Homogenization of a degenerate parabolic problem in a highly heterogeneous periodic medium

Abstract

We consider the homogenization of a time-dependent heat transfer problem in a highly heterogeneous periodic medium made of two connected components having heat capacities c α ( x ) and heat conductivities a α ( x ), α =1,2 of order one, separated by a third material with thickness of the same order ε than the basic periodicity cell having heat capacity c 3 ( x ) and conductivity λa 3 ( x ) where a 3 is of order one and λ tends to zero with the size ε of the elementary cell. We assume only that c i ( x )⩾0, i =1,2,3 almost everywhere, such that the problem can be degenerate (parabolic–elliptic). We show that the critical value of the problem is δ= lim ε→0 ε 2 λ and identify the homogenized problem depending on δ is zero, strictly positive finite or infinite.