Invasion into remnant instability: a case study of front dynamics

Grégory Faye (1) , Matt Holzer (2) , Arnd Scheel (3) , Lars Siemer

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 George Mason University [Fairfax]
3 UMN-MATH - School of Mathematics

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

Mots clés: traveling front; remnant instability; pointwise semigroup methods; absolute spectrum

Domaines: Equations aux dérivées partielles [math.AP]

Grégory Faye, Matt Holzer, Arnd Scheel, Lars Siemer. Invasion into remnant instability: a case study of front dynamics. Indiana University Mathematics Journal, In press. ⟨hal-02930171⟩