General decay result for an extensible Timoshenko system with nonlinear feedback

In this paper, we considered the following nonlinear mathematical model of an extensible thermoelastic Timoshenko system with nonlinear feedback.$ \begin{equation*} \left\lbrace \begin{array}{ll} \rho_1 p_{tt} -\kappa (p_x+q)_x-\left({\epsilon}_0+{\epsilon}_1\int_{0}^{L}\vert p_x\vert^2dx\right)p_{xx}+h(p_t) = 0, & \rm{in (0, L)\times(0, \infty) , }\ \\ \rho_2 q_{tt}-\alpha q_{xx}+\kappa (p_x+q)+\beta \theta_x = 0, & \rm{in (0, L)\times(0, \infty) , }\ \\ \rho_3{\theta}_t-\kappa_0 {\theta}_{xx}+\beta q_{xt} = 0, & \rm{in (0, L)\times(0, \infty) .} \end{array} \right. \end{equation*} $We established the existence of a solution to the above system by using the standard Faedo-Galerkin method. Then we proved explicit and general decay results for which the optimal exponential and polynomial decay rate estimates are only special cases. Our decay results, which depend on the nonlinear frictional damping $ h $, were obtained without any restrictive growth assumption on $ h $ at the origin. The multiplier method, the properties of the convex and the dual of the convex functions, Jensen's inequality, and the generalized Young inequality were used to establish the stability results. These results generalize the result of [6] from linear frictional damping to nonlinear feedback.