Homogenization for a nonlinear wave equation in domains with holes of small capacity

M. M. Cavalcanti; V.N. Domingos Cavalcanti; D. Andrade; T. F. Ma

doi:10.3934/dcds.2006.16.721

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2006, Volume 16, Issue 4: 721-743. Doi: 10.3934/dcds.2006.16.721

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Homogenization for a nonlinear wave equation in domains with holes of small capacity

1.
Department of Mathematics - State University of Maringá, 87020-900 Maringá, PR
2.
Department of Mathematics - State University of Maringá, Avenida Colombo, 5790, 87020-900 Maringá, PR

Received: December 31, 2004

Revised: March 31, 2006

Published: September 2006

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Abstract

This paper is concerned with the homogenization of the nonlinear wave equation

$\partial_{t t} u_{\varepsilon} - \Delta u_{\varepsilon} + \partial_t F(u_{\varepsilon}) = 0$ in $\Omega_{\varepsilon}\times(0,+\infty),$

where $\Omega_{\varepsilon}$ is a domain containing holes with small capacity. In the context of optimal control, this semilinear hyperbolic equation was studied by Lions (1980) through a theory of ultra-weak solutions. Combining his arguments with the abstract framework proposed by Cioranescu and Murat (1982), for the homogenization of elliptic problems, a new approach is presented to solve the above nonlinear homogenization problem. In the linear case, one improves early classical results by Cionarescu, Donato, Murat and Zuazua (1991).

Keywords:

Homogenization,
nonlinear wave equation.

Mathematics Subject Classification: Primary: 35B27; Secondary: 35B40.

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