Abstract
In this paper, we consider the Timoshenko–Ehrenfest beam models (Elishakoff 2019, Who developed the so-called Timoshenko beam theory? Math Mech Solids 25(1):97–116) and we established exponential decay results based on influence of the second spectrum of frequency and its damaging consequences for wave propagation speeds. For the classical case, having two wave speeds governing the stress waves and shear waves, we prove that the corresponding semigroup associated with the system decays exponentially under equal wave speeds assumption. On contrary, there is a lack of exponential stability and we prove its optimality based on Borichev-Tomilov approach. For the truncated case, we assure the well-posedness using the Faedo–Galerkin method and we prove that the total energy decays exponentially regardless any relationship between coefficients of the system using the energy method.
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Acknowledgements
The authors are grateful to the anonymous referees for their constructive remarks, which have enhanced the presentation of this paper.
Funding
D. S. Almeida Júnior thanks the CNPq for financial support through the project: “Impact of the second spectrum of frequency on the stabilization of partially dissipative Timoshenko type systems”, Grant 314273/2020-4; A. J. A. Ramos thanks the CNPq for financial support through the Grant 310729/2019-0.
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Communicated by Abimael Loula.
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Júnior, D.S.A., Freitas, M.M., Ramos, A.J.A. et al. Stabilization of Timoshenko–Ehrenfest type systems. Comp. Appl. Math. 41, 28 (2022). https://doi.org/10.1007/s40314-021-01723-z
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DOI: https://doi.org/10.1007/s40314-021-01723-z