Global Boundedness in a Logarithmic Keller–Segel System

In this paper, we propose a user-friendly integral inequality to study the coupled parabolic chemotaxis system with singular sensitivity under the Neumann boundary condition. Under a low diffusion rate, the classical solution of this system is uniformly bounded. Our proof replies on the construction of the energy functional containing ∫Ω|v|4v2 with v>0. It is noteworthy that the inequality used in the paper may be applied to study other chemotaxis systems.