A Stability Technique for Evolution Partial Differential Equations
A Dynamical Systems Approach
Authors: Galaktionov, Victor A., Vázquez, Juan Luis
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common feature is that these evolution problems can be formulated as asymptoti cally small perturbations of certain dynamical systems with betterknown 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 (timeindependent) 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 wellknown 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.
 Reviews

"The authors are famous experts in the field of PDEs and blowup techniques. In this book they present a stability theorem, the socalled Stheorem, 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
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Bibliographic Information
 Bibliographic Information

 Book Title
 A Stability Technique for Evolution Partial Differential Equations
 Book Subtitle
 A Dynamical Systems Approach
 Authors

 Victor A. Galaktionov
 Juan Luis Vázquez
 Series Title
 Progress in Nonlinear Differential Equations and Their Applications
 Series Volume
 56
 Copyright
 2004
 Publisher
 Birkhäuser Basel
 Copyright Holder
 Birkhäuser Boston
 eBook ISBN
 9781461220503
 DOI
 10.1007/9781461220503
 Hardcover ISBN
 9780817641467
 Softcover ISBN
 9781461273967
 Series ISSN
 14211750
 Edition Number
 1
 Number of Pages
 XIX, 377
 Topics