Entire solutions originating from monotone fronts to the Allen-Cahn equation

In this paper, we study entire solutions of the Allen-Cahn equation in one-dimensional Euclidean space. This equation is a scalar reaction-diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions originating from three and four fronts is shown by constructing some suitable pairs of super-sub-solutions. Moreover, we show that there are no entire solutions originating from more than four fronts.