## 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.

Original language | English (US) |
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Pages (from-to) | 2571-2598 |

Number of pages | 28 |

Journal | Transactions of the American Mathematical Society |

Volume | 363 |

Issue number | 5 |

DOIs | |

State | Published - May 2011 |