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

Abstract: In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^{2}(0,T;H_{0}^{1}(\Omega ))$, fulfilling a certain condition. We apply the results in the homogenization of the 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 that the homogenized problem is elliptic and that the matching for when the local problem is parabolic is shifted by $p$, compared to the standard matching that gives rise to local parabolic problems.