Inverse problem for the strongly degenerate heat equation in a domain with free boundary

Abstract

In a domain with free boundary, we establish conditions for the existence and uniqueness of a solution of the inverse problem of finding the time-dependent coefficient of heat conductivity. We study the case of strong degeneration where the unknown coefficient tends to zero as t → +0 as a power function t β , where β ≥ 1. Problems for degenerate parabolic equations, free-boundary problems, and inverse problems arise in the course of investigation of many processes important from the practical point of view. Each of these types of problems is fairly well investigated, but their combination within a single problem, in fact, has not been studied. The inverse problem for the one-dimensional heat equation in a domain with free boundary was investigated in [1]. Conditions for the existence and uniqueness of a solution of the inverse problem for a parabolic equation with strong power degeneration were established in [2]. The inverse problems for hyperbolic and elliptic equations with degeneration were studied in [3] and [4], respectively. The free-boundary problem for a degenerate parabolic system was investigated in [5]. In the present paper, we study the one-dimensional inverse problem for the heat equation with strong power degeneration in the case where a part of the boundary is unknown. An analogous problem for the case of weak degeneration was considered in [6].