Spectral stability of traveling fronts for nonlinear hyperbolic equations of bistable type

TL;DR

This paper proves the spectral stability of traveling fronts in certain nonlinear hyperbolic equations with bistable reactions, establishing spectral gaps and extending stability results to broader classes of equations.

Contribution

It provides a new proof of spectral stability for these fronts and extends the analysis to a wider class of hyperbolic equations with bistable reactions.

Findings

Spectrum is stable with negative real part, except for a simple zero eigenvalue.

Existence of a spectral gap prevents accumulation of essential spectrum near zero.

Introduces a transformation of eigenfunctions enabling energy estimates for stability analysis.

Abstract

This paper addresses the existence and spectral stability of traveling fronts for nonlinear hyperbolic equations with a positive "damping" term and a reaction function of bistable type. Particular cases of the former include the relaxed Allen-Cahn equation and the nonlinear version of the telegrapher's equation with bistable reaction term. The existence theory of the fronts is revisited, yielding useful properties such as exponential decay to the asymptotic rest states and a variational formula for the unique wave speed. The spectral problem associated to the linearized equation around the front is established. It is shown that the spectrum of the perturbation problem is stable, that is, it is located in the complex half plane with negative real part, with the exception of the eigenvalue zero associated to translation invariance, which is isolated and simple. In this fashion, it is shown that there exists an spectral gap precluding the accumulation of essential spectrum near the origin. To show that the point spectrum is stable we introduce a transformation of the eigenfunctions that allows to employ energy estimates in the frequency regime. This method produces a new proof of equivalent results for the relaxed Allen-Cahn case and extends the former to a wider class of equations. This result is a first step in a more general program pertaining to the nonlinear stability of the fronts under small perturbations, a problem which remains open.