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SUMMARY:Sharp macroscopic blow-up behavior for the parabolic-elliptic Kell
 er-Segel system in dimensions $n\\ge 3$
DTSTART:20260127T101500Z
DTEND:20260127T111500Z
DTSTAMP:20260424T050700Z
UID:indico-event-15248@indico.math.cnrs.fr
DESCRIPTION:Speakers: Loth Chabi\n\n(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 $3\\le n\\le 9$ (ass
 uming finite mass in the whole space case)\,there exists a nonflat backwar
 d self-similar solution $U$ such that$$u(x\,t)=(1+o(1))U(x\,t)\,\\quad\\hb
 ox{as $(x\,t)\\to (0\,T)$.}$$ This macroscopic behavior was previously kn
 own 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 consequ
 ence\, 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 w
 as 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\, w
 ith a different proof\, the recent result of Bai-Zhou by excluding the pos
 sibility $L=0$.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 a
 nd significantly simplify their proofs.\n\nhttps://indico.math.cnrs.fr/eve
 nt/15248/
URL:https://indico.math.cnrs.fr/event/15248/
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