On Self-Similar Solutions of a Multi-Phase Stefan Problem in a Moving Ray

We study self-similar solutions of a multi-phase Stefan problem for a heat equation on the moving ray $x>\alpha\sqrt{t}$ with Dirichlet or Neumann boundary conditions at the boundary $x=\alpha\sqrt{t}$. In the case of Dirichlet condition we prove that an algebraic system for determination of the free boundaries is gradient one and the corresponding potential is an explicitly written strictly convex and coercive function. Therefore, there exists a unique minimum point of the potential, which determines free boundaries and provides the solution. In the case of Neumann condition solutions with different numbers (called types) of phase transitions appear. For each fixed type the system for determination of the free boundaries is again gradient with a strictly convex potential. This allows to find precise conditions for existence and uniqueness of a solution. In the last section we study Stefan--Dirichlet problem on the half-line $x>0$ with infinitely many phase transitions. Using again a variational approach, we find sufficient conditions of existence and uniqueness of a solution to the problem under consideration.