Asymptotic profile of blow-up solutions of Keller-Segel systems in super-critical cases

We deal with (KS)_m below for the super-critical cases of q ≥ m+ 2/N with N ≥ 2, m ≥ 1; q ≥ 2. Based on an ε-regularity theorem in [20], we prove that the set S_u of blow-up points of the weak solution u consists of finitely many points if {equation presented}. Moreover, we show that {equation presented} forms a delta-function singularity at the blow-up time. Simultaneously, we give a suficient condition on u such that {equation presented}. Our condition exhibits a scaling invariant class associated with (KS)_m.