to the center of mass of $\omega$ at exponential rate $\exp(-2(m-1)t)$ for $m>1$, while for $0<m<1$, the value of $X(t,z)$ diverges to $+\infty$ or to $-\infty$ if $X(0,z)$ lies to the right or to the left of $\mu(0)$’s center of mass, respectively. In the borderline case $m=1$, the functions $X(t,\cdot)$ at different times $t\ge0$ only differ by a constant. In conclusion, we obtain the following.

\[thm:q2\] Assume $q_a=q_r=2$, and let $\mu:[0,\infty)\to{\mathcal{P}}({\mathbb{R}})$ be a transient solution.

-   If $m>1$, then $\mu(t)$ converges weakly to a Dirac measure concentrated at $\omega$’s center of mass.

-   If $0<m<1$, then $\mu(t)$ converges vaguely to zero.

-   If $m=1$, then every $\mu(t)$ is a translate of $\mu_0$, and the center of mass of $\mu(t)$ converges to that of $\omega$ with exponential rate $\exp(-2mt)$.

The case $q_r = 1$ {#sec:case-q-1}
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Next, we consider the case $ q_r = 1 $. Then $ \psi_r $ is a multiple of the Newtonian potential in 1D, [i.e.ifnextchar,]{}$\psi_r''=2\delta_0$. Definition \[dfn:steady\] of stationary solution $\mu^\star$ specializes to the following equation for its inverse distribution function $X^\star$: $$\label{eq:360}
  2z - 1 =
  \begin{cases}
    \int_0^m \psi_a'(X^\star(z) - Y(\zeta)) \, \mathrm{d} \zeta & \text{if $q_a>1$}, \\
    2G(X^\star(z))-m & \text{if $q_a=1$}.
  \end{cases}$$ In Proposition 