CORRECTOR RESULTS FOR A PARABOLIC PROBLEM WITH A MEMORY EFFECT

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 epsilon-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 epsilon(gamma.) We suppose that -1 < gamma <= 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < gamma < 1 and gamma = 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 Monsurro, 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 gamma = 1 is more interesting because of the presence of a memory effect in the homogenized problem.