our own result (see Theorem \[thm:q1\]) is the explicit characterization of the steady state $\mu^\star$.

Long-time asymptotics for the general case of power-type kernels with exponents $1<q_r\le q_a\le 2$ (see Theorem \[thm:1q2\]) have apparently not been addressed in the literature before.

Structure of the paper {#sec:structure-paper}
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This paper is organized as follows. In Section \[sec:pseudo-inverse-techn\] we recall some basic facts about the representation of positive finite measures by means of pseudo-inverse functions. This representation is our key technical tool, and it also motivates our notion of transient solutions for . Section \[sec:existence-theory-linfty\] is dedicated to the result of well-posedness of the initial-value problem; its proof is given in Appendix \[sec:proof-theor-refthm:l\]. The core of the paper is Section \[sec:asymptotics\], where we study the long time asymptotics of solutions. The abstract convergence argument for $ q_r = q_a \in [1, 4/3) $ relies on subtle moment bounds obtained in [@13-FornasierHuetter-VarProperties]; for the sake of self-containedness, the derivation of these bounds is sketched in Appendix \[sec:moment-bound-subl\].

Notations {#notations .unnumbered}
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We write ${\mathfrak{M}_+}({\mathbb{R}})$ for the space of non-negative finite measures on the real line ${\mathbb{R}}$, and ${\mathcal{P}}({\mathbb{R}})$, ${\mathcal{P}}_p({\mathbb{R}})$ denote the subspaces of probability measures, and of probability measures with finite $p$th moment, 