provided that it has mass larger or equal to one; if $\omega$ is of smaller mass, then there exists no steady state (of unit mass), and $\mu(t)$ converges vaguely to $\omega$, losing excess mass towards infinity, see Theorem \[thm:q1bis\]

-   If $1<q_r<q_a\le2$, or if $1<q_r=q_a<4/3$, and if $\omega$ has mass one, then we prove weak convergence of $\mu(t)$ to some stationary solution $\mu^\star$, see Theorem \[thm:1q2\]. Our proof is based on a compactness argument, and does not lead to an explicit characterization of the stationary state.

The detailed results in the special cases $q_r=1$ and $q_r=2$ are obtained by direct calculations with the inverse distribution function of $\mu(t)$. The proof of equilibration for the more general situations with $1<q_r<q_a\le2$ or $1<q_r=q_a<4/3$ relies on subtle moment estimates for measures $\mu$ of finite energy. These estimates have been recently derived by the second and third author, and are published in a companion paper [@13-FornasierHuetter-VarProperties].

Motivation: image dithering
---------------------------

Equation was introduced in [@FHS12] in the context of discrete variational methods for image dithering [@scgwbrwe11; @testgwscwe11; @grpost11]. In particular, in [@testgwscwe11], for $\psi_a(\cdot)=\psi_r(\cdot)=| \cdot|$ being the Euclidean norm, the authors considered the discrete energy functional $$\label{starting point}
  \mathcal E_N[p] := -\frac{1}{2 N^2} \sum_{i,j=1}^N  