Uniform exponential attractors for a singularly perturbed damped wave equation

Pages: 211 - 238, Volume 10, Issue 1/2, January/February 2004

doi:10.3934/dcds.2004.10.211       Abstract        Full Text (288.3K)       Related Articles

Pierre Fabrie - Université Bordeaux-I, Mathématiques Appliquées, 351 Cours de la Libération, 33405 Talence Cedex, France (email)
Cedric Galusinski - Université Bordeaux-I, Mathématiques Appliquées, 351 Cours de la Libération, 33405 Talence Cedex, France (email)
A. Miranville - Laboratoire d'Applications des Mathématiques - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, Chasseneuil Futuroscope Cedex, France (email)
Sergey Zelik - Université de Poitiers, Laboratoire d'Applications des Mathématiques - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex, France (email)

Abstract: Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.

Keywords:  Singularly perturbed damped wave equations, reaction-diffusion equations,uniform exponential attractors, time boundary layer.
Mathematics Subject Classification:  35B40, 35B45, 35L30.

Received: November 2001;      Revised: March 2003;      Published: October 2003.