Galaktionov V.A., Vázquez J.L. A Stability Technique For Evolution Partial Differential Equations. A Dynamical Systems Approach

Birkhauser, 2004. — 377 p. — (Progress in Nonlinear Differential Equations and Their Applications Volume 56) — ISBN13: 978-1-4612-7396-7.This book introduces a new, state-of-the-art method for the study of the asymptotic behavior of solutions to evolution partial differential equations; much of the text is dedicated to the application of this method to a wide class of nonlinear diffusion equations. The underlying theory hinges on a new stability result, formulated in the abstract setting of infinite-dimensional dynamical systems, which states that under certain hypotheses, the omega-limit set of a perturbed dynamical system is stable under arbitrary asymptotically small perturbations. The Stability Theorem is examined in detail in the first chapter, followed by a review of basic results and methods---many original to the authors---for the solution of nonlinear diffusion equations. Further chapters provide a self-contained analysis of specific equations, with carefully-constructed theorems, proofs, and references. In addition to the derivation of interesting limiting behaviors, the book features a variety of estimation techniques for solutions of semi- and quasilinear parabolic equations. Written by established mathematicians at the forefront of the field, this work is a blend of delicate analysis and broad application, appropriate for graduate students and researchers in physics and mathematics who have basic knowledge of PDEs, ordinary differential equations, functional analysis, and some prior acquaintance with evolution equations. It is ideal for a course or seminar in evolution equations and asymptotics, and the book's comprehensive index and bibliography will make it useful as a reference volume as well.Stability Theorem: A Dynamical Systems Approach
Nonlinear Heat Equations: Basic Models and Mathematical Techniques
Equation of Superslow Diffusion
Quasilinear Heat Equations with Absorption. The Critical Exponent
Porous Medium Equation with Critical Strong Absorption
The Fast Diffusion Equation with Critical Exponent
The Porous Medium Equation in an Exterior Domain
Blow-up Free-Boundary Patterns for the Navier-Stokes Equations
Equation ut = uxx + u ln2u: Regional Blow-up
Blow-up in Quasilinear Heat Equations Described by Hamilton—Jacobi Equations
A Fully Nonlinear Equation from Detonation Theory
Further Applications to Second- and Higher-Order Equations