Global existence and Asymptotic behavior for a system of wave equation in presence of distributed delay term

Abstract

In this paper, we consider the following viscoelastic coupled wave equation with a delay term: $$ \begin{gathered} u_{tt}(x,t)-Lu(x,t)-\int_0^t g_1(t-\sigma)L u(x,\sigma)d\sigma + \mu_{1}u_{t}(x,t) + \int_{\tau_1}^{\tau_2} \mu_2(s)u_{t}(x,t-s)ds + f_1(u,\upsilon)=0, \\ \upsilon_{tt}(x,t) - L\upsilon(x,t) - \int_0^t g_{2}(t-\sigma)L \upsilon(x,\sigma)d\sigma + \mu_3\upsilon_t(x,t) + \int_{\tau_1}^{\tau_2} \mu_4(s)\upsilon_{t}(x,t-s)ds + f_{2}(u,\upsilon)=0, \end{gathered} $$ in a bounded domain. Under appropriate conditions on $\mu_{1}$, $\mu_{2}$, $\mu_{3}$ and $\mu_{4}$, we prove global existence result by combining the energy method with the Faedo-Galerkin's procedure. In addition , we focus on asymptotic behavior by using an appropriate Lyapunov functional.