Blow-up Profiles for the Parabolic-Elliptic Keller-Segel System in Dimensions n3

We study the blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in space dimensions n3. 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 n3, we show that the final profile satisfies C1|x|-2u(x,T)C2|x|-2, with convergence in L-1 as tT.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) 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 n3 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.