  We give sharp conditions for the large time asymptotic simplification of
aggregation-diffusion equations with linear diffusion. As soon as the
interaction potential is bounded and its first and second derivatives decay
fast enough at infinity, then the linear diffusion overcomes its effect, either
attractive or repulsive, for large times independently of the initial data, and
solutions behave like the fundamental solution of the heat equation with some
rate. The potential $W(x) \sim \log |x|$ for $|x| \gg 1$ appears as the natural
limiting case when the intermediate asymptotics change. In order to obtain such
a result, we produce uniform-in-time estimates in a suitable rescaled change of
variables for the entropy, the second moment, Sobolev norms and the $C^\alpha$
regularity with a novel approach for this family of equations using modulus of
continuity techniques.
