On self-similar solutions of a multiphase Stefan problem on the half-line

We study self-similar solutions of a multi-phase Stefan problem for a heat equation on the half-line $x>0$ with a constant initial data and with Dirichlet or Neumann boundary conditions.
In the case of Dirichlet boundary condition we prove that a nonlinear 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, coordinates of this point determine free boundaries and provide the desired solution.
In the case of Neumann boundary condition we demonstrate that the problem can have solutions with different numbers (called types) of phase transitions. For each fixed type $n$ the system for determination of the free boundaries is again gradient and the corresponding potential is proved to be strictly convex and coercive, but in some wider non-physical domain. On the base of these properties we prove existence and uniqueness of a solution and provide precise conditions to specify the type of this solution.