On singular limits and exponential decay in full von Kármán beam with temperature and microtemperatures

In this article, we consider the recently derived mathematical model of the full von Kármán beam with temperature and microtemperature effects. The nonlinear governing equations were derived by using Hamilton principle in the framework of Euler–Bernoulli beam theory. The new aspect we propose here is to introduce the second sound law in the temperatures and the microtemperatures which turns to the Gurtin–Pipkin’s one. Hence the derived equations are physically more realistic since they overcome the property of infinite propagation speed and take into account the thermal memory. Even more so, the case of Fourier’s law, Cattaneo’s law and the Coleman–Gurtin’s law can be recovered from the derived system by considering a rescaled kernel in place of the original kernels through a proper singular limit procedure. Based on semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem under Gurtin–Pipkin’s law. By adding a frictional damping function acting on the transversal component and by using the multiplier method, we show that solutions decay exponentially. These results are obtained under condition which establishes the null solution as the only equilibrium of the derived model.