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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.20801 |
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| _version_ | 1866911404580143104 |
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| author | Martel, Yvan Pilod, Didier |
| author_facet | Martel, Yvan Pilod, Didier |
| contents | For any $ν\in(\frac 37,\frac12)$, we prove the existence of an $H^1$ solution $u$ of the mass critical generalized Korteweg-de Vries equation on the time interval $(0,T_0]$, for some $T_0>0$, which blows up at the time $t=0$ and at the point $x=0$ with the rate $\|\partial_x u (t,x)\|_{L^2} \approx t^{-ν}$. Such a blowup rate is associated to a blowup residue of the form $r_α(x)= x^{α-\frac 12}$ for $x>0$ close to the blowup point, where $α=\frac{3ν-1}{2-4ν}$. The condition $ν\in(\frac37,\frac12)$ is equivalent to $α>1$, which corresponds to the full range for which the residue $r_α$ belongs to $H^1$.
Such blowup at a finite point is in contrast with all the blowup solutions constructed for this equation, except the one constructed previously by the authors corresponding to the special value $ν=\frac 25$.
Finally, we present some open problems regarding the blowup phenomenon for the mass critical gKdV equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20801 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Continuum of finite point blowup rates for the critical generalized Korteweg-de Vries equation Martel, Yvan Pilod, Didier Analysis of PDEs For any $ν\in(\frac 37,\frac12)$, we prove the existence of an $H^1$ solution $u$ of the mass critical generalized Korteweg-de Vries equation on the time interval $(0,T_0]$, for some $T_0>0$, which blows up at the time $t=0$ and at the point $x=0$ with the rate $\|\partial_x u (t,x)\|_{L^2} \approx t^{-ν}$. Such a blowup rate is associated to a blowup residue of the form $r_α(x)= x^{α-\frac 12}$ for $x>0$ close to the blowup point, where $α=\frac{3ν-1}{2-4ν}$. The condition $ν\in(\frac37,\frac12)$ is equivalent to $α>1$, which corresponds to the full range for which the residue $r_α$ belongs to $H^1$. Such blowup at a finite point is in contrast with all the blowup solutions constructed for this equation, except the one constructed previously by the authors corresponding to the special value $ν=\frac 25$. Finally, we present some open problems regarding the blowup phenomenon for the mass critical gKdV equation. |
| title | Continuum of finite point blowup rates for the critical generalized Korteweg-de Vries equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.20801 |