Asymptotics for nonlinear damped wave equations with large initial data

We study the one dimensional nonlinear damped wave equation
\begin{equation} \begin{cases} u_{tt}+u_{t}-u_{xx}=\lambda|u|^{\sigma}u,&x\in\mathbf{R},\quad t>0,\\ u(0,x)=u_0(x),& x\in\mathbf{R},\\ u_t(0,x)=u_1(x),& x\in\mathbf{R}, \end{cases} \tag{0.1} \end{equation}
where $\sigma>0$, $\lambda\in\mathbf R$. Our aim is to prove the large time asymptotic formulas for solutions of the Cauchy problem (0.1) without any restriction on the size of the initial data.