Bazalii B. V.
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.
Classical Solvability of the First Initial Boundary-Value Problem for a Nonlinear Strongly Degenerate Parabolic Equation
Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1299-1320
We prove the existence of a classical solution global in time for the first initial boundary-value problem for a nonlinear strongly degenerate parabolic equation.
Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1446-1457
We study a nonstationary boundary-value problem for the Laplace equation in a plane angle with time derivative in a boundary condition. We obtain coercive estimates in weighted Hölder spaces.
Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1452–1462
We consider the Stefan problem for a parabolic equation with a small parameter as the coefficient of the derivative with respect to time. We justify the limit transition as the small parameter tends to zero, which enables us to prove the classical solvability of the Hele-Shaw problem with free boundary in the small with respect to time.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1299–1315
We consider a nonstationary problem with free boundary for an elliptic equation in the case where the value of the required function on an unknown boundary is proportional to the curvature of this boundary. We prove the existence of a solution in the small with respect to time in the spaces of smooth functions.
Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1155–1165
We formulate the filtration problem with free boundary as a problem with discontinuous nonlinearity for a degenerate elliptic or parabolic system. We prove that a solution of the Dirichlet problem exists in both cases. We study some qualitative properties of these solutions, e.g., the existence of “dead cores”.
Method for symmetrization and estimation of solutions of the Neumann problem for the equation of a porous medium in domains with noncompact boundary for infinitely increasing time
Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 147–157
We consider the initial boundary-value Neumann problem for the equation of a porous medium in a domain with noncompact boundary. By using a symmetrization method, we obtain exactL p-estimates, 1≤p≤∞, for solutions as t→∞.
Symmetrization and initial boundary-value problems for certain classes of nonlinear second order parabolic equations
Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 884–892
Initial boundary-value problems are considered for equations of porous medium and $p$-Laplace types. By using the Schwarz symmetrization method, the $L_p$-estimates for the solutions of the original problems are obtained in terms of the analogous estimates for the corresponding symmetric solutions.
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.
Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 295-297
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. - 1973. - 25, № 3. - pp. 332—336