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| This talk is devoted to the study on stability of oscillatory traveling waves to a kind of time-delayed reaction-diffusion equations including the Nicholson's blowflies equation. The equation under consideration losses its monotonicity, and its traveling waves are oscillatory when the time-delay $r$ or the wave speed $c$ is large.
This causes the study of stability of these non-monotone traveling waves not easy. For all non-critical traveling waves $\phi(x+ct)$ with $c>c_*>0$, where $c_*>0$ is the minimum wave speed, these oscillating waves are proved to be exponentially stable by Lin-Lin-Lin-Mei. But the stability on the critical oscillating waves $\phi(x+c_*t)$ are more challenging and still remain open due to some technical reasons. Here, we give a positive answer, that is, all critical traveling waves $\phi(x+c_*t)$ including the critical oscillating wavefronts are time-asymptotically stable, when the initial perturbations around the wavefronts in a certain weighted Sobolev space are small. Furthermore, when the initial perturbations exponentially decay much fast, we derive the algebraic convergence rate $O(t^{-1/2})$ for the solutions to the corresponding critical oscillating waves. Such a rate is optimal in $L^\infty$-sense. The adopted method is the technical weighted energy method, but with some new development so then we can treat the case of the critical oscillating waves. Finally, numerical simulations are also carried out, which further confirm and support our theoretical results. |
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