Diffusive stability of oscillations in reaction-diffusion systems

Gallay, Thierry; Scheel, Arnd

doi:10.1090/S0002-9947-2010-05148-7

Diffusive stability of oscillations in reaction-diffusion systems
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by Thierry Gallay and Arnd Scheel

Trans. Amer. Math. Soc. 363 (2011), 2571-2598

DOI: https://doi.org/10.1090/S0002-9947-2010-05148-7

Published electronically: December 20, 2010

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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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