LAUSR

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 u t t − M ∇ u t 2 Δ u + ∫ 0 t g t − s Δ u s d s + γ 1 u t + u t k x − 1 u t = u p x − 1 u in Ω × ℝ + $$ \begin{array}{@{}rcl@{}} &&u_{tt}-M\left( \left\vert \nabla u\left( t\right) \right\vert^{2}\right) {\Delta} u+{{\int}_{0}^{t}}g\left( t-s\right) {\Delta} u\left( s\right) ds+\gamma_{1}u_{t}+\left\vert u_{t}\right\vert^{k\left( x\right) -1}u_{t}\\ &=&\left\vert u\right\vert^{p\left( x\right) -1}u \text{ in }{\Omega} \times \mathbb{R}^{+} \end{array} $$ with simply supported boundary condition, where Ω is a bounded domain of ℝ n $\mathbb {R}^{n}$ , 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.