Critical mass for Keller–Segel systems with supercritical nonlinear sensitivity

Abstract

This paper is concerned with the following radially symmetric Keller–Segel systems with nonlinear sensitivity [Formula: see text] and [Formula: see text], posed on [Formula: see text] [Formula: see text] and subjected to homogeneous Neumann boundary conditions. It is well-known that [Formula: see text] is the critical exponent of the systems in the sense that all solutions exist globally if [Formula: see text] and there exist finite-time blowup solutions if [Formula: see text]. Here we consider the supercritical case [Formula: see text] and show a critical mass phenomenon. Precisely, we prove that there exists a critical mass [Formula: see text] such that (1) for arbitrary nonincreasing nonnegative initial data [Formula: see text] with [Formula: see text] and [Formula: see text], the corresponding solution blows up in finite time if [Formula: see text], and if [Formula: see text] we can only prove that the solution blows up in finite time or infinite time; (2) for some nonincreasing nonnegative initial data with [Formula: see text], the corresponding solutions are globally bounded. Our results extend that of Winkler’s paper [M. Winkler, How unstable is spatial homogeneity in Keller–Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic–elliptic cases, Math. Ann. 373 (2019) 1237–1282], where he proved similar results for the system with [Formula: see text].