Long-time asymptotics of 3-solitary waves for the damped nonlinear Klein-Gordon equation

We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-Δu+2α\partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in $\mathbb{R}$ \times $\mathbb{R}$^d, \end{align*} where $\alpha >0$, $1\le d\le 5$ and energy sub-critical exponents $p>2$. In this paper, we prove that 3-solitary waves behave as if the three solitons are on a line. Furthermore, the solitary waves have alternative signs and their distances are of order $\log t$.