Homogenization of a non-linear 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 comparable heat capacities and conductivities, separated by a third material with thickness of the same order ε as the basic periodicity cell but having a much lower conductivity such that the resulting interstitial heat flow is scaled by a factor λ tending to zero with a rate λ=λ(ε). The heat flux vectors aj, j=1,2,3 are non-linear, monotone functions of the temperature gradient. The heat capacities cj(x) are positive, but may vanish at some subsets such that the problem can be degenerate (parabolic–elliptic). We show that the critical value of the problem is δ=limε→0εp/λ and identify the homogenized problem depending on whether δ is zero, strictly positive finite or infinite. Copyright © 2005 John Wiley & Sons, Ltd.