On an attraction-repulsion chemotaxis model involving logistic source

This paper is concerned with the attraction-repulsion chemotaxis system involving logistic source: u(t) = triangle u - chi del <middle dot> (u del upsilon) + xi del <middle dot> (u del omega) + f(u), rho upsilon(t )= triangle upsilon - alpha 1 upsilon + beta(1)u, rho omega(t) = triangle omega - alpha(2)omega + beta(2)u under homogeneous Neumann boundary conditions with nonnegative initial data (u(0), upsilon(0), omega(0)) is an element of W-1,W-infinity (ohm))(3) , the parameters chi, xi, alpha(1), alpha(2), beta(1), beta(2) > 0, rho >= 0 subject to the non-flux boundary conditions in a bounded domain ohm subset of R (N) (N >= 3) with smooth boundary and f(u) <= au - mu u(2) with f(0) >= 0 and a >= 0, mu > 0 for all u > 0. Based on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a globally bounded classical solution provided that chi + xi < (<mu>)/(2) and there exists a constant beta(& lowast;) > 0 is sufficiently small for all beta(1), beta(2) < beta & lowast;.