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

• 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).

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

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