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A transmission problem for the Timoshenko system with one local Kelvin–Voigt damping and non-smooth coefficient at the interface

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Abstract

In this paper, we study the indirect stability of Timoshenko system with local or global Kelvin–Voigt damping, under fully Dirichlet or mixed boundary conditions. Unlike Zhao et al. (Acta Mathematica Sinica Engl Ser 21(3):655–666, 2004), Tian and Zhang (Mathematik und Physik 68(1), 2017), and Liu and Zhang (SIAM J Control Optim 56(6):3919–3947, 2018), in this paper, we consider the Timoshenko system with only one locally or globally distributed Kelvin–Voigt damping D [see System (1.1)]. Indeed, we prove that the energy of the system decays polynomially of type \(t^{-1}\) and that this decay rate is in some sense optimal. The method is based on the frequency domain approach combining with multiplier method.

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Acknowledgements

The authors would like to thank the referees for their valuable comments and useful suggestions.

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Authors and Affiliations

  1. Faculty of Sciences 1, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Lebanese University, Hadath-Beirut, Lebanon

    Ali Wehbe & Mouhammad Ghader

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  1. Ali Wehbe
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  2. Mouhammad Ghader
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Communicated by Eduardo Cerpa.

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Wehbe, A., Ghader, M. A transmission problem for the Timoshenko system with one local Kelvin–Voigt damping and non-smooth coefficient at the interface. Comp. Appl. Math. 40, 297 (2021). https://doi.org/10.1007/s40314-021-01446-1

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  • DOI: https://doi.org/10.1007/s40314-021-01446-1

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