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Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.ORCID iD: 0000-0003-2318-1716
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
2018 (English)In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 63, no 5, p. 503-521Article in journal (Refereed) Published
##### Abstract [en]

This paper is devoted to the study of the linear parabolic problem $\varepsilon\partial_tu_{\varepsilon}\left(x,t\right)-\nabla\cdot\left(a\left(x/\varepsilon,t/\varepsilon^3\right)\nabla u_{\varepsilon}\left(x,t\right)\right)=f\left(x,t\right)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $\varepsilon$ in front of the timederivative. First, we have an elliptic homogenized problem although the problem studiedis parabolic. Secondly, we get a parabolic local problem even though the problem has adifferent 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 $\left{u_{\varepsilon}\right}$ different from the standard setting are used, which means that these results are also of independent interest.

##### Place, publisher, year, edition, pages
2018. Vol. 63, no 5, p. 503-521
##### Keywords [en]
homogenization, parabolic problem, multiscale convergence, very weak multiscale convergence, two-scale convergence
Mathematics
##### Identifiers
ISI: 000448719900002Scopus ID: 2-s2.0-85055682025OAI: oai:DiVA.org:miun-35061DiVA, id: diva2:1268383
Available from: 2018-12-05 Created: 2018-12-05 Last updated: 2019-03-25Bibliographically approved

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Publisher's full textScopushttps://articles.math.cas.cz/?type=A&v=63&n=5

#### Authority records BETA

Johnsen, PernillaLobkova, Tatiana

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Department of Mathematics and Science Education
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Applications of Mathematics
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Cite
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