Global attractor for a parabolic-hyperbolic PenroseFife phase field system

A singular nonlinear parabolic-hyperbolic PDE’s system describing 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 associated dynamical system from the point of view of global attractors. The physical variables involved in the process are the absolute temperature ϑ (whose evolution is governed by a parabolic singular equation coming from the Penrose-Fife theory) and the order parameter χ (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 different sets of hypotheses and have to be settled in slightly different functional spaces