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This paper is devoted to the study of the linear parabolic problem $\epsilon {\partial }_t{u}_{\epsilon }(x,t)-\nabla ·\left(a(x/\epsilon ,t/{\epsilon }^3)\nabla u_{\epsilon }(x,t)\right)=f(x,t)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $\epsilon$ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic local problems....

In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^2(0,T;H_0^1\left(\Omega \right))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation ${\epsilon }^p{\partial }_t{u}_{\epsilon }(x,t)-\nabla ·\left(a(x{\epsilon }^{-1},x{\epsilon }^{-2},t{\epsilon }^{-q},t{\epsilon }^{-r})\nabla u_{\epsilon }(x,t)\right)=f(x,t)$, 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...

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 the scale parameter $\epsilon$. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems.