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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.05731 |
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| _version_ | 1866908355884220416 |
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| author | Liu, Xuan Xu, Chengbin |
| author_facet | Liu, Xuan Xu, Chengbin |
| contents | In this paper, we investigate the global well-posedness and scattering theory for the defocusing energy supcritical inhomogeneous nonlinear Schrödinger equation $iu_t + Δu =|x|^{-b} |u|^αu$ in four space dimension, where $s_c := 2- \frac{2-b}α \in (1, 2)$ and $0<b<\min \{ (s_c-1)^2+1,3-s_c\}$.
We prove that if the solution has a prior bound in the critical Sobolev space, that is, $u \in L_t^\infty(I; \dot{H}_x^{s_c}(\mathbb{R}^4))$, then $u$ is global and scatters. The proof of the main results is based on the concentration-compactness/rigidity framework developed by Kenig and Merle [Invent. Math. 166 (2006)], together with a long-time Strichartz estimate, a spatially localized Morawetz estimate, and a frequency-localized Morawetz estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05731 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The defocusing energy-supercritical inhomogeneous NLS in four space dimension Liu, Xuan Xu, Chengbin Analysis of PDEs In this paper, we investigate the global well-posedness and scattering theory for the defocusing energy supcritical inhomogeneous nonlinear Schrödinger equation $iu_t + Δu =|x|^{-b} |u|^αu$ in four space dimension, where $s_c := 2- \frac{2-b}α \in (1, 2)$ and $0<b<\min \{ (s_c-1)^2+1,3-s_c\}$. We prove that if the solution has a prior bound in the critical Sobolev space, that is, $u \in L_t^\infty(I; \dot{H}_x^{s_c}(\mathbb{R}^4))$, then $u$ is global and scatters. The proof of the main results is based on the concentration-compactness/rigidity framework developed by Kenig and Merle [Invent. Math. 166 (2006)], together with a long-time Strichartz estimate, a spatially localized Morawetz estimate, and a frequency-localized Morawetz estimate. |
| title | The defocusing energy-supercritical inhomogeneous NLS in four space dimension |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.05731 |