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.

Keywords: Periodic traveling waves, Nonlinear stability, Bloch decomposition

@article{AIHPC_2011__28_4_471_0, author = {Johnson, Mathew A. and Zumbrun, Kevin}, title = {Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction--diffusion equations}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, publisher = {Elsevier}, volume = {28}, number = {4}, year = {2011}, pages = {471-483}, doi = {10.1016/j.anihpc.2011.05.003}, zbl = {1246.35034}, mrnumber = {2823880}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2011__28_4_471_0} }

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, Volume 28 (2011) no. 4, pp. 471-483. doi : 10.1016/j.anihpc.2011.05.003. http://www.numdam.org/item/AIHPC_2011__28_4_471_0/

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