A Two-Phase Free Boundary Problem for the Nonlinear Heat Equation

Free Boundary Problems (FBP) motivated several studies in the past due to their relevance in applications [1 − 4]. From the mathematical point of view FBP are initial/ boundary value problems with a moving boundary [5]. The motion of the boundary is unknown (free boundary) and has to be determined together with the solution of the given partial differential equation. As a consequence the solution of FBP is in most cases equivalent to the solution of a nonlinear system. In recent studies [6 − 13] some free boundary problems for nonlinear evolution equations relevant in applications have been considered. In particular in [12] the solution of a one-phase Stefan Problem in nonlinear conduction is proved to exist and to be unique for short intervals of time. Furthemore a particular travelling wave solution was obtained. On the other hand two-phase Stefan Problems are more complicated than their one-phase counterparts and the theory is more elaborate. It is the aim of this paper to analyse a two–phase Stefan Problem for the nonlinear heat equation considered in [12]. Such an equation arises as a model of heat conduction in solid crystalline hydrogen [14]. It admits an exact linearization into the heat equation and therefore belongs to the class of C-integrable equations [16]. In the following we show that the two-phase Stefan Problem for the nonlinear heat equation admits a linearization into two distinct one-phase moving boundary problems for the linear heat equation. The two linearized problems are connected through a constraint on the relative motion of their moving boundaries. Such a constraint is induced by the free boundary motion of the nonlinear problem. We start our analysis with the following system of nonlinear heat equations