  We analyse qualitative properties of the solutions to a mean-field equation
for particles interacting through a pairwise potential while diffusing by
Brownian motion. Interaction and diffusion compete with each other depending on
the character of the potential. We provide sufficient conditions on the
relation between the interaction potential and the initial data for diffusion
to be the dominant term. We give decay rates of Sobolev norms showing that
asymptotically for large times the behavior is then given by the heat equation.
Moreover, we show an optimal rate of convergence in the $L^1$-norm towards the
fundamental solution of the heat equation.
