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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.07787 |
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| _version_ | 1866915439537291264 |
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| author | Park, Jeongheon Kwon, Soonsik Kim, Taegyu |
| author_facet | Park, Jeongheon Kwon, Soonsik Kim, Taegyu |
| contents | We study the blow-up dynamics for the $L^2$-critical focusing half-wave equation on the real line, a nonlocal dispersive PDE arising in various physical models. As in other mass-critical models, the ground state solution becomes a threshold between the global well-posedness and the existence of a blow-up. The first blow-up construction is due to Krieger, Lenzmann and Raphaël, in which they constructed the minimal mass blow-up solution at the threshold mass. In this paper, we construct finite-time blow-up solutions with mass slightly exceeding the threshold. This is inspired by similar results in the mass-critical NLS by Bourgain and Wang, and their instability by Merle, Raphaël and Szeftel. We exhibit a blow-up profile driven by the rescaled ground state, with a decoupled dispersive radiation component. We rigorously describe the asymptotic behavior of such solutions near the blow-up time, including sharp modulation dynamics. Furthermore, we demonstrate the instability of these solutions by constructing non-blow-up solutions that are arbitrarily close to the blow-up solutions. The main contribution of this work is to overcome the nonlocal setting of half-wave and to extend insights from the mass-critical NLS to a setting lacking pseudo-conformal symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07787 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass Park, Jeongheon Kwon, Soonsik Kim, Taegyu Analysis of PDEs We study the blow-up dynamics for the $L^2$-critical focusing half-wave equation on the real line, a nonlocal dispersive PDE arising in various physical models. As in other mass-critical models, the ground state solution becomes a threshold between the global well-posedness and the existence of a blow-up. The first blow-up construction is due to Krieger, Lenzmann and Raphaël, in which they constructed the minimal mass blow-up solution at the threshold mass. In this paper, we construct finite-time blow-up solutions with mass slightly exceeding the threshold. This is inspired by similar results in the mass-critical NLS by Bourgain and Wang, and their instability by Merle, Raphaël and Szeftel. We exhibit a blow-up profile driven by the rescaled ground state, with a decoupled dispersive radiation component. We rigorously describe the asymptotic behavior of such solutions near the blow-up time, including sharp modulation dynamics. Furthermore, we demonstrate the instability of these solutions by constructing non-blow-up solutions that are arbitrarily close to the blow-up solutions. The main contribution of this work is to overcome the nonlocal setting of half-wave and to extend insights from the mass-critical NLS to a setting lacking pseudo-conformal symmetry. |
| title | Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.07787 |