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

This paper is concerned with the stability of critical traveling waves for a kind of non-monotone timedelayed 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. © 2015 Elsevier Inc. All rights reserved. MSC: 35K57; 35B35; 35C07; 35K15; 35K58; 92D25 * Corresponding author at: Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada. E-mail addresses: [email protected] (I-L. Chern), [email protected], [email protected] (M. Mei), [email protected] (X. Yang), [email protected] (Q. Zhang). http://dx.doi.org/10.1016/j.jde.2015.03.003 0022-0396/© 2015 Elsevier Inc. All rights reserved. 1504 I-L. Chern et al. / J. Differential Equations 259 (2015) 1503–1541