Finite propagation in fourth-order aggregation--diffusion

We prove finite propagation for a fourth-order aggregation--diffusion equation with linear mobility and quadratic attraction on $\R^d$. For every $d\ge2$, every positive attraction coefficient and every compactly supported nonnegative $H^1$ probability density, there is one global weak solution whose support remains in a bounded ball on each finite time interval. The solution has the same $L^\infty H^1\cap L^2H^2$ regularity and integrated weak formulation as the variational existence theory. A localized power-entropy estimate is proved on the same minimizing movements by a bounded-derivative auxiliary flow. The resulting superlinear inequality for the exterior spacetime tail forces that tail to vanish at a finite radius, in every finite dimension.