A nonlocal memory strange term arising in the critical scale homogenization of diffusion equations with dynamic boundary conditions

Our main interest in this article is the study of homogenized limit of a parabolic equation with a nonlinear dynamic boundary condition of the micro-scale model set on a domain with periodically place particles. We focus on the case of particles (or holes) of critical diameter with respect to the period of the structure. Our main result proves the weak convergence of the sequence of solutions of the original problem to the solution of a reaction-diffusion parabolic problem containing a "strange term". The novelty of our result is that this term is a nonlocal memory solving an ODE. We prove that the resulting system satisfies a comparison principle