Properties of Solutions of the Inverse Stefan Problem

Abstract

Inverse problems for parabolic equations in domains with free boundaries (inverse Stefan problems) arise in the modeling and control of processes with phase transitions in thermal physics and continuum mechanics. Unlike inverse problems in domains with given boundaries, these problems are not sufficiently well studied; this is especially the case for quasilinear equations with unknown dependence of the phase transition front on time. The present paper continues the investigation of inverse Stefan problems in [1–3]. We consider the inverse problem of finding an unknown boundary mode for a two-phase quasilinear Stefan problem with additional information given on another boundary of the domain. We suggest a new approach based on the reduction of the original problem to a sequence of auxiliary inverse boundary value problems with smooth coefficients without phase transitions. We introduce the notion of a generalized exact solution of an inverse Stefan problem and establish its relationship with the exact solutions of auxiliary inverse problems in Holder classes. We prove the uniqueness of such a generalized solution as well as of the exact solution of the inverse Stefan problem in Holder classes without any restrictions on the number of points of intersection of the phase fronts corresponding to distinct boundary modes. 1. We shall deal with one of the widely used versions of the direct statement of the Stefan problem. Consider the two-phase quasilinear problem of finding a function u(x, t) in the domain Q = Q1 ∪ Q2 and a phase front ξ(t) for 0 ≤ t ≤ T from the conditions