## Differential and Integral Equations

- Differential Integral Equations
- Volume 23, Number 7/8 (2010), 601-618.

### 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)$_m$.

#### Article information

**Source**

Differential Integral Equations, Volume 23, Number 7/8 (2010), 601-618.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019186

**Mathematical Reviews number (MathSciNet)**

MR2654260

**Zentralblatt MATH identifier**

1240.35263

**Subjects**

Primary: 35K57: Reaction-diffusion equations 35K45: Initial value problems for second-order parabolic systems

#### Citation

Sugiyama, Yoshie. Asymptotic profile of blow-up solutions of Keller-Segel systems in super-critical cases. Differential Integral Equations 23 (2010), no. 7/8, 601--618. https://projecteuclid.org/euclid.die/1356019186