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

• Published in: Journal of Differential Equations Vol. 259; no. 4; pp. 1503 - 1541
• Main Authors: Chern, I-Liang, Mei, Ming, Yang, Xiongfeng, Zhang, Qifeng
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.08.2015
• Subjects: Critical traveling waves; Nicholson's blowflies equation; Oscillation; Stability; Time-delayed reaction–diffusion equation; Critical traveling waves; Stability; 92D25; 35B35; Nicholson's blowflies equation; Oscillation; 35K15; Time-delayed reaction–diffusion equation; 35K58; 35K57; 35C07
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2015.03.003

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 ϕ(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 ϕ(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.