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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Diffusive stability of oscillations in reaction-diffusion systems


Authors: Thierry Gallay and Arnd Scheel
Journal: Trans. Amer. Math. Soc. 363 (2011), 2571-2598
MSC (2010): Primary 35B35, 35B40, 35B10, 35K57
Published electronically: December 20, 2010
MathSciNet review: 2763727
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Abstract: We study nonlinear stability of spatially homogeneous oscillations in reaction-diffusion systems. Assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase, we prove that spatially localized perturbations decay algebraically with the diffusive rate $ t^{-n/2}$ in space dimension $ n$. We also compute the leading order term in the asymptotic expansion of the solution and show that it corresponds to a spatially localized modulation of the phase. Our approach is based on a normal form transformation in the kinetics ODE which partially decouples the phase equation at the expense of making the whole system quasilinear. Stability is then obtained by a global fixed point argument in temporally weighted Sobolev spaces.


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Additional Information

Thierry Gallay
Affiliation: Institut Fourier, UMR CNRS 5582, BP 74, Université de Grenoble I, 38402 Saint-Martin-d’Hères, France
Email: Thierry.Gallay@ujf-grenoble.fr

Arnd Scheel
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
Email: scheel@umn.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05148-7
PII: S 0002-9947(2010)05148-7
Received by editor(s): June 24, 2008
Received by editor(s) in revised form: June 26, 2009
Published electronically: December 20, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.