AIMS Mathematics, 2021, 6(1): 333-361. doi: 10.3934/math.2021021

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Stability rate of a thermoelastic laminated beam: Case of equal-wave speed and nonequal-wave speed of propagation

1 Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin 31991, Saudi Arabia
2 Department of Mathematics and Statistics University of Sharjah Sharjah, United Arab Emirates
3 Department of Mathematics and Statistics, KFUPM, Dhahran 31261, Saudi Arabia

In this article, we investigate a one-dimensional thermoelastic laminated beam system with viscoelastic dissipation on the effective rotation angle and through heat conduction in the interfacial slip equations. Under general conditions on the relaxation function and the relationship between the coefficients of the wave propagation speed of the first two equations, we show that the solution energy has an explicit and general decay rate from which the exponential and polynomial stability are just particular cases. Moreover, we establish a weaker decay result in the case of non-equal wave of speed propagation and give some examples to illustrate our results. This new result improves substantially many other results in the literature.
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1. T. A. Apalara, On the stability of a thermoelastic laminated beam, Acta Mathematica Scientia, 39 (2019), 1-8.

2. J. M. Wang, G. Q. Xu, S. P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J. Control Optim., 44 (2005), 1575-1597.

3. V. I. Arnold, Mathematical Methods of Classical Mechanics, New York: Springer-Verlag, 1989.

4. R. Spies, Structural damping in a laminated beams due to interfacial slip, J. Sound Vib., 204 (1997), 183-202.

5. X. Cao, D. Liu, G. Xu, Easy test for stability of laminated beams with structural damping and boundary feedback controls, J. Dyn. Control Syst., 13 (2007), 313-336.

6. S. W. Hansen, A model for a two-layered plate with interfacial slip. In: Control and Estimation of Distributed Parameter Systems, Int. Series Numer. Math., 118 (1993), 143-170.

7. J. M. Wang, G. Q Xu, S. P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J. Control Optim., 44 (2005), 1575-1597.

8. B. Feng, T. F. Ma, R. N. Monteiro, C. A. Raposo, Dynamics of Laminated Timoshenko Beams, J. Dyn. Diff Equat., 30 (2018), 1489-1507.

9. G. Li, X. Kong, W. Liu, General decay for a laminated beam with structural damping and memory, the case of non-equal-wave, J. Integral. Equations Appl., 30 (2018), 95-116.

10. M. I. Mustafa, General decay result for nonlinear viscoelastic equations J. Math. Anal. Appl., 457 (2018), 134-152.

11. A. Guesmia, S. A. Messaoudi, On the stabilization of Timoshenko systems with memory and different speeds of wave propagation, Appl. Math. Comput., 219 (2013), 9424-9437.

12. J. L. Jensen, Sur les fonctions convexes et les inégualités entre les valeurs moyennes, Acta Math., 30 (1906), 175-193.

13. S. E. Mukiawa, T. A. Apalara, S. A. Messaoudi, A general and optimal stability result for a laminated beam, J. Integral Equations Appl., 32 (2020), 341-359.

14. T. A. Apalara, S. A. Messaoudi, An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay, Appl. Math. Optim., 71 (2015), 449-472.

15. M. I. Mustafa, Laminated Timoshenko beams with viscoelastic damping, J. Math. Anal. Appl., 466 (2018), 619-641.

16. J. H. Hassan, S. A. Messaoudi, M. Zahri, Existence and new general decay result for a viscoleastic-type Timoshenko system, J. Anal. Appl., 39 (2020), 185-222.

17. T. A. Apalara, S. A. Messaoudi, A. A. Keddi, On the decay rates of Timoshenko system with second sound, Math. Methods Appl. Sci., 39 (2016), 2671-2684.

18. M. M. Cavalcanti, A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Differ. Integral Equ., 18 (2005), 583-600.

19. S. E. Mukiawa, T. A. Apalara, S. A. Messaoudi, A stability result for a memory-type Laminatedthermoelastic system with Maxwell-Cattaneo heat conduction, J. Thermal Stresses, 43 (2020), 1437-1466,

20. C. D. Enyi, S. E. Mukiawa, Dynamics of a thermoelastic-laminated beam problem, AIMS Mathematics, 5 (2020), 5261-5286.

21. E. H. Dill, Continuum mechanics: elasticity, plasticity, viscoelasticity, CRC Press, Taylor and Francis Group, New York, (2006).

© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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