Sharp macroscopic blow-up behavior for the parabolic-elliptic Keller-Segel system in dimensions $n\ge 3$

TL;DR

The paper provides a sharp macroscopic description of finite-time blow-up for the parabolic-elliptic Keller-Segel system in dimensions $3\le n\le 9$ with radially decreasing data. It proves the existence of nonflat backward self-similar profiles $U\in\mathcal S\setminus\{1\}$ and shows that solutions satisfy $u(x,t)=(1+o(1))\frac{1}{T-t}U\left(\frac{|x|}{\sqrt{T-t}}\right)$ as $(x,t)\to(0,T)$, together with a two-sided bound $\frac{C_1}{T-t+|x|^2}\le u(x,t)\le\frac{C_2}{T-t+|x|^2}$ in a neighborhood of blow-up and a sharp final profile $u(x,T)\sim L|x|^{-2}$ with $L>0$. The work extends to type I and time-monotone solutions, refines known estimates, and provides a streamlined, robust proof strategy based on the averaged mass, similarity variables, and zero-number arguments to yield the macroscopic concentration behavior in the original space-time variables. These results significantly sharpen the understanding of chemotactic blow-up by delivering macroscopic-scale asymptotics and a precise final blow-up profile, complementing prior microscopic descriptions.

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 solution in dimensions $3\le n\le 9$ (assuming finite mass in the whole space case), there exists a nonflat backward self-similar solution $U$ such that $$u(x,t)=(1+o(1))U(x,t),\quad\hbox{as $(x,t)\to (0,T)$.}$$ 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 (\cite{GMS}) to hold only in the microscopic scale $|x|\le O(\sqrt{T-t})$ as $t\to T$ (and in the whole space case only). As a consequence, we obtain the two-sided global estimate $$C_1\le (T-t+|x|^2)u(x,t)\le C_2\quad\hbox{in $B_R\times(T/2,T)$},$$ whose upper part only was known before (\cite{Soup-Win}), as well as the sharp final profile: $$\lim_{x\to 0} |x|^2u(x,T)=L\in(0,\infty).$$ The latter improves, with a different proof, the recent result of \cite{BZ} 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.

Theorems & Definitions (43)

• Theorem 2.1
• Remark 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 2.4
• Corollary 2.5
• Remark 3.1
• Remark 3.2
• Proposition 4.1
• Remark 4.1