no. 4
Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction–diffusion equations
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 471-483

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.

DOI : 10.1016/j.anihpc.2011.05.003
Keywords: Periodic traveling waves, Nonlinear stability, Bloch decomposition 
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     title = {Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction{\textendash}diffusion equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     year = {2011},
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Johnson, Mathew A.; Zumbrun, Kevin. Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction–diffusion equations. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 471-483. doi: 10.1016/j.anihpc.2011.05.003

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