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

TL;DR

This paper establishes the global existence and analyzes the long-term behavior of solutions for a coupled wave system with distributed delay terms, using energy methods, Galerkin approximations, and Lyapunov functionals.

Contribution

It introduces new techniques to prove global existence and asymptotic stability for a viscoelastic wave system with delays, extending previous results to more complex models.

Findings

Proved global existence of solutions under certain conditions.

Analyzed asymptotic decay using Lyapunov functionals.

Established stability results for the delayed wave system.

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.