traveling wave solutions of parabolic systems

Translations of M ATHEMATICAL M ONOGRAPHS ΑΓΕΩΜΕ ΕΙΣΙΤΩ ΤΡΗΤΟΣ ΜΗ F O U N D E D 1 8 8 8 A M E R I C A N M A T H E M A T I C A L S O C I E T Y American Mathematical Society Providence, Rhode Island Volume 140 Traveling Wave Solutions of Parabolic Systems Aizik I. Volpert Vitaly A. Volpert Vladimir A. Volpert A. I. Volpert, Vit. A. Volpert, Vl. A. Volpert 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.(A˘ızik Isaakovich) [Begushchie volny, opisyvaemye parabolicheskimi sistemami. English] Traveling wave solutions of parabolic systems/Aizik I. Volpert, Vitaly A. Volpert, Vladimir A. Volp ert. 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 c  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 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 coefficients 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 Preface The theory of traveling wave solutions of parabolic equations is one of the fast developing areas of modern mathematics. The history of this theory begins with the famous mathematical work by Kolmogorov, Petrovski˘ı, and Piskunov and with works in chemical physics, the best known among them by Zel  dovich and Frank-Kamenetski˘ı in combustion theory and by Semenov, who discovered branching chain flames. Traveling wave solutions are solutions of special type. They can be usually characterized as solutions invariant with respect to translation in space. The existence of traveling waves appears to be very common in nonlinear equations, and, in addition, they often determine the behavior of the solutions of Cauchy-type problems. From the physical point of view, traveling waves usually describe transition processes. Transition from one equilibrium to another is a typical case, although more complicated situations can arise. These transition processes usually “forget” their initial conditions and reflect the properties of the medium itself. Among the basic questions in the theory of traveling waves we mention the problem of wave existence, stability of waves with respect to small perturbations and global stability, bifurcations of waves, determination of wave speed, and systems of waves (or wave trains). The case of a scalar equation has been rather well studied, basically due to applicability of comparison theorems of a special kind for parabolic equations and of phase space analysis for the ordinary differential equations. For systems of equations, comparison theorems of this kind are, in general, not appliΓ cable, and the phase space analysis becomes much more complicated. This is why systems of equations are much less understood and require new approaches. In this book, some of these approaches are presented, together with more traditional approaches adapted for specific classes of systems of equations and for a more complete analysis of scalar equations. From our point of view, it is very important that these mathematical results find numerous applications, first and foremost in chemical kinetics and combustion. The authors understand that the theory of traveling waves is far from being complete and hope that this book will help in its development. This book was basically written when the authors worked at the Institute of Chemical Physics of the Soviet Academy of Sciences. This scientific school, created by N. N. Semenov, Director of the Institute for a long time, by Ya. B. Zeldovich, who worked there, and by other outstanding personalities, has a strong tradition xi xii PREFACE of collaboration among physicists, chemists, and mathematicians. This special atmosphere had a strong influence on the scientific interests of the authors and was very useful to us. We would like to thank all our colleagues with whom we worked for many years and without whom this book could not have been written. Aizik Volpert Department of Mathematics, Technion, Haifa, 32000, Israel Vitaly Volpert Universite Lyon 1, CNRS, Villeurbanne Cedex, 69622 France Vladimir Volpert Northwestern University, Evanston, Illinois 60208 June 1993 INTRODUCTION Traveling Waves Described by Parabolic Systems Propagation of waves, described by nonlinear parabolic equations, was first considered in a paper by A. N. Kolmogorov, I. G. Petrovski˘ı, and N. S. Piskunov [Kolm 1]. These mathematical investigations arose in connection with a model for the propagation of dominant genes, a topic also considered by R. A. Fisher [Fis 1]. Moreover, when [Kolm 1] appeared in 1937, the fact that waves can be described not only by hyperbolic equations, but also by parabolic equations, did not receive the proper attention of mathematicians. This is indicated by the fact that sub- sequent mathematical papers in this direction (Ya. I. Kanel  [Kan1,2,3]) did not appear until more than twenty years later, although mathematical models, which form a basis for these papers, models of combustion, were formulated by Ya.B.Zel  dovich somewhat earlier (see, for example, [Zel 4, 5]). Itwasnotuntil the seventies, under the influence of a great number of the most diverse problems of physics, chemistry, and biology, that an intensive development of this theme began. At the present time a large number of papers is devoted to wave solutions of parabolic