\[thm:existence\]) is not new. Our motivation to include it here is that we also provide a short and quite elementary proof, based on the use of the inverse distribution function. Our strategy of proof builds on that of [@BDi08]. Several non-trivial modifications are necessary to deal with the lack of smoothness of $\psi_r(x)$ at $x=0$.

Equilibration of solutions to equations of type in one spatial dimension has been investigated in a series of papers [@FR2; @FR1; @ra10]; an extremely rich theory in higher space dimensions is currently developing as well, see e.g. [@BCLR]. Mainly, the shape and nonlinear stability of stationary states for have been analyzed for combined attractive-repulsive kernels $K$. Under suitable hypotheses, there exist non-trivial steady states $\mu^\star$, even in the absence of external attraction forces.

The only result with direct relation to our own work is [@FR2 Theorem 1.2]. There, weak convergence of $\mu(t)$ towards a stationary state has been established (essentially) under the hypothesis that the kernel function $K:{\mathbb{R}}\to{\mathbb{R}}$ is convex on ${\mathbb{R}}_+$ and can be written as the sum of a negative multiple of $|\cdot|$ and a $C^2$-smooth function. This covers in particular the situation $\psi_r(z)=|z|$, i.e., $q_r=1$, considered here. The essential improvement contained in 