Entire solutions in monostable reaction–diffusion equations with delayed nonlinearity

Entire solutions for monostable reaction–diffusion equations with nonlocal delay in one-dimensional spatial domain are considered. A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions. Some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations, and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory. Key techniques include the characterization of the asymptotic behaviors of solutions as t→−∞ in term of appropriate subsolutions and supersolutions. Two models of reaction–diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results.