  Analytic solutions $F(v,t)$ of the nonlinear Boltzmann equation in
$d$-dimensions are studied for a new class of dissipative models, called
inelastic repulsive scatterers, interacting through pseudo-power law
repulsions, characterized by a strength parameter $\nu$, and embedding
inelastic hard spheres ($\nu=1$) and inelastic Maxwell models ($\nu=0$). The
systems are either freely cooling without energy input or driven by
thermostats, e.g. white noise, and approach stable nonequilibrium steady
states, or marginally stable homogeneous cooling states, where the data,
$v^d_0(t) F(v,t)$ plotted versus $c=v/v_0(t)$, collapse on a scaling or
similarity solution $f(c)$, where $v_0(t)$ is the r.m.s. velocity. The
dissipative interactions generate overpopulated high energy tails, described
generically by stretched Gaussians, $f(c) \sim \exp[-\beta c^b]$ with $0 < b <
2$, where $b=\nu$ with $\nu>0$ in free cooling, and $b=1+{1/2} \nu$ with $\nu
\geq 0$ when driven by white noise. Power law tails, $f(c) \sim 1/c^{a+d}$, are
only found in marginal cases, where the exponent $a$ is the root of a
transcendental equation. The stability threshold depend on the type of
thermostat, and is for the case of free cooling located at $\nu=0$. Moreover we
analyze an inelastic BGK-type kinetic equation with an energy dependent
collision frequency coupled to a thermostat, that captures all qualitative
properties of the velocity distribution function in Maxwell models, as
predicted by the full nonlinear Boltzmann equation, but fails for harder
interactions with $\nu>0$.
