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

Abstract

In this paper, we study a class of time-delayed reaction–diffusion equation with local nonlinearity for the birth rate. For all wavefronts 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 → − ∞ , 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.P. 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 D m ( ε 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 ≈ 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.P. 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 .