Properties of global and exponential attractors for nonlinear thermo-diffusion Timoshenko system

Abstract

In this paper, we consider a new Timoshenko beam model with thermal and mass diffusion effects according to the Gurtin-Pinkin model. Heat and mass exchange with the environment during thermodiffusion in Timoshenko beam, depending on the past history of the temperature and the chemical potential gradients through memory kernels. We analyze the longtime properties for such a model with linear frictional damping and nonlinear source terms. We prove the global well-posedness of the system by using the C0-semigroup theory of linear operators. Then, we show, without assuming the well-known equal wave speeds condition, that the thermal and chemical potential coupling is strong enough to guarantee the quasistability. By showing that the system is gradient and asymptotically compact, we prove the existence of a global attractor with a finite fractal dimension and with a smoothness property. Furthermore, the existence of a fractal exponential attractor is also derived.