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We present a nonlinear stability theory for periodic wave trains inreaction-diffusion systems, which relies on pure $L^\infty$-estimates only. Ouranalysis shows that localization or periodicity requirements on perturbations,as present in the current literature, can be completely lifted. Inspired byprevious works considering localized perturbations, we decompose the semigroupgenerated by the linearization about the wave train and introduce aspatio-temporal phase modulation to capture the most critical dynamics, whichis governed by a viscous Burgers' equation. We then aim to close a nonlinearstability argument by iterative estimates on the corresponding Duhamelformulation, where, hampered by the lack of localization, we must rely ondiffusive smoothing to render decay of the semigroup. Yet, this decay is notstrong enough to control all terms in the Duhamel formulation. We address thisdifficulty by applying the Cole-Hopf transform to eliminate the criticalBurgers'-type nonlinearities. Ultimately, we establish nonlinear stability ofdiffusively spectrally stable wave trains against$C_{\mathrm{ub}}^2$-perturbations. Moreover, we show that the perturbedsolution converges to a modulated wave train, whose phase and wavenumber areapproximated by solutions to the associated viscous Hamilton-Jacobi andBurgers' equation, respectively.

Nonlinear stability and asymptotic behavior of periodic wave trains in reaction-diffusion systems against $C_{\mathrm{ub}}^2$-perturbations