On a multi-phase Stefan problem in the half-line with different boundary conditions at the fixed boundary
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, Neumann or Robin boundary condition at the fixed boundary $x=0$. 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 the study is complicated by the fact that number of phase transitions undergone by a solution (called its type) cannot be directly determined from the boundary data. 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 it is proved that the Dirichlet-to-Neumann map is a strictly increasing continuous function. This allows to establish existence and uniqueness of a solution to Stefan-Neumann problem, and to specify the type of this solution. The same technique is further applied to Stefan-Robin problem with a positive connection coefficient. In the last section we also study some particular ill-posed Stefan-Robin problem with a negative connection coefficient, using again the variational approach developed in the previous sections.