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Title Stability of traveling waves for time-delayed reaction-diffusion equations: (I) Non-critical oscillating waves

Name Ming Mei
Country Canada
Email ming.mei8@gmail.com
Co-Author(s) Chi-Kun Lin, Chi-Tien Lin, Yaping Lin and Ming Mei
Submit Time 2014-02-18 15:13:18
Special Session 12: Complexity in reaction-diffusion systems
This talk is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equations. It is known that, when the ratio of birth rate coefficient and death rate coefficient satisfies $1e$, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay $r$ or the wave speed $c$ is large, which causes the study of stability of these non-monotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that, when $ec_*>0$ are exponentially stable, where $c_*>0$ is the minimum wave speed. Here, we allow the traveling wave to be any, monotone or non-monotone with any speed $c>c_*$, and any size of the time-delay $r>0$; while, when $\frac{p}{d}> e^2$ with a small time-delay $rc_*>0$ are exponentially stable, too. As a corollary, we also prove the uniqueness of traveling waves in the case of $\frac{p}{d}> e^2$, which was open as we know. Finally, some numerical simulations are carried out. When $e e^2$, if the time-delay is small, then the solution numerically behaves like a monotone/non-monotone traveling wave, but if the time-delay is big, then the solution is numerically demonstrated to be chaotically oscillatory but not an oscillatory traveling wave. These either confirm and support our theoretical results, or open up some new phenomena for future research.