Pointwise nonlinear stability of nonlocalized modulated periodic reaction-diffusion waves

TL;DR

This paper proves the pointwise nonlinear stability of nonlocalized modulated periodic reaction-diffusion waves under spectral stability assumptions, extending previous results to include nonlocalized perturbations with algebraic decay.

Contribution

It extends the stability analysis of reaction-diffusion waves to nonlocalized perturbations, providing pointwise estimates under spectral stability assumptions.

Findings

Established pointwise nonlinear stability for nonlocalized modulated waves.

Derived estimates for perturbations combining localized decay and nonlocalized modulation.

Extended previous stability results to broader classes of perturbations.

Abstract

In this paper, extending previous results of \cite{J1}, we obtain pointwise nonlinear stability of periodic traveling reaction-diffusion waves, assuming spectral linearized stability, under nonlocalized perturbations. More precisely, we establish pointwise estimate of nonlocalized modulational perturbation under a small initial perturbation consisting of a nonlocalized modulation plus a localized perturbation decaying algebraically.