systems and this number continues to increase. In recent years, along with the study of one-dimensional waves, an interest in multi-dimensional waves has developed. This interest was stimulated by observation of spinning waves in combustion, spiral waves in chemical kinetics, etc. The overwhelming number of natural science problems mentioned above leads to wave solutions of the parabolic system of equations (0.1) ∂u ∂t = A∆u + F(u), where u =(u 1 , ,u m ) is a vector-valued function, A is a symmetric nonnegative- definite matrix, ∆ is the Laplace operator, and F (u) is a given vector-valued function, which we will sometimes refer to as a source. System (0.1) is considered in a domain Ω of space R n on whose boundary, assuming Ω does not coincide with R n , boundary conditions are specified. We attempt in the present introduction to give a general picture of current results concerning wave solutions of system (0.1) (see also [Vol 47]). Later on in the text we present in detail results of a general character, i.e., results connected with general methods of analysis and with sufficiently general classes of systems. In the remaining cases we limit ourselves to a brief exposition or to references to original papers. However, in selecting material for a detailed exposition interests of the authors are dominant. Numbering of formulas and various propositions are carried out according to sections, the first digit indicating the section number. If in references the chapter is not indicated, it may be assumed that reference is being made to a section within the current chapter. 1 2 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS §1. Classification of waves Waves described by parabolic systems can be divided into several classes. The most conventional is the class of waves referred to as stationary. By a stationary wave we mean a solution u(x, t) of system (0.1) of the form (1.1) u(x, t)=w(x 1 − ct, x  ), where w(x) is a function of n variables, x =(x 1 , ,x n ), x  =(x 2 , ,x n ), and c is a constant (speed of the wave). We assume here that Ω is a cylinder and that the system of coordinates is chosen so that axis x 1 is directed along the axis of the cylinder. In recent years a large body of experimental material has accumulated and, in addition, a number of mathematical models connected with it have been studied in which not just stationary waves can be observed. In particular, we can observe periodic waves , defined as solutions u(x, t) of system (0.1) of the form (1.2) u(x, t)=w(x 1 − ct, x  ,t), where the function w(x, t)isperiodicint; Ω, as defined above, is a cylinder; and x 1 is directed along the axis of the cylinder. Other forms of waves also occur, some of which we indicate below. 1.1. Stationary waves. We present a classification of stationary waves cur- rently being studied. Part I of the present text is devoted to stationary waves. 1.1.1. One-dimensional planar waves. We consider system (0.1) with the fol- lowing boundary condition on the surface of cylinder Ω: (1.3) ∂u ∂ν =0, where ν is the normal to the surface. We refer to a solution of the form (1.4) u(x, t)=w(x 1 − ct) as a planar wave. This, obviously, corresponds to the definition given above of a stationary wave, one-dimensional in space, i.e., a solution of the system (1.5) ∂u ∂t = A ∂ 2 u ∂x 2 1 + F (u). Function w of the variable ξ = x 1 − ct is a solution of the following system of ordinary differential equations over the whole axis: (1.6) Aw  + cw  + F (w)=0. Obviously, the system of equations (1.6) can be reduced to the system of first order equations (1.7) w  = p, Ap  = −cp − F(w). Thus, the problem of classifying planar waves can be reduced to the study of the trajectories of system (1.7). Apparently, however, not all trajectories are of interest. Solutions of system (1.6) are stationary solutions of system (1.5), written in coordinates connected with the front of a wave; of most interest are those waves which are stable stationary solutions. We present a classification of planar waves encountered in applications. §1. CLASSIFICATION OF WAVES 3 ξ c w Figure 1.1. A monotone wave front By wave fronts we mean solutions w(ξ) of system (1.6), having limits as ξ → ±∞, (1.8) lim ξ→±∞ w(ξ)=w ± , where (1.9) w + = w − . Typical representatives of such waves are waves of combustion and waves in chemical kinetics, in particular, frontal polymerization, concentrational waves in Belousov- Zhabotinsky reactions, cold flames, etc. A characteristic form of a monotone wave front for each component of the vector-valued function w is shown in Figure 1.1. If we return to the initial coordinate x 1 , the wave front is then the profile shown in this figure moving along the x 1 -axis at constant speed c. It is readily seen that we have the equalities (1.10) F (w + )=0,F(w − )=0 if the function w(ξ), together with its first derivative, is bounded on the whole axis and if the limits (1.8) exist. Actually, in this case it is easy to show that w  (ξ) → 0andw  (ξ) → 0as|ξ|→∞, and, passing to the limit in (1.6), we obtain (1.10). Thus, w + and w − are stationary points of the nondistributed system (1.11) du dt = F (u), corresponding to system (1.5). It turns out to be the case that in studying wave fronts connecting points w + and w − (i.e., solutions of system (1.6) satisfying conditions (1.8)) it is very important to have information concerning stability of the stationary points w + and w − . Obviously, only the following three types of sources F (u) are possible: A. Both points w + and w − are stable stationary points of equation (1.11). B. One of the points w + or w − is stable, the other is unstable. C. Both points w + and w − are unstable. As we shall show below, answers to questions concerning the existence of waves, their uniqueness, and a number of other questions, depend on the source type for F (u). [...]