Decay rate of the solutions to the Lord Shulman thermoelastic Timoshenko model

Abdelbaki Choucha; Sofian Abuelbacher Adam Saad; Rashid Jan; Salah Boulaaras; Abdelbaki Choucha; Sofian Abuelbacher Adam Saad; Rashid Jan; Salah Boulaaras

doi:10.3934/math.2023881

AIMS Mathematics

2023, Volume 8, Issue 7: 17246-17258. doi: 10.3934/math.2023881

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Research article

Decay rate of the solutions to the Lord Shulman thermoelastic Timoshenko model

1.
Department of Material Sciences, Faculty of Sciences, Amar Teleji Laghouat University, Algeria
2.
Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia
3.
University of Nyala, Faculty of Economics & Commercial Studies, dept. of Economics, Sudan
4.
Department of Mathematics, University of Swabi, Swabi 23561, KPK Pakistan

Received: 19 March 2023 Revised: 16 April 2023 Accepted: 25 April 2023 Published: 18 May 2023
MSC : 35B37, 35L55, 74D05, 93D15, 93D20

In this work, we deal with a one-dimensional Cauchy problem in Timoshenko system with thermal effect and damping term. The heat conduction is given by the theory of Lord-Shulman. We prove that the dissipation induced by the coupling of the Timoshenko system with the heat conduction of Lord-Shulman's theory alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow.
- partial differential equations,
- mathematical operators,
- decay rate,
- Lord-Shulman,
- thermoelasticity,
- Fourier transform
Citation: Abdelbaki Choucha, Sofian Abuelbacher Adam Saad, Rashid Jan, Salah Boulaaras. Decay rate of the solutions to the Lord Shulman thermoelastic Timoshenko model[J]. AIMS Mathematics, 2023, 8(7): 17246-17258. doi: 10.3934/math.2023881

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Abstract

In this work, we deal with a one-dimensional Cauchy problem in Timoshenko system with thermal effect and damping term. The heat conduction is given by the theory of Lord-Shulman. We prove that the dissipation induced by the coupling of the Timoshenko system with the heat conduction of Lord-Shulman's theory alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow.

References

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