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-Pr & uuml;ss-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.