Nonlinear Parabolic and Elliptic Equations

Nonlinear Parabolic and Elliptic Equations

Nonlinear Parabolic and Elliptic Equations c. V. Pao North Carolina State University Raleigh, North Carolina Plenum Press New York and London

Library of Congress Cataloging in Publication Data Pao, C. V. (Chia-Ven), 1933- Nonlinear parabolic and elliptic equations / C. V. Pao. p. cm. Includes bibliographical references and index. ISBN 0-306-44343-0 1. Differential equations, Nonlinear. I. Title. QA377.P275 1992 515'.353-dc20 92-30342 CIP ISBN 0-306-44343-0 1992 Plenum Press, New York A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

To my wife Mei-Shan

Preface The recent development of reaction diffusion systems in biology, ecology and biochemistry, and the traditional importance of these systems in physics, heat-mass transfer, and engineering lead to extensive study in various aspects of nonlinear parabolic and elliptic partial differential equations. A large amount of mathematically rich and physically interesting work, including several excellent books, has been published in the literature since the middle of the 1960s. Because of the fast growth of reaction diffusion type of problems in various different fields it is desirable to have a unified mathematical treatment and practical methods for solving these problems. This book is intended to give a systematic treatment of the basic mathematical theory and constructive methods for a class of nonlinear parabolic and elliptic differential equations as well as their applications to various reaction diffusion problems. The mathematical problems under consideration include scalar boundary-value problems of parabolic and elliptic equations, integroparabolic and integroelliptic boundary-value problems, and coupled systems of parabolic and elliptic equations. The boundary conditions for these equations may be linear or nonlinear, including nonlinear boundary conditions of integral type. The fundamental approach to all of these problems is the method of upper and lower solutions and the associated monotone iterations. This approach leads not only to the basic results of existence, uniqueness, and multiplicity of solutions but also to various qualitative properties of the solution through suitable construction of upper and lower solutions. Moreover, since the book is concerned primarily with classical solutions, the monotone iteration processes for various types of nonlinear problems are adaptable to numerical solutions of the corresponding discrete system. Some of these methods for finite difference parabolic and elliptic equations have already appeared in the current literature. Extensive discussion is given to the stability analysis and the asymptotic behavior of the time-dependent solution for both scalar boundary-value problems and coupled systems of equations. This includes the stability or instability of a steady-state solution, the asymptotic limit of the time-dependent solution, and decay or growth property of the solution. This qualitative property of the solution leads to an intrinsic relationship between the solutions of a parabolic boundary-value problem and its corresponding elliptic problem, and is a major concern in many physical, ecological, and engineering problems. Attention is given to various model problems in ecology, biochemistry, enzyme kinetics, combustion theory, and chemical and nuclear engineering. A special topic of the analysis is the finite-time blow-up probvii

viii Preface lem for parabolic equations. Several methods for determining the blow-up behavior of a solution are used for both scalar equations under linear or nonlinear boundary conditions and coupled systems of parabolic equations, including a discussion on the problem of quenching. A chapter is devoted to parabolic and elliptic equations in unbounded domains using the method of upper and lower solutions. This chapter covers the Cauchy problem in jrn, a half-space problem, and problems in the exterior of a bounded domain. Both linear and nonlinear boundary conditions are considered and various qualitative analyses, such as the finite-time blow-up problem, are discussed. The book consists of twelve chapters; the first seven chapters treat the scalar parabolic and elliptic boundary-value problems and the remaining five chapters are concerned with coupled systems of parabolic and elliptic equations. The problems with unbounded spatial domains are considered in Chapter 7, while all the other chapters involve only bounded domains. Chapter 12 is devoted to applications of coupled systems of parabolic and elliptic equations to various model problems in different fields, including a coupled system of neutron transport and heat equations. These model problems have motivated much of the discussions in the chapters involving coupled systems. An overview of each chapter is given at the beginning of the chapter, and notes and remarks are added at the end of the chapter to provide historical comments and references. Theorems, Lemmas, Equations etc. are ordered in standard form. For example, the first theorem in section 4 of Chapter 2 is referred to as Theorem 4.1 when it appears in Chapter 2 and is referred to as Theorem 2.4.1 when it appears in a different chapter. Because of the limitation of scope many interesting topics, such as traveling wave solutions, periodic solutions, and Lyapunov method for stability problems, are not discussed in the book. Also omitted is the class of equations where the nonlinear reaction function depends on the gradient of the unknown function. The references given in the book are mostly related to comparison methods and the method of upper and lower solutions, and is not intended to be complete. I apologize for the incompleteness and omissions in the list of references. The book, in part, is an outgrowth of my research on the subject during the past two decades, and much of the development is motivated by some kind of applied problems. Although effort is made to make the presentation self-contained it is necessary to use some basic theory for linear parabolic and elliptic equations. The book can be used as a reference for mathematicians, engineers and scientists, and can also be used as a text for graduate students who are interested in applied partial differential equations or reaction diffusion systems. Portions of the book have been used as the text in a special course on "Nonlinear Reaction Diffusion Equations" at North Carolina State University.

