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Lack of Exponential Stability in Timoshenko Systems with Flat Memory Kernels

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Abstract

We analyze the decay properties of the solution semigroup generated by the linear Timoshenko system

$$\begin{aligned} \left\{ \begin{array}{l} \rho _1 \varphi _{tt} -\kappa (\varphi _x +\psi )_x = 0\\ \displaystyle \rho _2 \psi _{tt} -b\psi _{xx} +\int _0^\infty \mu (s)\psi _{xx}(t-s)\,\mathrm{d}s+\kappa (\varphi _x +\psi )= 0 \end{array}\right. \end{aligned}$$

in presence of a flat (i.e. piecewise constant) memory kernel \(\mu \). In this situation, the uniform decay of the solutions does not occur, also when the two equations exhibit the same propagation speed.

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Notes

  1. A linear semigroup \(S(t)\) acting on a Hilbert space \({\mathcal H}\) is said to be exponentially stable if there are \(\omega >0\) and \(K\ge 1\) such that

    $$\begin{aligned} \Vert S(t)z\Vert _{{\mathcal H}}\le K\mathrm{e}^{-\omega t}\Vert z\Vert _{\mathcal H},\quad \forall z\in {\mathcal H}. \end{aligned}$$

  2. We agree to include also trivial linear combinations.

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Authors and Affiliations

  1. DICATAM, Università di Brescia, Via Valotti 9, 25133 , Brescia, Italy

    Filippo Dell’Oro

  2. Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 , Milan, Italy

    Vittorino Pata

Authors
  1. Filippo Dell’Oro
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  2. Vittorino Pata
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Correspondence to Vittorino Pata.

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Dell’Oro, F., Pata, V. Lack of Exponential Stability in Timoshenko Systems with Flat Memory Kernels. Appl Math Optim 71, 79–93 (2015). https://doi.org/10.1007/s00245-014-9253-5

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  • DOI: https://doi.org/10.1007/s00245-014-9253-5

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