  This paper studies the large time behavior of aggregation-diffusion
equations. For one spatial dimension with certain assumptions on the
interaction potential, the diffusion index $m$, and the initial data, we prove
the convergence to the unique steady state as time goes to infinity
(equilibration), with an explicit algebraic rate. The proof is based on a
uniform-in-time bound on the first moment of the density distribution, combined
with an energy dissipation rate estimate. This is the first result on the
equilibration of aggregation-diffusion equations for a general class of weakly
confining potentials $W(r)$: those satisfying
$\lim_{r\rightarrow\infty}W(r)<\infty$.
