Global Well-Posedness and Blow-Up for the fifth order $L^2-$critical KP-I equation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913765673402368 |
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| author | Bozgan, Francisc |
| author_facet | Bozgan, Francisc |
| contents | In the current paper, we investigate the fifth order modified KP-I eqaution, namely
\begin{equation*}
\partial_t u-\partial_{x}^{5}u-\partial_{x}^{-1}\partial_{y}u+\partial_{x}(u^3)=0.
\end{equation*}
This equation is $L^2$ critical and we prove on $\mathbb{R}\times\mathbb{R}$ that it is globally well posed in the natural energy space if the $L^2$ norm of the initial data is less the $L^2$ norm of the ground state associated to this equation. We also find a subspace of the natural energy space associated to this equation where we have local well-posedness, nevertheless if the initial data is sufficiently localized we obtain blow-up. On $\mathbb{R}\times \mathbb{T},$ we prove global well-posedness in the energy space for small data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Well-Posedness and Blow-Up for the fifth order $L^2-$critical KP-I equation Bozgan, Francisc Analysis of PDEs 35A02, 35Q53, 35E15 In the current paper, we investigate the fifth order modified KP-I eqaution, namely \begin{equation*} \partial_t u-\partial_{x}^{5}u-\partial_{x}^{-1}\partial_{y}u+\partial_{x}(u^3)=0. \end{equation*} This equation is $L^2$ critical and we prove on $\mathbb{R}\times\mathbb{R}$ that it is globally well posed in the natural energy space if the $L^2$ norm of the initial data is less the $L^2$ norm of the ground state associated to this equation. We also find a subspace of the natural energy space associated to this equation where we have local well-posedness, nevertheless if the initial data is sufficiently localized we obtain blow-up. On $\mathbb{R}\times \mathbb{T},$ we prove global well-posedness in the energy space for small data. |
| title | Global Well-Posedness and Blow-Up for the fifth order $L^2-$critical KP-I equation |
| topic | Analysis of PDEs 35A02, 35Q53, 35E15 |
| url | https://arxiv.org/abs/2503.22778 |