Homogenization of degenerate parabolic equations with nonlinear Robin reactions in perforated domains

Abstract

We study periodic homogenization of a parabolic equation with linear, degenerate bulk terms in a perforated domain, with a monotone nonlinear Robin reaction of order $\epsilon$ on the pore walls. The storage coefficient may vanish on sets of positive measure, and the diagonal diffusion weights are not bounded below but satisfy an inverse-integrability condition. Under positivity of the mean storage, a quantitative cell-detection estimate yields strong compactness of the states and of the scaled traces. The effective equation has positive storage, a positive-definite diffusion matrix given by a weighted cell problem, and a nonlinear volume reaction obtained by averaging over the pore surface. We also prove a strong weighted-gradient corrector and uniform-in-time convergence of the dissipated and stored energies and of the states in the storage-weighted $L^2$ norm.