Homogenization for a nonlinear wave equation in domains with holes of small capacity
Marcelo M. Cavalcanti, V. N. Domingos Cavalcanti, D. Andrade, To Fu Ma · Discrete and Continuous Dynamical Systems · 2006
This paper is concerned with the homogenization of the nonlinear waveequation $\partial_{t t} u_{\varepsilon} - \Delta u_{\varepsilon} + \partial_t F(u_{\varepsilon}) = 0$ in $\Omega_{\varepsilon}\times(0,+\infty),$where $\Omega_{\varepsilon}$ is a domain containing holes withsmall 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 frameworkproposed by Cioranescu and Murat (1982), for thehomogenization of elliptic problems, a new approach ispresented to solve the above nonlinear homogenization problem.In the linear case, one improves early classical resultsby Cionarescu, Donato, Murat and Zuazua (1991).