Homogenization of parabolic problems with dynamical boundary conditions of reactive-diffusive type in perforated media

Abstract

This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size $\varepsilon$, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes $$ \nabla u_\varepsilon \cdot \nu+\varepsilon\,\displaystyle\frac{\partial u_\varepsilon}{\partial t}=\varepsilon\,\delta \Delta_{\Gamma}u_\varepsilon-\varepsilon\,g(u_\varepsilon), $$ where $\Delta_{\Gamma}$ denotes the Laplace-Beltrami operator on the surface of the holes, $\nu$ is the outward normal to the boundary, $\delta>0$ plays the role of a surface diffusion coefficient and $g$ is the nonlinear term. We generalize our previous results established in the case of a dynamical boundary condition of pure-reactive type, i.e., with $\delta=0$. We prove the convergence of the homogenization process to a nonlinear reaction-diffusion equation whose diffusion matrix takes into account the reactive-diffusive condition on the surface of the holes.