Homogenization of a hyperbolic equation with highly contrasting diffusivity coefficients

Abstract

We study a hyperbolic problem in the framework of periodic homogenization assuming a high contrast between the diffusivity coefficients of the two components $M_{\varepsilon}$ and $ B_{\varepsilon}$ of the heterogeneous medium. There are three regimes depending on the ratio between the size of the period and the amplitude ${\alpha_{\varepsilon}}$ of the diffusivity in $ B_{\varepsilon}$. For the critical regime $ \alpha_{\varepsilon} \simeq {\varepsilon}$, the limit problem is a strongly coupled system involving both the macroscopic and the microscopic variables. We also include the results in the non critical case.