New Decay Results for a Partially Dissipative Viscoelastic Timoshenko System with Infinite Memory

Abstract

In this paper, we consider the following dissipative viscoelastic with memory-type Timoshenko system \begin{equation*} \begin{gathered} \begin{cases} \rho_1 \phi_{tt} - \kappa ( \phi _{x} + \psi) _x + \kappa \int_0^\infty g(s) (\phi_x +\psi)_x(t-s) ~ds =0 & \text{in}~ \left( {0,L} \right) \times \mathbb{R}^+ , \\ \rho_2 \psi_{tt} - b \psi_{xx} + \kappa ( \phi _{x} + \psi)-\kappa \int_0^\infty g(s) (\phi_x +\psi)(t-s)~ ds=0 & \text{in}~ \left( {0,L} \right) \times \mathbb{R}^+ , \\ \end{cases} \end{gathered} \end{equation*} with Dirichlet boundary conditions, where $g$ is a positive non-increasing function satisfying, for some nonnegative functions $\xi$ and $H$, \[g'(t)\leq-\xi(t)H(g(t)),\qquad\forall~ t\geq0.\] Under appropriate conditions on $\xi$ and $H$, we establish some new decay results for the case of equal-speeds of propagation that generalize and improve many earlier results in the literature.