5,884 research outputs found

    Global attractors for the one dimensional wave equation with displacement dependent damping

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    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

    Timoshenko systems with fading memory

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    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

    A New Approach to Equations with Memory

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    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

    Steady states of elastically-coupled extensible double-beam systems

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    Given βR\beta\in\mathbb{R} and ϱ,k>0\varrho,k>0, we analyze an abstract version of the nonlinear stationary model in dimensionless form {u""(β+ϱ01u(s)2ds)u"+k(uv)=0v""(β+ϱ01v(s)2ds)v"k(uv)=0\begin{cases} u"" - \Big(\beta+ \varrho\int_0^1 |u'(s)|^2\,{\rm d} s\Big)u" +k(u-v) = 0 v"" - \Big(\beta+ \varrho\int_0^1 |v'(s)|^2\,{\rm d} s\Big)v" -k(u-v) = 0 \end{cases} 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

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    We consider an abstract system of Timoshenko type {ρ1φ¨+aA12(A12φ+ψ)=0ρ2ψ¨+bAψ+a(A12φ+ψ)δAγθ=0ρ3θ˙+cAθ+δAγψ˙=0 \begin{cases} \rho_1{{\ddot \varphi}} + a A^{\frac12}(A^{\frac12}\varphi + \psi) =0\\ \rho_2{{\ddot \psi}} + b A \psi + a (A^{\frac12}\varphi + \psi) - \delta A^\gamma {\theta} = 0\\ \rho_3{{\dot \theta}} + c A\theta + \delta A^\gamma {{\dot \psi}} =0 \end{cases} where the operator AA is strictly positive selfadjoint. For any fixed γR\gamma\in\mathbb{R}, the stability properties of the related solution semigroup S(t)S(t) are discussed. In particular, a general technique is introduced in order to prove the lack of exponential decay of S(t)S(t) when the spectrum of the leading operator AA is not made by eigenvalues only.Comment: Corrected typo

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

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    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

    A quantitative Riemann-Lebesgue lemma with application to equations with memory

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    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
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