Nonlinear systems describing phase transition models compatible with thermodynamics

This paper is devoted to analyzing solutions of a nonlinear evolution system describing the phase transition in a rigid heat conductor in the presence of phase relaxation. First, in a general framework, a rate type constitutive law for the phase variable is considered and matched with the Helmholtz free energy involving the state of the material. Thermodynamic compatibility of the resulting models is scrutinized. Moreover, a comparison with a different phase change modelling is performed. Under proper assumptions, a nonlinear system in the (absolute) temperature and phase variable is achieved. For it, existence and uniqueness of the solution is proved and positivity of temperature is recovered.