\mu(t,.)\right)(x) \, \mathrm{d} \mu(t,x) \, \mathrm{d} t \\
      &- \int_{0}^{\infty} \int_{\mathbb{R}} \partial_{x} \varphi(t,x) \cdot \left(\psi_a' \ast \omega\right)(x)\, \mathrm{d} \mu(t,x) \, \mathrm{d} t. 
    \end{split}
  \end{aligned}$$

We stress that Theorem \[thm:existence\] is not the main result of the paper. In fact, it can be obtained by combining results from the recent literature: the case $1<q_r\le2$ is covered by [@BCLR Theorem 7], and $q_r=1$ is contained in [@BS Theorem 4.3.1]. However, we provide a simple and short unifying proof of this result (see Appendix \[sec:proof-theor-refthm:l\]) which is based on the use of the inverse distribution function.

Asymptotic behavior {#sec:asymptotics}
===================

This section contains our main contribution: an analysis of the long time asymptotics in various regimes of $q_a$ and $q_r$. First, we give a definition of *steady states* in the spirit of Section \[sec:existence-theory-linfty\].

\[dfn:steady\] A probability measure $\mu^\star \in{\mathcal{P}}({\mathbb{R}})$ with density in $L^\infty_c({\mathbb{R}})$ is called a *steady state* for the equation and exponents $1\leq q_r, q_a \leq 2$, if its inverse distribution function $X^\star=X_{\mu^\star}$ satisfies $$\label{eq:evoloo}
        \begin{cases}
          \int_0^m\psi_a'\big(X^\star(z)-Y(\zeta)\big){\, \mathrm{d}}\zeta & \text{if $q_a>1$}, \\
          2G\big(X^\star(z)\big)-m & \text{if $q_a=1$},
        \end{cases}
        =
        \begin{cases}
          \int_0^1\psi_r'\big(X^\star(z)-X^\star(\zeta)\big){\, \mathrm{d}}\zeta & \text{if $q_r>1$}, \\
          2z-1 & \text{if $q_r=1$}
        \end{cases}$$ at every $z\in[0,1)$.

We shall now study various regimes of the 