Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type

Yoshie Sugiyama, Yumi Yahagi

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

We consider the quasi-linear Keller-Segel system of singular type, where the principal part Δum represents a fast diffusion like 0<m<1. We first construct a global weak solution with small initial data in the scaling invariant norm LN(q≥m)2 for all dimensions Nq≥2 and all exponents qq≥2. As for the large initial data, we show that there exists a blow-up solution in the case of N=2. In the second part, the decay property in Lr with 1<r<q≥ for 1?2Nmq≥<1 with the mass conservation is shown. On the other hand, in the case of 0<m<<;12N, the extinction phenomenon of solution is proved. It is clarified that the case of m=1<2N exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across 12N.< For the borderline case of m=1<2N, our solution decays in Lr exponentially as <.

Original languageEnglish
Pages (from-to)3047-3087
Number of pages41
JournalJournal of Differential Equations
Volume250
Issue number7
DOIs
Publication statusPublished - Apr 1 2011

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Fast Diffusion
Extinction
Blow-up
Decay
Global Weak Solutions
Mass Conservation
Blow-up Solution
Conservation
Exponent
Scaling
Norm
Invariant

All Science Journal Classification (ASJC) codes

  • Analysis

Cite this

Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type. / Sugiyama, Yoshie; Yahagi, Yumi.

In: Journal of Differential Equations, Vol. 250, No. 7, 01.04.2011, p. 3047-3087.

Research output: Contribution to journalArticle

Sugiyama, Yoshie ; Yahagi, Yumi. / Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type. In: Journal of Differential Equations. 2011 ; Vol. 250, No. 7. pp. 3047-3087.
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