An Introduction to Nonlinear Partial Differential Equations

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1 An Introduction to Nonlinear Partial Differential Equations

2 PURE AND APPLIED MATHEMATICS A Wiley-Interscience Series of Texts, Monographs, and Tracts Founded by RICHARD COURANT Editors Emeriti: MYRON B. ALLEN 111, DAVID A. COX, PETER HILTON, HARRY HOCHSTADT, PETER LAX, JOHN TOLAND A complete list of the titles in this series appears at the end of this volume.

3 An Introduction to Nonlinear Partial Differential Equations Second Edition J. David Logan Willa Cather Professor of Mathematics University of Nebraska, Lincoln Department of Mathematics Lincoln, NE WI LEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION

4 Copyright 'C 2008 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, h'ew Jersey Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act. without either the prior Ivritten permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., I 1 1 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at Limit of Liability/Disclairner of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) , outside the United States at (317) or fax (317) Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic format. For information about Wiley products, visit our web site at Library of Congress Cataloging-in-Publication Data: Logan, J. David (John David) An introduction to nonlinear partial differential equations / J. David Logan. - 2nd ed. p. cm. Includes bibliographical references and index. ISBN (cloth : acid-free paper) 1. Differential equations, Nonlinear. 2. Differential equations, Partial. I. Title. QA377.L '.353-d~ Printed in the United States of America

5 To Tess, for all her asfection und support

6 Contents Preface 1. Introduction to Partial Differential Equations 1.1 Partial Differential Equations Equations and Solutions Classification Linear versus Nonlinear Linear Equations 1.2 Conservation Laws One Dimension Higher Dimensions 1.3 Constitutive Relations 1.4 Initial and Boundary Value Problems 1.5 Waves Traveling Waves Plane TVaves Plane JYaves and Transforms Nonlinear Dispersion 2. First-Order Equations and Characteristics 2.1 Linear First-Order Equations Advection Equation Variable Coefficients 2.2 Nonlinear Equations 2.3 Quasilinear Equations The General Solution xi

7 Vlll Contents 2.4 Propagation of Singularities 2.5 General First-Order Equation Complete Integral 2.6 A Cniqueness Result 2.7 Models in Biology Age Structure Structured Predator-Prey Model Chemotherapy i\iass Structure Size-Dependent Predation 3. Weak Solutions to Hyperbolic Equations 3.1 Discontinuous Solutions 3.2 Jump Conditions Rarefaction WTaves Shock Propagation 3.3 Shock Formation 3.4 Applications Traffic Flow Plug Flow Chemical Reactors 3.5 Weak Solutions: A Formal Approach 3.6 Asymptotic Behavior of Shocks Equal-Area Principle Shock Fitting Asymptotic Behavior 4. Hyperbolic Systems 4.1 Shallow-FYater Waves: Gas Dynamics Shallow-Water Waves Small-Amplitude Approximation Gas Dynamics 4.2 Hyperbolic Systems and Characteristics Classification 4.3 The Riemann Method Jump Conditions for Systems Breaking Dam Problem Receding Wall Problem Formation of a Bore Gas Dynamics 4.4 Hodographs and Wavefronts Hodograph Transformation LVavefront Expansions

8 Contents ix 4.5 IVeakly Konlinear Approximations Derivation of Burgers Equation 5. Diffusion Processes 5.1 Diffusion and Random Motion 5.2 Similarity AIethods 5.3 Nonlinear Diffusion Alodels 5.4 Reaction-Diffusion: Fisher s Equation Traveling JYave Solutions Perturbation Solution Stability of Traveling JJ-aves Nagumo s Equation 5.5 Advection-Diffusion: Burgers Equation Traveling IYave Solution Initial Value Problem 5.6 Asymptotic Solution to Burgers Equation Evolution of a Point Source Appendix: Dynamical Systems 6. Reaction-Diffusion Systems 6.1 Reaction-Diffusion Models Predator-Prey Model Combustion Chemotaxis 6.2 Ttaveling IJ1Bve Solutions Model for the Spread of a Disease Contaminant Transport in Groundmter 6.3 Existence of Solutions Fixed-Point Iteration Semilinear Equations Kormed Linear Spaces General Existence Theorem 6.4 ;\laximum Principles and Comparison Theorems Naximum Principles Comparison Theorems 6.5 Energy Estimates and Asymptotic Behavior Calculus Inequalities Energy Estimates Invariant Sets 6.6 Pattern Formation

9 X Contents 7. Equilibrium Models 7.1 Elliptic Afodels 7.2 Theoretical Results Maximum Principle Existence Theorem 7.3 Eigenvalue Problems Linear Eigenvalue Problenis Konlinear Eigenvalue Problems 7.4 Stability and Bifurcation Ordinary Differential Equations Partial Differential Equations References 387 Index 395

