Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales
Pernilla Johnsen, Tatiana Lobkova · Applications of Mathematics · 2018
summary:This paper is devoted to the study of the linear parabolic problem $\varepsilon \partial _{t}u_{\varepsilon }( x,t) - abla \cdot ( a( {x}/{\varepsilon },{t}/{\varepsilon ^{3}}) abla u_{\varepsilon }( x,t)) =f( x,t) $ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $\varepsilon $ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence $\{ u_{\varepsilon }\} $ different from the standard setting are used, which means that these results are also of independent interest.