A generalization of the Keller-Segel system to higher dimensions from a structural viewpoint

Abstract

We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than (8\pi)^2, whereas the solution emanating from some initial data of large magnitude may blows up.This result can be regarded as a generalization of the well-known 8\pi​ problem in the Keller--Segel system to higher dimensions. We will compare mathematical structures of the Keller--Segel system and our system and discuss the difference.