A Stability Technique for Evolution Partial Differential Equations

1. Front Matter

   Pages i-xix

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2. Stability Theorem: A Dynamical Systems Approach

   • Victor A. Galaktionov, Juan Luis Vázquez

   Pages 1-12

3. Nonlinear Heat Equations: Basic Models and Mathematical Techniques

   • Victor A. Galaktionov, Juan Luis Vázquez

   Pages 13-55

4. Equation of Superslow Diffusion

   • Victor A. Galaktionov, Juan Luis Vázquez

   Pages 57-79

5. Quasilinear Heat Equations with Absorption. The Critical Exponent

   • Victor A. Galaktionov, Juan Luis Vázquez

   Pages 81-125

6. Porous Medium Equation with Critical Strong Absorption

   • Victor A. Galaktionov, Juan Luis Vázquez

   Pages 127-167

7. The Fast Diffusion Equation with Critical Exponent

   • Victor A. Galaktionov, Juan Luis Vázquez

   Pages 169-187

8. The Porous Medium Equation in an Exterior Domain

   • Victor A. Galaktionov, Juan Luis Vázquez

   Pages 189-215

9. Blow-up Free-Boundary Patterns for the Navier-Stokes Equations

   • Victor A. Galaktionov, Juan Luis Vázquez

   Pages 217-236

10. Equation ut = uxx + u ln2u: Regional Blow-up

    • Victor A. Galaktionov, Juan Luis Vázquez

    Pages 237-263

11. Blow-up in Quasilinear Heat Equations Described by Hamilton—Jacobi Equations

    • Victor A. Galaktionov, Juan Luis Vázquez

    Pages 265-298

12. A Fully Nonlinear Equation from Detonation Theory

    • Victor A. Galaktionov, Juan Luis Vázquez

    Pages 299-325

13. Further Applications to Second- and Higher-Order Equations

    • Victor A. Galaktionov, Juan Luis Vázquez

    Pages 327-357

14. Back Matter

    Pages 359-377

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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.

• Navier-Stokes equation
• continuum mechanics
• differential equation
• fluid dynamics
• functional analysis
• nonlinear analysis
• ordinary differential equation
• partial differential equation
• pdes
• partial differential equations

"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 Mathématiques Pures et Appliquées

• Department of Mathematical Sciences, University of Bath, Bath, UK

  Victor A. Galaktionov

• Keldysh Institute of Applied Mathematics, Moscow, Russia

  Victor A. Galaktionov

• Department of Mathematics, Universidad Autónoma de Madrid, Madrid, Spain

  Juan Luis Vázquez

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