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

TL;DR

This paper proves the existence of global, bounded solutions for a Keller-Segel type system with supercritical initial data, addressing a previously unresolved case in chemotaxis models.

Contribution

It establishes the first known global existence result for supercritical initial data in a Keller-Segel system.

Findings

Global solutions exist for certain supercritical initial data.

Solutions remain uniformly bounded over time.

Addresses a gap in the understanding of chemotaxis models.

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 ($\|\rho_0\|_1<8\pi$) initial data, or blow--up in finite time for suitably chosen supercritical ($\|\rho_0\|_1>8\pi$) 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.