On the existence of global solutions for the 3D chemorepulsion system

In this paper, we give sufficient conditions for global-in-time existence of classical solutions for the fully parabolic chemorepulsion system posed on a convex, bounded three-dimensional domain. Our main result establishes global-in-time existence of regular nonnegative solutions provided that $\mathrm{\nabla }\sqrt{u}\in L^4(0,T;L^2(\mathrm{\Omega }))$. Our method is related to the Bakry–Émery calculation and appears to be new in this context.