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 \begin{align*} \partial_{t}^2u-\partial_{x}^2u+2α\partial_{t}u+u-γδ_0u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}, \end{align*} where $p>2$, $α>0,\ γ<2$, and $δ_0=δ_0 (x)$ denotes the Dirac delta with the mass at the origin. When $γ=0$, Côte, 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 $γ<0$ and construct an even 2-solitary wave solution when $γ\leq -2$. 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.

Theorems & Definitions (56)

• Remark 1.1
• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.2
• Remark 1.5
• Definition 1.1
• Theorem 1.3
• Lemma 2.1