| A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach 2004 Edition Contributor(s): Galaktionov, Victor A. (Author), Vázquez, Juan Luis (Author) |
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ISBN: 0817641467 ISBN-13: 9780817641467 Publisher: Birkhauser OUR PRICE: $52.24 Product Type: Hardcover - Other Formats Published: December 2003 Annotation: 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. |
| Additional Information |
| BISAC Categories: - Mathematics | Differential Equations - Partial - Mathematics | Mathematical Analysis |
| Dewey: 515.353 |
| LCCN: 2003064059 |
| Series: Progress in Nonlinear Differential Equations and Their Applications |
| Physical Information: 0.9" H x 6.4" W x 9.48" (1.55 lbs) 377 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |