Diffusive stability against nonlocalized perturbations of planar wave trains in reaction-diffusion systems
arXiv:1803.09812 · doi:10.1016/j.jde.2020.10.027
Abstract
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 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.
arXiv comment32 pages, no figures
References in corpus (7)
- Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
- Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability
- Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations
- Nonlinear stability of source defects in the complex Ginzburg-Landau equation
- Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions
- Diffusive stability of Turing patterns via normal forms
- Diffusive mixing of periodic wave trains in reaction-diffusion systems