compact support. Further, since is mass-preserving and invariant under simultaneous multiplication of $\mu$ and $\omega$ by a positive constant, we may assume without loss of generality that $\mu(t)$ is a *probability* measure. By abuse of notation, we shall frequently identify $\mu$ with its density function.

The heuristics behind the long-time behavior of $\mu(t)$ is the following. In the *attraction dominated case* $q_r<q_a$, the attracting force $\nabla\psi_a\ast\omega$ generated by the reference profile is stronger than the repulsion force $\nabla\psi_r\ast\mu$ between the particles on large space scales. It is thus expected that particles remain at a bounded distance from $\omega$, and that the system equilibrates at a localized stationary solution. In the *repulsion dominated case* $q_r>q_a$ the repulsive force dominates the attracting one at large distances and will cause initially “sufficiently delocalized” solutions to diverge as $t\to\infty$; however, since attraction is stronger than repulsion at short distances, there might be solutions that remain localized for all times. In this paper, we are concerned with the *attraction dominated* case $q_r<q_a$, and with the *balanced* case $q_r=q_a$.

As a preliminary result, we prove well-posedness of the dynamics inside the set of probability measures with compactly supported and bounded density. Since we are working in 