
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 phasefield 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 wellknown phasefield model
of Caginalp type. The goal of the present paper is an asymptotic
analysis from the viewpoint of infinitedimensional dynamical
systems. We first prove that the model, endowed with appropriate
boundary conditions, generates a strongly continuous semigroup on
a suitable phasespace $\mathcal V_0$, 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 phasespace
$\mathcal V_1$. 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.
Mathematics Subject Classification: Primary 35B40, 35B41, 35L05, 35Q40, 37L25, 37L30, 80A22.
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