Traveling wavefronts for time-delayed reaction-diffusion equation: (I) Local nonlinearity

被引:142
作者
Mei, Ming [1 ,2 ]
Lin, Chi-Kun [3 ]
Lin, Chi-Tien [4 ]
So, Joseph W. -H. [5 ]
机构
[1] Champlain Coll St Lambert, Dept Math, St Lambert, PQ J4P 3P2, Canada
[2] McGill Univ, Dept Math & Stat, Montreal, PQ H3G 1M8, Canada
[3] Natl Chiao Tung Univ, Dept Appl Math, Hsinchu 30010, Taiwan
[4] Providence Univ, Dept Appl Math, Taichung 43301, Taiwan
[5] Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada
基金
加拿大自然科学与工程研究理事会;
关键词
Reaction-diffusion equation; Time-delay; Traveling waves; Stability; NICHOLSONS BLOWFLIES EQUATION; ASYMPTOTIC STABILITY; POPULATION-MODEL;
D O I
10.1016/j.jde.2008.12.026
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In this paper, we study a class of time-delayed reaction-diffusion equation with local nonlinearity for the birth rate. For all wave-fronts with the speed c > c(*), where c(*) > 0 is the critical wave speed, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the traveling waves decays exponentially as x -> -infinity, but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results obtained by Mei. So. Li and Shen [M. Mei, J.W.-H. So, M.Y. Li, S.S.R Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594] for the speed c > 2 root D-m (epsilon p - d(m)) with small initial perturbation and by Lin and Mei [C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson's blowflies equations with diffusion, submitted for publication] for c > c(*) with sufficiently small delay time r approximate to 0. The approach adopted in this paper is the technical weighted energy method used in [M. Mei, J.W.-H. So, M.Y. Li, S.S.R Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594], but inspired by Gourley [S.A. Gourley, Linear stability of travelling fronts in an age-structured reaction-diffusion Population model, Quart. J. Mech. Appl. Math. 58 (2005) 257-268] and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any c > c(*) and for an arbitrary time-delay r > 0. (C) 2009 Elsevier Inc. All rights reserved.
引用
收藏
页码:495 / 510
页数:16
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