ASYMPTOTIC BEHAVIOR FOR A VISCOELASTIC WAVE EQUATION WITH A DELAY TERM

Abstract

The following viscoelastic wave equation with a delay term in internal feedback: \begin{equation*} \left\vert u_{t}\right\vert ^{\rho }u_{tt}-\Delta u-\Delta u_{tt} + \int_{0}^{t} g(t-s) \Delta u(s) ds + \mu _{1} u_{t}(x,t) + \mu _{2} u_{t}(x,t-\tau ) = 0, \end{equation*} is considered in a bounded domain. Under appropriate conditions on $\mu_{1}$, $\mu_{2}$ and on the kernel $g$, we prove the local existence result by Faedo-Galerkin method and establish the decay result by suitable Lyapunov functionals.