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

Abstract

We deal with (KS)$_m$ below for the super-critical cases of $q \ge m+\frac{2}{N}$ with $N \ge 2, \ m \ge 1, \ q \ge 2$. Based on an $\varepsilon$-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 $u^{\frac{N(q-m)}{2}} \in C_w([0,T]; L^1(\mathbb R^N))$. Moreover, we show that $u^{\frac{N(q-m)}{2}}$ forms a delta-function singularity at the blow-up time. Simultaneously, we give a sufficient condition on $u$ such that $u^{\frac{N(q-m)}{2}} \in C_w([0,T]; L^1(\mathbb R^N))$. Our condition exhibits a scaling invariant class associated with (KS)$_m$.