Traveling wavefronts for time-delayed reaction–diffusion equation: (II) Nonlocal nonlinearity

This is the second part of a series of study on the stability of traveling wavefronts of reaction–diffusion equations with time delays. In this paper we will consider a nonlocal time-delayed reaction–diffusion equation. When the initial perturbation around the traveling wave decays exponentially as x→−∞ (but the initial perturbation can be arbitrarily large in other locations), we prove the asymptotic stability of all traveling waves for the reaction–diffusion equation, including even the slower waves whose speed are close to the critical speed. This essentially improves the previous stability results by Mei and So [M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction–diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551–568] for the speed with a small initial perturbation. The approach we use here is the weighted energy method, but the weight function is more tricky to construct due to the property of the critical wavefront, and the difficulty arising from the nonlocal nonlinearity is also overcome. Finally, by using the Crank–Nicholson scheme, we present some numerical results which confirm our theoretical study.