Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916117924020224 |
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| author | Campos, Luccas Correia, Simão Farah, Luiz Gustavo |
| author_facet | Campos, Luccas Correia, Simão Farah, Luiz Gustavo |
| contents | We consider the initial value problem associated to the inhomogeneous nonlinear Schrö\-din\-ger equation,
\begin{equation}
iu_t + Δu +μ|x|^{-b}|u|^αu=0, \quad u_0\in H^s(\mathbb R^N) \text{ or } u_0 \in\dot H ^s(\mathbb R^N),
\end{equation} with $μ=\pm 1$, $b > 0$, $s\geq 0$ and $0 < α\leq \frac{4-2b}{N-2s}$. By means of an adapted version of the fractional Leibniz rule, we prove new local well-posedness results in Sobolev spaces for a large range of parameters. We also prove an ill-posedness result for this equation, through a delicate analysis of the associated Duhamel operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_07060 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation Campos, Luccas Correia, Simão Farah, Luiz Gustavo Analysis of PDEs We consider the initial value problem associated to the inhomogeneous nonlinear Schrö\-din\-ger equation, \begin{equation} iu_t + Δu +μ|x|^{-b}|u|^αu=0, \quad u_0\in H^s(\mathbb R^N) \text{ or } u_0 \in\dot H ^s(\mathbb R^N), \end{equation} with $μ=\pm 1$, $b > 0$, $s\geq 0$ and $0 < α\leq \frac{4-2b}{N-2s}$. By means of an adapted version of the fractional Leibniz rule, we prove new local well-posedness results in Sobolev spaces for a large range of parameters. We also prove an ill-posedness result for this equation, through a delicate analysis of the associated Duhamel operator. |
| title | Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2210.07060 |