Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with $d\ge3$ and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass $M_c$ such that if $M \in (0,M_c]$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for $M \in (0,M_c)$. While characterising the eventual infinite time blowing-up profile for $M=M_c$, we observe that the long time asymptotics are much more complicated than in the classical Patlak-Keller-Segel system in dimension two.