Homogenization of Evolution Problems in a Fiber Reinforced Structure

We study the homogenization of parabolic or hyperbolic equations like $$\rho_\epsilon(x){\partial^n u_\epsilon \over \partial t^n}- div(a_\epsilon(x) \nabla u_\epsilon) =f$$ on $\Omega\times (0, T)$ plus boundary conditions, $n \in \{1,2\}$, where the coefficients $a_\epsilon$ and $\rho_\epsilon$ takes values of very different order on an $\epsilon$-periodic subset $T_\epsilon \subset \Omega$ (fibered structure) and elsewhere. We find a non local effective equation deduced from a homogenized system of several equations.