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On travelling wavefronts of Nicholson’s blowflies equation with diffusion. (English) Zbl 1194.35094
Summary: This paper is devoted to the study of Nicholson’s blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c>c * (c * is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x-, but it can be large in other locations. The result develops and improves the previous wave stability obtained by M. Mei et al. [Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 3, 579–594 (2004; Zbl 1059.34019)]. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.
35C07Traveling wave solutions of PDE
35R10Partial functional-differential equations
35K15Second order parabolic equations, initial value problems
35K58Semilinear parabolic equations