Global boundedness in an attraction-repulsion chemotaxis system with logistic source

We study the attraction-repulsion chemotaxis system of parabolic-elliptic type with logistic source: ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u), 0=Δv−βv+αu, 0=Δw−δw+γu, subject to the non-flux boundary conditions in a bounded domain Ω⊂Rn(n≥1) with smooth boundary, f(s)≤a−bsη for all s≥0, where constants χ,ξ,η,α,δ,γ,b>0, a≥0. The global boundedness of solutions to this problem was established by Li and Xiang (2016) for the repulsion domination case χα<ξγ with η≥1, the attraction domination case χα>ξγ with η>2 (or η=2, b properly large), and the balance case χα=ξγ with η>12(n2+4n−n+2), respectively. In the present paper we prove for the balance case χα=ξγ that the weakened restriction η>2n+2n+2 is sufficient to ensure the global boundedness of solutions.