Invasion into remnant instability: a case study of front dynamics

We study the invasion of an unstable state by a propagating front in peculiar but generic situation where process exhibits remnant instability. Here, instability refers to fact that spatially constant invaded is linearly any exponentially weighted space frame moving with linear speed. Our main result nonlinear asymptotic stability selected for prototypical model coupling spatio-temporal oscillations and monotone dynamics. establish through decomposition perturbation into two pieces: one bounded second unbounded which converges uniformly zero unweighted at exponential rate. Interestingly, long-time numerical simulations reveal apparent some cases. exhibit how this caused round-off errors introduce resonant otherwise non-resonant modes, we determine accelerated