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April  2000, 6(2): 393-418. doi: 10.3934/dcds.2000.6.393

Uniform inertial sets for damped wave equations

P. Fabrie 1, , C. Galusinski 2, and  A. Miranville 3,

1. 

Université de Bordeaux I, Laboratoire de Mathématiques Appliquées de Bordeaux, 351 cours de la libération, 33400 Talence, France
2. 

Université de Bordeaux I, Laboratoire de Mathématiques Appliquées de Bordeaux, 351 Cours de la Libération, 33405 Talence Cedex, France
3. 

Université de Poitiers, Département de Mathématiques, 40 Avenue du Recteur Pineau, 86022 Poitiers Cedex, France

Received  March 1999 Revised  June 1999 Published  January 2000

In this paper, we establish the existence of inertial sets for a class of wave equations in which the coefficient of the second order time derivative is $\varepsilon$. We show that the fractal dimension of these inertial sets does not depend on $\varepsilon$ for $\varepsilon$ small enough. We then compare the asymptotic behavior of the problem (as $\varepsilon\to 0$) through a continuity like property of the inertial sets. The autonomous case and nonautonomous case are studied.
Keywords: Exponential attractors, wave equations, nonautonomous system..
Mathematics Subject Classification: 35B40, 35B45, 35L0.
Citation: P. Fabrie, C. Galusinski, A. Miranville. Uniform inertial sets for damped wave equations. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 393-418. doi: 10.3934/dcds.2000.6.393
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