Stability and dynamics of planar fronts in reaction-diffusion systems under nonlocalized perturbations

TL;DR

This paper studies the stability and dynamics of planar fronts in reaction-diffusion systems under nonlocalized perturbations, revealing conditions for stability, effective modulation equations, and challenges posed by continuous spectrum.

Contribution

It establishes Lyapunov stability for nonlocalized perturbations, derives a viscous Hamilton-Jacobi modulation equation, and introduces novel techniques to handle linear and nonlinear challenges.

Findings

Lyapunov stability holds under spectral assumptions.

Front dynamics are governed by a viscous Hamilton-Jacobi equation.

Asymptotic stability requires localized perturbations in transverse directions.

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.