L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves

Assuming spectral stability conditions of periodic reaction-diffusion waves u¯(x), we consider L1(R)-nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initial modulations. Lp(R)-nonlinear stability of such waves has been studied in Johnson et al. (2013) for p≥2 by using Hausdorff-Young inequality. In this note, by using the pointwise estimates obtained in Jung, (2012) and Jung and Zumbrun (2016), we extend Lp(R)-nonlinear stability (p≥2) in Johnson et al. (2013) to L1(R)-nonlinear stability. More precisely, we obtain L1(R)-estimates of modulated perturbations u~(x-ψ(x,t),t)-u¯(x) of u¯ with a phase function ψ(x,t) under small initial perturbations consisting of localized initial perturbations u~(x-h0(x),0)-u¯(x) and nonlocalized initial modulations h0(x)=ψ(x,0).