Investigation of the Solvability in a Weighted Hölder Spaces of the Linearized Problem

Abstract

In the study, the linear problem for the second order general parabolic equations in the bounded domain is considered. This problem arises in the linearization of multidimensional two-phase problem with the condition on the free boundary in which the unknown temperature at the free surface depending on the curvature and the velocity of the free surface. The proof of the existence and uniqueness of solutions of the linear problem and derivation of estimate of its solution is realized with Schauder method and construction of regularizer. The unique solvability of the linear problem in weighted Hölder spaces is proved, the coercive estimates of the solutions is established.Key words: Second order parabolic equation, weighted hölder space, existence, uniqueness of the solutions, coercive estimate INTRODUCTIONBazaliy and Degtyarev (1992) investigated the three-dimensional Stefan problem for the heat equation with the condition u 1 = u 2 = αk-βV v on the free boundary, where k-the curvature of free boundary and V v -velocity of free boundary on the direction of the normal ν. Radkevich (1992) studied such a problem for a divergent parabolic equation with conormal derivative in the boundary condition for the space of dimensions n = 2, 3. In these works in the classical Hölder space were considered the problems in domains of dimension 2 and 3 at overstated smoothness of the given surface and the initial data.The weighted Hölder spaces , introduced by Belonosov and Zelenyak (1975), permits