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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.