Sharp macroscopic blow-up behavior for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3
Abstract
We study the space-time concentration or blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel system in the whole space or in a ball. We show that, for any { blow-up} solution in dimensions 3≤ n≤ 9 (assuming finite mass in the whole space case), there exists a nonflat backward self-similar solution U such that u(x,t)=(1+o(1))U(x,t), as $(x,t)→ (0,T)$, { where T is the blow-up time.} This macroscopic behavior is important from the physical point of view, since it gives a sharp description of the concentration phenomenon in the scale of the original space-time variables~(x,t). It strongly improves on existing results, since such behavior was previously known ([Giga-Mizoguchi-Senba]) to hold only in the microscopic scale |x|≤ O(√(T−t)) as t→ T (and in the whole space case only). As a consequence, we obtain the two-sided global estimate C1≤ (T−t+|x|2)u(x,t)≤ C2 in $BR×(T/2,T)$, whose upper part only was known before ([Souplet-Winkler]), as well as the sharp final profile: limx→ 0 |x|2u(x,T)=L∈(0,∞). The latter improves, with a different proof, the recent result of [Bai-Zhou] by excluding the possibility L=0. We also give extensions of these results, in higher dimensions, to type~I and to time monotone solutions. Moreover, we extend the known results on type I estimates and on convergence in similarity variables, and significantly simplify their proofs.
Related
- CLASSIFICATION OF THE DYNAMICS OF RADIAL SOLUTIONS TO THE 2D PARABOLIC-ELLIPTIC KELLER-SEGEL SYSTEM
- Well-posedness of 2D stochastic parabolic-parabolic Keller-Segel system with logistic source
- Boundedness of radial solutions to the parabolic-parabolic Keller--Segel system on the plane with subcritical mass
- Blow-up suppression for the 2D parabolic-parabolic Patlak-Keller-Segel-Navier-Stokes system
- Infinite time blow-up of solutions to the classical Keller-Segel model of chemotaxis in higher dimensions
BibTeX
@misc{chabi2026sharp,
title = {{Sharp macroscopic blow-up behavior for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3}},
author = {Loth Damagui Chabi and Philippe Souplet},
howpublished = {HAL},
year = {2026},
month = oct,
eprint = {2601.14469},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05801885v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp macroscopic blow-up behavior for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3
AU - Loth Damagui Chabi
AU - Philippe Souplet
PY - 2026
DA - 2026/10/15
PB - HAL
UR - https://hal.science/hal-05801885v1
AB - We study the space-time concentration or blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel system in the whole space or in a ball. We show that, for any { blow-up} solution in dimensions 3≤ n≤ 9 (assuming finite mass in the whole space case), there exists a nonflat backward self-similar solution U such that u(x,t)=(1+o(1))U(x,t), as $(x,t)→ (0,T)$, { where T is the blow-up time.} This macroscopic behavior is important from the physical point of view, since it gives a sharp description of the concentration phenomenon in the scale of the original space-time variables~(x,t). It strongly improves on existing results, since such behavior was previously known ([Giga-Mizoguchi-Senba]) to hold only in the microscopic scale |x|≤ O(√(T−t)) as t→ T (and in the whole space case only). As a consequence, we obtain the two-sided global estimate C_1≤ (T−t+|x|^2)u(x,t)≤ C_2 in $B_R×(T/2,T)$, whose upper part only was known before ([Souplet-Winkler]), as well as the sharp final profile: lim_x→ 0 |x|^2u(x,T)=L∈(0,∞). The latter improves, with a different proof, the recent result of [Bai-Zhou] by excluding the possibility L=0. We also give extensions of these results, in higher dimensions, to type~I and to time monotone solutions. Moreover, we extend the known results on type I estimates and on convergence in similarity variables, and significantly simplify their proofs.
ER -