Boundedness, blowup and critical mass phenomenon in competing chemotaxis

Hai Yang Jin, Zhian Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

47 Citations (Scopus)

Abstract

We consider the following attraction–repulsion Keller–Segel system: {ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω, with homogeneous Neumann boundary conditions in a bounded domain Ω⊂R2with smooth boundary. The system models the chemotactic interactions between one species (denoted by u) and two competing chemicals (denoted by v and w), which has important applications in Alzheimer's disease. Here all parameters χ, ξ, α, β, γ and δ are positive. By constructing a Lyapunov functional, we establish the global existence of uniformly-in-time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction (i.e., ξγ≥αχ). If the attraction dominates (i.e., ξγ<αχ), a critical mass phenomenon is found. Specifically speaking, we find a critical mass m⁎=4παχ−ξγ such that the solution exists globally with uniform-in-time bound if M<m⁎and blows up if M>m⁎and M∉{4πmθ:m∈N+} where N+denotes the set of positive integers and M=∫Ωu0dx the initial cell mass.
Original languageEnglish
Pages (from-to)162-196
Number of pages35
JournalJournal of Differential Equations
Volume260
Issue number1
DOIs
Publication statusPublished - 1 Jan 2016

Keywords

  • Attraction–repulsion
  • Blowup
  • Boundedness
  • Chemotaxis
  • Lyapunov functional

ASJC Scopus subject areas

  • Analysis

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