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—X▽ ·(u▽u)+ ζ▽ ·(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 x,ζ,α,δ,γ,b>0,a>0.The global boundedness of solutions to this problem was established in Li and Xiang(2016)for the repulsion domination case Xα<ζγ with η≥ 1,the attraction domination case xa>ζy withη>2(or η = 2,b properly large),and the balance case Xα。ζ with η>1/2((?)-n+2),respectively.In the present paper we prove for the balance case Xa ＝ ζγ that the weakened restriction η>2n+2/n+2 is sufficient to ensure the global boundedness of solutions.