Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 55, pp. 1-19.
Title: Homogenization and uniform stabilization for a nonlinear
hyperbolic equation in domains with holes of small capacity
Authors: Marcelo M. Cavalcanti (Univ. Estadual de Maringa, Brasil)
Valeria N. Domingos Cavalcanti (Univ. Estadual de Maringa, Brasil)
Juan A. Soriano (Univ. Estadual de Maringa, Brasil)
Joel S. Souza (Univ. Federal de Santa Catarina, Brasil)
Abstract:
In this article we study the homogenization and
uniform decay of the nonlinear hyperbolic equation
$$
\partial_{tt} u_{\varepsilon} -\Delta u_{\varepsilon}
+F(x,t,\partial_t u_{\varepsilon},\nabla u_{\varepsilon})=0
\quad\hbox{in }\Omega_{\varepsilon}\times(0,+\infty)
$$
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].
Submitted September 23, 2003. Published April 9, 2004.
Math Subject Classifications: 35B27, 35B40, 35L05
Key Words: Homogenization; asymptotic stability; wave equation.