  This paper is devoted to the analysis of non-negative solutions for the
chemotaxis model with nonlocal source in bounded domain. The qualitative
behavior of solutions is determined by the nonlinearity from the aggregation
and the reaction. The nonlocal nonlinear Fisher-KPP reaction helps preventing
blow-up phenomena in chemotaxis system. For appropriately chosen exponents, the
global existence of classical solutions is proved with arbitrary initial data,
where a modified Moser-Alikakos iteration method plays a key role in the a
priori estimates.
