Asymptotic behavior of a parabolic-hyperbolic system

We consider a parabolic equation nonlinearly coupled with a damped semilinear wave equation. This system describes the evolution of the relative temperature nu and of the order parameter X in a material subject to phase transitions in the framework of phase-field theories. The hyperbolic dynamics is characterized by the presence of the inertial term mupartial derivativettchi with mu > 0. When mu = 0, we reduce to the well-known phase-field model of Caginalp type. The goal of the present paper is an asymptotic analysis from the viewpoint of infinite-dimensional dynamical systems. We first prove that the model, endowed with appropriate boundary conditions, generates a strongly continuous semigroup on a suitable phase-space v(0), which possesses a universal attractor A(mu). Our main result establishes that A, is bounded by a constant independent of mu in a smaller phase-space v(1). This bound allows us to show that the lifting A(0) of the universal attractor of the parabolic system (corresponding to p = 0) is upper semicontinuous at 0 with respect to the family {A(mu), mu > 0}. We also construct an exponential attractor; that is, a set of finite fractal dimension attracting all the trajectories exponentially fast with respect to the distance in v(0). The existence of an exponential attractor is obtained in the case p = 0 as well. Finally, a noteworthy consequence is that the above results also hold for the damped semilinear wave equation, which is obtained as a particular case of our system when the coupling term vanishes. This provides a generalization of a number of theorems proved in the last two decades.