Dissipation for a non-convex gradient flow problem of a Patlack-Keller-Segel type for densities on ${\mathbb{R}}^n$, $n\ge 3$

We study an evolution equation that is the gradient flow in the $2$-Wasserstien metric of a non-convex functional for densities in ${\mathbb{R}}^n$ with $n\ge 3$. Like the Patlack-Keller-Segel system on ${\mathbb{R}}^2$, this evolution equation features a competition between the dispersive effects of diffusion, and the accretive effects of a concentrating drift. We determine a parameter range in which the diffusion dominates, and all mass leaves any fixed compact subset of ${\mathbb{R}}^n$ at an explicit polynomial rate.