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

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.