Energy decay of damped Timoshenko–type beams on two-parameter elastic foundations

Marwa Jomaa; Toufic El Arwadi; Samer Israwi; Dilberto da S. Almeida Júnior; Marwa Jomaa; Toufic El Arwadi; Samer Israwi; Dilberto da S. Almeida Júnior

doi:10.3934/cam.2026012

Communications in Analysis and Mechanics

2026, Volume 18, Issue 2: 296-328. doi: 10.3934/cam.2026012

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Research article

Energy decay of damped Timoshenko–type beams on two-parameter elastic foundations

1.
Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, P.O Box 11-5020, Beirut, Lebanon
2.
Arts, Sciences, and Technology University in Lebanon (AUL), CRAMS: Center for Research in Applied Mathematics and Statistics, Beirut, Lebanon
3.
Laboratoire de mathématiques-EDST, Faculté des sciences et EDST, Université Libanaise, Hadat, Lebanon
4.
PhD Program in Mathematics, Federal University of Pará, Belém, Pará, Brazil

Received: 22 August 2025 Revised: 23 October 2025 Accepted: 15 December 2025 Published: 10 April 2026
35L53, 35B40

This work is devoted to the analysis of well-posedness and stabilization properties of both the classical and truncated Timoshenko beam models resting on a two-parameter elastic foundation. We rigorously investigate the effects of the elastic subgrade on the dynamic behavior of the systems and consider the influence of viscous damping mechanisms acting on the angular rotation. For each model, we establish the existence, uniqueness, and continuous dependence of solutions. Moreover, we analyze the long-time behavior of the solutions, proving exponential or polynomial energy decay depending on the parameters of the wave speeds. The results highlight the differences in stability between the classical and truncated models and contribute to the mathematical theory of damped elastic structures interacting with elastic media.
- Timoshenko beam,
- two-parameter elastic foundation,
- energy decay,
- exponential stability,
- polynomial stability,
- viscous damping,
- well-posedness,
- stabilization
Citation: Marwa Jomaa, Toufic El Arwadi, Samer Israwi, Dilberto da S. Almeida Júnior. Energy decay of damped Timoshenko–type beams on two-parameter elastic foundations[J]. Communications in Analysis and Mechanics, 2026, 18(2): 296-328. doi: 10.3934/cam.2026012

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Abstract

This work is devoted to the analysis of well-posedness and stabilization properties of both the classical and truncated Timoshenko beam models resting on a two-parameter elastic foundation. We rigorously investigate the effects of the elastic subgrade on the dynamic behavior of the systems and consider the influence of viscous damping mechanisms acting on the angular rotation. For each model, we establish the existence, uniqueness, and continuous dependence of solutions. Moreover, we analyze the long-time behavior of the solutions, proving exponential or polynomial energy decay depending on the parameters of the wave speeds. The results highlight the differences in stability between the classical and truncated models and contribute to the mathematical theory of damped elastic structures interacting with elastic media.

References

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