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Diffusive stability of Turing patterns via normal forms

Authors
  • Scheel, Arnd
  • Wu, Qiliang
Type
Preprint
Publication Date
Mar 10, 2013
Submission Date
Mar 10, 2013
Identifiers
DOI: 10.1007/s10884-013-9335-0
Source
arXiv
License
Yellow
External links

Abstract

We investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a "normal form" coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction-diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove nonlinear stability of Turing patterns with respect to perturbations that are small in $L^1\cap L^\infty$, with sharp rates.

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