General and optimal decay in a memory-type Timoshenko system

Abstract

This paper is concerned with the following memory-type Timoshenko system \[ \rho _1\varphi _{tt}-K(\varphi _x+\psi )_x=0 \] \[ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+ \displaystyle \int _0^tg(t-s)\psi _{xx}(s)\,ds=0, \] $(x,t)\in (0,L)\times (0,\infty )$, with Dirichlet boundary conditions, where $g$ is a positive non-increasing function satisfying, for some constant $1\leq p\lt {3}/{2}$, \[ g'(t)\leq -\xi (t)g^p(t),\quad \mbox {for all }t\geq 0. \] We prove some decay results which generalize and improve many earlier results in the literature. In particular, our result gives the optimal decay for the case of polynomial stability.