Brezis-Merle inequalities and application to the global existence of the cauchy problem of the Keller-Segel system

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 L^∞_loc(0, ∞ L ^∞(ℝ²)). The proof is based upon the BrezisMerle type inequalities of the elliptic and parabolic equations. The proof can be applied to the Cauchy problem which is describing the self-interacting system.