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: u(t) = Delta u - chi del center dot (u del v) + xi del center dot (u del w) + f(u), 0 = del v - beta v + alpha u, 0 = Delta w-delta w+gamma u, subject to the non-flux boundary conditions in a bounded domain Omega subset of R-n (n >= 1) with smooth boundary, f (s) <= a-bs(eta) for all s <= 0, where constants chi,xi,eta,alpha,delta,gamma,b > 0, a >= 0. The global boundedness of solutions to this problem was established by Li and Xiang (2016) for the repulsion domination case chi alpha < xi gamma with eta >= 1, the attraction domination case chi alpha > xi gamma with eta > 2 (or eta = 2, b properly large), and the balance case chi alpha = xi gamma with eta > 1/2(root n(2) + 4n - n + 2), respectively. In the present paper we prove for the balance case chi alpha = xi gamma that the weakened restriction eta > 2n+2/n+2 is sufficient to ensure the global boundedness of solutions. (C) 2018 Elsevier Ltd. All rights reserved.