EECT

We analyze
an abstract version of the evolution system
ruling the dynamics of a memory relaxation of a type III thermoelastic
extensible beam or Berger plate
occupying a volume $\Omega$
\begin{equation}
\begin{cases}
u_{tt}-ωΔ u_{tt}+Δ^2 u-[b
+||\nabla u\|^2_{L^2(\Omega)}]\Delta u+Δ α_t=g\\
α_{tt}-Δ α-∫_0^\infty
u(s)Δ[α(t)-α(t-s)]d s-Δ u_t=0
\end{cases}
\end{equation}
subject to hinged boundary conditions for $u$
and to the Dirichlet boundary condition for $\alpha$,
where the dissipation is entirely contributed by the convolution term in the second equation.
The study of the asymptotic properties of the related solution semigroup
is addressed.

DCDS-S

This paper is concerned with the integrodifferential equation
$\partial_{t} u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\d s +
\varphi(u)=f$

arising in the Coleman-Gurtin's theory of heat conduction with
hereditary memory,
in presence of a nonlinearity $\varphi$
of critical growth.
Rephrasing the equation within the history space framework,
we prove the existence of global and exponential attractors
of optimal regularity and finite fractal dimension for the related solution
semigroup, acting both on the basic
weak-energy space and on a more regular phase space.

CPAA

This article is concerned with the incompressible Navier-Stokes equations
in a three-dimensional domain.
A criterion of Prodi-Serrin type up to the boundary for
global existence of strong solutions is established.

DCDS

A material with heterogeneous structure at microscopic level is
considered. The microscopic mechanical behavior is described by a
stress-strain law of Kelvin-Voigt type. It has been
shown that a homogenization process leads to a macroscopic stress-strain
relation containing a time convolution term which accounts for memory effects.
Consequently, the displacement field $\mathbf{u}$ obeys to a Volterra
integrodifferential motion equation. The longtime behavior of $\mathbf{u}$ is
here investigated proving the existence of a uniform attractor when the body
forces vary in a suitable metric space.

PROC

We provide an optimal regularity result for the universal attractor of the
weakly damped semilinear wave equation, when the nonlinearity satisfies the critical
growth condition. This allows us to prove an upper semicontinuity result as well as
the existence of an exponential 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.

CPAA

This article is focused
on the solution semigroup in the history space framework
arising from an abstract version
of the boundary value problem with memory
$\partial_{t t} u(t)-\Delta [u(t)+\int_0^\infty
\mu(s)[u(t)-u(t-s)] ds ]=0,\quad u(t)_{|\partial\Omega}=0,$

modelling linear viscoelasticity.
The exponential stability of the semigroup is
discussed, establishing a necessary and sufficient condition
involving the memory
kernel $\mu$.

DCDS

A strongly damped semilinear wave equation on the whole space is considered.
Existence and uniqueness results are provided, together with the existence
of an absorbing set, which is uniform as the external force is allowed to run in a
certain functional set. In the autonomous case, the equation is shown to possess a
universal attractor.

DCDS

We consider a Timoshenko model of a
viscoelastic beam fixed at the endpoints and subject to nonlinear
external forces. The model consists of two coupled second order
linear integrodifferential hyperbolic equations that govern the
evolution of the lateral displacement $u$ and the total rotation
angle $\phi$. We prove that these equations generate a dissipative
dynamical system, whose trajectories are eventually confined in a
uniform absorbing set, the dissipativity being due to the memory
mechanism solely. This fact allows us to state the existence of a
uniform compact attractor.

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.