A dynamic contact problem for a thermoelastic diffusion beam with the rotational inertia

We study the dynamic behaviour of a thermoelastic diffusion beam with the rotational inertia, clamped at one end and free to move between two stops at the other. The contact with the stops is modelled with the normal compliance condition. The system, recently derived by Aouadi (2015) [2], describes the behaviour of thermoelastic diffusion thin plates. This problem poses new mathematical difficulties due to the nonlinear boundary conditions. We prove the existence and uniqueness of weak solution using the Faedo-Galerkin method as well as the exponential stability at a rate proportional to the rotational inertia parameter. We propose a finite element approximation and we prove that the associated discrete energy decays to zero. Finally, we give an error estimate assuming extra regularity on the solution and we present some results of our numerical experiments.