Memory relaxation of type III thermoelastic extensible beams and Berger plates
Filippo Dell'Oro Vittorino Pata
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.
keywords: Type III thermoelastic beams and plates equations with memory global attractor Lyapunov functional regularity.
Asymptotics of the Coleman-Gurtin model
Mickaël D. Chekroun Francesco di Plinio Nathan Glatt-Holtz Vittorino Pata
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.

keywords: exponential attractor. history space framework solution semigroup Heat conduction with memory global attractor
On the regularity of solutions to the Navier-Stokes equations
Vittorino Pata
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.
keywords: weak solutions Navier-Stokes equations regularity criteria. strong solutions blow-up
Longtime behavior of a homogenized model in viscoelastodynamics
M. Grasselli Vittorino Pata
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.
keywords: Integro-differential equation translation compact functions asymptotic behavior uniform absorbing set uniform attractor.
On the damped semilinear wave equation with critical exponent
Maurizio Grasselli Vittorino Pata
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.
keywords: exponential attractors. upper semicontinuity Damped wave equation universal attractors
On the regularity of global attractors
Monica Conti Vittorino Pata
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.
keywords: regularity global attractor absorbing set semigroup strongly damped wave equation. Solution operator
Exponential stability in linear viscoelasticity with almost flat memory kernels
Vittorino Pata
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 $\mu$.

keywords: contraction semigroup exponential stability. Linear viscoelasticity memory kernel
Attractors for semilinear strongly damped wave equations on $\mathbb R^3$
Veronica Belleri Vittorino Pata
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.
keywords: uniform absorbing sets Kuratowski measure of non-compactness universal attractors. Strongly damped wave equations
Longtime behavior of a viscoelastic Timoshenko beam
M. Grasselli Vittorino Pata Giovanni Prouse
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.
keywords: Timoshenko beam attractors. absorbing sets viscoelasticity
A minimal approach to the theory of global attractors
Vladimir V. Chepyzhov Monica Conti Vittorino Pata
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.
keywords: absorbing and attracting sets Semigroups global attractors invariant sets.

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