On the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential
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| Format: | Preprint |
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2024
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| _version_ | 1866916298741514240 |
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| author | An, JinMyong Kim, JinMyong Kim, OkByol |
| author_facet | An, JinMyong Kim, JinMyong Kim, OkByol |
| contents | In this paper, we study the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential \[iu_{t} +Δu-c|x|^{-a}u=\pm |x|^{-b} |u|^{σ} u,\;\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\in \mathbb N$, $c\in \mathbb R$, $a,b>0$ and $σ>0$. First, we establish the local well-posedness in the fractional Sobolev spaces $H^s(\mathbb R^d)$ with $s\ge 0$ by using contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, the global existence and blow-up of $H^1$-solution are investigated. Our results extend the known results in several directions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_16365 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential An, JinMyong Kim, JinMyong Kim, OkByol Analysis of PDEs 2020 Mathematics Subject Classification. 35Q55, 35A01, 35B44 In this paper, we study the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential \[iu_{t} +Δu-c|x|^{-a}u=\pm |x|^{-b} |u|^{σ} u,\;\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\in \mathbb N$, $c\in \mathbb R$, $a,b>0$ and $σ>0$. First, we establish the local well-posedness in the fractional Sobolev spaces $H^s(\mathbb R^d)$ with $s\ge 0$ by using contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, the global existence and blow-up of $H^1$-solution are investigated. Our results extend the known results in several directions. |
| title | On the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential |
| topic | Analysis of PDEs 2020 Mathematics Subject Classification. 35Q55, 35A01, 35B44 |
| url | https://arxiv.org/abs/2406.16365 |