Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system

A singular nonlinear parabolic-hyperbolic PDE s system describ- ing the evolution of a material subject to a phase transition is considered. The goal of the present paper is to analyze the asymptotic behaviour of the associ- ated dynamical system from the point of view of global attractors. The physical variables involved in the process are the absolute temperature T (whose evolu- tion is governed by a parabolic singular equation coming from the Penrose-Fife theory) and the order parameter p (whose evolution is ruled by a nonlinear damped hyperbolic relation coming from a hyperbolic relaxation of the Allen- Cahn equation). Dissipativity of the system and the existence of a global attractor are proved. Due to questions of regularity, the one space dimensional case (1D) and the 2D - 3D cases require di erent sets of hypotheses and have to be settled in slightly di erent functional spaces.