Behaviour of mushy regions under the action of a volumetric heat source

In this paper we consider solutions to Stefan problems in spatial dimension $N\ge 1$. We find necessary conditions on the heat source for the appearance of a ``mushy region'' (i.e., a region where temperature coincides identically with the temperature of change of phase) inside a purely liquid (or solid) phase. For sources depending on energy, such conditions are connected only with the local behaviour of the source near the energy level corresponding to the beginning of the change of phase. Both weak and smooth solutions are considered; in the latter case the behaviour of the solution at the free boundary is investigated in detail.