Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source

In this paper, we study the attraction-repulsion chemotaxis system with logistic source: u(t) = u-delta(u delta v)+delta(u delta w)+f(u), 0 = v-v+u, 0 = w-w+u, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain < subset of>R4, where ,,,,, and are positive constants, and f:RR is a smooth function satisfying f(s) a-bs(3/2) for all s 0 with a 0 and b > 0. It is proved that when the repulsion cancels the attraction (ie, =), for any nonnegative initial data u0C0(Omega), the solution is globally bounded. This result corresponds to the one in the classical 2-dimensional Keller-Segel model with logistic source bearing quadric growth restrictions.