CLASSIFICATION OF THE DYNAMICS OF RADIAL SOLUTIONS TO THE 2D PARABOLIC-ELLIPTIC KELLER-SEGEL SYSTEM
Abstract
This note gives a complete classification of the asymptotic behavior of radial solutions to the two-dimensional parabolic-elliptic Keller-Segel system on the whole space, for general initial data in the large. We review previous separate results, and unify them within a single classification framework. Depending on the mass, the flow exhibits three distinct asymptotic regimes. For a subcritical mass, solutions converge toward the unique self-similar expander of same mass. At the critical mass 8π, solutions concentrate in infinite time around the stationary state with a universal logarithmic rate. The determination of this behaviour was the last missing step for achieving a complete radial classification, and we prove it in a companion paper (in fact without radial assumption). For a supercritical mass, solutions undergo type II finite-time blow-up with an explicit universal asymptotic rate and the stationary state as profile. This trichotomy holds for all radial initial data with finite second momentum. For non-radial data or infinite second momentum, it is known that other dynamics can be possible; for each of these three universal regimes we review the known results showing how they persist.
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BibTeX
@misc{buseghin2026classification,
title = {{CLASSIFICATION OF THE DYNAMICS OF RADIAL SOLUTIONS TO THE 2D PARABOLIC-ELLIPTIC KELLER-SEGEL SYSTEM}},
author = {Federico Buseghin and Charles Collot},
howpublished = {HAL},
year = {2026},
month = oct,
eprint = {2606.20870},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05781866v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - CLASSIFICATION OF THE DYNAMICS OF RADIAL SOLUTIONS TO THE 2D PARABOLIC-ELLIPTIC KELLER-SEGEL SYSTEM
AU - Federico Buseghin
AU - Charles Collot
PY - 2026
DA - 2026/10/06
PB - HAL
UR - https://hal.science/hal-05781866v1
AB - This note gives a complete classification of the asymptotic behavior of radial solutions to the two-dimensional parabolic-elliptic Keller-Segel system on the whole space, for general initial data in the large. We review previous separate results, and unify them within a single classification framework. Depending on the mass, the flow exhibits three distinct asymptotic regimes. For a subcritical mass, solutions converge toward the unique self-similar expander of same mass. At the critical mass 8π, solutions concentrate in infinite time around the stationary state with a universal logarithmic rate. The determination of this behaviour was the last missing step for achieving a complete radial classification, and we prove it in a companion paper (in fact without radial assumption). For a supercritical mass, solutions undergo type II finite-time blow-up with an explicit universal asymptotic rate and the stationary state as profile. This trichotomy holds for all radial initial data with finite second momentum. For non-radial data or infinite second momentum, it is known that other dynamics can be possible; for each of these three universal regimes we review the known results showing how they persist.
ER -