Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation

Abstract

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