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In this paper we establish compactness results of multiscale and very weakmultiscale type for sequences bounded in $L^{2}(0,T;H_{0}^{1}(\Omega ))$,fulfilling a certain condition. We apply the results in the homogenization ofthe parabolic partial differential equation $\varepsilon^{p}\partial_{t}u_{\varepsilon }\left( x,t\right) -\nabla \cdot \left( a\left(x/\varepsilon ,x/\varepsilon ^{2},t/\varepsilon^{q},t/\varepsilon ^{r}\right)\nabla u_{\varepsilon }\left( x,t\right)\right) = f\left( x,t\right) $, where$0<p<q<r$. The homogenization result reveals two special phenomena, namely thatthe homogenized problem is elliptic and that the matching for when the localproblem is parabolic is shifted by $p$, compared to the standard matching thatgives rise to local parabolic problems.

Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales