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Birkhäuser - Birkhäuser Mathematics | A Stability Technique for Evolution Partial Differential Equations - A Dynamical Systems Approach

A Stability Technique for Evolution Partial Differential Equations

A Dynamical Systems Approach

Galaktionov, Victor A., Vázquez, Juan Luis

2004, XIX, 377 p.

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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.

Content Level » Research

Keywords » Navier-Stokes equation - continuum mechanics - differential equation - fluid dynamics - functional analysis - nonlinear analysis - ordinary differential equation - partial differential equation - pdes

Related subjects » Birkhäuser Engineering - Birkhäuser Mathematics

Table of contents 

1. Stability Theorem: A Dynamical Systems Approach.- 1.1 Perturbed dynamical systems.- 1.2 Some concepts from dynamical systems.- 1.3 The three hypotheses.- 1.4 The S-Theorem: Stability of omega-limit sets.- 1.5 Practical stability assumptions.- 1.6 A result on attractors.- Remarks and comments on the literature.- 2. Nonlinear Heat Equations: Basic Models and Mathematical Techniques.- 2.1 Nonlinear heat equations.- 2.2 Basic mathematical properties.- 2.3 Asymptotics.- 2.4 The Lyapunov method.- 2.5 Comparison techniques.- 2.5.1 Intersection comparison and Sturm’s theorems.- 2.5.2 Shifting comparison principle (SCP).- 2.5.3 Other comparisons.- Remarks and comments on the literature.- 3. Equation of Superslow Diffusion.- 3.1 Asymptotics in a bounded domain.- 3.2 The Cauchy problem in one dimension.- Remarks and comments on the literature.- 4. Quasilinear Heat Equations with Absorption. The Critical Exponent.- 4.1 Introduction: Diffusion-absorption with critical exponent.- 4.2 First mass analysis.- 4.3 Sharp lower and upper estimates.- 4.4 ?-limits for the perturbed equation.- 4.5 Extended mass analysis: Uniqueness of stable asymptotics.- 4.6 Equation with gradient-dependent diffusion and absorption.- 4.7 Nonexistence of fundamental solutions.- 4.8 Solutions with L1 data.- 4.9 General nonlinearity.- 4.10 Dipole-like behaviour with critical absorption exponents in a half line and related problems.- Remarks and comments on the literature.- 5. Porous Medium Equation with Critical Strong Absorption.- 5.1 Introduction and results: Strong absorption and finite-time extinction.- 5.2 Universal a priori bounds.- 5.3 Explicit solutions on two-dimensional invariant subspace.- 5.4 L?-estimates on solutions and interfaces.- 5.5 Eventual monotonicity and on the contrary.- 5.6 Compact support.- 5.7 Singular perturbation of first-order equation.- 5.8 Uniform stability for semilinear Hamilton-Jacobi equations.- 5.9 Local extinction property.- 5.10 One-dimensional problem: first estimates.- 5.11 Bernstein estimates for singularly perturbed first-order equations.- 5.12 One-dimensional problem: Application of the S-Theorem.- 5.13 Empty extinction set: A KPP singular perturbation problem.- 5.14 Extinction on a sphere.- Remarks and comments on the literature.- 6. The Fast Diffusion Equation with Critical Exponent.- 6.1 The fast diffusion equation. Critical exponent.- 6.2 Transition between different self-similarities.- 6.3 Asymptotic outer region.- 6.4 Asymptotic inner region.- 6.5 Explicit solutions and eventual monotonicity.- Remarks and comments on the literature.- 7. The Porous Medium Equation in an Exterior Domain.- 7.1 Introduction.- 7.2 Preliminaries.- 7.3 Near-field limit: The inner region.- 7.4 Self-similar solutions.- 7.5 Far-field limit: The outer region.- 7.6 Self-similar solutions in dimension two.- 7.7 Far-field limit in dimension two.- Remarks and comments on the literature.- 8. Blow-up Free-Boundary Patterns for the Navier-Stokes Equations.- 8.1 Free-boundary problem.- 8.2 Preliminaries, local existence.- 8.3 Blow-up: The first, stable monotone pattern.- 8.4 Semiconvexity and first estimates.- 8.5 Rescaled singular perturbation problem.- 8.6 Free-boundary layer.- 8.7 Countable set of nonmonotone blow-up patterns on stable manifolds.- 8.8 Blow-up periodic and globally decaying patterns.- Remarks and comments on the literature.- 9. Equation ut = uxx + u ln2u: Regional Blow-up.- 9.1 Regional blow-up via Hamilton-Jacobi equation.- 9.2 Exact solutions: Periodic global blow-up.- 9.3 Lower and upper bounds: Method of stationary states.- 9.4 Semiconvexity estimate.- 9.5 Lower bound for blow-up set and asymptotic profile.- 9.6 Localization of blow-up.- 9.7 Minimal asymptotic behaviour.- 9.8 Minimal blow-up set.- 9.9 Periodic blow-up solutions.- Remarks and comments on the literature.- 10. Blow-up in Quasilinear Heat Equations Described by Hamilton-Jacobi Equations.- 10.1 General models with blow-up degeneracy.- 10.2 Eventual monotonicity of large solutions.- 10.3 L?-bounds: Method of stationary states.- 10.4 Gradient bound and single-point blow-up.- 10.5 Semiconvexity estimate and global blow-up.- 10.6 Singular perturbation problem.- 10.7 Uniform stability for Hamilton-Jacobi equation. Asymptotic profile.- 10.8 Blow-up final-time profile.- Remarks and comments on the literature.- Remarks and comments on the literature.- 11. A Fully Nonlinear Equation from Detonation Theory.- 11.1 Mathematical formulation of the problem.- 11.2 Outline of results.- 11.3 On local existence, regularity and quenching.- 11.4 Single-point quenching and first sharp estimate.- 11.5 Fundamental estimates: Dynamical system of inequalities.- 11.6 Asymptotic profile near the quenching time.- Remarks and comments on the literature.- 12. Further Applications to Second- and Higher-Order Equations.- 12.1 A homogenization problem for heat equations.- 12.2 Stability of perturbed nonlinear parabolic equations with Sturmian property.- 12.3 Global solutions of a 2mth-order semilinear parabolic equation in the supercritical range.- 12.4 The critical exponent for 2mth-order semilinear parabolic equations with absorption.- 12.5 Regional blow-up for 2mth-order semilinear parabolic equations….- Remarks and comments on the literature.- References.

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