one space dimension, the evolution equation can be reformulated as an integro-differential equation for the inverse distribution function of $\mu$. In that framework, we obtain existence and uniqueness using a fixed point argument, see Theorem \[thm:existence\].

The main part of the paper is then devoted to the long-time behavior of these solutions. Depending on the value of the parameters $q_a$ and $q_r$, we obtain a more or less complete picture.

-   For $q_r=q_a=2$, the behavior is threefold, see Theorem \[thm:q2\]: if $\omega$ has mass larger than one, then $\mu(t)$ converges weakly to a Dirac measure concentrated at $\omega$’s center of mass; if $\omega $ is of mass smaller than one, then $\mu(t)$ converges vaguely to zero; finally, if $\omega$ is of unit mass, then each $\mu(t)$ is a translate of $\mu_0$, and $\mu(t)$’s center of mass converges to that of $\omega$ exponentially fast.

-   For $1=q_r<q_a\le2$, the solution $\mu(t)$ converges weakly to a steady state $\mu^\star$ which is obtained by cutting off the positive function $\psi_a''\ast\omega$ in a symmetric way around its median to normalize its mass to one. See Theorem \[thm:q1\] for details. In the more delicate case $q_a=1$, $\mu^\star$ is given by a symmetric cut-off of $\omega$ itself, 