6,332 research outputs found
Global attractors for the one dimensional wave equation with displacement dependent damping
We study the long-time behavior of solutions of the one dimensional wave
equation with nonlinear damping coefficient. We prove that if the damping
coefficient function is strictly positive near the origin then this equation
possesses a global attractor
A New Approach to Equations with Memory
In this work, we present a novel approach to the mathematical analysis of
equations with memory based on the notion of a state, namely, the initial
configuration of the system which can be unambiguously determined by the
knowledge of the future dynamics. As a model, we discuss the abstract version
of an equation arising from linear viscoelasticity. It is worth mentioning that
our approach goes back to the heuristic derivation of the state framework,
devised by L.Deseri, M.Fabrizio and M.J.Golden in "The concept of minimal state
in viscoelasticity: new free energies and applications to PDEs", Arch. Ration.
Mech. Anal., vol. 181 (2006) pp.43-96. Starting from their physical
motivations, we develop a suitable functional formulation which, as far as we
know, is completely new.Comment: 39 pages, no figur
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
Steady states of elastically-coupled extensible double-beam systems
Given and , we analyze an abstract version
of the nonlinear stationary model in dimensionless form describing the equilibria of an elastically-coupled extensible double-beam
system subject to evenly compressive axial loads. Necessary and sufficient
conditions in order to have nontrivial solutions are established, and their
explicit closed-form expressions are found. In particular, the solutions are
shown to exhibit at most three nonvanishing Fourier modes. In spite of the
symmetry of the system, nonsymmetric solutions appear, as well as solutions for
which the elastic energy fails to be evenly distributed. Such a feature turns
out to be of some relevance in the analysis of the longterm dynamics, for it
may lead up to nonsymmetric energy exchanges between the two beams, mimicking
the transition from vertical to torsional oscillations
Stability analysis of abstract systems of Timoshenko type
We consider an abstract system of Timoshenko type where the operator is strictly positive
selfadjoint. For any fixed , the stability properties of
the related solution semigroup are discussed. In particular, a general
technique is introduced in order to prove the lack of exponential decay of
when the spectrum of the leading operator is not made by eigenvalues
only.Comment: Corrected typo
A quantitative Riemann-Lebesgue lemma with application to equations with memory
An elementary proof of a quantitative version of the Riemann-Lebesgue lemma
for functions supported on the half line is given. Applications to differential
models with memory are discussed
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
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.Comment: 38 pages, published versio
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