Pointwise nonlinear stability of nonlocalized modulated periodic   reaction-diffusion waves
  In this paper, extending previous results of \cite{J1}, we obtain pointwise nonlinear stability of periodic traveling reaction-diffusion waves, assuming spectral linearized stability, under nonlocalized perturbations. More precisely, we establish pointwise estimate of nonlocalized modulational perturbation under a small initial perturbation consisting of a nonlocalized modulation plus a localized perturbation decaying algebraically. 
Soyeun Jung, Kevin Zumbrun
math.AP