Application of an Adams type inequality to a two-chemical substances chemotaxis system

Kentarou Fujie, Takasi Senba

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)

Abstract

This paper deals with positive solutions of the fully parabolic system, {ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞), under homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded domain Ω⊂Rn (n≤4) with positive parameters τ12,χ<0 and nonnegative smooth initial data (u0,v0,w0). In the lower dimensional case (n≤3), it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded. In the case n=4, it is shown that in the radially symmetric setting solutions to the Neumann boundary value problem of the system exist globally in time and remain bounded if ‖u0L1(Ω)>(8π)2/χ; as to the mixed boundary value problem, we will establish global existence and boundedness of solutions if ‖u0L1(Ω)>(8π)2/χ without radial symmetry. The key ingredients are a Lyapunov functional and an Adams type inequality. A Lyapunov functional of the above problems will be constructed and the constant (8π)2/χ is deduced from the critical constant in the Adams type inequality. This result is regarded as a generalization of the well-known 8π problem in the Keller–Segel system to higher dimensions.

Original languageEnglish
Pages (from-to)88-148
Number of pages61
JournalJournal of Differential Equations
Volume263
Issue number1
DOIs
Publication statusPublished - 2017 Jul 5
Externally publishedYes

Keywords

  • Adams' inequality
  • Chemotaxis
  • Global existence
  • Lyapunov functional

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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