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1072

Franz Rothe

Global Solutions of Reaction-Diffusion Systems

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Author

Franz Rothe Lehrstuhl fur Biomathematik, Universitat Tlibingen Auf der Morgenstelle 28,7400 Tlibingen, Federal Republic of Germany

AMS Subject Classifications (1980): 35B35, 35B40, 35B45, 35B65, 35K55, 92A 17, 92A40

ISBN 3-540-13365-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13365-8 Springer-Verlag New York Heidelberg Berlin Tokyo Library of Congress Cataloging in Publication Data Rothe, Franz, 1947 - Global solutions of reaction-diffusion systems. (Lecture notes in mathematics; 1072) Bibliography: p.lncludes index. 1. Differential equations, Partial-Numerical solutions. 2. Differential equations, Parabolic-Numerical solutions. 3. Biomathematics. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1072. OA3.L28 no. 1072 [OA377] 515.3'53 84-13887 ISBN 0-387-13365-8 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach I Bergstr. 2146/3140-543210

Preface

This monograph is motivated by some problems from Mathematical Biology. Although there exists an extensive literature about nonlinear parabolic differential equations, none of the known results could be used to prove global existence of solutions for the reaction-diffusion systems considered in this monograph. In this situation, I gave an ad hoc proof of global existence for the one-dimensional reaction-diffusion system with reaction A + B

C subject to the mass action law. Afterwards it turned

out that the method used could be generalized and applied to other problems as well. For the time being, the subject is not yet exhausted. Further interesting examples from applications are needed in order to build a substantial theory which is not just an unnecessarily abstract disguise of some specific problem. The author hopes that this monograph will be useful to stimulate research in this direction. I take the opportunity to thank Prof. Dr. K.-P. Hadeler from the Lehrstuhl of Biomathematik, University of Ttibingen, for his intensive personal and scientific support as well as my colleagues Dr. W. Ebel and Dr. H. Munz for reading the manuscript and giving much constructive critisism. With this help, I hope, the manuscript became more rigorous and more readable at the same time.

May 1984

Franz Rothe

Contents

Introduction .••....• " .••..•.•••••.••••••••.••••..• , ••.•••••......• Part I

Existence and A Priori Estimates for 5

Reaction-Diffusion Equations

Basic Notations and Definitions.......................