Homogenized Models with Memory Effect for Heterogeneous Periodic Media

The homogenization of initial boundary value problems for heat conduction equations with asymptotically degenerate rapidly oscillating periodic coefficients are considered. Such problems model thermal processes in heterogeneous periodic media. Homogenized problems (whose solutions determine approximate asymptotics for solutions of the original problems) are presented. Estimates for the accuracy of the asymptotics and relevant convergence theorem are discussed. The homogenized problems have the form of initial boundary value problems for integro-differential equations in convolutions. The presence of convolutions in models for media is called the memory effect. Statements about the solvability and regularity for the problems and the homogenized problems are proved. These results are optimal even in the case of zero convolutions, when the homogenized problems coincide with the classical heat conduction problems