Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $n\ge 3$
Philippe Souplet, Michael Winkler
TL;DR
In dimensions $n\ge 3$, the paper analyzes blow-up for radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system. By transforming to an averaged-mass equation for $w(r,t)$, the authors obtain sharp upper bounds $u(r,t)\le\bigl(\frac{1}{u(0,t)}+Kr^2\bigr)^{-1}$, leading to the final profile bound $U(x)\sim O(|x|^{-2})$ and a space-time estimate $u(x,t)\le C(T-t+|x|^2)^{-1}$ for type I blowup. They show the final blow-up profile satisfies $C_1|x|^{-2}\le U(x)\le C_2|x|^{-2}$ away from $0$, with $L^1$ convergence as $t\to T$, and under additional hypotheses on the initial data a matching lower bound $U(x)\ge c|x|^{-2}$ near the origin, indicating the final profile is governed by a $|x|^{-2}$ singularity rather than a Dirac mass. These results highlight a dimension-dependent distinction from the 2D case and connect to self-similar solutions, clarifying the asymptotic aggregation behavior in higher dimensions.
Abstract
We study the blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in space dimensions $n\ge 3$. In view of the biological background of this system and of its mass conservation property, blowup is usually interpreted as a phenomenon of concentration or aggregation of the bacterial population. Understanding the asymptotic behavior of solutions at the blowup time is thus meaningful for the interpretation of the model. Under mild assumptions on the initial data, for $n\ge 3$, we show that the final profile satisfies $C_1|x|^{-2}\le u(x,T)\le C_2|x|^{-2}$, with convergence in $L^1$ as $t\to T$. This is in sharp contrast with the two-dimensional case, where solutions are known to concentrate to a Dirac mass at the origin (plus an integrable part). We also obtain refined space-time estimates of the form $u(x,t)\le C(T-t+|x|^2)^{-1}$ for type~I blowup solutions. Previous work had shown that radial, self-similar blowup solutions (which satisfy the above estimates) exist in dimensions $n\ge 3$ and do not exist in dimension $2$. Our results thus reveal that the final profile displayed by these special solutions actually corresponds to a much more general phenomenon.