Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results

Abstract

In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ1φtt−κ(φx+ψ)x+(g∗φxx)(t)=0,−bψxx+κ(φx+ψ)=0,\begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array} where the convolution memory function gg belongs to a class of L1(0,∞){L}^{1}\left(0,\infty ) functions that satisfies g′(t)≤−ξ(t)ϒ(g(t)),∀t≥0,g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0, where ξ\xi is a positive nonincreasing differentiable function and ϒ\Upsilon is an increasing and convex function near the origin. Using just this general assumptions on the behavior of gg at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ(s)=sp\Upsilon \left(s)={s}^{p} and pp covers the full admissible range [1,2)\left[1,2). Given this degree of generality, our results improve some of earlier related results in the literature.

Keywords

• timoshenko system
• shear beam models
• multiplier method
• general and optimal decay
• memory
• 35b40
• 93d20