Construction of type I-Log blowup for the Keller-Segel system in dimensions $3$ and $4$
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915082247602176 |
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| author | Nguyen, V. T. Nouaili, N. Zaag, H. |
| author_facet | Nguyen, V. T. Nouaili, N. Zaag, H. |
| contents | We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system
$\partial_t u = Δu - \nabla \cdot (u \nabla \mathcal{K}_u), \quad -Δ\mathcal{K}_u = u \quad \textup{in}\;\; \mathbb{R}^d,\; d = 3,4,$
and derive the final blowup profile
$ u(r,T) \sim c_d \frac{|\log r|^\frac{d-2}{d}}{r^2} \quad \textup{as}\;\; r \to 0, \;\; c_d > 0.$
To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner, Constantin, Kadanoff, Schenkel, and Venkataramani (Nonlinearity, 1999). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13932 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Construction of type I-Log blowup for the Keller-Segel system in dimensions $3$ and $4$ Nguyen, V. T. Nouaili, N. Zaag, H. Analysis of PDEs We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system $\partial_t u = Δu - \nabla \cdot (u \nabla \mathcal{K}_u), \quad -Δ\mathcal{K}_u = u \quad \textup{in}\;\; \mathbb{R}^d,\; d = 3,4,$ and derive the final blowup profile $ u(r,T) \sim c_d \frac{|\log r|^\frac{d-2}{d}}{r^2} \quad \textup{as}\;\; r \to 0, \;\; c_d > 0.$ To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner, Constantin, Kadanoff, Schenkel, and Venkataramani (Nonlinearity, 1999). |
| title | Construction of type I-Log blowup for the Keller-Segel system in dimensions $3$ and $4$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2309.13932 |