Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains

Abstract

The viscoelastic Euler-Bernoulli equation with nonlinear and nonlocal damping $$u_{tt}+\Delta^2u-\int_0^tg(t-\tau )\Delta^2u(\tau )\,d\tau +a(t )u_t=0\,\,\,\hbox{in}\,\,\,\Omega\times {\bf R}^{+},$$ where $a(t)=M\left(\int_{\Omega}\left|\nabla u(x,t)\right|^2dx\right )$, is considered in bounded or unbounded domains $\Omega$ of ${\bf R}^ n$. The existence of global solutions and decay rates of the energy are proved.