Front propagation near the onset of instability

Avery, Montie

doi:10.1090/proc/17074

Front propagation near the onset of instability
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by Montie Avery;

Proc. Amer. Math. Soc. 153 (2025), 1093-1108

DOI: https://doi.org/10.1090/proc/17074

Published electronically: January 24, 2025

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Abstract:

We describe the resulting spatiotemporal dynamics when a homogeneous equilibrium loses stability in a spatially extended system. More precisely, we consider reaction-diffusion systems, assuming only that the reaction kinetics undergo a transcritical, saddle-node, or supercritical pitchfork bifurcation as a parameter passes through zero. We construct traveling front solutions which describe the invasion of the now-unstable state by a nearby stable state. We show that these fronts are marginally spectrally stable near the bifurcation point, which, together with recent advances in the theory of front propagation into unstable states, establishes that these fronts govern the dynamics of localized perturbation to the unstable state. Our proofs are based on functional analytic tools to study the existence and eigenvalue problems for fronts, which become singularly perturbed after a natural rescaling.

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Proceedings of the American Mathematical Society

Front propagation near the onset of instability
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Abstract: