The propagation of regularity and dispersive blow-up phenomenon to higher-order generalized KdV equations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912002717253632 |
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| author | Shan, Minjie |
| author_facet | Shan, Minjie |
| contents | Some special properties of smoothness and singularity concerning to the initial value problem associated with higher-order generalized KdV equations are investigated. On one hand, we show the propagation of regularity phenomena. More precisely, the regularity of initial data on the right-hand side of the real line is propagated to the left-hand side with infinite speed under the higher-order KdV flow. On the other hand, we show that the dispersive blow-up phenomenon will occur by constructing a class of smoothing initial data such that global solutions with the given initial data keep smooth at positive generic irrational times, while global solutions display singularity at each time-space positive rational point. The blow-up phenomenon is exclusively caused by the linear part of solutions due to the focusing of short or long waves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14741 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The propagation of regularity and dispersive blow-up phenomenon to higher-order generalized KdV equations Shan, Minjie Analysis of PDEs 35Q53, 35B44 Some special properties of smoothness and singularity concerning to the initial value problem associated with higher-order generalized KdV equations are investigated. On one hand, we show the propagation of regularity phenomena. More precisely, the regularity of initial data on the right-hand side of the real line is propagated to the left-hand side with infinite speed under the higher-order KdV flow. On the other hand, we show that the dispersive blow-up phenomenon will occur by constructing a class of smoothing initial data such that global solutions with the given initial data keep smooth at positive generic irrational times, while global solutions display singularity at each time-space positive rational point. The blow-up phenomenon is exclusively caused by the linear part of solutions due to the focusing of short or long waves. |
| title | The propagation of regularity and dispersive blow-up phenomenon to higher-order generalized KdV equations |
| topic | Analysis of PDEs 35Q53, 35B44 |
| url | https://arxiv.org/abs/2408.14741 |