TY - JOUR
T1 - Global Stability of Critical Traveling Waves with Oscillations for Time-Delayed Reaction-Diffusion Equations
AU - Mei , Ming
AU - Zhang , Kaijun
AU - Zhang , Qifeng
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 375
EP - 397
PY - 2018
DA - 2018/11
SN - 16
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/12874.html
KW - Nicholson's blowflies equation, time-delayed reaction-diffusion equation, critical traveling waves, oscillation, stability, numerical simulations.
AB - <p style="text-align: justify;">For a class of non-monotone reaction-diffusion equations with time-delay, the large
time-delay usually causes the traveling waves to be oscillatory. In this paper, we are interested in
the global stability of these oscillatory traveling waves, in particular, the challenging case of the
critical traveling waves with oscillations. We prove that, the critical oscillatory traveling waves
are globally stable with the algebraic convergence rate $t$<sup>−1/2</sup>, and the non-critical traveling waves
are globally stable with the exponential convergence rate $t$<sup>−1/2</sup>$e$<sup>−$µt$</sup> for some positive constant $µ$,
where the initial perturbations around the oscillatory traveling wave in a weighted Sobolev can be
arbitrarily large. The approach adopted is the technical weighted energy method with some new
development in establishing the boundedness estimate of the oscillating solutions, which, with the
help of optimal decay estimates by deriving the fundamental solutions for the linearized equations,
can allow us to prove the global stability and to obtain the optimal convergence rates. Finally,
numerical simulations in different cases are carried out, which further confirm our theoretical
stability for oscillatory traveling waves, where the initial perturbations can be large.</p>