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.
Categories: math.AP