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 = ?u—X? ·(u?u)+ ?? ·(u?w)+ f(u),0 = ?v—?v + ?u,0 = ?w—?w + ?u,subject to the non-flux boundary conditions in a bounded domain ?(?)Rn(n? 1)with smooth boundary,f(s)? a—bs? for all s>0,where constants x,?,?,?,?,b>0,a>0.The global boundedness of solutions to this problem was established in Li and Xiang(2016)for the repulsion domination case X?<?? with ?? 1,the attraction domination case xa>?y with?>2(or ? = 2,b properly large),and the balance case X??? with ?>1/2((?)-n+2),respectively.In the present paper we prove for the balance case Xa ? ?? that the weakened restriction ?>2n+2/n+2 is sufficient to ensure the global boundedness of solutions.