Communications on Pure and Applied Analysis (CPAA)

Asymptotic behavior of a parabolic-hyperbolic system

Pages: 849 - 881, Volume 3, Issue 4, December 2004      doi:10.3934/cpaa.2004.3.849

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M. Grasselli - Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, I-20133 Milano, Italy (email)
V. Pata - Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, I-20133 Milano, Italy (email)

Abstract: We consider a parabolic equation nonlinearly coupled with a damped semilinear wave equation. This system describes the evolution of the relative temperature $\vartheta$ and of the order parameter $\chi$ 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 $\mu\partial_{t t}\chi$ 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 $\mathcal V_0$, which possesses a universal attractor $\mathcal A_\mu$. Our main result establishes that $\mathcal A_\mu$ is bounded by a constant independent of $\mu$ in a smaller phase-space $\mathcal V_1$. This bound allows us to show that the lifting $\mathcal A_0$ of the universal attractor of the parabolic system (corresponding to $\mu=0$) is upper semicontinuous at $0$ with respect to the family $\{\mathcal A_\mu,\mu>0\}$. 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 $\mathcal V_0$. The existence of an exponential attractor is obtained in the case $\mu=0$ 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.

Keywords:  Phase-field models, absorbing sets, universal attractors, upper semi- continuity, exponential attractors, damped semilinear wave equation.
Mathematics Subject Classification:  Primary 35B40, 35B41, 35L05, 35Q40, 37L25, 37L30, 80A22

Received: December 2003;      Revised: June 2004;      Available Online: September 2004.