Nonlocal nonlinear reaction preventing blow-up in supercritical case of chemotaxis system

This paper is devoted to the analysis of non-negative solutions for the chemotaxis model with nonlocal nonlinear source in bounded domain. The qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction. When the growth factor is stronger than the dampening effect, with the help of the nonlocal nonlinear term in the reaction, for appropriately chosen exponents and arbitrary initial data, the model admits a classical solution which is uniformly bounded. Moreover, when the growth factor has the same order with the dampening effect, the nonlocal nonlinear exponents can prevent the chemotactic collapse.