Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity

Kentarou Fujie, Michael Winkler, Tomomi Yokota

Research output: Contribution to journalArticle

32 Citations (Scopus)

Abstract

This paper is concerned with the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic source,{ut=Δu-χ(u/ vv)+ru-μu2,xεΩ,t>0,0=Δv-v+u,xεΩ, t>0, under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊃ R2, where χ>0,rεR,μ>0, with nonnegative initial data 0≢ u0εC0(Ω ̄). It is shown that in this two-dimensional setting, the absorptive character of the logistic kinetics is sufficient to enforce global existence of classical solutions even for arbitrarily large χ>0 and any μ>0 and rεR. It is moreover shown that if in addition r>0 is sufficiently large then all these solutions are uniformly bounded. A main step in the derivation of these results consists of establishing appropriate positive a priori bounds from below for the mass functional ℓΩu, which due to the presence of logistic kinetics is not preserved. These in turn provide pointwise lower bounds for v, which then allow for the choice of p>1, explicitly depending inter alia on infv, such that ℓΩu p(x,t)dx can be suitably bounded from above.

Original languageEnglish
Pages (from-to)56-71
Number of pages16
JournalNonlinear Analysis, Theory, Methods and Applications
Volume109
DOIs
Publication statusPublished - Nov 2014

Keywords

  • Boundedness
  • Chemotaxis
  • Global existence
  • Logarithmic sensitivity

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