Preface ix I am grateful to my students and colleagues for their reading the manuscript and helpful comments, and to the editorial staff of Plenum Publishing Corporation for editing the manuscript. I also owe a special debt of gratitude to Ms. Dionne Wilson for her cordial cooperation and excellent typing of the manuscript. Raleigh, North Carolina C. v. Pao

Contents Chapter 1 Reaction Diffusion Equations 1.1 Derivation of Reaction Diffusion Equations... 1 1.2 Boundary Conditions... 4 1.3 Derivation of Some Specific Models... 7 1.4 Linear Reaction Diffusion Equations... 14 1.5 Monotone Method for Time-Dependent Problems... 20 1.6 Nonuniqueness of Time-Dependent Solutions... 26 1. 7 Monotone Method for Steady-State Problems... 31 1.8 Applications to Specific Models... 36 1.9 Notes and Comments... 45 Chapter 2 Parabolic Boundary-Value Problems 2.1 A Review of the Linear Parabolic Problem... 47 2.2 A Positivity Lemma... 54 2.3 Upper and Lower Sequences... 56 2.4 Existence-Comparison Theorems... 62 2.5 Positivity and Boundedness of Solutions... 65 2.6 Integroparabolic Equations of Volterra Type... 69 2.7 Integroparabolic Equations of Fredholm Type... 80 2.8 Parabolic Boundary-Value Problems with Time Delay... 86 2.9 Notes and Comments... 92 Chapter 3 Elliptic Boundary-Value Problems 3.1 The Linear Boundary-Value Problem... 93 3.2 The Method of Upper and Lower Solutions... 98 3.3 The Uniqueness Problem... 104 3.4 Positive Steady-State Solutions... 108 3.5 The Spectrum Problem... 112 3.6 Multiple Steady-State Solutions... 117 3.7 Integroelliptic Boundary-Value Problems... 125 xi

xii Contents 3.8 Applications... 133 3.9 Notes and Comments... 138 Chapter 4 Equations with Nonlinear Boundary Conditions 4.1 Parabolic Boundary-Value Problems... 139 4.2 An Application to the Linear Problem... 145 4.3 Boundary Conditions of Integral Type... 149 4.4 Elliptic Boundary-Value Problems... 154 4.5 Existence Theorems for Holder-Continuous Functions... 161 4.6 Uniqueness of Positive Solution... 168 4.7 Spectrum for Problems with Nonlinear Boundary Conditions... 171 4.8 Applications... 175 4.9 Notes and Comments... 180 Chapter 5 Stability Analysis 5.1 Lyapunov Stability... 183 5.2 Stability of Uniform Steady-State Solutions... 188 5.3 Stability of Nonuniform Steady-State Solutions... 192 5.4 Monotone Convergence of Time-Dependent Solutions... 198 5.5 Stability of Maximal and Minimal Solutions... 202 5.6 Problems with Nonlinear Boundary Conditions... 206 5.7 Application to Models with Nonlinear Reaction Functions.. 214 5.8 Application to Models with Nonlinear Boundary Conditions 219 5.9 Notes and Comments... 226 Chapter 6 Blowing-Up Behavior of Solutions 6.1 Growth Property of Solutions... 229 6.2 Blowing-Up Property of the Solution... 234 6.3 Estimate of the Finite Blowing-Up Time... 240 6.4 The Blowing-Up Point of the Solution... 243 6.5 Nonlinear Boundary Functions... 249

