Hayashi N., Kaikina E.I., Naumkin, P.I., Shishmarev I.A. Asymptotics for Dissipative Nonlinear Equations

Berlin: Springer-Verlag. – 2006. – 569 p. (Lecture Notes in Mathematics) Modern mathematical physics is almost exclusively a mathematical theory of nonlinear partial differential equations describing various physical processes. Since only a few partial differential equations have succeeded in being solved explicitly, different qualitative methods play a very important role. One of the most effective ways of qualitative analysis of differential equations are asymptotic methods, which enable us to obtain an explicit approximate representation for solutions with respect to a large parameter time. Asymptotic formulas allow us to know such basic properties of solutions as how solutions grow or decay in different regions, where solutions are monotonous and where they oscillate, which information about initial data is preserved in the asymptotic representation of the solution after large time, and so on. It is interesting to study the influence of the nonlinear term in the asymptotic behavior of solutions. This book is the first attempt to give a systematic approach for obtaining the large time asymptotic representations of solutions to the nonlinear evolution equations with dissipation. We restrict our attention to the investigation of the Cauchy problems (initial value problems) leaving outside the wide and important class of the initial-boundary value problems (in some respects the reader can fill this gap by consulting a recent book Hayashi and Kaikina [2004]). In our book we pay much attention to typical well-known equations which have huge applications: the nonlinear heat equation, Burgers equation, Korteweg-de Vries-Burgers equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes equations and others. Certainly we do not claim Preface VII that we could embrace all equations and all cases. However we succeeded in selecting a sufficiently wide class of equations, which could be treated by a unified approach and which fall into the same theory. Many of the methods proposed in this book have been developed by a great number of authors. The results and proofs presented throughout the book are mainly based on the research articles of the authors.Preliminary results.
Weak Nonlinearity.
Critical Nonconvective Equations.
Critical Convective Equations.
Subcritical Nonconvective Equations.
Subcritical Convective Equations.