Long-time asymptotics of the damped nonlinear Klein–Gordon equation with a delta potential

Abstract

We consider the damped nonlinear Klein–Gordon equation with a delta potential: ${\partial }_t^{2}u-{\partial }_x^{2}u+2\alpha {\partial }_{t}u+u-\gamma {\delta }_{0}u-{\left|u\right|}^{p-1}u=0,(t,x)\in R\times R,$where $p>1$, $\alpha >0,\gamma <2$, and ${\delta }_0={\delta }_0\left(x\right)$ denotes the Dirac delta with the mass at the origin. When $\gamma =0$, Côte et al. (2021) proved that any global solution either converges to 0 or to the sum of $K\ge 1$ decoupled solitary waves which have alternative signs. In this paper, we first prove that any global solution either converges to 0 or to the sum of $K\ge 1$ decoupled solitary waves. We then construct a single solitary wave solution that moves away from the origin when $\gamma <0$ and construct an even 2-solitary wave solution when $\gamma \le -2$. Last we give single solitary wave solutions and even 2-solitary wave solutions an upper bound for the distance between the origin and the solitary wave.