Mid Sweden University

In this paper we prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of epsilon. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems. At the end of the paper an example is given to concretize the use of the main result.

We study the homogenization of a certain linear hyperbolic-parabolic problem exhibiting two rapid spatial scales {ε; ε²}. The homogenization is performed by means of evolution multiscale convergence, a generalization of the concept of two-scale convergence to include any number of scales in both space and time. In particular we apply a compactness result for gradients. The outcome of the homogenization procedure is that we obtain a homogenized problem of hyperbolic-parabolic type together with two elliptic local problems, one for each rapid scale, for the correctors.

We study the homogenization of a hyperbolic-parabolic PDE with oscillations in one fast spatial scale. Moreover, the first order time derivative has a degenerate coefficient passing to infinity when ϵ→0. We obtain a local problem which is of elliptic type, while the homogenized problem is also in some sense an elliptic problem but with the limit for ϵ-1∂tuϵ as an undetermined extra source term in the right-hand side. The results are somewhat surprising and work remains to obtain a fully rigorous treatment. Hence the last section is devoted to a discussion of the reasonability of our conjecture including numerical experiments.

In this article, we study homogenization of a parabolic linear problem governed by a coefficient matrix with rapid spatial and temporal oscillations in periodically perforated domains with homogeneous Neumann data on the boundary of the holes. We prove results adapted to the problem for a characterization of multiscale limits for gradients and very weak multiscale convergence.