linear Parabolic Equations ..........................................................................35
2.1.1. Initial Value Problems......................................................................35
2.1.2. Initial Boundary Value Problems ....................................................37
2.2. Linear Thermoelastic Systems .....................................................................39
2.2.1. Initial Value Problems......................................................................39
2.2.2. Initial Boundary Value Problems ....................................................53
2.3. Linear Thermoviscoelastic Systems ............................................................61
2.3.1. Initial Value Problems......................................................................61
2.3.2. Initial Boundary Value Problems .................................................64
3. Global Existence for Small Initial Data...................................................................69
3.1. Fully Nonlinear Parabolic Equations: IVP ................................................70
3.2. Fully Nonlinear Parabolic Equations: IBVP .............................................84
3.3. Nonlinear Hyperbolic-Parabolic Coupled Systems: IVP ..........................93
3.3.1. Global Existence and Uniqueness .................................................95
3.3.2. Applications .....................................................................................115
3.4. Nonlinear Hyperbolic-Parabolic Coupled Systems: IBVP......................119
3.5. Nonexistence of Global Solutions ..............................................................137
4. Global Existence for Large Initial Data.................................................................145
4.1. Phase-field Equations ................................................................................146
4.1.1. The Caginalp Model .......................................................................146
4.1.2. The Penrose-Fife Model.................................................................154
Contents ix
4.2, Systems of PDEs from Shape Memory Alloys .......................................169
4.3. The Coupled Cahn-Hilliard Equations .....................................................184
5. Stationary Problems and Asymptotic Behaviour ...............................................201
5.1. Nonlinear BVP with Nonlocal Terms and Constraints ..........................202
5.2. Convergence to Equilibrium ......................................................................216
5.3. Global Attractors and Inertial Manifolds.................................................219
Bibliography....................................................................................................................231
Index................................................................................................................................251