Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder’s estimates for a degenerate parabolic problem with dynamic boundary conditions

Abstract

We consider multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type in classes of smooth functions. First we find a natural Hölder class for the Dirichlet boundary conditions in the initial boundary boundary problem for a degenerate parabolic equation of second order. This class then is used to obtain the Schauder estimates for a degenerate parabolic equation with dynamic boundary conditions. As a result we prove the existence locally in time of a smooth solution for Stefan problem for degenerate parabolic equations.