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It is known that, for the parabolic-elliptic Keller-Segel system withcritical porous-medium diffusion in dimension $\RR^d$, $d \ge 3$ (also referredto as the quasilinear Smoluchowski-Poisson equation), there is a critical valueof the chemotactic sensitivity (measuring in some sense the strength of thedrift term) above which there are solutions blowing up in finite time and belowwhich all solutions are global in time. This global existence result is shownto remain true for the parabolic-parabolic Keller-Segel system with criticalporous-medium type diffusion in dimension $\RR^d$, $d \ge 3$, when thechemotactic sensitivity is below the same critical value. The solution isconstructed by using a minimising scheme involving the Kantorovich-Wassersteinmetric for the first component and the $L^2$-norm for the second component. Thecornerstone of the proof is the derivation of additional estimates which relieson a generalisation to a non-monotone functional of a method due to Matthes,McCann, & Savar\'e (2009).

The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\RR^d$, $d \ge 3$