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 ε, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes

where denotes the Laplace–Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, 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 . 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.