Mid Sweden University

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.

This paper is devoted to the study of the linear parabolic problem  by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient  in front of the timederivative. First, we have an elliptic homogenized problem although the problem studiedis parabolic. Secondly, we get a parabolic local problem even though the problem has adifferent relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence  different from the standard setting are used, which means that these results are also of independent interest.

In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in L2 (0, T; H10 (Ω)), fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation εp ∂t uε(x, t) − ∇ · (a(xε−1, xε−2, tε−q, tε−r)∇uε(x, t)) = f(x, t), where 0 < p < q < r. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by p, compared to the standard matching that gives rise to local parabolic problems.

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.