Thermoelastic Extensible Timoshenko Beam with Symport Term: Singular Limits, Lack of Differentiability and Optimal Polynomial Decay

Moncef Aouadi (), Taoufik Moulahi and Najmeddine Attia ()
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Moncef Aouadi: UR Systèmes Dynamiques et Applications, 17ES21, Ecole Nationale d’Ingénieurs de Bizerte, Université de Carthage, P.O. Box 66, Bizerte 7035, Tunisia
Taoufik Moulahi: Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, P.O. Box 173, Al-Kharj 11942, Saudi Arabia
Najmeddine Attia: Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Mathematics, 2025, vol. 13, issue 5, 1-30

Abstract: In this article, we consider the equations of the nonlinear model of a thermoelastic extensible Timoshenko beam, recently derived by Aouadi in the context of Fourier’s law. The new aspect we propose here is to introduce a second sound model in the temperatures which turns into a Gurtin–Pipkin’s model. Thus, the derived equations are physically more realistic since they overcome the property of infinite propagation speed (Fourier’s law property). They are also characterized by the presence of a symport term. Moreover, it is possible to recover the Fourier, Cattaneo and Coleman–Gurtin laws from the derived system by considering a scaled kernel instead of the original kernel through an appropriate singular limit method. The well-posedness of the derived problem is proved by means of the semigroups theory. Then, we show that the associated linear semigroup (without extensibility and with a constant symport term) is not differentiable by an approach based on the Gearhart–Herbst–Prüss–Huang theorem. The lack of analyticity and impossibility of localization of the solutions in time are immediate consequences. Then, by using a resolvent criterion developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the same associated linear semigroup under a condition on the physical coefficients. In particular, we show that the considered problem is not exponentially stable. Moreover, by following a result according to Arendt–Batty, we show that the linear semigroup is strongly stable.

Keywords: thermoelastic extensible Timoshenko beam; Gurtin–Pipkin’s law; well-posedness; differentiability; optimal polynomial decay (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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