Homogenization and uniform stabilization of the wave equation in perforated domains

In this article we study the homogenization and uniform decay rates estimates of the energy associated to the damped nonlinear wave equation∂ttuε−Δuε+f(uε)+a(x)g(∂tuε)=0in Ωε×(0,∞) where Ωε is a domain containing holes with small capacity (i.e. the holes are smaller than a critical size). The homoge...

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Bibliographic Details
Published in:Journal of Differential Equations 2024-09, Vol.402, p.218-249
Main Authors: Cavalcanti, Marcelo M., Domingos Cavalcanti, Valéria N., Vicente, André
Format: Article
Language:English
Subjects:
Homogenization
Uniform decay
Wave equation
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Summary:In this article we study the homogenization and uniform decay rates estimates of the energy associated to the damped nonlinear wave equation∂ttuε−Δuε+f(uε)+a(x)g(∂tuε)=0in Ωε×(0,∞) where Ωε is a domain containing holes with small capacity (i.e. the holes are smaller than a critical size). The homogenization's proofs are based on the abstract framework introduced by Cioranescu and Murat [14] for the study of homogenization of elliptic problems. The main goal of this article is to prove, in one shot, uniform decay rate estimates of the energy associated to solutions of the problem posed in the perforated domain Ωε as well as for the limit case Ω when ε→0 by using refined arguments of microlocal analysis.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2024.04.035