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The main focus in this paper is on homogenization of the parabolic problem ${\partial }_t{u}^{\epsilon }-\nabla ·\left(a(x/\epsilon ,t/\epsilon ,t/{\epsilon }^r)\nabla u^{\epsilon }\right)=f$. Under certain assumptions on $a$, there exists a $G$-limit $b$, which we characterize by means of multiscale techniques for $r>0$, $r\ne 1$. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.

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.