Progress in Nonlinear Differential Equations and Their Applications

A Stability Technique for Evolution Partial Differential Equations

A Dynamical Systems Approach

Authors: Galaktionov, Victor A., Vázquez, Juan Luis

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About this book

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.

Reviews

"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


Table of contents (12 chapters)

  • Stability Theorem: A Dynamical Systems Approach

    Galaktionov, Victor A. (et al.)

    Pages 1-12

  • Nonlinear Heat Equations: Basic Models and Mathematical Techniques

    Galaktionov, Victor A. (et al.)

    Pages 13-55

  • Equation of Superslow Diffusion

    Galaktionov, Victor A. (et al.)

    Pages 57-79

  • Quasilinear Heat Equations with Absorption. The Critical Exponent

    Galaktionov, Victor A. (et al.)

    Pages 81-125

  • Porous Medium Equation with Critical Strong Absorption

    Galaktionov, Victor A. (et al.)

    Pages 127-167

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-1-4612-2050-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $99.00
price for USA
  • ISBN 978-0-8176-4146-7
  • Free shipping for individuals worldwide
  • This title is currently reprinting. You can pre-order your copy now.
Softcover $139.00
price for USA
  • ISBN 978-1-4612-7396-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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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
Series Title
Progress in Nonlinear Differential Equations and Their Applications
Series Volume
56
Copyright
2004
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Boston
eBook ISBN
978-1-4612-2050-3
DOI
10.1007/978-1-4612-2050-3
Hardcover ISBN
978-0-8176-4146-7
Softcover ISBN
978-1-4612-7396-7
Series ISSN
1421-1750
Edition Number
1
Number of Pages
XIX, 377
Topics