Abstract:
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.
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