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Corrector results for a parabolic problem with a memory effect

Published online by Cambridge University Press:  04 February 2010

Patrizia Donato  and
Editha C. Jose
Patrizia Donato
Affiliation:
Université de Rouen, Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Avenue de l'Université, BP 12, 76801 St Étienne du Rouvray, France. Patrizia.Donato@univ-rouen.fr
Editha C. Jose
Affiliation:
Math Division, Institute of Mathematical Sciences and Physics, College of Arts and Sciences, University of the Philippines Los Baños, Los Baños, Laguna, 4031, Philippines. ecjose@uplb.edu.ph
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Abstract

The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of order εγ. We suppose that -1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. As seen in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] and [Faella and Monsurrò, Topics on Mathematics for Smart Systems, World Sci. Publ., Hackensack, USA (2007) 107–121] (also in [Donato et al., J. Math. Pures Appl.87 (2007) 119–143]), the case γ = 1 is more interesting because of the presence of a memory effect in the homogenized problem.

Keywords

Periodic homogenizationcorrectorsheat equationinterface problems
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 3 , May 2010 , pp. 421 - 454
Copyright
© EDP Sciences, SMAI, 2010

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