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We study the propagation of a Newtonian shock in a spherically symmetric,homologously expanding ejecta. We focus on media with a steep power-law densityprofile 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 areexpected in the leading edge of supernovae ejecta and sub-relativistic outflowsfrom binary neutron star mergers. We find that such shocks always accelerate inthe observer frame and lose causal contact with the bulk of the driver gas,owing to the steep density profile. However, the prolonged shock evolutionexhibits two distinct pathways: In one, the shock strength diminishes with timeuntil the shock eventually dies out. In the other, the shock strength steadilyincreases, and the solution approaches the self-similar solution of a shock isa static medium. By mapping the parameter space of shock solutions, we findthat the evolutionary pathways are dictated by $\alpha$ and by the initialratio between the shock velocity and the local upstream velocity. We find thatfor $\alpha<\omega_c$ ($\omega_c \approx 8$), the shock always decays, and thatfor $\alpha>\omega_c$ the shock may decay or grow stronger depending on theinitial value of the velocity ratio. These two branches bifurcate from aself-similar solution derived analytically for a constant velocity ratio. Weanalyze properties of the solutions that may have an impact on theobservational signatures of such systems, and assess the conditions requiredfor decaying shocks to break out from a finite medium.

Spherical shocks in a steep density gradient of expanding media