Mathematica Bohemica, Vol. 146, No. 4, pp. 483-511, 2021
Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales
Tatiana Danielsson, Pernilla Johnsen
Received June 13, 2019. Published online March 3, 2021.
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_tu_{\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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Affiliations: Tatiana Danielsson, Pernilla Johnsen, Department of Mathematics and Science Education, Mid Sweden University, Kunskapens väg 8, S-83125 Östersund, Sweden, e-mail: tatiana.danielsson@miun.se, pernilla.johnsen@miun.se