... of system (0.1) is obtained as t → ∞ Systems of waves, or in other terminology wave trains, or minimal decomposition of waves, were studied first from the physical point of view in combustion theory [Kha 3, Mer 6] and mathematically in [Fife 7] §2 Existence of waves 2.1 Methods of proof for the existence of waves At the present time there is a large number of papers concerned with the existence of waves... made of wave systems for a scalar equation We introduce here the concept of a minimal system of waves, which describes the asymptotic behavior of solutions of a Cauchy problem and which, as will be shown later, exists for arbitrary sources §3 Stability of waves 3.1 Stability and spectrum One of the most widely used methods for studying the stability of stationary solutions of nonlinear evolutionary systems. .. Target type waves Waves of this kind are observed experimentally in chemical kinetics as concentric waves diverging from a center with simultaneous generation of new waves at the center References concerned with these waves are [Erm 4, Fife 4, 5, Hag 2, Kop 1, 2, 5, 6, Tys 1] 1.4 Systems of waves A study of the behavior of solutions of a Cauchy problem for system (0.1) for large values of t shows that... birth of a separatrix loop from the stationary points may be proved One of the general approaches to proving the existence of periodic waves also yields a theory of bifurcations The question concerns birth of periodic waves of small amplitude from constant stationary solutions under a change of parameters (see below) Another approach to proving the existence of periodic waves is §2 EXISTENCE OF WAVES... facts that waves exist 14 INTRODUCTION TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS for a whole half-interval (half-axis) of speeds and form an entire family of solutions It can be expected that the degree can be successfully introduced with a proper selection of weighted norms, identifying a single wave (a single speed) of a family of waves Having defined the degree, we see that the possibility of obtaining... equation of higher order, is presented in [Gard 6] Analysis of the stability of traveling waves for scalar viscous conservation laws appears in [Jon 4] and [Goo 1, 2] Along with the problem of stability of waves, there is the closely related problem of the approach to a wave solution of a Cauchy problem with initial conditions “far” from the wave We distinguish three types of approach to a wave By a... such waves can be considered in a section of the cylinder by plane x1 = 0 and a corresponding body of revolution about the axis of the cylinder can be obtained As bifurcation analysis shows, such waves can exist with a various number of spots In the case of a one-spot mode, motion of the spots takes place 10 INTRODUCTION TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS Figure 1.15 One-spot radial wave. .. CLASSIFICATION OF WAVES 7 Figure 1.9 Two-spot mode of wave propagation in a strip basic general methods for studying periodic waves are the methods of bifurcation theory (see below) We now present some forms of periodic waves encountered in applications It is convenient to describe the character of wave propagation by considering the motion of certain characteristic points, for example, maximum points of solutions. .. cross-section of the cylinder By a 8 INTRODUCTION TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS Figure 1.10 One-spot spinning mode of wave propagation in a circular cylinder Figure 1.11 Two-spot spinning mode of wave propagation in a circular cylinder spinning wave we mean a solution of system (0.1) of the form u(x, t) = w(x1 − ct, r, ϕ − σt), where c and σ are constants; c is the speed of propagation... boundary of a ball of sufficiently large radius; and to construct a continuous deformation of the initial system to the model system such that there are a priori estimates of solutions §2 EXISTENCE OF WAVES 13 Rotation of a vector field for completely continuous vector fields is well defined and widely applied, in particular, in proving existence of solutions by the LeraySchauder method Systems of equations of . 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. sistemami. English] Traveling wave solutions of parabolic systems/ Aizik I. Volpert, Vitaly A. Volpert, Vladimir A. Volp ert. p. cm. — (Translations of mathematical monographs, ISSN 006 5-9 282; v. 140) Includes. 1.15. One-spot radial wave Figure 1.16. A mode of wave propagation in a cylinder of rect- angular cross-section Figure 1.17. A mode of wave propagation in a cylinder of rect- angular cross-section as