Multiphase homogenized diffusion models for problems with several parameters

G. V. Sandrakov

Izvestiya. Mathematics , Tome 71 (2007) no. 6, pp. 1193-1252

Résumé

We deal with the homogenization of initial-boundary-value problems for parabolic equations with asymptotically degenerate rapidly oscillating periodic coefficients, which are models for diffusion processes in a strongly inhomogeneous medium. The solutions of these problems depend on a finite positive parameter and two small positive parameters. We obtain homogenized initial-boundary-value problems (whose solutions determine approximate asymptotics for solutions of the problems under consideration) and prove estimates for the accuracy of these approximations. The homogenized problems are initial-boundary-value problems for integro-differential equations whose solutions depend on additional positive parameters: the intensity of diffusion exchange and the impulse exchange. In the general case, the homogenized equations form a system of equations coupled through the exchange coefficients and define multiphase mathematical models of diffusion for a homogenized (limiting) medium. We consider the spectral properties of some homogenized problems. We also prove assertions on asymptotic reductions of the homogenized problems under additional hypothesis on the limiting behaviour of the exchange parameters.