The Approach of Solutions for the Nonlocal Diffusion Equation to Traveling Fronts

The paper is concerned with the asymptotic behavior as t → ±∞ of an entire solution u(x,t) for the nonlocal diffusion equation. With bistable assumption, it is well known that the model has three different types of travel- ing fronts. Under certain conditions on the wave speeds, and by some auxiliary rational functions with certain properties to construct appropriate super- and sub solutions of the model, we establish two new types of entire solutions u(x,t) which approach to three travelling fronts or the positive equilibrium as t → ±∞.