Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling waves of systems of reaction–diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneiderʼs renormalization techniques do not appear to apply.
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Mathew A. Johnson, Kevin Zumbrun, Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction–diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 4, pp. 471–483DOI 10.1016/J.ANIHPC.2011.05.003