$L^2$-Stability of Traveling Wave Solutions to Nonlocal Evolution Equations

TL;DR

This paper establishes $L^2$-stability of traveling wave solutions to nonlocal evolution equations using Lyapunov methods, offering an alternative to spectral analysis and applicable to stochastic perturbations.

Contribution

It introduces an $L^2$-based stability analysis for traveling waves in nonlocal equations, extending stability results to stochastic perturbations and small wave speeds.

Findings

Proves $L^2( ho)$ stability for traveling waves.

Establishes stability under stochastic perturbations.

Provides stability results for small wave speeds.

Abstract

Stability of the traveling wave solution to a general class of one-dimensional nonlocal evolution equations is studied in $L^2$-spaces, thereby providing an alternative approach to the usual spectral analysis with respect to the supremum norm. We prove that the linearization around the traveling wave solution satisfies a Lyapunov-type stability condition in a weighted space $L^2(\rho)$ for a naturally associated density $\rho$. The result can be applied to obtain stability of the traveling wave solution under stochastic perturbations of additive or multiplicative type. For small wave speeds, we also prove an alternative Lyapunov-type stability condition in $L^2(\mathfrak{m})$, where $\mathfrak{m}$ is the symmetrizing density for the traveling wave operator, which allows to derive a long-term stochastic stability result.