Diffusive stability of oscillations in reaction-diffusion systems

TL;DR

This paper proves the nonlinear stability of spatially homogeneous oscillations in reaction-diffusion systems, showing localized perturbations decay algebraically and identifying the asymptotic behavior of solutions.

Contribution

It introduces a normal form transformation approach to analyze the stability of oscillations, accounting for phase modulation and quasilinear effects.

Findings

Localized perturbations decay as t^{-n/2} in space dimension n

Asymptotic expansion reveals phase modulation as leading order behavior

Stability proven via fixed point argument in weighted Sobolev spaces

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.