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.