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Using spatial domain techniques developed by the authors and Myunghyun Oh inthe context of parabolic conservation laws, we establish under a natural set ofspectral stability conditions nonlinear asymptotic stability with decay atGaussian rate of spatially periodic traveling-waves of systems of reactiondiffusion equations. In the case that wave-speed is identically zero for allperiodic solutions, we recover and slightly sharpen a well-known result ofSchneider obtained by renormalization/Bloch transform techniques; by the samearguments, we are able to treat the open case of nonzero wave-speeds to whichSchneider's renormalization techniques do not appear to apply

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations