Long-time behavior for a nonlinear Timoshenko system: Thermal damping versus weak damping of variable-exponents type

Adel M. Al-Mahdi; Adel M. Al-Mahdi

doi:10.3934/math.20231515

AIMS Mathematics

2023, Volume 8, Issue 12: 29577-29603. doi: 10.3934/math.20231515

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Research article

Long-time behavior for a nonlinear Timoshenko system: Thermal damping versus weak damping of variable-exponents type

Adel M. Al-Mahdi ^{1,2
,
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1.
The Preparatory Year Program, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
2.
The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received: 13 September 2023 Revised: 26 October 2023 Accepted: 26 October 2023 Published: 01 November 2023
MSC : 35L20, 45K05, 74F05, 93D23

In this work, we consider a nonlinear thermoelastic Timoshenko system with a time-dependent coefficient where the heat conduction is given by Coleman-Gurtin ^[1]. Consequently, the Fourier and Gurtin-Pipkin laws are special cases. We prove that the system is exponentially and polynomially stable. The equality of the wave speeds is not imposed unless the system is not fully damped by the thermoelasticity effect. In other words, the thermoelasticity is only coupled to the first equation in the system. By constructing a suitable Lyapunov functional, we establish exponential and polynomial decay rates for the system. We noticed that the decay sometimes depends on the behavior of the thermal kernel, the variable exponent, and the time-dependent coefficient. Our results extend and improve some earlier results in the literature especially the recent results by Fareh ^[2], Mustafa ^[3] and Al-Mahdi and Al-Gharabli ^[4].
- thermoelastic Timoshenko system,
- variable exponents,
- embedding theory,
- Coleman-Gurtin's law,
- general decay,
- energy method
Citation: Adel M. Al-Mahdi. Long-time behavior for a nonlinear Timoshenko system: Thermal damping versus weak damping of variable-exponents type[J]. AIMS Mathematics, 2023, 8(12): 29577-29603. doi: 10.3934/math.20231515

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Abstract

In this work, we consider a nonlinear thermoelastic Timoshenko system with a time-dependent coefficient where the heat conduction is given by Coleman-Gurtin ^[1]. Consequently, the Fourier and Gurtin-Pipkin laws are special cases. We prove that the system is exponentially and polynomially stable. The equality of the wave speeds is not imposed unless the system is not fully damped by the thermoelasticity effect. In other words, the thermoelasticity is only coupled to the first equation in the system. By constructing a suitable Lyapunov functional, we establish exponential and polynomial decay rates for the system. We noticed that the decay sometimes depends on the behavior of the thermal kernel, the variable exponent, and the time-dependent coefficient. Our results extend and improve some earlier results in the literature especially the recent results by Fareh ^[2], Mustafa ^[3] and Al-Mahdi and Al-Gharabli ^[4].

References

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