  In this paper, we prove the nonlinear stability under localized perturbations
of spectrally stable time-periodic source defects of reaction-diffusion
systems. Consisting of a core that emits periodic wave trains to each side,
source defects are important as organizing centers of more complicated flows.
Our analysis uses spatial dynamics combined with an instantaneous
phase-tracking technique to obtain detailed pointwise estimates describing
perturbations to lowest order as a phase-shift radiating outward at a linear
rate plus a pair of localized approximately Gaussian excitations along the
phase-shift boundaries; we show that in the wake of these outgoing waves the
perturbed solution converges time-exponentially to a space-time translate of
the original source pattern.
