The paper is concerned with a parabolic-hyperbolic free boundary problem for change of phase processes in which the melting temperature is space- dependent. A weak formulation is given, pointing out that relevant differences occur with respect to the case of melting temperature. For the case in which weak solutions are suitably smooth it is established that: 1) mushy regions may appear even if volumetric heat sources are absent; 2) a differential system is satisfied, consisting of parabolic equations and first order hyperbolic equations coupled through free boundaries; 3) the interface conditions have the form of ''unilateral constraints''. The existence of a smooth solution for a mathematical model is proved.