We consider a two-phase Stefan problem in cylindrical symmetry with supercooling of the liquid phase, when the melting temperature is supposed to be a constant and zero flux conditions are imposed on the fixed boundaries. We perform an a priori analysis of the possibility of continuing the solution to arbitrarily large time intervals, and we relate the occurrence of each possible case with the value of an energy integral involving the initial data. Analogous results are achieved considering superheating of the solid phase instead of supercooling of the liquid one, or spherical instead of cylindrical symmetry.
