Homogenization and uniform stabilization for a nonlinear hyperbolic equation in domains with holes of small capacity

Abstract

In this article we study the homogenization and uniform decay of the nonlinear hyperbolic equation $$ partial_{tt} u_{varepsilon} -Delta u_{varepsilon} +F(x,t,partial_t u_{varepsilon}, abla u_{varepsilon})=0 quadhbox{in }Omega_{varepsilon}imes(0,+infty) $$ where $Omega_{varepsilon}$ 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 [8] for the study of homogenization of elliptic problems. Moreover, uniform decay rates are obtained by considering the perturbed energy method developed by Haraux and Zuazua [10].

Keywords

• Homogenization
• asymptotic stability
• wave equation.