On the uniform stability of a thermoelastic Timoshenko system with infinite memory

Hasan Almutairi; Soh Edwin Mukiawa; Hasan Almutairi; Soh Edwin Mukiawa

doi:10.3934/math.2024787

AIMS Mathematics

2024, Volume 9, Issue 6: 16260-16279. doi: 10.3934/math.2024787

Previous Article Next Article

Research article

On the uniform stability of a thermoelastic Timoshenko system with infinite memory

Hasan Almutairi ^,,
Soh Edwin Mukiawa

Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin 31991, Saudi Arabia

Received: 12 February 2024 Revised: 15 April 2024 Accepted: 18 April 2024 Published: 08 May 2024
MSC : 35B35, 35B40, 35D30, 35D35, 93D20

The present research is aim at investigating a thermoelastic Timoshenko system with an infinite memory term on the shear force while the bending moment is under the influence of a thermoelastic dissipation governed by Fourier's law. We prove that the system's stability holds for a broader class of relaxation functions. Under this class of relaxation functions $ h $ at infinity, we establish a relation between the decay rate of the solution and the growth of $ h $ at infinity. Moreover, we drop the boundedness assumptions on the history data. We employ Neumann-Dirichlet-Neumann boundary conditions for our result. In comparison to the bulk of results in the literature, which frequently enforce the "equal-wave-speed" constraint, the present result shows that the infinite memory of the beam and the thermal damping are strong enough to guarantee stability without any conditions on the parameters.
- Timoshenko,
- infinite memory,
- thermoelasticity,
- stability analysis
Citation: Hasan Almutairi, Soh Edwin Mukiawa. On the uniform stability of a thermoelastic Timoshenko system with infinite memory[J]. AIMS Mathematics, 2024, 9(6): 16260-16279. doi: 10.3934/math.2024787

Related Papers:

Abstract

The present research is aim at investigating a thermoelastic Timoshenko system with an infinite memory term on the shear force while the bending moment is under the influence of a thermoelastic dissipation governed by Fourier's law. We prove that the system's stability holds for a broader class of relaxation functions. Under this class of relaxation functions $ h $ at infinity, we establish a relation between the decay rate of the solution and the growth of $ h $ at infinity. Moreover, we drop the boundedness assumptions on the history data. We employ Neumann-Dirichlet-Neumann boundary conditions for our result. In comparison to the bulk of results in the literature, which frequently enforce the "equal-wave-speed" constraint, the present result shows that the infinite memory of the beam and the thermal damping are strong enough to guarantee stability without any conditions on the parameters.

References

[1]	S. P. Timoshenko, LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars, London, Edinburgh, Dublin Philosophical Mag. J. Sci., 41 (1921), 744–746. https://doi.org/10.1080/14786442108636264 doi: 10.1080/14786442108636264
[2]	S. P. Timoshenko, Vibration problems in engineering, New York: Van Nostrand, 1955.
[3]	F. Amar-Khodja, A. Benabdallah, J. E. Muñoz Rivera, Energy decay for Timoshenko systems of memory type, J. Differ. Equ., 194 (2003), 82–115. https://doi.org/10.1016/S0022-0396(03)00185-2 doi: 10.1016/S0022-0396(03)00185-2
[4]	J. E. Muñoz Rivera, R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248–278. https://doi.org/10.1016/S0022-247X(02)00436-5 doi: 10.1016/S0022-247X(02)00436-5
[5]	M. O. Alves, E. H. G. Tavares, M. A. J. Silva, J. H. Rodrigues, On modeling and of a partially dissipative viscoelastic Timoshenko System, SIAM J. Math. Anal., 51 (2019), 4520–4543. https://doi.org/10.1137/18M1191774 doi: 10.1137/18M1191774
[6]	M. M. Chen, W. J. Liu, W. C. Zhou, Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type Ⅲ with frictional damping and delay terms, Adv. Nonlinear Anal., 7 (2018), 547–569. https://doi.org/10.1515/anona-2016-0085 doi: 10.1515/anona-2016-0085
[7]	M. Conti, F. Dell'Oro, V. Pata, Timoshenko systems with fading memory, Dyn. Partial Differ. Equ., 10 (2013), 367–377. https://doi.org/10.4310/DPDE.2013.v10.n4.a4 doi: 10.4310/DPDE.2013.v10.n4.a4
[8]	B. Feng, Uniform decay of energy for a porous thermoelasticity system with past history, Appl. Anal., 97 (2018), 210–229. https://doi.org/10.1080/00036811.2016.1258116 doi: 10.1080/00036811.2016.1258116
[9]	S. E. Mukiawa, On the stability of a viscoelastic Timoshenko system with Maxwell-Cattaneo heat conduction J. Differ. Equ. Appl., 14 (2022), 393–415. https://doi.org/10.7153/dea-2022-14-28 doi: 10.7153/dea-2022-14-28
[10]	S. E. Mukiawa, Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws, Open Math., 21 2023, 20230127. https://doi.org/10.1515/math-2023-0127 doi: 10.1515/math-2023-0127
[11]	S. E. Mukiawa, Well-posedness and stabilization of a type three layer beam system with Gurtin-Pipkin's thermal law, AIMS Math., 8 (2023), 28188–28209. https://doi.org/10.3934/math.20231443 doi: 10.3934/math.20231443
[12]	S. E. Mukiawa, Y. Khan, H. Al Sulaimani, M. E. Omaba, C. D. Enyi, Thermal Timoshenko beam system with suspenders and Kelvin-Voigt damping, Front. Appl. Math. Stat., 9 (2023), 1153071. https://doi.org/10.3389/fams.2023.1153071 doi: 10.3389/fams.2023.1153071
[13]	A. M. Al-Mahdi, M. Al-Gharabli, A. Guesmia, S. A. Messaoudi, New decay results for a viscoelastic-type Timoshenko system with infinite memory, Z. Angew. Math. Phys., 72 (2021), 22. https://doi.org/10.1007/s00033-020-01446-x doi: 10.1007/s00033-020-01446-x
[14]	A. M. Al-Mahdi, General stability result for a viscoelastic plate equation with past history and general kernel, J. Math. Anal. Appl., 490 (2020), 124216, https://doi.org/10.1016/j.jmaa.2020.124216 doi: 10.1016/j.jmaa.2020.124216
[15]	A. M. Al-Mahdi, M. Al-Gharabli, S. A. Messaoudi, New general decay result for a system of viscoelastic wave equations with past history, Commun. Pure Appl. Math., 20 (2021), 389–404. https://doi.org/10.3934/cpaa.2020273 doi: 10.3934/cpaa.2020273
[16]	G. Xu, H. Wang, Stabilisation of Timoshenko system with delay in the boundary control, Int. J. Control, 86 (2013), 1165–1178. https://doi.org/10.1080/00207179.2013.787494 doi: 10.1080/00207179.2013.787494
[17]	M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134–152. https://doi.org/10.1016/j.jmaa.2017.08.019 doi: 10.1016/j.jmaa.2017.08.019
[18]	J. L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications, Vol. 1, Springer, 1972. https://doi.org/10.1007/978-3-642-65161-8
[19]	V. I. Arnold, Mathematical methods of classical mechanics, New York: Springer-Verlag, 1989. https://doi.org/10.1007/978-1-4757-2063-1

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)