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