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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.02158 |
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| _version_ | 1866908515379970048 |
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| author | Cui, Zhi-Yuan Li, Yuan Zhao, Dun |
| author_facet | Cui, Zhi-Yuan Li, Yuan Zhao, Dun |
| contents | We consider the defocusing inhomogeneous nonlinear Schrödinger equation
$i\partial_tu+Δu= |x|^{-b}|u|^αu,$
where $0<b<1$ and $0<α<\infty$. This problem has been extensively studied for initial data in $H^1(\R^N)$ with $N\geq 2$. However, in the one-dimensional setting, due to the difficulty in dealing with the singularity factor $|x|^{-b}$, the well-posedness and scattering in $H^1(\R)$ are scarce, and almost known results have been established in $H^s(\R)$ with $s<1$. In this paper, we focus on the odd initial data in $H^1(\R)$. For this case, we establish local well-posedness for $0<α<\infty$, as well as global well-posedness and scattering for $4-2b<α<\infty$, which corresponds to the mass-supercritical case. The key ingredient is the application of the one-dimensional Hardy inequality for odd functions to overcome the singularity induced by $|x|^{-b}$. Our proof is based on the Strichartz estimates and employs the concentration-compactness/rigidity method developed by Kenig-Merle as well as the technique for handling initial data living far from the origin, as proposed by Miao-Murphy-Zheng. Our results fill a gap in the theory of well-posedness and energy scattering for the inhomogeneous nonlinear Schrödinger equation in one dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02158 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness and scattering of odd solutions for the defocusing INLS in one dimension Cui, Zhi-Yuan Li, Yuan Zhao, Dun Analysis of PDEs We consider the defocusing inhomogeneous nonlinear Schrödinger equation $i\partial_tu+Δu= |x|^{-b}|u|^αu,$ where $0<b<1$ and $0<α<\infty$. This problem has been extensively studied for initial data in $H^1(\R^N)$ with $N\geq 2$. However, in the one-dimensional setting, due to the difficulty in dealing with the singularity factor $|x|^{-b}$, the well-posedness and scattering in $H^1(\R)$ are scarce, and almost known results have been established in $H^s(\R)$ with $s<1$. In this paper, we focus on the odd initial data in $H^1(\R)$. For this case, we establish local well-posedness for $0<α<\infty$, as well as global well-posedness and scattering for $4-2b<α<\infty$, which corresponds to the mass-supercritical case. The key ingredient is the application of the one-dimensional Hardy inequality for odd functions to overcome the singularity induced by $|x|^{-b}$. Our proof is based on the Strichartz estimates and employs the concentration-compactness/rigidity method developed by Kenig-Merle as well as the technique for handling initial data living far from the origin, as proposed by Miao-Murphy-Zheng. Our results fill a gap in the theory of well-posedness and energy scattering for the inhomogeneous nonlinear Schrödinger equation in one dimension. |
| title | Well-posedness and scattering of odd solutions for the defocusing INLS in one dimension |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.02158 |