Large mass self-similar solutions of the parabolic-parabolic Keller--Segel model of chemotaxis

In two space dimensions, the parabolic-parabolic Keller--Segel system shares many properties with the parabolic-elliptic Keller--Segel system. In particular, solutions globally exist in both cases as long as their mass is less than 8?. However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above 8? always blow up. Here we study forward self-similar solutions of the parabolic-parabolic Keller--Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above 8?, which is forbidden in the parabolic-elliptic case.