Stability of non-monotone critical traveling waves for reaction diffusion equations with time-delay

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

Chern, I-Liang^1,2  Mei, Ming^3,4  Yang, Xiongfeng^5,6  Zhang, Qifeng⁷ 

[1] Natl Cent Univ, Dept Math, Taoyuan 32001, Taiwan

[2] Natl Taiwan Univ, Inst Appl Math Sci, Taipei 10764, Taiwan

[3] Champlain Coll St Lambert, Dept Math, St Lambert, PQ J4P 3P2, Canada

[4] McGill Univ, Dept Math & Stat, Montreal, PQ H3A 2K6, Canada

[5] Shanghai Jiao Tong Univ, Dept Math, Shanghai 200240, Peoples R China

[6] Shanghai Jiao Tong Univ, MOE LSC, Shanghai 200240, Peoples R China

[7] Zhejiang Sci Tech Univ, Sch Sci, Hangzhou 310018, Zhejiang, Peoples R China

关键词: Nicholson's blowflies equation;    Time-delayed reaction-diffusion equation;    Critical traveling waves;    Oscillation;    Stability;   

DOI  :  10.1016/j.jde.2015.03.003

来源: Elsevier

【 摘 要 】

This paper is concerned with the stability of critical traveling waves for a kind of non-monotone time-delayed reaction diffusion equations including Nicholson's blowflies equation which models the population dynamics of a single species with maturation delay. Such delayed reaction diffusion equations possess monotone or oscillatory traveling waves. The latter occurs when the birth rate function is non-monotone and the time-delay is big. It has been shown that such traveling waves phi (x + ct) exist for all c >= c(*) and are exponentially stable for all wave speed c > c(*) [13], where c* is called the critical wave speed. In this paper, we prove that the critical traveling waves phi (x c(*)t) (monotone or oscillatory) are also time-asymptotically stable, when the initial perturbations are small in a certain weighted Sobolev norm. The adopted method is the technical weighted-energy method with some new flavors to handle the critical oscillatory waves. Finally, numerical simulations for various cases are carried out to support our theoretical results. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free