Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions

This paper studies the attraction-repulsion chemotaxis system with logistic source ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u), vt=Δv−α1v+β1u, wt=Δw−α2w+β2u in a smooth bounded convex domain Ω⊂R3, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where χ, ξ, αi and βi (i=1,2) are positive parameters and the logistic source function f fulfills f(s)=s−μsγ+1,s≥0,μ>0andγ≥1. It is shown that this system possesses a unique global bounded classical solution under the conditions αi≥12 and μ≥max⁡{(412χβ1+9ξβ2)γ,(9χβ1+412ξβ2)γ}. Furthermore, whenever u0≢0 and for any γ∈N, the solution of the system approaches to the steady state ((1μ)1γ,(1μ)1γβ1α1,(1μ)1γβ2α2) in the norm of L∞(Ω) as t→∞.