Self-similar solutions of a multi-phase Stefan problem on the half-line

In this paper, we study self-similar solutions of the multiphase Stefan problem for the heat equation on the half-line x > 0 with constant initial data and Dirichlet or Neumann boundary conditions. In the case of the Dirichlet boundary condition, we prove that the nonlinear algebraic system for determining the free boundaries is a gradient system, and the corresponding potential is an explicitly written strictly convex and coercive function. Consequently, there exists a unique minimum point of the potential, the coordinates of which determine the free boundaries and give the desired solution. In the case of the Neumann boundary condition, we show that the problem can have solutions with different numbers (types) of phase transitions. For each fixed type n, the system for determining the free boundaries is again a gradient system, and the corresponding potential turns out to be strictly convex and coercive, but in some wider nonphysical domain. In particular, a solution of type n is unique and can exist only if the minimum point of the potential belongs to the physical domain. We give an explicit criterion for the existence of solutions of any type n. Due to the rather complicated structure of the set of solutions, neither the existence nor the uniqueness of a solution to the Stefan-Neumann problem is guaranteed.