Global Stability of Traveling Waves for a More General Nonlocal Reaction-Diffusion Equation

Rui Yan,Guirong Liu

doi:10.1155/2018/6910491

Abstract

The purpose of this paper is to investigate the global stability of traveling front solutions with noncritical and critical speeds for a more general nonlocal reaction-diffusion equation with or without delay. Our analysis relies on the technical weighted energy method and Fourier transform. Moreover, we can get the rates of convergence and the effect of time-delay on the decay rates of the solutions. Furthermore, according to the stability results, the uniqueness of the traveling front solutions can be proved. Our results generalize and improve the existing results.

Highlights

In the study of biology and other subject fields, the reactiondiffusion equations with delays are usually utilized to depict the population distribution and physical evolution process and so forth, for instance, [1,2,3,4,5,6]
The purpose of this paper is to investigate the global stability of traveling front solutions with noncritical and critical speeds for a more general nonlocal reaction-diffusion equation with or without delay
Where Δ is the Laplacian operator on R, D > 0, r ≥ 0

Summary

Introduction

In the study of biology and other subject fields, the reactiondiffusion equations with delays are usually utilized to depict the population distribution and physical evolution process and so forth, for instance, [1,2,3,4,5,6]. For some nonlocal timedelayed reaction-diffusion equations, Mei, Ou and Zhao [21] and Wang [22] proved the globally exponential stability of traveling front solutions with noncritical speeds and globally algebraical stability of traveling front solutions with critical speed by using the weighted energy method and Green’s function method. Mei and Wang [23] considered a class of nonlocal time-delayed Fisher-KPP type reactiondiffusion equations in n-dimensional space They obtained the exponential stability of all noncritical planar wavefronts and the algebraic stability of the critical planar wavefronts by using the weighted energy method coupled with Fourier transform. Chern et al [24] studied the stability of critical traveling waves for a kind of nonmonotone time-delayed reaction-diffusion equations by using the technical weighted energy method with some new developments. We apply our results to some models

Preliminaries

The Stability of Traveling Front Solutions

The Uniqueness of the Traveling Front Solutions

Applications