Instability of multi-spot patterns in shadow systems of reaction-diffusion equations

Abstract

Our aim in this paper is to prove the instability of multi-spot patterns in a shadow system, which is obtained as a limiting system of a reaction-diffusion model as one of the diffusion coefficients goes to infinity. Instead of investigating each eigenfunction for a linearized operator, we characterize the eigenspace spanned by unstable eigenfunctions

The spatially heterogeneous diffusive rabies model and its shadow system

“…The concept of the "shadow system" was first proposed by Keener [17] in 1978. Since then, it has been extensively studied, and most of the research focuses on the existence of non-constant steady states (see, e.g., [14,15,22,23,24,28,29,32,40]), the stability and the instability of the non-constant positive steady states (see, e.g., [10,22,24,28,29,31,40]), the Hopf bifurcations (see, e.g., [28,39]), the compact attractors and the existence of monotone solutions (see, e.g., [12,18]), the global existence and boundedness, finite time blow-ups of the in-time solutions (see, e.g., [9,19,21]), the spectra of the linear diffusion operators related to the shadow system (see, e.g., [20]), and the dynamics of the stochastic shadow system (see, e.g., [41]).…”