A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach (Progress in Nonlinear Differential Equations and Their Applications (56))

Victor A. Galaktionov, Juan Luis Vasquez

ISBN 10: 0817641467 / ISBN 13: 9780817641467

Published by Birkhäuser, 2003

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Bibliographic Details

• Title: A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach (Progress in Nonlinear Differential Equations and Their Applications (56))
• Publisher: Birkhäuser
• Publication Date: 2003
• Language: English
• ISBN 10: 0817641467
• ISBN 13: 9780817641467
• Binding: Hardcover
• Condition: Like New
• Book Type: book

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Synopsis

common feature is that these evolution problems can be formulated as asymptoti­ cally small perturbations of certain dynamical systems with better-known behaviour. Now, it usually happens that the perturbation is small in a very weak sense, hence the difficulty (or impossibility) of applying more classical techniques. Though the method originated with the analysis of critical behaviour for evolu­ tion PDEs, in its abstract formulation it deals with a nonautonomous abstract differ­ ential equation (NDE) (1) Ut = A(u) + C(u, t), t > 0, where u has values in a Banach space, like an LP space, A is an autonomous (time-independent) operator and C is an asymptotically small perturbation, so that C(u(t), t) ~ ° as t ~ 00 along orbits {u(t)} of the evolution in a sense to be made precise, which in practice can be quite weak. We work in a situation in which the autonomous (limit) differential equation (ADE) Ut = A(u) (2) has a well-known asymptotic behaviour, and we want to prove that for large times the orbits of the original evolution problem converge to a certain class of limits of the autonomous equation. More precisely, we want to prove that the orbits of (NDE) are attracted by a certain limit set [2* of (ADE), which may consist of equilibria of the autonomous equation, or it can be a more complicated object.

Review

"The authors are famous experts in the field of PDEs and blow-up techniques. In this book they present a stability theorem, the so-called S-theorem, and show, with several examples, how it may be applied to a wide range of stability problems for evolution equations. The book [is] aimed primarily aimed at advanced graduate students."

―Mathematical Reviews

"The book is very interesting and useful for researchers and students in mathematical physics...with basic knowledge in partial differential equations and functional analysis. A comprehensive index and bibliography are given" ---Revue Roumaine de Mathématiques Pures et Appliquées