Long-time Asymptotics of the One-dimensional Damped Nonlinear Klein–Gordon Equation

For the one-dimensional nonlinear damped Klein-Gordon equation \[ \partial_{t}^{2}u+2\alpha\partial_{t}u-\partial_{x}^{2}u+u-|u|^{p-1}u=0 \quad \mbox{on $\mathbb{R}\times\mathbb{R}$,}\] with $\alpha>0$ and $p>2$, we prove that any global finite energy solution either converges to $0$ or behaves asymptotically as $t\to \infty$ sum of $K\geq 1$ decoupled solitary waves. In multi-soliton case 2$, waves have alternate signs their distances are order $\log t$.