pseudo-inverse $X_\nu$, then $$\begin{aligned}
    \label{eq:2}
    \big(\psi\ast\nu\big)(x) := \int_{\mathbb{R}}\psi(x-y){\, \mathrm{d}}\nu(y) = \int_0^{\nu({\mathbb{R}})} \psi(x-X_\nu(\zeta)){\, \mathrm{d}}\zeta \quad \text{for every $x\in{\mathbb{R}}$}.
  \end{aligned}$$

\[lem:17\] [@07_Carillo_Toscani_prob-metrics Section 2.2] Let $\mu,\nu\in{\mathfrak{M}_+}({\mathbb{R}})$ with $m:=\mu({\mathbb{R}})=\nu({\mathbb{R}})$. Then, for $p \in [1,\infty]$, the *$p$th Wasserstein distance* between $\mu$ and $\nu$ equals $$W_p(\mu,\nu) = \left\| X_\mu-X_\nu\right\|_{L^p} =
    \begin{cases}
      \left( \int_0^m \lvert X_\mu(z) - X_\nu(z)\rvert^p \, \mathrm{d} z \right)^{1/p} & 1 \leq p < \infty,\\
      \sup_{0<z<m} \left| X_\mu(z) - X_\nu(z) \right| & p = \infty.
    \end{cases}$$

We refer to [@piro13] for a recent notion of generalized Wasserstein-distance between measures of [*different*]{} mass. Some of the results we obtain below can be restated in terms of these generalized distances.

The transformed equation {#sec:pseudo-inverse-trans}
------------------------

In order to transform equation in terms of the pseudo-inverse, we introduce further notations. Denote by $\mu:[0,\infty)\to{\mathcal{P}}_2({\mathbb{R}})$ one of its solutions and by $\omega$ the given datum, as well as by $F(t,\cdot)=F_{\mu(t)}$ and $G=F_\omega$ the respective CDFs and by $X(t,\cdot)=X_{\mu(t)}$ and $Y=X_\omega$ their pseudo-inverses. Recall that we required $\mu(t)$ to be a probability measure at every $t\ge0$, but we allow $m:=\omega({\mathbb{R}})>0$ to be an arbitrary positive real, at least for a part of our results, so $X(t;\cdot)$ and $Y$ are defined on different domains in general.

Let us further assume 