Global Stability of Critical Traveling Waves with Oscillations for Time-Delayed Reaction-Diffusion Equations

Abstract

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 $t$^−1/2, and the non-critical traveling waves are globally stable with the exponential convergence rate $t$^−1/2$e$^−$µt$ 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.