\(L^1(\mathbb{R})\)-nonlinear stability of nonlocalized modulated periodic reaction-diffusion waves
type: Item
instance of: scholarly article
title: \(L^1(\mathbb{R})\)-nonlinear stability of nonlocalized modulated periodic reaction-diffusion waves
zbMATH Open document ID: 1400.35022
DOI: 10.1155/2017/3824501
author: Soyeun Jung
published in: Advances in Mathematical Physics
publication date: 2018-10-23T00:00:00Z
review text: 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 \(h_0(x) = \psi(x, 0)\).
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 6962617
zbMATH Keywords: localized small initial perturbations
zbMATH Keywords: nonlocalized initial modulations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2017/3824501
OpenAlex ID: W2765949671
cites work: Nonlocalized modulation of periodic reaction diffusion waves: nonlinear stability
cites work: Pointwise asymptotic behavior of modulated periodic reaction-diffusion waves
cites work: Pointwise nonlinear stability of nonlocalized modulated periodic reaction-diffusion waves
cites work: Q3841913
cites work: Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation
cites work: Diffusive mixing of periodic wave trains in reaction-diffusion systems
cites work: Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction-diffusion equations
cites work: Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case
cites work: Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations
cites work: Pointwise stability estimates for periodic traveling wave solutions of systems of viscous conservation laws
cites work: The dynamics of modulated wave trains
cites work: Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
cites work: Nonlocalized modulation of periodic reaction diffusion waves: the Whitham equation
cites work: Spectral and dynamical stability of nonlinear waves
description: scientific article; zbMATH DE number 6962617