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

Soyeun Jung

Advances in Mathematical Physics, 2017, vol. 2017, issue 1

Abstract: 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).

Date: 2017
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https://doi.org/10.1155/2017/3824501

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlamp:v:2017:y:2017:i:1:n:3824501

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