Ebook: A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach

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.

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This book introduces a new, state-of-the-art method for the study of the asymptotic behavior of solutions to evolution partial differential equations; much of the text is dedicated to the application of this method to a wide class of nonlinear diffusion equations. The underlying theory hinges on a new stability result, formulated in the abstract setting of infinite-dimensional dynamical systems, which states that under certain hypotheses, the omega-limit set of a perturbed dynamical system is stable under arbitrary asymptotically small perturbations.

The Stability Theorem is examined in detail in the first chapter, followed by a review of basic results and methods---many original to the authors---for the solution of nonlinear diffusion equations. Further chapters provide a self-contained analysis of specific equations, with carefully-constructed theorems, proofs, and references. In addition to the derivation of interesting limiting behaviors, the book features a variety of estimation techniques for solutions of semi- and quasilinear parabolic equations.

Written by established mathematicians at the forefront of the field, this work is a blend of delicate analysis and broad application, appropriate for graduate students and researchers in physics and mathematics who have basic knowledge of PDEs, ordinary differential equations, functional analysis, and some prior acquaintance with evolution equations. It is ideal for a course or seminar in evolution equations and asymptotics, and the book's comprehensive index and bibliography will make it useful as a reference volume as well.

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This book introduces a new, state-of-the-art method for the study of the asymptotic behavior of solutions to evolution partial differential equations; much of the text is dedicated to the application of this method to a wide class of nonlinear diffusion equations. The underlying theory hinges on a new stability result, formulated in the abstract setting of infinite-dimensional dynamical systems, which states that under certain hypotheses, the omega-limit set of a perturbed dynamical system is stable under arbitrary asymptotically small perturbations.

The Stability Theorem is examined in detail in the first chapter, followed by a review of basic results and methods---many original to the authors---for the solution of nonlinear diffusion equations. Further chapters provide a self-contained analysis of specific equations, with carefully-constructed theorems, proofs, and references. In addition to the derivation of interesting limiting behaviors, the book features a variety of estimation techniques for solutions of semi- and quasilinear parabolic equations.

Written by established mathematicians at the forefront of the field, this work is a blend of delicate analysis and broad application, appropriate for graduate students and researchers in physics and mathematics who have basic knowledge of PDEs, ordinary differential equations, functional analysis, and some prior acquaintance with evolution equations. It is ideal for a course or seminar in evolution equations and asymptotics, and the book's comprehensive index and bibliography will make it useful as a reference volume as well.

Content:
Front Matter....Pages i-xix
Stability Theorem: A Dynamical Systems Approach....Pages 1-12
Nonlinear Heat Equations: Basic Models and Mathematical Techniques....Pages 13-55
Equation of Superslow Diffusion....Pages 57-79
Quasilinear Heat Equations with Absorption. The Critical Exponent....Pages 81-125
Porous Medium Equation with Critical Strong Absorption....Pages 127-167
The Fast Diffusion Equation with Critical Exponent....Pages 169-187
The Porous Medium Equation in an Exterior Domain....Pages 189-215
Blow-up Free-Boundary Patterns for the Navier-Stokes Equations....Pages 217-236
Equation ut = uxx + u ln2u: Regional Blow-up....Pages 237-263
Blow-up in Quasilinear Heat Equations Described by Hamilton—Jacobi Equations....Pages 265-298
A Fully Nonlinear Equation from Detonation Theory....Pages 299-325
Further Applications to Second- and Higher-Order Equations....Pages 327-357
Back Matter....Pages 359-377

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This book introduces a new, state-of-the-art method for the study of the asymptotic behavior of solutions to evolution partial differential equations; much of the text is dedicated to the application of this method to a wide class of nonlinear diffusion equations. The underlying theory hinges on a new stability result, formulated in the abstract setting of infinite-dimensional dynamical systems, which states that under certain hypotheses, the omega-limit set of a perturbed dynamical system is stable under arbitrary asymptotically small perturbations.

The Stability Theorem is examined in detail in the first chapter, followed by a review of basic results and methods---many original to the authors---for the solution of nonlinear diffusion equations. Further chapters provide a self-contained analysis of specific equations, with carefully-constructed theorems, proofs, and references. In addition to the derivation of interesting limiting behaviors, the book features a variety of estimation techniques for solutions of semi- and quasilinear parabolic equations.

Written by established mathematicians at the forefront of the field, this work is a blend of delicate analysis and broad application, appropriate for graduate students and researchers in physics and mathematics who have basic knowledge of PDEs, ordinary differential equations, functional analysis, and some prior acquaintance with evolution equations. It is ideal for a course or seminar in evolution equations and asymptotics, and the book's comprehensive index and bibliography will make it useful as a reference volume as well.

Content:
Front Matter....Pages i-xix
Stability Theorem: A Dynamical Systems Approach....Pages 1-12
Nonlinear Heat Equations: Basic Models and Mathematical Techniques....Pages 13-55
Equation of Superslow Diffusion....Pages 57-79
Quasilinear Heat Equations with Absorption. The Critical Exponent....Pages 81-125
Porous Medium Equation with Critical Strong Absorption....Pages 127-167
The Fast Diffusion Equation with Critical Exponent....Pages 169-187
The Porous Medium Equation in an Exterior Domain....Pages 189-215
Blow-up Free-Boundary Patterns for the Navier-Stokes Equations....Pages 217-236
Equation ut = uxx + u ln2u: Regional Blow-up....Pages 237-263
Blow-up in Quasilinear Heat Equations Described by Hamilton—Jacobi Equations....Pages 265-298
A Fully Nonlinear Equation from Detonation Theory....Pages 299-325
Further Applications to Second- and Higher-Order Equations....Pages 327-357
Back Matter....Pages 359-377
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