Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system

Hideo Kozono, Yoshie Sugiyama

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31 Citations (Scopus)

Abstract

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every u0 ∈ L1 (ℝ2). The local existence time is characterized for u 0 ∈ L1 ∩ Lq* (ℝ2) with 1 < q* < 2. Next, we prove the finite time blow-up of strong solution under the assumption ∥u0L1 > 8π and ∥x|2u0L1 < 1/γ·g (∥u0L1/8π), where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.

Original languageEnglish
Pages (from-to)353-378
Number of pages26
JournalJournal of Evolution Equations
Volume8
Issue number2
DOIs
Publication statusPublished - 2008 May 1

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Keywords

  • Blow up
  • Blow-up rate
  • Keller-Segel system
  • Local and global existence

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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