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: $$\begin{array}{c}{u}_{tt}(x,t)-Lu(x,t)-{\int }_0^t{g}_1(t-\sigma )Lu(x,\sigma )d\sigma +{\mu }_1{u}_t(x,t)+{\int }_{{\tau }_1}^{{\tau }_2}{\mu }_2(s)u_t(x,t-s)ds+f_1(u,\upsilon )=0,\\ {\upsilon }_{tt}(x,t)-L\upsilon (x,t)-{\int }_0^t{g}_2(t-\sigma )L\upsilon (x,\sigma )d\sigma +{\mu }_3{\upsilon }_t(x,t)+{\int }_{{\tau }_1}^{{\tau }_2}{\mu }_4(s){\upsilon }_t(x,t-s)ds+f_2(u,\upsilon )=0,\end{array}$$ 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.

for more revision