Asymptotic Analysis in Dynamical Heat Transfer Problems in Heterogeneous Periodic Media

The effective behavior of the solution of a nonlinear dynamical boundary-value problem modeling thermal diffusion in some heterogeneous periodic media is analyzed. We deal, at the microscale, with an -periodic structure, consisting of two parts: a fluid phase and a solid skeleton (reactive obstacles). In such a domain, we consider a heat equation, with nonlinear sources and with a dynamical condition imposed on the heterogeneous boundaries of the reactive obstacles. We are interested in describing the asymptotic behavior, as the small parameter e which characterizes the size of the reactive obstacles tends to zero, of the temperature field in this periodic structure. Using an homogenization procedure, we prove that the effective behavior of the solution of our problem is governed by a new parabolic equation, with extra-terms coming from the influence of the nonhomogeneous dynamical boundary condition.