Pyramidal traveling fronts in a nonlocal delayed diffusion equation

Recently, there have been great progresses on the study of nonplanar traveling wave solutions of reaction-diffusion equations. In this paper, we study the pyramidal traveling fronts of nonlocal delayed diffusion equation in RN with N≥2 by using the comparison principle and establishing super- and subsolution. Since the effect of nonlocal delay, we show that nonlocal delayed diffusion equation in N-dimensional space has a pyramidal traveling front u(t,x)=V(x′,xN+st) toward XN-axis for each s>c>0. In particular, two-dimensional traveling curved front and three-dimensional pyramidal fronts for nonlocal delayed diffusion equation in R2 and R3 are also established, respectively. Moreover, we also extend our results to generally nonlocal delayed reaction-diffusion equation.