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Keywords:
homogenization; parabolic problem; multiscale convergence; very weak multiscale convergence; two-scale convergence
homogenization; parabolic problem; multiscale convergence; very weak multiscale convergence; two-scale convergence
Summary:
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 }(x,t) -\nabla \cdot ( a( x\varepsilon ^{-1} ,x\varepsilon ^{-2},t\varepsilon ^{-q},t\varepsilon ^{-r}) \nabla u_{\varepsilon }(x,t) ) = 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 problems.
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