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

TL;DR

This paper studies the long-term behavior of solutions to a damped nonlinear Klein-Gordon equation with a delta potential, showing convergence to solitary waves or zero, and constructing specific solitary wave solutions with bounds on their positions.

Contribution

It extends previous results by proving convergence to solitary waves with delta potential, and constructs explicit solitary wave solutions for certain parameter ranges.

Findings

Solutions converge to zero or sums of solitary waves.

Constructed solitary wave solutions moving away from the origin.

Provided bounds on the distance between solitary waves and the origin.

Abstract

We consider the damped nonlinear Klein-Gordon equation with a delta potential \begin{align*} \partial_{t}^2u-\partial_{x}^2u+2\alpha \partial_{t}u+u-\gamma {\delta}_0u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}, \end{align*} where $p>2$, $\alpha>0,\ \gamma<2$, and $\delta_0=\delta_0 (x)$ denotes the Dirac delta with the mass at the origin. When $\gamma=0$, C\^{o}te, Martel and Yuan proved that any global solution either converges to 0 or to the sum of $K\geq 1$ 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 $K\geq 1$ decoupled solitary waves. Next we construct a single solitary wave solution that moves away from the origin when $\gamma<0$ and construct an even 2-solitary wave solution when $\gamma\leq -2$. 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.