Quantization of radial finite-time collapse in fully parabolic Keller–Segel
Let $\Omega = B_R(0) \subset \mathbb{R}^2$ be a disk and consider the fully parabolic Keller–Segel system $$u_t = \Delta u - \nabla \cdot (u \nabla v), \qquad v_t = \Delta v - v + u \qquad \text{in } \Omega \times (0, T),$$ with $\partial_\nu u = \partial_\nu v = 0$ on $\partial\Omega$ and radially symmetric, nonnegative, smooth initial data satisfying the compatibility conditions. Nagai, Senba and Suzuki proved that a radial solution can blow up only at the origin and that, if it blows up at a finite time $T$, then $u(\cdot, t) \rightharpoonup^* m \delta_0 + f$ as $t \uparrow T$ with $m \ge 8\pi$. Herrero and Velázquez constructed radial solutions that blow up in finite time with $m = 8\pi$.
Problem. Suppose the maximal classical solution blows up at a finite time $T$ and $$u(\cdot, t) \, dx \rightharpoonup^* m \delta_0 + f(x) \, dx \qquad (t \uparrow T),$$ where $f \in L^1(\Omega)$ is nonnegative and $m \ge 8\pi$. Must $m = 8\pi$?
Source: Stated as open by Soga (2025), §1, after the collapse theorem of Nagai, Senba and Suzuki (2000); stated for radial solutions in a disk by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, chemotactic collapse, finite-time blow-up, Keller–Segel system
References
@article{soga2026,
author = {Soga, Yuri},
title = {A sufficient condition for absence of mass quantization in a chemotaxis system with local sensing},
journal = {Nonlinear Anal. Real World Appl.},
volume = {88},
year = {2026},
pages = {104504},
doi = {10.1016/j.nonrwa.2025.104504},
eprint = {2505.10912},
archivePrefix = {arXiv},
}
@article{nagai2000,
author = {Nagai, Toshitaka and Senba, Takasi and Suzuki, Takashi},
title = {Chemotactic collapse in a parabolic system of mathematical biology},
journal = {Hiroshima Math. J.},
volume = {30},
number = {3},
year = {2000},
pages = {463--497},
doi = {10.32917/hmj/1206124609},
}
@article{herrero1997,
author = {Herrero, Miguel A. and Vel{\'a}zquez, Juan J. L.},
title = {A blow-up mechanism for a chemotaxis model},
journal = {Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)},
volume = {24},
number = {4},
year = {1997},
pages = {633--683},
url = {http://www.numdam.org/item/ASNSP_1997_4_24_4_633_0/},
}
@misc{aim-keller-segel-collapse-quantization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Quantization of radial finite-time collapse in fully parabolic Keller–Segel}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/373-keller-segel-collapse-quantization.md},
note = {Entry 373 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00604, "Quantization of radial finite-time collapse in fully parabolic Keller–Segel", clopen.solutions.