In this paper, we investigate the existence of a local solution in time and discuss the exponential asymptotic behavior to a weakly damped wave equation involving the variable-exponents

\begin{array}{@{}rcl@{}} &&u_{tt}-M\left( \left\vert \nabla u\left( t\right) \right\vert^{2}\right) {\Delta} u+{{\int}_{0}^{t}}g\left( t-s\right) {\Delta} u\left( s\right) ds+\gamma_{1}u_{t}+\left\vert u_{t}\right\vert^{k\left( x\right) -1}u_{t}\\ &=&\left\vert u\right\vert^{p\left( x\right) -1}u \text{ in }{\Omega} \times \mathbb{R}^{+} \end{array}

with simply supported boundary condition, where $\Omega$ is a bounded domain of $\mathbb{R}^n, g>0$ is a memory kernel that decays exponentially, and $M(s)$ is a locally Lipschitz function. This kind of problem without the memory term when $k(.)$ and $p(.)$ are constants models viscoelastic Kirchhoff equation.