Stability and dynamics of planar fronts in reaction-diffusion systems under nonlocalized perturbations
Björn de Rijk, Joris van Winden
TL;DR
This work analyzes stability of bistable planar fronts in multi-component reaction-diffusion systems on ${\mathbb R}^d$ ($d\ge 2$) under fully nonlocalized perturbations. Under natural spectral stability assumptions, the planar front is Lyapunov stable, but asymptotic orbital stability generally fails; the leading-front dynamics are governed by a transverse modulation $\sigma(t,y)$ solving the forced viscous Hamilton–Jacobi equation with coefficient $d_{\perp}>0$. A nonlinear tracking scheme, combining a Fourier-based linear analysis, a Cole–Hopf transform to tame the quadratic nonlinearity, and forcing to cancel slow decays, yields sharp decay estimates and a precise description of the interface evolution. Asymptotic stability can be recovered when perturbations are localized transversely, while the effective front dynamics provide a principled framework for understanding front oscillations and interface motion in multidimensional RD systems. The results extend prior localized perturbation theory and generalize stability analyses from scalar to multicomponent systems with general diffusion matrices.
Abstract
We analyze the stability and dynamics of bistable planar fronts in multicomponent reaction-diffusion systems on $\mathbb{R}^{d}$. Under standard spectral stability assumptions, we establish Lyapunov stability of the front against fully nonlocalized perturbations. Such perturbations could previously be treated only for scalar equations via comparison principles. We also prove that the leading-order dynamics of the perturbed front are governed by a modulation that tracks the motion of the front interface and evolves according to a viscous Hamilton-Jacobi equation. This effective description reveals that asymptotic orbital stability does not hold in general. However, asymptotic stability can be recovered by imposing localization of perturbations in the transverse spatial directions. The treatment of nonlocalized perturbations on $\mathbb{R}^{d}$ poses significant challenges, both at the linear and nonlinear level. At the linear level, the neutral translational mode gives rise to continuous spectrum which touches the origin and cannot be projected out by conventional means, resulting in merely algebraic decay rates for the residual. Our linear estimates are necessarily $L^{\infty}$-based, yielding significantly weaker decay rates than those available for $L^p$-localized perturbations. At the nonlinear level, quadratic gradient terms decay at a critical rate and cannot be treated perturbatively. We overcome these challenges by carefully decomposing the linearized dynamics, blending semigroup methods with ideas from the stability analysis of viscous shock waves, and introducing a novel nonlinear tracking scheme that combines spatiotemporal modulation with forcing techniques and the Cole-Hopf transform.