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

This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size $\epsilon$, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes $$\mathrm{\nabla }{u}_{\epsilon }\cdot \nu +\epsilon {\displaystyle \frac{\mathrm{\partial }{u}_{\epsilon }}{\mathrm{\partial }t}=\epsilon \delta {\mathrm{\Delta }}_{\mathrm{\Gamma }}{u}_{\epsilon }-\epsilon g(u_{\epsilon }),}$$ where ${\mathrm{\Delta }}_{\mathrm{\Gamma }}$ 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.