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Journal of Evolution Equations

, Volume 8, Issue 2, pp 353–378 | Cite as

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

  • Hideo KozonoEmail author
  • Yoshie Sugiyama
Article

Abstract.

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every \(u_0 \in L^1 (\mathbb {R}^2)\). The local existence time is characterized for \(u_0 \in L^1 \cup L^{q*}(\mathbb {R}^2)\) with 1 < q * < 2. Next, we prove the finite time blow-up of strong solution under the assumption \(||u_0||_{L^{1}} > 8 \pi\) and \(||x|^2u_0||{L^1} < \frac {1}{\gamma}.g (||u_0||{L^1}/8\pi)\), 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.

Mathematics Subject Classifications (2000):

35K45 35K57 

Keywords:

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

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Copyright information

© Birkhaueser 2008

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan
  2. 2.Department of MathematicsTsuda CollegeTokyoJapan

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