Volpert, Volpert, Volpert - Traveling Wave Solutions of Parabolic Systems.pdf

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Translations of
M ATHEMATICAL
M ONOGRAPHS
Volume 140
Traveling Wave Solutions
of Parabolic Systems
Aizik I. Volpert
Vitaly A. Volpert
Vladimir A. Volpert
E
Τ Ρ Η Τ Ο Σ Μ Η
American Mathematical Society
Providence, Rhode Island
E
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A. I. Volpert, Vit. A. Volpert, Vl. A. Volpert
BEGUWIE VOLNY, OPISYVAEMYE
PARABOLIQESKIMI SISTEMAMI
Translated by James F. Heyda from an original Russian manuscript
2000 Mathematics Subject Classification . Primary 35K55, 80A30;
Secondary 92E10, 80A25.
Abstract. Traveling wave solutions of parabolic systems describe a wide class of phenomena in
combustion physics, chemical kinetics, biology, and other natural sciences. The book is devoted to
the general mathematical theory of such solutions. The authors describe in detail such questions as
existence and stability of solutions, properties of the spectrum, bifurcations of solutions, approach
of solutions of the Cauchy problem to waves and systems of waves. The final part of the book is
devoted to applications to combustion theory and chemical kinetics.
The book can be used by graduate students and researchers specializing in nonlinear differential
equations, as well as by specialists in other areas (engineering, chemical physics, biology), where
the theory of wave solutions of parabolic systems can be applied.
Library of Congress Cataloging-in-Publication Data
Vol pert,A.I.(Aızik Isaakovich)
[Begushchie volny, opisyvaemye parabolicheskimi sistemami. English]
Traveling wave solutions of parabolic systems/Aizik I. Volpert, Vitaly A. Volpert, Vladimir A.
Volpert.
p. cm. — (Translations of mathematical monographs, ISSN 0065-9282; v. 140)
Includes bibliographical references.
ISBN 0-8218-4609-4 (acid-free)
1. Differential equations, Parabolic. 2. Differential equations, Nonlinear. 3. Chemical
kinetics—Mathematical models. I. Volpert, Vitaly A., 1958– . II. Volpert, Vladimir A., 1954– .
III. Title. IV. Series.
QA377.V6413 1994
515 .353—dc20
94-16518
1994 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
Reprinted with corrections, 2000
∞ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Information on copying and reprinting can be found in the back of this volume.
This volume was typeset by the author using A M S -T E X,
the American Mathematical Society’s T E X macro system.
Visit the AMS home page at URL: http://www.ams.org/
1098765432 050403020100
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Contents
Preface
xi
Introduction. Traveling Waves Described by Parabolic Systems
1
§
1. Classification of waves
2
§
2. Existence of waves
11
§
3. Stability of waves
16
§
4. Wave propagation speed
22
§
5. Bifurcations of waves
23
§
6. Traveling waves in physics, chemistry, and biology
32
Part I. Stationary Waves
Chapter 1. Scalar Equation
39
§ 1. Introduction
39
§ 2. Functionals ω ∗ and ω ∗
45
§ 3. Waves and systems of waves
51
§
4. Properties of solutions of parabolic equations
72
§
5. Approach to waves and systems of waves
85
§
6. Supplement (Additions and bibliographic commentaries)
111
Chapter 2. Leray-Schauder Degree
121
§
1. Introduction. Formulation of results
121
§
2. Estimate of linear operators from below
128
§ 3. Functional c ( u ) and operator A ( u )
134
§ 4. Leray-Schauder degree
138
§
5. Linearized operator
141
§
6. Index of a stationary point
144
7. Supplement. Leray-Schauder degree in the multidimensional
case
149
Chapter 3. Existence of Waves
153
§
1. Introduction. Formulation of results
153
§ 2. A priori estimates
159
§ 3. Existence of monotone waves
173
§ 4. Monotone systems
176
§
5. Supplement and bibliographic commentaries
183
Chapter 4. Structure of the Spectrum
187
§
1. Elliptic problems with a parameter
189
§
2. Continuous spectrum
192
§
3. Structure of the spectrum
198
vii
§
viii
CONTENTS
§
4. Examples
208
§
5. Spectrum of monotone systems
212
Chapter 5. Stability and Approach to a Wave
217
§
1. Stability with shift and its connection with the spectrum
218
§
2. Stability of planar waves to spatial perturbations
225
§
3. Conditions of instability
237
§
4. Stability of waves for monotone systems
238
§
5. On the solutions of nonstationary problems
242
§
6. Approach to a monotone wave
250
§
7. Minimax representation of the speed
254
Part II. Bifurcation of Waves
Chapter 6. Bifurcation of Nonstationary Modes of Wave Propagation
259
§
1. Statement of the problem
259
2. Representation of solutions in series form. Stability of
solutions
263
§
3. Examples
268
Chapter 7. Mathematical Proofs
273
§
1. Statement of the problem and linear analysis
273
2. General representation of solutions of the nonlinear problem.
Existence of solutions
285
§
3. Stability of branching-off solutions
295
Part III. Waves in Chemical Kinetics and Combustion
Chapter 8. Waves in Chemical Kinetics
299
§
1. Equations of chemical kinetics
299
§
2. Monotone systems
306
§ 3. Existence and stability of waves
312
§ 4. Branching chain reactions
316
§ 5. Other model systems
333
Bibliographic commentaries
335
Chapter 9. Combustion Waves with Complex Kinetics
337
§
1. Introduction
337
2. Existence of waves for kinetic systems with irreversible
reactions
338
3. Stability of a wave in the case of equality of transport
coe=cients
362
§
4. Examples
366
Bibliographic commentaries
375
Chapter 10. Estimates and Asymptotics of the Speed of Combustion
Waves
377
1. Estimates for the speed of a combustion wave in a condensed
medium
377
§
2. Estimates for the speed of a gas combustion wave
392
3. Determination of asymptotics of the speed by the method of
successive approximations
400
Bibliographic commentaries
409
§
§
§
§
§
§
CONTENTS
ix
Supplement. Asymptotic and Approximate Analytical Methods in
Combustion Problems
411
1. Narrow reaction zone method. Speed of a stationary
combustion wave
411
§
2. Stability of a stationary combustion wave
415
§
3. Nonadiabatic combustion
416
§
4. Stage combustion
418
§
5. Transformations in a combustion wave
423
6. Application of the methods of bifurcation theory to the study
of nonstationary modes of propagation of combustion waves
426
§
7. Surveys and monographs
431
Bibliography
433
§
§

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