Degtyarev S. P.
On the instantaneous shrinking of the support of a solution to the Cauchy problem for an anisotropic parabolic equation
Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 625-640
We study the phenomenon of instantaneous shrinking of the support of solution to the Cauchy problem for the parabolic equation with anisotropic degeneration, double nonlinearity, and strong absorption. In terms of the behavior of locally integrable initial data, we formulate necessary and sufficient conditions for the realization of instantaneous shrinking and establish the exact (in order) bilateral estimates for the size of the support of solution.
On one boundary-value problem for a strongly degenerate second-order elliptic equation in an angular domain
Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 867–883
We prove the existence and uniqueness of a classical solution of a singular elliptic boundary-value problem in an angular domain. We construct the corresponding Green function and obtain coercive estimates for the solution in the weighted Hölder classes.
Bilateral estimates for the support of a solution of the Cauchy problem for an anisotropic quasilinear degenerate equation
Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1477–1486
We establish exact-order bilateral estimates for the size of the support of a solution of the Cauchy problem for a doubly nonlinear parabolic equation with anisotropic degeneration in the case where the initial data are finite and have finite mass.
Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem
Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1035–1044
We study the phenomenon of instantaneous compactification and the initial behavior of the support of solution of the filtration equation for inhomogeneous porous media.
Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 155–166
The Stefan problem is considered with the kinetic condition u+=u−=ɛk(y, τ)-ɛv at the phase interface, where k(y, τ) is the half-sum of the principal curvatures of the free boundary and v is the speed of its shifting in the direction of a normal. The solvability of a modified Stefan problem in spaces of smooth functions and the convergence of its solutions as ɛ → 0 to a solution of the classical Stefan problem are proved.
Solvability of a problem with an unknown boundary between the domains of a parabolic and an elliptic equations
Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1343–1349
Ukr. Mat. Zh. - 1989. - 41, № 9. - pp. 1192–1198