Well-posedness and stability result for Timoshenko system with thermodiffusion effects and time-varying delay term

The main aim of the present paper is to investigate a new Timoshenko beam model with thermal and mass diffusion effects combined with a time-varying delay. Heat and mass exchange with the environment during a thermodiffusion in the Timoshenko beam, where the heat conduction is given by the classical Fourier law and acts on both the rotation angle and the transverse displacements. The heat conduction is given by the Cattaneo law. Under an appropriate assumption on the weights of the delay and the damping, we prove a well-posedness result, more precisely, we prove the existence of the weak solution. Then we proceed to the strong solution using the classical elliptic regularity and we get the result by applying the Lax-Milgram theorem, the Lumer-Phillips corollary and the Hille-Yosida theorem. We show the exponential stability result of the system in the case of nonequal speeds of wave propagation by using a multiplier technique combined with an appropriate Lyapunov functions. Our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical PDEs theory. To achieve our goals, we employ of the semigroup method and the energy method.