Goncharenko M. V.
Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1201-1216
We consider a boundary-value problem used to describe the process of stationary diffusion in a porous medium with nonlinear absorption on the boundary. We study the asymptotic behavior of the solution when the medium becomes more and more porous and denser located in a bounded domain $Q$. A homogenized equation for the description of the main term of the asymptotic expansion is constructed.
Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1443-1459
We consider a homogenized system of equations that is a macroscopic model of nonstationary vibrations of an elastic medium with a large number of small cavities filled with viscous incompressible liquid (wet elastic medium). It is proved that the solution of the initial boundary-value problem for this system in a bounded domain $\Omega$ tends to zero in the metric of $L_2(\Omega)$ exponentially with time.
Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1309–1329
We consider an initial boundary-value problem used to describe the nonstationary vibration of an elastic medium with large number of small cavities filled with a viscous incompressible fluid. We study the asymptotic behavior of the solution in the case where the diameters of the cavities tend to zero, their number tends to infinity, and the cavities occupy a three-dimensional region. We construct an averaged equation to describe the leading term of the asymptotics. This equation serves as a model of propagation of waves in various media, such as damped soil, rocks, and some biological tissues.