A Stability Technique for Evolution Partial Differential Equations : A Dynamical Systems Approach (Progress in Nonlinear Differential Equations and Their Applications Vol.56) (2004. XIX, 377 p.)

A Stability Technique for Evolution Partial Differential Equations : A Dynamical Systems Approach (Progress in Nonlinear Differential Equations and Their Applications Vol.56) (2004. XIX, 377 p.)

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  • 製本 Hardcover:ハードカバー版/ページ数 377 p.
  • 商品コード 9780817641467

Full Description


* 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.

Table of Contents

Introduction: Stability Approach and Nonlinear     xi
Models
The S-Theorem xi
Asymptotics of nonlinear evolution PDEs xii
Description of the applications xiv
Prerequisites and use xix
1 Stability Theorem: A Dynamical Systems 1 (12)
Approach
1.1 Perturbed dynamical systems 1 (1)
1.2 Some concepts from dynamical systems 2 (2)
1.3 The three hypotheses 4 (2)
1.4 The S-Theorem: Stability of omega-limit 6 (3)
sets
1.5 Practical stability assumptions 9 (1)
1.6 A result on attractors 10 (1)
Remarks and comments on the literature 11 (2)
2 Nonlinear Heat Equations: Basic Models and 13 (44)
Mathematical Techniques
2.1 Nonlinear heat equations 13 (3)
2.2 Basic mathematical properties 16 (9)
2.3 Asymptotics 25 (12)
2.4 The Lyapunov method 37 (5)
2.5 Comparison techniques 42 (10)
2.5.1 Intersection comparison and Sturm's 42 (6)
theorems
2.5.2 Shifting comparison principle (SCP) 48 (2)
2.5.3 Other comparisons 50 (2)
Remarks and comments on the literature 52 (5)
3 Equation of Superslow Diffusion 57 (24)
3.1 Asymptotics in a bounded domain 57 (9)
3.2 The Cauchy problem in one dimension 66 (12)
Remarks and comments on the literature 78 (3)
4 Quasilinear Heat Equations with Absorption. 81 (46)
The Critical Exponent
4.1 Introduction: Diffusion-absorption with 81 (5)
critical exponent
4.2 First mass analysis 86 (1)
4.3 Sharp lower and upper estimates 87 (2)
4.4 cv-limits for the perturbed equation 89 (1)
4.5 Extended mass analysis: Uniqueness of 90 (3)
stable asymptotics
4.6 Equation with gradient-dependent 93 (3)
diffusion and absorption
4.7 Nonexistence of fundamental solutions 96 (2)
4.8 Solutions with L 1 data 98 (2)
4.9 General nonlinearity 100(2)
4.10 Dipole-like behaviour with critical 102(21)
absorption exponents in a half line and
related problems
Remarks and comments on the literature 123(4)
5 Porous Medium Equation with Critical Strong 127(42)
Absorption
5.1 Introduction and results: Strong 127(6)
absorption and finite-time extinction
5.2 Universal a priori bounds 133(2)
5.3 Explicit solutions on two-dimensional 135(3)
invariant subspace
5.4 L infinity-estimates on solutions and 138(3)
interfaces
5.5 Eventual monotonicity and on the 141(3)
contrary
5.6 Compact support 144(1)
5.7 Singular perturbation of first-order 144(1)
equation
5.8 Uniform stability for semilinear 145(6)
Hamilton-Jacobi equations
5.9 Local extinction property 151(1)
5.10 One-dimensional problem: first 152(2)
estimates
5.11 Bernstein estimates for singularly 154(4)
perturbed first-order equations
5.12 One-dimensional problem: Application 158(2)
of the S-Theorem
5.13 Empty extinction set: A KPP singular 160(1)
perturbation problem
5.14 Extinction on a sphere 161(4)
Remarks and comments on the literature 165(4)
6 The Fast Diffusion Equation with Critical 169(20)
Exponent
6.1 The fast diffusion equation. Critical 169(2)
exponent
6.2 Transition between different 171(2)
self-similarities
6.3 Asymptotic outer region 173(7)
6.4 Asymptotic inner region 180(4)
6.5 Explicit solutions and eventual 184(2)
monotonicity
Remarks and comments on the literature 186(3)
7 The Porous Medium Equation in an Exterior 189(28)
Domain
7.1 Introduction 189(3)
7.2 Preliminaries 192(1)
7.3 Near-field limit: The inner region 193(2)
7.4 Self-similar solutions 195(7)
7.5 Far-field limit: The outer region 202(4)
7.6 Self-similar solutions in dimension two 206(2)
7.7 Far-field limit in dimension two 208(6)
Remarks and comments on the literature 214(3)
8 Blow-up Free-Boundary Patterns for the 217(20)
Navier-Stokes Equations
8.1 Free-boundary problem 217(3)
8.2 Preliminaries, local existence 220(1)
8.3 Blow-up: The first, stable monotone 221(2)
pattern
8.4 Semiconvexity and first estimates 223(2)
8.5 Rescaled singular perturbation problem 225(6)
8.6 Free-boundary layer 231(1)
8.7 Countable set of nonmonotone blow-up 232(3)
patterns on stable manifolds
8.8 Blow-up periodic and globally decaying 235(1)
patterns
Remarks and comments on the literature 236(1)
9 Equation ut = uxx + u ln2u: Regional Blow-up 237(28)
9.1 Regional blow-up via Hamilton-Jacobi 237(4)
equation
9.2 Exact solutions: Periodic global blow-up 241(2)
9.3 Lower and upper bounds: Method of 243(4)
stationary states
9.4 Semiconvexity estimate 247(1)
9.5 Lower bound for blow-up set and 248(2)
asymptotic profile
9.6 Localization of blow-up 250(3)
9.7 Minimal asymptotic behaviour 253(5)
9.8 Minimal blow-up set 258(2)
9.9 Periodic blow-up solutions 260(2)
Remarks and comments on the literature 262(3)
10 Blow-up in Quasilinear Heat Equations 265(34)
Described by Hamilton-Jacobi Equations
10.1 General models with blow-up degeneracy 265(4)
10.2 Eventual monotonicity of large 269(3)
solutions
10.3 Linfinity-bounds: Method of stationary 272(4)
states
10.4 Gradient bound and single-point blow-up 276(6)
10.5 Semiconvexity estimate and global 282(4)
blow-up
10.6 Singular perturbation problem 286(1)
10.7 Uniform stability for Hamilton-Jacobi 287(8)
equation. Asymptotic profile
10.8 Blow-up final-time profile 295(2)
Remarks and comments on the literature 297(2)
11 A Fully Nonlinear Equation from Detonation 299(28)
Theory
11.1 Mathematical formulation of the problem 299(1)
11.2 Outline of results 300(1)
11.3 On local existence, regularity and 301(5)
quenching
11.4 Single-point quenching and first sharp 306(4)
estimate
11.5 Fundamental estimates: Dynamical 310(6)
system of inequalities
11.6 Asymptotic profile near the quenching 316(6)
time
Remarks and comments on the literature 322(5)
12 Further Applications to second- and 327(32)
Higher-Order Equations
12.1 A homogenization problem for heat 328(6)
equations
12.2 Stability of perturbed nonlinear 334(9)
parabolic equations with Sturmian property
12.3 Global solutions of a 2mth-order 343(4)
semilinear parabolic equation in the
supercritical range
12.4 The critical exponent for 2mth-order 347(2)
semilinear parabolic equations with
absorption
12.5 Regional blow-up for 2mth-order 349(7)
semilinear parabolic equations
Remarks and comments on the literature 356(3)
References 359(16)
Index 375