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Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction-diffusion equations. (English) Zbl 1246.35034
The authors consider stability of periodic travelling waves (wavetrains) in systems of reaction diffusion equations. Under spectral stability assumptions they prove nonlinear stability in the sense that perturbations that are sufficiently small in \(L^1\cap H^k\) decay in \(H^k\) as \(t^{- 1/4}\). Towards this they construct (by integral representations) suitable co-moving frames (which decay) that enjoy \(L^p\) estimates.
The approach appears to provide an alternative and extension to convection of the classical results of G. Schneider.

35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
35B10 Periodic solutions to PDEs
35C07 Traveling wave solutions
35K45 Initial value problems for second-order parabolic systems
Full Text: DOI arXiv
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