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

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.

arXiv commentin comparison to first version setting in subsection 3.3 changed to stochastic neural field equations with multiplicative noise

References in corpus (4)

• Front Propagation in Stochastic Neural Fields: A Rigorous Mathematical Framework
• Stability of travelling waves in stochastic Nagumo equations
• Stability of travelling waves in stochastic bistable reaction-diffusion equations
• A Multiscale Analysis of Traveling Waves in Stochastic Neural Fields

Cited by in corpus (7)

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• Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations
• Metastability of Waves and Patterns Subject to Spatially-Extended Noise
• Solvability and microlocal analysis of the fractional Eringen wave equation
• Quasi-Ergodicity of Transient Patterns in Stochastic Reaction-Diffusion Equations
• Stability of Travelling Waves on Exponentially Long Timescales in Stochastic Reaction-Diffusion Equations
• Stability of travelling wave solutions to reaction-diffusion equations driven by additive noise with Hölder continuous paths