Diffusive stability against nonlocalized perturbations of planar wave trains in reaction-diffusion systems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:265

Article

de Rijk, Bjoen¹  Sandstede, Bjorn² 

[1] Univ Stuttgart, Inst Anal Dynam & Modellierung, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

[2] Brown Univ, Div Appl Math, 182 George St, Providence, RI 02912 USA

关键词: Nonlinear stability;    Pointwise estimates;    Nonlocalized perturbations;    Planar reaction-diffusion systems;    Periodic traveling waves;   

DOI  :  10.1016/j.jde.2018.07.011

来源: Elsevier

【 摘 要 】

Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction-diffusion systems. We establish nonlinear diffusive stability against perturbations that are bounded along a line in R-2 and decay exponentially in the distance from this line. Our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation. We also consider perturbations that are fully localized and establish nonlinear stability with better decay rates, suggesting a trade-off between spatial localization of perturbations and temporal decay rate. Our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations. The nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme; instead, we track the perturbed solution in two different coordinate systems. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free