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.

DCDS

This note is focused on a novel
technique to establish
the boundedness in more regular spaces
for global attractors
of dissipative dynamical systems,
without appealing to uniform-in-time
estimates.
As an application,
we consider the semigroup
generated by the
strongly damped wave equation
with critical nonlinearity,
whose attractor is shown to possess
the optimal regularity.

DCDS

For a semigroup $S(t):X\to X$ acting
on a metric space $(X,d)$, we give a notion of global attractor
based only on the minimality with respect to the attraction property.
Such an attractor is shown to be invariant whenever $S(t)$ is
*asymptotically closed*.
As a byproduct, we generalize earlier results on the existence of
global attractors in the classical sense.

DCDS-S

This paper deals with the longtime behavior of the Caginalp phase-field system
with coupled dynamic boundary conditions on both state variables.
We prove that the system generates a dissipative semigroup in a suitable phase-space
and possesses the finite-dimensional smooth global attractor and an exponential attractor.

PROC

We consider a class of weakly damped
semilinear hyperbolic equations with memory, expressed by a
convolution integral. We study the passage to the singular limit
when the memory kernel collapses into the Dirac mass at zero, and
we establish a convergence result for a proper family of
exponential attractors.

DCDS

We consider a phase-field system modeling phase transition phenomena, where the Cahn-Hilliard-Oono equation for the order parameter is coupled with the Coleman-Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases.

DCDS-B

We establish a necessary and sufficient condition
of exponential stability for the contraction semigroup generated
by an abstract version of the linear differential equation
$$∂_t u(t)-\int_0^\infty k(s)\Delta u(t-s)ds = 0
$$
modeling hereditary heat conduction of Gurtin-Pipkin type.

DCDS

A semilinear integrodifferential equation of hyperbolic type
is studied,
where the dissipation
is entirely contributed by the convolution term accounting for
the past history of the variable.
Within a novel abstract framework,
based on the notion of * minimal state*,
the existence of a regular global attractor is proved.

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.

CPAA

We discuss the existence of the global attractor for a family of processes $U_\sigma(t,\tau)$ acting on
a metric space $X$ and depending on a symbol $\sigma$ belonging to some other metric space $\Sigma$.
Such an attractor
is uniform with respect to $\sigma\in\Sigma$, as well as
with respect to the choice of the initial time $\tau\in R$. The existence of the attractor is established
for totally dissipative processes
without any continuity assumption. When the process satisfies some additional (but rather mild)
continuity-like hypotheses, a characterization of the attractor is given.