On self-similar solutions of a multi-phase Stefan problem

Abstract

We study self-similar solutions of a multi-phase Stefan problem, first in the case of one space variable, and then in the radial multidimensional case. In both these cases, we prove that a nonlinear algebraic system for determination of the free boundaries is gradient one and the corresponding potential is an explicitly written coercive function. Therefore, there exists a minimum point of the potential, coordinates of this point determine free boundaries and provide the desired solution. Moreover, in the one-dimensional case the potential is proved to be strictly convex and this implies the uniqueness of the solution. On the contrary, in the multidimensional case the potential is not convex but the uniqueness of our solution remains true, it follows from the general theory.