Fault zones are often surrounded by a damage zone that exhibits lower seismic velocities than the wall-rock, influencing earthquake propagation and arrest. We present a one-dimensional minimal model of frictional rupture, that approximates the elastodynamics of a fault embedded within a low velocity damage zone. This model predicts two families of steady-state rupture solutions: an overdamped regime, describing a crack-like rupture, and an underdamped regime with oscillating slip-rate in the wake of the rupture, which promotes pulse-like dynamics. The minimal model contains two free parameters: pre-stress on the fault, and seismic velocity reduction in the damage zone. The 1D model results are validated by two-dimensional elastodynamics simulations of earthquake rupture. We discuss the applicability of our model results to natural observations, identifying the preferred rupture style as function of structure of the fault zone, and the geological consequences of oscillatory slip in the wake of pulse-like ruptures.