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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2402.14381 |
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| _version_ | 1866911783619395584 |
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| author | Ishizuka, Kenjiro |
| author_facet | Ishizuka, Kenjiro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14381 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time asymptotics of the damped nonlinear Klein-Gordon equation with a delta potential Ishizuka, Kenjiro Analysis of PDEs 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. |
| title | Long-time asymptotics of the damped nonlinear Klein-Gordon equation with a delta potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.14381 |