New general decay result of the laminated beam system with infinite history

Abstract

This paper is concerned with the asymptotic behavior of the solution of a laminated Timoshenko beam system with viscoelastic damping. We extend the work known for this system with finite memory to the case of infinite memory. We use minimal and general conditions on the relaxation function and establish explicit energy decay formula, which gives the best decay rates expected under this level of generality. We assume that the relaxation function $g$ satisfies, for some nonnegative functions $\xi$ and $H$, $g^{\prime }\left(t\right)\le -\xi \left(t\right)H\left(g\left(t\right)\right)$, $\forall t\ge 0$. Our decay results generalize and improve many earlier results in the literature. Moreover, we remove some assumptions on the boundedness of initial data used in many earlier papers in the literature.