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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.11334 |
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| _version_ | 1866909394426396672 |
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| author | Zheng, Bowen Ozawa, Tohru |
| author_facet | Zheng, Bowen Ozawa, Tohru |
| contents | In this paper, we study a class of variable coefficient Schrödinger equations with a linear potential \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)-V(x)u=-|x|^c|u|^pu,\] where $2-n<b\leq0,\ c\geq b-2$ and $0<\textbf{p}_c\leq(2-b)(p+2)$, where $\textbf{p}_c:=np-2c$. In the radial or finite variance case, we firstly prove the global existence and blow-up below the ground state threshold for the mass-critical and inter-critical nonlinearities. Next, adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a sufficient condition on the nonradial initial data, under which the global behavior of the general solution is established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11334 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global existence and blow-up for the variable coefficient Schrödinger equations with a linear potential Zheng, Bowen Ozawa, Tohru Analysis of PDEs 35Q55, 35B44 In this paper, we study a class of variable coefficient Schrödinger equations with a linear potential \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)-V(x)u=-|x|^c|u|^pu,\] where $2-n<b\leq0,\ c\geq b-2$ and $0<\textbf{p}_c\leq(2-b)(p+2)$, where $\textbf{p}_c:=np-2c$. In the radial or finite variance case, we firstly prove the global existence and blow-up below the ground state threshold for the mass-critical and inter-critical nonlinearities. Next, adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a sufficient condition on the nonradial initial data, under which the global behavior of the general solution is established. |
| title | Global existence and blow-up for the variable coefficient Schrödinger equations with a linear potential |
| topic | Analysis of PDEs 35Q55, 35B44 |
| url | https://arxiv.org/abs/2411.11334 |