Electron. J. Diff. Eqns., Vol. 2004(2004), No. 55, pp. 1-19.

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

Marcelo M. Cavalcanti, Valeria N. Domingos Cavalcanti, Juan A. Soriano, Joel S. Souza

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.

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Marcelo M. Cavalcanti
Universidade Estadual de Maringa
87020-900 Maringa - PR, Brasil
email: mmcavalcanti@uem.br
Valeria N. Domingos Cavalcanti
Universidade Estadual de Maringa
87020-900 Maringa - PR, Brasil
email: vndcavalcanti@uem.br
Juan Amadeo Soraino
Universidade Estadual de Maringa
87020-900 Maringa - PR, Brasil
email: jaspalomino@uem.br
  Joel S. Souza
Departamento de Matematica
Universidade Federal de Santa Catarina
80040-900 Florianopolis - SC, Brasil
email: cido@dme.ufpb.br

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