is not directly applicable. In that situation, we assume in addition that $\mu(t)$ is absolutely continuous. Since for absolutely continuous $\nu\in{\mathfrak{M}_+}({\mathbb{R}})$, one has $$\begin{aligned}
    \int_{\mathbb{R}} \psi'(x - y) \, \mathrm{d} \nu(y) = \int_\mathbb{R} \operatorname{sgn}(x - y) \, \mathrm{d} \nu (y) 
  = \nu((-\infty,x]) - \nu((x,\infty))
  = 2F_\nu(x) - \nu(\mathbb{R})\label{eq:328}\end{aligned}$$ at every $x\in{\mathbb{R}}$, it follows that the evolution equation attains the special form $$\begin{aligned}
  \label{eq:evolq1}
  \partial_t X(t,z) = - \int_0^m \psi_a'(X(t,z) - Y(\zeta)) \, \mathrm{d} \zeta + 2z - 1.\end{aligned}$$ And in the particular case $q_a=q_r=1$, the evolution equation simplifies to $$\label{eq:229}
  \partial_{t}X(t,z) = 2 \left[z - G(X(t,z))\right]+m-1.$$

These formal calculations motivate our definition of solutions for in the next section.

Existence of solutions {#sec:existence-theory-linfty}
======================

Recall our choice for $\psi_a$, $\psi_r$, and let $\omega\in{\mathfrak{M}_+}({\mathbb{R}})$ be a given profile with total mass $m:=\omega({\mathbb{R}})>0$. We further assume that $\omega$ is absolutely continuous with a density function in $L^\infty_c$, so that $Y:=X_\omega\in{\mathrm{D}}([0,m))$.

\[dfn:transient\] A map $\mu:[0,\infty)\to{\mathcal{P}}_2({\mathbb{R}})$ is called a *transient solution* to , if it satisfies the following:

-   $\mu(t)$ has a density in $L^\infty_c$ at every time $t\ge0$,

-   the map $t\mapsto X_{\mu(t)}$ is continuously differentiable from $[0,\infty)$ into ${\mathrm{D}}([0,1))$,

-   the evolution equation $$\begin{aligned}
          \label{eq:evol}
          \begin{split}
            \partial_t X_{\mu(t)}(z) = 
            & - 
            \begin{cases}
              