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