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

TL;DR

This paper derives new decay results for a dissipative viscoelastic Timoshenko system with infinite memory, improving upon previous findings by considering equal-speed propagation and specific memory kernel conditions.

Contribution

It introduces generalized decay results for the system with infinite memory, extending prior work and providing sharper conditions for energy decay.

Findings

Established new decay rates under specific kernel conditions

Generalized previous results to systems with equal propagation speeds

Improved understanding of energy dissipation in viscoelastic systems

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.