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

par Loth Chabi

mardi 27 janv. 2026, 11:15 → 12:15 Europe/Paris

(Joint work with Philippe Souplet): We study the space-time 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 was previously known to hold only in the microscopic scale $|x|\le C\sqrt{T-t}$ as $t\to T$ (and in the whole space only [Giga-Mizogouchi-Senba]). 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 [Souplet-Winkler], 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 Bai-Zhou by excluding the possibility $L=0$.
We give extensions of our 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.