arXiv subjects

Vittorino Pata

Publications and source records attributed to Vittorino Pata.

28 records · Page 2Linked to original sources

arXiv2015

Stability analysis of abstract systems of Timoshenko type

We consider an abstract system of Timoshenko type $$ \begin{cases} ρ_1{\ddot φ} + a A^{\frac12}(A^{\frac12}φ+ ψ) =0\\ ρ_2{\ddot ψ} + b A ψ+ a (A^{\frac12}φ+ ψ) - δA^γθ = 0\\ ρ_3{\dot θ} + c Aθ+ δA^γ{\dot ψ} =0 \end{cases} $$ where the operator $A$ is strictly positive selfadjoint. For any fixed $γ\in\mathbb{R}$, the stability properties of the related solution semigroup $S(t)$ are discussed. In particular, a general technique is introduced in order to prove the lack of exponential decay of $S(t)$ when the spectrum of the leading operator $A$ is not made by eigenvalues only.

math.AP↗

arXiv2014

Exponential attractors for abstract equations with memory and applications to viscoelasticity

We consider an abstract equation with memory of the form $$\partial_t \boldsymbol{x}(t)+\int_{0}^\infty k(s) \boldsymbol{A}\boldsymbol{x}(t-s){\rm d} s+\boldsymbol{B}\boldsymbol{x}(t)=0$$ where $\boldsymbol{A},\boldsymbol{B}$ are operators acting on some Banach space, and the convolution kernel $k$ is a nonnegative convex summable function of unit mass. The system is translated into an ordinary differential equation on a Banach space accounting for the presence of memory, both in the so-called history space framework and in the minimal state one. The main theoretical result is a theorem providing sufficient conditions in order for the related solution semigroups to possess finite-dimensional exponential attractors. As an application, we prove the existence of exponential attractors for the integrodifferential equation $$\partial_{tt} u - h(0)Δu - \int_{0}^\infty h'(s) Δu(t-s){\rm d} s+ f(u) = g$$ arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel $h(s)=k(s)+1$.

math.DS↗

arXiv2013

Global attractors for nonlinear viscoelastic equations with memory

We study the asymptotic properties of the semigroup S(t) arising from a nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain written in the past history framework of Dafermos. We establish the existence of the global attractor of optimal regularity for S(t) for a wide class of nonlinearities as well as within the most general condition on the memory kernel.

math.AP↗

arXiv2013

Timoshenko systems with fading memory

The decay properties of the semigroup generated by a linear Timoshenko system with fading memory are discussed. Uniform stability is shown to occur within a necessary and sufficient condition on the memory kernel.

math.AP↗

arXiv2013

Totally dissipative dynamical processes and their uniform global attractors

We discuss the existence of the global attractor for a family of processes $U_σ(t,τ)$ acting on a metric space $X$ and depending on a symbol $σ$ belonging to some other metric space $Σ$. Such an attractor is uniform with respect to $σ\inΣ$, as well as with respect to the choice of the initial time $τ\in\R$. 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.

math.DS↗

arXiv2013

On the stability of Timoshenko systems with Gurtin-Pipkin thermal law

We analyze a differential system describing a Timoshenko beam coupled with a temperature evolution of Gurtin-Pipkin type. A necessary and sufficient condition for exponential stability is established in terms of the structural parameters of the equations. In particular, we generalize previously known results on the Fourier-Timoshenko and the Cattaneo-Timoshenko beam models.

math.AP↗

arXiv2012

Attractors for processes on time-dependent spaces. Applications to wave equations

For a process U(t,s) acting on a one-parameter family of normed spaces, we present a notion of time-dependent attractor based only on the minimality with respect to the pullback attraction property. Such an attractor is shown to be invariant whenever the process is T-closed for some T>0, a much weaker property than continuity (defined in the text). As a byproduct, we generalize the recent theory of attractors in time-dependent spaces developed in [10]. Finally, we exploit the new framework to study the longterm behavior of wave equations with time-dependent speed of propagation.

math.DS↗

arXiv2010

Asymptotics of the Coleman-Gurtin model

This paper is concerned with the integrodifferential equation $$\partial_t u-Δu -\int_0^\infty κ(s)Δu(t-s)\,\d s + φ(u)=f$$ arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity $φ$ 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.

math.AP↗

arXiv2009

On the regularity of global attractors

This note is focused on a novel technique in order 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 of the abstract result, the semigroup generated by the strongly damped wave equation $$u_{tt}-Δu_t-Δu+ϕ(u)=f$$ with critical nonlinearity is considered, whose attractor is shown to possess the optimal regularity.

math.DS↗

arXiv1999

Stable laws and domains of attraction in free probability theory

In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.

math.OA↗