Springer (part of Springer Nature), Journal of Mathematical Biology, 1(63), p. 1-32
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 a critical threshold M(c). 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 M(c) 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 M(c), which is forbidden in the parabolic-elliptic case.