Large-time behavior of an attraction–repulsion chemotaxis system

In this study, we consider an attraction–repulsion chemotaxis system{ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,wt=Δw+γu−δw,x∈Ω,t>0 with homogeneous Neumann boundary conditions, where χ, ξ, α, β, γ and δ   are positive parameters. Under the critical condition that χα−ξγ=0χα−ξγ=0, we prove that the system possesses a unique global solution, which is uniformly bounded in the physical domain Ω⊂RnΩ⊂Rn (n=2,3n=2,3). In addition, we assert that we can find ϵ0>0ϵ0>0 such that for all of the initial data u0u0 that satisfy ∫Ωu0<ϵ0∫Ωu0<ϵ0, the solution of the system approaches the steady state (u¯0,αβu¯0,γδu¯0) exponentially as time tends to infinity, where u¯0:=1|Ω|∫Ωu0.