A stability result of the nonlinear thermodiffusion for a Timoshenko system with thermal effect, infinite history and distributed delay term

In this work, we consider a one-dimensional linear thermoelastic system of Timoshenko type with a delay term in the feedback with thermal and mass diffusion effects according to the Gurtin-Pinkin model. There is heat and mass exchange with the environment during thermodiffusion in Timoshenko beam, depending on the past history of the temperatureand the chemical potential gradients through memory kernels. We prove the existence and the uniqueness of the solution by using the C0-semigroup theory of linear operators, and we study the exponential stability for the case of equal speeds of wave propagation.