Perturbation of the spectra for asymptotically constant differential operators and applications

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.