rapidly growing literature. Below, we only mention very few selected contributions that are related to our own analysis.

In [@CDFLS], a well-posedness theory has been developed mainly for attractive kernels $K$ that are symmetric and $C^1$-regular except possibly for the origin, and $\lambda$-convex (see also [@CLM] for an extension to barely $\lambda$-convex potentials without any regularity assumption). Existence and uniqueness are discussed for measure-valued solutions $\mu(t)$. In fact, it is one of the key observations that despite the relatively high regularity imposed on $K$ by the $\lambda$-convexity assumption, solutions $\mu$ to may generate point masses in finite time. In [@belaro11], convexity and smoothness hypotheses on $K$ have been relaxed, and additional “Osgood” criteria have been formulated under which solutions remain absolutely continuous. In this context, a solution theory in $L^p$ was developed. This theory has been further extended in [@BCLR], leading e.g. to existence and uniqueness of classical solutions in the cases $K(x)\approx-|x|^\alpha$ with $\alpha>1$ near $x=0$ — which corresponds to $\psi_r$ in with $1<q_r\le 2$ in our context. The important special case $K(x)\approx-|x|$ has been analyzed in [@BS], see also [@BCDP].

In view of [@BCLR; @BS], our result on well-posedness of for the considered range of kernels (see Theorem 