Critical mass for finite-time chemotactic collapse in the critical dimension via comparison
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909617475289088 |
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| author | Mao, Xuan Liu, Meng Li, Yuxiang |
| author_facet | Mao, Xuan Liu, Meng Li, Yuxiang |
| contents | We study the Neumann initial-boundary value problem for the parabolic-elliptic chemotaxis system, proposed by Jäger and Luckhaus (1992). We confirm that their comparison methods can be simplified and refined, applicable to seek the critical mass $8π$ concerning finite-time blowup in the unit disk. As an application, we deal with a parabolic-elliptic-parabolic chemotaxis model involving indirect signal production in the unit ball of $\mathbb R^4$, proposed by Tao and Winkler (2025). Within the framework of radially symmetric solutions, we prove that if initial mass is less than $64π^2$, then solution is globally bounded; for any $m$ exceeding $64π^2$, there exist nonnegative initial data with prescribed mass $m$ such that the corresponding classical solutions exhibit a formation of Dirac-delta type singularity in finite time, termed a chemotactic collapse. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical mass for finite-time chemotactic collapse in the critical dimension via comparison Mao, Xuan Liu, Meng Li, Yuxiang Analysis of PDEs We study the Neumann initial-boundary value problem for the parabolic-elliptic chemotaxis system, proposed by Jäger and Luckhaus (1992). We confirm that their comparison methods can be simplified and refined, applicable to seek the critical mass $8π$ concerning finite-time blowup in the unit disk. As an application, we deal with a parabolic-elliptic-parabolic chemotaxis model involving indirect signal production in the unit ball of $\mathbb R^4$, proposed by Tao and Winkler (2025). Within the framework of radially symmetric solutions, we prove that if initial mass is less than $64π^2$, then solution is globally bounded; for any $m$ exceeding $64π^2$, there exist nonnegative initial data with prescribed mass $m$ such that the corresponding classical solutions exhibit a formation of Dirac-delta type singularity in finite time, termed a chemotactic collapse. |
| title | Critical mass for finite-time chemotactic collapse in the critical dimension via comparison |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.14278 |