A global existence result for a Keller-Segel type system with supercritical initial data

Abstract

We consider a parabolic-elliptic Keller-Segel type system, which is related to a simplified model of chemotaxis. Concerning the maximal range of existence of solutions, there are essentially two kinds of results: either global existence in time for general subcritical ($\Vert {\rho }_0{\Vert }_1<8\pi$) initial data, or blow--up in finite time for suitably chosen supercritical ($\Vert {\rho }_0{\Vert }_1>8\pi$) initial data with concentration around finitely many points. As a matter of fact there are no results claiming the existence of global solutions in the supercritical case. We solve this problem here and prove that, for a particular set of initial data which share large supercritical masses, the corresponding solution is global and uniformly bounded.