Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations

Yicheng Jiang, ,Kaijun Zhang

doi:10.3934/krm.2018048

Abstract

This paper is concerned with the stability of noncritical/critical traveling waves for nonlocal time-delayed reaction-diffusion equation. When the birth rate function is non-monotone, the solution of the delayed equation is proved to converge time-exponentially to some (monotone or non-monotone) traveling wave profile with wave speed $c>c_*$, where $c_*>0$ is the minimum wave speed, when the initial data is a small perturbation around the wave. However, for the critical traveling waves ($c = c_*$), the time-asymptotical stability is only obtained, and the decay rate is not gotten due to some technical restrictions. The proof approach is based on the combination of the anti-weighted method and the nonlinear Halanay inequality but with some new development.

Highlights

The object of this paper is to consider a nonlocal time-delayed reaction-diffusion equation∂u(t, x) ∂2u(t, x) ∂t − D ∂x2 + d(u(t, x)) = ∞fα(y)b(u(t − r, x − y))dy, (t, x) ∈ R+ × R, −∞ (1)with the following initial data u(s, x) = u0(s, x), (s, x) ∈ [−r, 0] × R. (2)Here u(t, x) denotes the total mature population of the species at time t and position x, D > 0 is the spatial diffusion rate for the mature population, r > 0 is the maturation delay, the time required for a newborn to become matured
For the non-critical traveling wave with wave speed c > c∗, Mei et al [19] obtained the exponential stability by using combination of the weighted energy method and the companison principle for the monotone equation
The stability of critical traveling waves was obtained by Chern-Mei-Yang-Zhang [2] by using the anti-weighted technique and energy estimates

Summary

Introduction

For the non-critical traveling wave with wave speed c > c∗, Mei et al [19] obtained the exponential stability by using combination of the weighted energy method and the companison principle for the monotone equation. The stability of critical traveling waves was obtained by Chern-Mei-Yang-Zhang [2] by using the anti-weighted technique and energy estimates. Using the anti-weighted method, the stability of oscillating traveling wave for time-delayed nonlocal dispersion equations was obtained by Huang-Mei-Zhang-Zhang [11].

Results

Conclusion