  We present a nonlinear stability theory for periodic wave trains in
reaction-diffusion systems, which relies on pure $L^\infty$-estimates only. Our
analysis shows that localization or periodicity requirements on perturbations,
as present in the current literature, can be completely lifted. Inspired by
previous works considering localized perturbations, we decompose the semigroup
generated by the linearization about the wave train and introduce a
spatio-temporal phase modulation to capture the most critical dynamics, which
is governed by a viscous Burgers' equation. We then aim to close a nonlinear
stability argument by iterative estimates on the corresponding Duhamel
formulation, where, hampered by the lack of localization, we must rely on
diffusive smoothing to render decay of the semigroup. Yet, this decay is not
strong enough to control all terms in the Duhamel formulation. We address this
difficulty by applying the Cole-Hopf transform to eliminate the critical
Burgers'-type nonlinearities. Ultimately, we establish nonlinear stability of
diffusively spectrally stable wave trains against
$C_{\mathrm{ub}}$-perturbations. Moreover, we show that the perturbed solution
converges to a modulated wave train, whose phase and wavenumber are
approximated by solutions to the associated viscous Hamilton-Jacobi and
Burgers' equation, respectively.
