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### Open Access Journals

CPAA

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.

DCDS

We consider a one-dimensional reaction-diffusion type equation
with memory, originally proposed by W.E. Olmstead

*et al*. to model the velocity $u$ of certain viscoelastic fluids. More precisely, the usual diffusion term $u_{x x}$ is replaced by a convolution integral of the form $\int_0^\infty k(s) u_{x x}(t-s)ds$, whereas the reaction term is the derivative of a double-well potential. We first reformulate the equation, endowed with homogeneous Dirichlet boundary conditions, by introducing the integrated past history of $u$. Then we replace $k$ with a time-rescaled kernel $k_\varepsilon$, where $\varepsilon>0$ is the relaxation time. The obtained initial and boundary value problem generates a strongly continuous semigroup $S_\varepsilon(t)$ on a suitable phase-space. The main result of this work is the existence of the global attractor for $S_\varepsilon(t)$, provided that $\varepsilon$ is small enough.
CPAA

We consider an integro-partial
differential equation of hyperbolic type
with a cubic nonlinearity,
in which no dissipation mechanism is present,
except for the convolution term accounting for
the past memory of the variable.
Setting the equation in the history space framework,
we prove the existence of a regular global attractor.

CPAA

In this note, we establish a general result on the existence of global attractors
for semigroups $S(t)$ of operators acting on a Banach space $\mathcal X$, where the strong
continuity $S(t)\in C(\mathcal X,\mathcal X)$ is replaced by the much weaker requirement that
$S(t)$ be a closed map.

DCDS

We consider a family of phase-field systems with memory effects in
the temperature $\vartheta$, depending on a parameter $\omega\geq 0$.
Setting the problems in a suitable phase-space accounting for the
past history of $\vartheta$, we prove the existence of a
family of exponential attractors $\mathcal E_\omega$ which is robust as
$\omega\to 0$.

CPAA

In this short note we present a direct method to establish
the optimal regularity of the attractor for
the semilinear damped wave equation with
a nonlinearity of
critical growth.

CPAA

We study the asymptotic properties of the semigroup $S(t)$ arising from the
nonlinear viscoelastic equation with hereditary memory
on a bounded three-dimensional domain
\begin{eqnarray}
|\partial_t u|^\rho \partial_{t t} u-\Delta \partial_{t t} u-\Delta \partial_t u\\
-\Big(1+\int_0^\infty \mu(s)\Delta s \Big)\Delta u
+\int_0^\infty \mu(s)\Delta u(t-s)\Delta s +f(u)=h
\end{eqnarray}
written in the past history framework of Dafermos [10].
We establish the existence of the global attractor of optimal regularity for $S(t)$
when $\rho\in [0,4)$
and $f$ has polynomial growth of (at most) critical order 5.

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