A Stability Technique for Evolution Partial Differential Equations - Progress in Nonlinear Differential Equations and Their Applications Softcover Reprint of the Original 1st Ed. 2004 edition

Description for Sales People: This book introduces a new, state-of-the-art method for the study of asymptotic behavior of solutions for evolution equations. The underlying theory hinges on a new stability result, which is presented in detail; also included is a review of basic techniques---many original to the authors---for the solution of nonlinear diffusion equations. Subsequent chapters feature a self-contained analysis of specific equations whose solutions depend on the stability theorem; a variety of estimation techniques for solutions of semi- and quasilinear parabolic equations are provided as well. With its carefully-constructed theorems, proofs, and references, the text is appropriate for students and researchers in physics and mathematics who have basic knowledge of PDEs and some prior acquaintance with evolution equations. Written by established mathematicians at the forefront of their field, this blend of delicate analysis and broad application is ideal for a course or seminar in asymptotic analysis and nonlinear partial differential equations. Review Quotes: "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 Mathematiques Pures et AppliqueesReview Quotes:"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 Mathematiques Pures et Appliquees"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."Publisher Marketing: 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."