Perturbation of the spectra for asymptotically constant differential operators and applications

TL;DR

This paper develops a unified framework using exponential dichotomies and Brouwer degree theory to analyze spectral perturbations of asymptotically constant differential operators, with applications to stability and bifurcation problems.

Contribution

It introduces a novel approach to spectral perturbation analysis for asymptotically constant operators, applicable to various nonlinear PDE stability problems.

Findings

Spectral perturbation results for asymptotically constant operators.

Application to stability analysis of nonlinear PDE solutions.

Analysis of bifurcations in reaction-diffusion and fluid models.

Abstract

We study the spectra for a class of differential operators with asymptotically constant coefficients.These operators widely arise as the linearizations of nonlinear partial differential equations about patterns or nonlinear waves. We present a unified framework to prove the perturbation results on the related spectra. The proof is based on exponential dichotomies and the Brouwer degree theory. As applications, we employ the developed theory to study the stability of quasi-periodic solutions of the Ginzburg-Landau equation,fold-Hopf bifurcating periodic solutions of reaction-diffusion systems coupled with ordinary differential equations, and periodic annulus of the hyperbolic Burgers-Fisher model.