Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay

In this paper, we consider initial-boundary value problem of viscoelastic wave equation with a delay term in the interior feedback. Namely, we study the following equation $$u_{tt}(x,t)-\mathrm{\Delta }u(x,t)+{\int }_0^{t}g(t-s)\mathrm{\Delta }u(x,s)ds+{\mu }_1{u}_t(x,t)+{\mu }_2{u}_t(x,t-\tau )=0$$ together with initial-boundary conditions of Dirichlet type in $\mathrm{\Omega }\times (0,+\mathrm{\infty })$, and prove that for arbitrary real numbers ${\mu }_1$ and ${\mu }_2$, the above mentioned problem has a unique global solution under suitable assumptions on the kernel $g$. This improve the results of the previous literature such as [6] and [13] by removing the restriction imposed on ${\mu }_1$ and ${\mu }_2$. Furthermore, we also get an exponential decay results for the energy of the concerned problem in the case ${\mu }_1=0$ which solves an open problem proposed by M. Kirane and B. Said-Houari in [13].