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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.17457 |
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| _version_ | 1866910074837925888 |
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| author | Lan, Yang Yuan, Xu |
| author_facet | Lan, Yang Yuan, Xu |
| contents | For the mass-critical generalized Korteweg-de Vries equation, $$ \partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}.$$ We prove the existence of a global solution that blows up in infinite time and approaches the sum of two decoupled bubbles with opposite signs. The proof is inspired by the techniques developed for the two-dimensional mass-critical NLS equation in a similar context by Martel-Raphaël [37]. The main difficulty originates from the fact that the unstable directions related to scaling are excited by the nonlinear interactions. To overcome this difficulty, a refined approximate solution that involves some non-localized profiles is needed. In particular, a sharp understanding for the interactions between solitons and such profiles is also required. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17457 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Construction of two-bubble blow-up solutions for the mass-critical gKdV equations Lan, Yang Yuan, Xu Analysis of PDEs For the mass-critical generalized Korteweg-de Vries equation, $$ \partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}.$$ We prove the existence of a global solution that blows up in infinite time and approaches the sum of two decoupled bubbles with opposite signs. The proof is inspired by the techniques developed for the two-dimensional mass-critical NLS equation in a similar context by Martel-Raphaël [37]. The main difficulty originates from the fact that the unstable directions related to scaling are excited by the nonlinear interactions. To overcome this difficulty, a refined approximate solution that involves some non-localized profiles is needed. In particular, a sharp understanding for the interactions between solitons and such profiles is also required. |
| title | Construction of two-bubble blow-up solutions for the mass-critical gKdV equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.17457 |