Contents xiii 6.6 The Method of Concavity... 257 6.7 The Quenching Problem... 267 6.8 Thermal Explosion Problems in Combustion Theory... 277 6.9 Applications to Reactor Dynamics and Nonlinear Polarization... 284 6.10 Notes and Comments... 287 Chapter 7 Parabolic and Elliptic Equations in Unbounded Domains 7.1 The Linear Parabolic and Elliptic Equations... 290 7.2 The Cauchy Problem in ~n.. 296 7.3 A Half-Space Problem... 302 7.4 Parabolic Problem in General Unbounded Domains... 312 7.5 Elliptic Equations in ~n 319 7.6 Radially Symmetric Solutions... 325 7.7 A Model Problem from Geometry and Applied Physics... 334 7.8 Elliptic Equations in Exterior Domains... 340 7.9 Elliptic Equations in General Unbounded Domains... 345 7.10 Exterior Problem with Nonlinear Boundary Conditions... 354 7.11 Stability and Asymptotic Behavior of Solutions... 361 7.12 Blowing-Up Behavior of the Solution in ~n 371 7.13 Notes and Comments... 379 Chapter 8 Coupled Systems of Reaction Diffusion Equations 8.1 Quasimonotone Reaction Functions... 381 8.2 Monotone Sequences for Coupled Parabolic Equations... 384 8.3 Existence-Comparison Theorems for Coupled Parabolic System... 392 8.4 Existence-Comparison Theorems for Coupled Elliptic System... 401 8.5 Elliptic Systems with Mixed Quasimonotone Functions... 408 8.6 Uniqueness of Steady-State Solution... 413 8.7 Positive Invariant Rectangles... 417

xiv Contents 8.8 Finite Parabolic Systems with Quasimonotone Functions... 424 8.9 Finite Parabolic Systems with Nonquasimonotone Functions... 431 8.10 Finite Elliptic Systems... 438 8.11 Finite Parabolic-Ordinary Systems... 445 8.12 Finite Integroparabolic and Integroelliptic Systems... 449 8.13 Notes and Comments... 458 Chapter 9 Systems with Nonlinear Boundary Conditions 9.1 Quasimonotone Boundary Functions... 459 9.2 Construction of Monotone Sequences... 462 9.3 Existence-Comparison Theorems for Parabolic System... 468 9.4 Elliptic Systems with Coupled Boundary Conditions... 471 9.5 Positive Solution for Coupled Systems... 479 9.6 Global Existence Theorems for Bounded Reaction Functions 485 9.7 Finite Parabolic Systems with Coupled Boundary Conditions... 491 9.8 Finite Elliptic Systems with Coupled Boundary Conditions. 498 9.9 Finite Systems with Nonlocal Boundary Conditions... 503 9.10 Notes and Comments... 509 Chapter 10 Stability and Asymptotic Behavior of Solutions 10.1 Stability of the Zero Solution... 511 10.2 Stability of Nontrivial Steady-State Solutions... 516 10.3 Instability of Steady-State Solutions... 522 10.4 Monotone Convergence of Time-Dependent Solutions... 529 10.5 Asymptotic Stability in a Sector... 536 10.6 Spatially Homogeneous Upper and Lower Solutions... 542 10.7 Stability of Solutions for Nonautonomous Systems... 546 10.8 Systems with Nonlinear Boundary Conditions... 555 10.9 Finite Coupled Systems with Quasimonotone Nondecreasing Functions... 563 10.10 Notes and Comments... 568

Contents xv Chapter 11 Asymptotic Limit and Blowing-Up Behavior of Solutions 11.1 Nonisolated Steady-State Solutions... 570 11.2 Asymptotic Limit of Time-Dependent Solutions... 573 11.3 Coupled Parabolic and Ordinary Equations... 581 11.4 A Special Model... 588 11.5 Blowing-Up of Solution for Neumann Boundary Problems. 593 11.6 Blowing-Up of Solution for Robin Boundary Problems... 598 11.7 Blowing-Up of Solution for Dirichlet Boundary Problems. 604 11.8 Nonquasimonotone Functions... 609 11.9 Coupled Nonlinear Boundary Conditions... 613 11.10 Notes and Comments... 619 Chapter 12 Applications of Coupled Systems to Model Problems 12.1 A Gas-Liquid Interaction Problem... 622 12.2 The Belousov-Zhabotinskii Reaction Diffusion System... 628 12.3 Enzyme-Substrate Reaction Diffusion Problems... 641 12.4 The Volterra-Lotka Competition Model in Ecology... 654 12.5 Some Prey-Predator Models in Ecology... 666 12.6 A Cooperating Model in Ecology... 674 12.7 The FitzHugh-Nagumo Equations in Neurophysiology... 681 12.8 Heat-Mass Transfer in Chemical Reactors and Combustion Theory... 687 12.9 Epidemic Problems with Diffusion... 695 12.10 Coupled Systems in Nuclear Reactor Dynamics... 713 12.11 Neutron Transport Problems with Temperature Feedback 723 12.12 Notes and Comments... 744 References.............................................. 747 Index...,... 769