NONLINEAR CONVECTIVE STABILITY OF A CRITICAL PULLED FRONT UNDERGOING A TURING BIFURCATION AT ITS BACK: A CASE STUDY

Overview & Synopsis

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.