Large time behavior of solutions to Penrose-Fife phase change models

A non-conserved phase transition model of Penrose Fife type is considered where Dirichlet boundary conditions for the temperature are taken. A sketch of the proof of existence and uniqueness of the solution is given. Then, the large time behaviour of such a solution is studied. By using the Simon-Lojasiewicz inequality it is shown that the whole solution trajectory converges to a single stationary state. Due to the non-coercive character of the energy functional, the main di culty in the proof is to control the large values of the temperature. This is achieved by means of non-standard a priori estimates.