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

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December 2006, 16(4): 721-743. doi: 10.3934/dcds.2006.16.721

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

M. M. Cavalcanti ^1, , V.N. Domingos Cavalcanti ^2, , D. Andrade ^1, and T. F. Ma ^1,

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

Received January 2005 Revised April 2006 Published September 2006

Abstract
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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: nonlinear wave equation., Homogenization.

Mathematics Subject Classification: Primary: 35B27; Secondary: 35B4.

Citation: M. M. Cavalcanti, V.N. Domingos Cavalcanti, D. Andrade, T. F. Ma. Homogenization for a nonlinear wave equation in domains with holes of small capacity. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 721-743. doi: 10.3934/dcds.2006.16.721

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