We consider an initial boundary-value problem for a parabolic equation describing non-stationary diffusion in porous media with non-linear absorption on the boundary and the transfer of the diffusing substance by fluid. We prove the existence of the unique solution for this problem. We study the asymptotic behavior of a sequence of solutions when the scale of microstructure tends to zero and obtain the homogenized model of the diffusion process.

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We consider a boundary-value problem for the Poisson equation in a strongly perforated domain Ω ε = Ω \ F ε ⊂ R n (n 2) with non-linear Robin's condition on the boundary of the perforating set F ε . The domain Ω ε depends on the small parameter ε > 0 such that the set F ε becomes more and more loosened and distributes more densely in the domain Ω as ε → 0. We study the asymptotic behavior of the solution u ε (x) of the problem as ε → 0. A homogenized equation for the main term u(x) of the asymptotics of u ε (x) is constructed and the integral conditions for the convergence of u ε (x) to u(x) are formulated.