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
* Introduces a state-of-the-art method for the study of the asymptotic behavior of solutions to evolution partial differential 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 PDEs.
* Well-organized text with detailed index and bibliography, suitable as a course text or reference volume.
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.