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

TL;DR

This paper proves the global existence and exponential decay of solutions for a viscoelastic wave equation with delay, removing previous restrictions on parameters and solving an open problem in the field.

Contribution

It establishes the existence and decay results for a viscoelastic wave equation with delay without restrictions on parameters, improving prior results and addressing an open problem.

Findings

Global existence of solutions for arbitrary parameters

Exponential decay of energy when =0

Removes previous restrictions on  and 

Abstract

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 $$u_{tt}(x,t)-\Delta u(x,t)+\int_0^t g(t-s)\Delta u(x,s)ds +\mu_1 u_t(x,t)+ \mu_2 u_t(x,t-\tau)=0$$ together with initial-boundary conditions of Dirichlet type in $\Omega\times (0,+\infty)$, 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].