Long Time Behavior for a Wave Equation with Time Delay

Abstract

In this paper, we consider the wave equation with internal time delay and source terms\[ u_{tt}(x,t) - \triangle u(x,t) + \mu_1 u_t(x,t) + \mu_2 u_t(x,t-\tau) + f(x,u) = h(x)\]in a bounded domain. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution under initial-boundary data for the above equation. Moreover, under suitable conditions on the forcing term $f(x,u)$ and $\mu_1$, $\mu_2$, the existence of a compact global attractor is proved. Further, the asymptotic behavior and the decay property of global solution are discussed.