  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 $n \geq 3$. 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.
