Boundedness, blowup and critical mass phenomenon in competing chemotaxis

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⊂R2Ω⊂R2 with 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≥αχξγ≥αχ). If the attraction dominates (i.e., ξγ<αχξγ<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 Mm⁎M>m⁎ and M∉{4πmθ:m∈N+} where N+N+ denotes the set of positive integers and M=∫Ωu0dxM=∫Ωu0dx the initial cell mass.