BREZIS–MERLE INEQUALITIES AND APPLICATION TO THE GLOBAL EXISTENCE OF THE CAUCHY PROBLEM OF THE KELLER–SEGEL SYSTEM

Abstract

We discuss the existence of the global solution for two types of nonlinear parabolic systems called the Keller–Segel equation and attractive drift–diffusion equation in two space dimensions. We show that the system admits a unique global solution in [Formula: see text]. The proof is based upon the Brezis–Merle type inequalities of the elliptic and parabolic equations. The proof can be applied to the Cauchy problem which is describing the self-interacting system.