Differential and Integral Equations

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

Yoshie Sugiyama

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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


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