  We study the propagation of a Newtonian shock in a spherically symmetric,
homologously expanding ejecta. We focus on media with a steep power-law density
profile of the form $\rho \propto t^{-3}v^{-\alpha}$, with $\alpha>5$, where
$v$ is the velocity of the expanding medium and $t$ is time. Such profiles are
expected in the leading edge of supernovae ejecta and sub-relativistic outflows
from binary neutron star mergers. We find that such shocks always accelerate in
the observer frame and lose causal contact with the bulk of the driver gas,
owing to the steep density profile. However, the prolonged shock evolution
exhibits two distinct pathways: In one, the shock strength diminishes with time
until the shock eventually dies out. In the other, the shock strength steadily
increases, and the solution approaches the self-similar solution of a shock is
a static medium. By mapping the parameter space of shock solutions, we find
that the evolutionary pathways are dictated by $\alpha$ and by the initial
ratio between the shock velocity and the local upstream velocity. We find that
for $\alpha<\omega_c$ ($\omega_c \approx 8$), the shock always decays, and that
for $\alpha>\omega_c$ the shock may decay or grow stronger depending on the
initial value of the velocity ratio. These two branches bifurcate from a
self-similar solution derived analytically for a constant velocity ratio. We
analyze properties of the solutions that may have an impact on the
observational signatures of such systems, and assess the conditions required
for decaying shocks to break out from a finite medium.