10 Preface Nonlinear partial differential equations (PDEs) is a vast area. and practitioners include applied mathematicians. analysts. and others in the pure and applied sciences. This introductory text on nonlinear partial differential equations evolved from a graduate course I have taught for many years at the University of Nebraska at Lincoln. It emerged as a pedagogical effort to introduce. at a fairly elementary level. nonlinear PDEs in a format and style that is accessible to students with diverse backgrounds and interests. The audience has been a mixture of graduate students from mathematics. physics, and engineering. The prerequisites include an elementary course in PDEs emphasizing Fourier series and separation of variables. and an elementary course in ordinary differential equations. There is enough independence among the chapters to allow the instructor considerable flexibility in choosing topics for a course. The text may be used for a second course in partial differential equations. a first course in nonlinear PDEs, a course in PDEs in the biological sciences. or an advanced course in applied mathematics or mathematical modeling. The range of applications include biology. chemistry. gas dynamics, porous media. combustion. traffic flow. water waves. plug flow reactors. heat transfer. and other topics of interest in applied mathematics. There are three major changes from the first edition, which appeared in Because the original chapter on chemically reacting fluids was highly specialized for an introductory text. it has been removed from the new edition. Additionally. because of the surge of interest in mathematical biology. considerable material on that topic has been added; this includes linear and nonlinear age structure. spatial effects. and pattern formation. Finally. the text has been reorganized with the chapters on hyperbolic equations separated from

xii Preface the chapters on diffusion processes. rat,her than int,ermixirig them. The references have been updated and. as in the previous edition.

11 xii Preface the chapters on diffusion processes. rat,her than int,ermixirig them. The references have been updated and. as in the previous edition. are selected to suit, t,he needs of an introductory text. point'ing the reader to parallel treatments and resources for further study. Finally, many new exercises have been added. The exercises are intermediate-level and are designed to build t,he students' problem solving techniques beyond what is experienced in a beginning course. Chapter 1 develops a perspective on how to understand problems involving PDEs and horn the subject, interrelakes wit'li physical phenomena. The subject is developed from the basic conservation law. which, when appended to constitutive relations, gives rise to the fundamental models of diffusion. advection, and reaction. There is emphasis on understanding that nonlinear hyperbolic and parabolic PDEs describe evolutionary processes: a solution is a signal that is propagated int,o a spacetime domain from the boundaries of that domain. Also. there is focus on the structure of the various equations arid what the terms describe physically. Chapters 2-3 deal with wave propagation and hyperbolic problems. In Chapter 2 we assume that the equations have smooth solutions and we develop algoritlinis to solve the equat,ions analytically. In Chapter 3 we study discontinuous solutions and shock format,ion. and we introduce the concept of a weak solution. In keeping with our strategy of thinking about initial waveforms evolving in time. we focus on the initial 1-alue problem rather than the general Cauchy problem. The idea of characteristics is central and forms the thread that, weaves through t,hese two chapters. Next. Chapter 4 introduces the shallow-water equations as the prototype of a hyperbolic system. arid those equations are taken t'o illustrate basic concepts associat,ed wit,h hyperbolic system: characteristics. Riemann's method. the hodograph transformation. and asyrnpt'otic behavior. Also. the general classification of systems of first-order PDEs is developed. and weakly nonlinear methods of analysis are described: the latter are illustrated by a derivation of Burgers' equation. Chapters 1-4 can form t,he basis of a one-semester course focusing on wave propagation. characteristics, and hyperbolic equations. Chapter 5 introduces diffusion processes. After establishing a probabilist,ic basis for diffusion, we examine methods that are useful in studying the solution structure of diffusion problems. including phase plane analysis. similarity methods. and asymptotic expansions. The prototype equations for reaction-diffusion and advection-diffusion. Fisher's equation and Burgers' equation. respectively. are studied in detail with emphasis 011 traveling wave solutions. the st,abilit,y of those solutions. arid the asymptotic behavior of solutions. The Appendix to Chapt,er 5 reviews phase plane analysis. In Chapter 6 we discuss systems of reaction-diffusion equations, emphasizing applications and model building, especially in t,he biological sciences. \Ye expend some effort addressing theoret-

12 Preface... Xlll ical concepts such as existence, uniqueness, comparison and maximum principles. energy estimates, blowup. and invariant sets: a key application includes pattern forniation. Finally, elliptic equations are introduced in Chapter 7 as a asymptotic limit of reaction-diffusion equations: nonlinear eigenvalue problems, stability. and bifurcation phenomena forin the core of this chapter. Chapter 1, along with Chapters 5-8. can form the basis of a one-semester course in diffusion and reaction-diffusion processes. with emphasis on PDEs in mathematical biology. I want to acknowledge many users of the first edition who suggested impro.\ ements, corrections. and new topics. Their excitement for a second edition. along with the unwavering encouragement of my editor Susanne Steitz-Filler at JYiley. provided the stimulus to actually complete it. My own interest in nonlinear PDEs was spawned over many years by collaboration with those with whom I have had the privilege of working: Kane Yee at Kansas State. John Bdzil at Los Alamos. *4sh Kapila at Rensselaer Polytechnic Institute. and several of my colleagues at Nebraska (Professors Steve Cohn. Steve Dunbar. Tony Joern in biology. Glenn Ledder. Tom Shores. Vital! Zlotnik in geology, and my former student Bill \Volesensky. now at the College of Saint Rlary). Readers of this text \\-ill see the influence of the classic books of G. B. IVhitham (Lznear and Nonlznear Waues) and J. Smoller (Shock Waves and Reactaon-Dzffuszon Equatzons). R. Courant and K. 0. Friedrichs (Supersonzc Flow and Shock Waves). and the text on mathematical biology by J. D. Murray (Mathematzeal Bzology). Finally, I express niy gratitude to the National Science Foundation and to the Department of Energy for supporting my research efforts over the last several years J. David Logan Lincoln. Kebraska

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