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A Stability Technique for Evolution Partial Differential Equations : A Dynamical Systems Approach

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概要 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 pert...urbation 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.続きを見る
目次 Introduction: A Stability Approach and Nonlinear Models
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
The Equation ut = uxx + uln2u: 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
References
Index.
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登録日 2020.06.27
更新日 2020.06.28