\(L^1(\mathbb{R})\)-nonlinear stability of nonlocalized modulated periodic reaction-diffusion waves
Summary: Assuming spectral stability conditions of periodic reaction-diffusion waves \(\overline{u}(x)\), we consider \(L^1(\mathbb{R})\)-nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initial modulations. \(L^p(\mathbb{R})\)-nonlinear stability of such waves has been studied in Johnson et al. (2013) for \(p \geq 2\) by using Hausdorff-Young inequality. In this note, by using the pointwise estimates obtained in [the author, J. Differ. Equations 253, No. 6, 1807--1861 (2012; Zbl 1268.35015); with \textit{K. Zumbrun}, ibid. 261, No. 7, 3941--3963 (2016; Zbl 1353.35053)], we extend \(L^p(\mathbb{R})\)-nonlinear stability (\(p \geq 2\)) in [\textit{M. A. Johnson} et al., Arch. Ration. Mech. Anal. 207, No. 2, 693--715 (2013; Zbl 1276.35031)] to \(L^1(\mathbb{R})\)-nonlinear stability. More precisely, we obtain \(L^1(\mathbb{R})\)-estimates of modulated perturbations \(\widetilde{u}(x - \psi(x, t), t) - \overline{u}(x)\) of \(\overline{u}\) with a phase function \(\psi(x, t)\) under small initial perturbations consisting of localized initial perturbations \(\widetilde{u}(x - h_0(x), 0) - \overline{u}(x)\) and nonlocalized initial modulations \(h_0(x) = \psi(x, 0)\).
scientific article; zbMATH DE number 6962617
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