We will study the asymptotic solutions for a nonlinear damped wave problem showing the existence and uniqueness of solutions by contraction mapping \(\begin{aligned} {\left\{ \begin{array}{ll} v_{tt} + 2\kappa v_{t}-\eta v_{xx}=-\delta \partial _{x}\left( \left| v\right| ^{\rho }v\right) \\ v\left( 0,x\right) =\gamma \left( x\right) , v_{t}\left( 0,x\right) =\chi \left( x\right) , x\in \end{array}\right. } \end{aligned}\) where \(\Omega =\left[ -\pi ,\pi \right] \) , \(\kappa , \eta , \delta , \rho >0\) . We prove that if initial data \(\gamma \in H^1\) and \(\chi \in \textbf{L}^2\) , then there exists” a unique solution \(u(t,x) \in C([0,+\infty ):\textbf{H}^1)\) of the periodic problem for all \(t>0\) .