A melting problem with a mushy region: qualitative properties

We consider a model problem of heat conduction in a slab where melting is produced by distributed heat sources. The same problem was considered by Atthey (1974) who produced numerical evidence of the appearance of a region (called a mushy region) where die temperature is identically equal to the melting temperature. Atthey's results were based on a weak formulation. Following Primiccrio (1982) this problem can be given a classical formulation as a three-phase problem with specific conditions at the interfaces (free boundaries). Here, referring to this classical formulation, we investigate some relevant properties of the solution and we give a fairly accurate description of the boundary of the mushy region