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Nonlinear partial differential equations. Asymptotic behavior of solutions and self-similar solutions. (English) Zbl 1215.35001
Progress in Nonlinear Differential Equations and Their Applications 79. Basel: Birkhäuser (ISBN 978-0-8176-4173-3/pbk; 978-0-8176-4651-6/ebook). xviii, 294 p. (2010).
This very nice and useful book is an extended and revised version of a Japanese book written by the first two authors in 1999. The book presents typical methods (including rescaling methods) for the examination of the behavior of solutions of nonlinear partial differential equations of diffusion type. In particular, the authors examine such equations by analyzing special so-called self-similar solutions which are, roughly speaking, solutions invariant under a scaling transformation that does not change the equation. For several typical equations they give mathematical proofs that certain self-similar solutions asymptotically approximate the typical behavior of a wide class of solutions.
Instead of explaining fundamental universal theory for PDE analysis, such as functional analysis, and discussing general issues for PDEs (like solvability) in that framework, the authors study directly the behavior of solutions of particular equations without preparing the fundamental theory. The fundamental tools used in the study (calculus inequalities, for example) are discussed in the second part of the book. The aim of the authors was to teach the readers to deal with such tools during the study of PDEs and to give them a strong motivation for their study.
Part I of the book consists of three chapters. In Chapter 1 the authors use two different methods to show that the large-time behavior of solutions of the heat equation is asymptotically self-similar. The first method is based on a representation formula of the solution. The second method is based on the study of convergence of suitable rescaled solutions and applies to a wide class of problems. In particular, it is also used in Chapter 2 in order to analyze the two-dimensional vorticity equations (obtained from the Navier-Stokes equations). This detailed analysis also includes an application concerning the formation of the Burgers vortex in three dimensions and the proof of uniqueness of weak solutions. Chapter 3 deals mainly with the porous medium equation and the mean curvature flow equation (and briefly with nondiffusion-type equations like KdV or nonlinear Schrödinger equations).
Part II is devoted to useful analytic tools and consists of four chapters: Chapters 4, 5, 6 and 7 deal with various properties of the heat equation, compactness theorems, calculus inequalities and convergence theorems in theory of integration, respectively.
The book is written in a very pedagogical way. Each chapter contains plenty of exercises whose detailed solutions can be found at the end of the book.

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
35C06 Self-similar solutions to PDEs
35K55 Nonlinear parabolic equations
35K93 Quasilinear parabolic equations with mean curvature operator
35Q30 Navier-Stokes equations
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