Nonlinear convective stability of a critical pulled front undergoing a Turing bifurcation at its back: a case study

TL;DR

This paper studies the nonlinear stability of a critical pulled front in a reaction-diffusion system undergoing a Turing bifurcation, revealing algebraic decay and bounded pattern formation despite instability.

Contribution

It demonstrates the asymptotic stability of a monostable pulled front undergoing a Turing bifurcation, using point-wise semigroup estimates and mode-filter techniques.

Findings

Front remains stable with algebraic decay rate t^{-3/2}

Turing pattern behind the front remains bounded in time

Front persists despite both equilibrium states being unstable

Abstract

We investigate a specific reaction-diffusion system that admits a monostable pulled front propagating at constant critical speed. When a small parameter changes sign, the stable equilibrium behind the front destabilizes, due to essential spectrum crossing the imaginary axis, causing a Turing bifurcation. Despite both equilibrium states are unstable, the front continues to exist, and is shown to be asymptotically stable, against suitably-localized perturbations, with algebraic temporal decay rate $t^{-3/2}$. To obtain such decay, we rely on point-wise semigroup estimates, and show that the Turing pattern behind the front remains bounded in time, by use of mode-filters.