A problem with free boundary for a two-dimensional parabolic equation

Abstract

UDC 517.95 We establish conditions for the existence and uniqueness of a smooth solution of a problem with free boundary for a two-dimensional parabolic equation in a curvilinear rectangle, for which the position of the curvilinear part is determined by a function that is a product of an unknown time-dependent function and the given function of the spatial variable. Inverse problems and problems with unknown (or free) boundaries arise in a number of important technological processes related to the phase transitions in bodies and to the determination of new formations in bodies with known properties [13–16]. In the one-dimensional case, the motion of the unknown boundary is described by an unknown time-dependent function. Such problems were quite completely investigated in the case of linear and quasilinear equations of the parabolic type [5–8, 12]. The results obtained for them were extended to the two-dimensional case where the domain is a curvilinear rectangle, whose position is determined by an unknown time-dependent function [1–4]. In this case, the task consisted of the determination of the unknown boundary and the qualitative properties of new formations. Since the most important question in problems of such type is frequently the dynamics of the process, i.e., the temporal variation of the location of the unknown boundary, this allows one to model the motion of the unknown boundary by assuming that the change of the location of every point of the boundary is determined by the same time-dependent unknown function. In the present work, we used such an approach to solve a problem with free boundary for the two-dimensional parabolic equation.