Invasion into Remnant Instability: A Case Study of Front Dynamics

Grégory Faye, Matt Holzer, Arnd Scheel, Lars Siemer

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2 Scopus citations


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 show 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.

Original languageEnglish (US)
Pages (from-to)1819-1896
Number of pages78
JournalIndiana University Mathematics Journal
Issue number5
StatePublished - 2022

Bibliographical note

Funding Information:
3.1. Acknowledgments G. Faye acknowledges support from Labex CIMI under grant agreement ANR-11-LABX-0040. The research of M. Holzer was partially supported by the National Science Foundation through DMS-2007759. A. Sheel was supported by the National Science Foundation through NSF DMS-1907391. L.S. acknowledges support by the Deutsche Forschungsgemeinschaft (German Research Foundation), Projektnummer 281474342/GRK2224/1.

Publisher Copyright:
© 2022 Department of Mathematics, Indiana University. All rights reserved.


  • Traveling front
  • absolute spectrum
  • pointwise semigroup methods
  • remnant instability


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