Homogenization of a degenerate linear parabolic problem in a highly heterogeneous thin structure

We consider the homogenization of a time-dependent heat transfer problem in a thin domain of thickness e   in RNRN whose heat capacity and conductivity vary periodically in the cross-section with periodicity cell of size ϵ>0ϵ>0. The structure is highly heterogeneous so that the conductivity is of order 1 in two connected parts of the basic cell, separated by a third one having a much lower conductivity of the scale λλ. We assume that e   (resp. λλ) tends to zero with a rate e=e(ϵ)e=e(ϵ) (resp. λ=λ(ϵ)λ=λ(ϵ)). The heat capacities cjcj of the three components are positive, but may vanish at some subsets such that the problem can be degenerate (parabolic–elliptic). We show that the critical values of the problem are ρ=limϵ↓0eϵ and δ=limϵ↓0ϵ2λ, and we identify the homogenized problem depending on whether ρρ and δδ are zero, strictly positive finite or infinite.