Diffusive stability of oscillations in reaction-diffusion systems

Thierry Gallay, Arnd Scheel

Research output: Contribution to journalArticlepeer-review

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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 languageEnglish (US)
Pages (from-to)2571-2598
Number of pages28
JournalTransactions of the American Mathematical Society
Issue number5
StatePublished - May 2011


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