Blow-up criteria for the classical Keller-Segel model of chemotaxis in higher dimensions (Q2042674)

The author derives criteria for global-in-time existence versus finite time blow-up of solutions for radially symmetric solutions of the minimal parabolic-elliptic Keller-Segel model of chemotaxis in the whole space \(\mathbb R^d\). They are expressed in terms of averaged initial data compared to averaged stationary solutions. The proofs are based on delicate monotonicity arguments. It is worth noting that conditions for the blow-up in dimensions \(3\le d\le 9\) differ from those for \(d\ge 10\) which reflects a different structure of the set of stationary solutions for such \(d\)'s.