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

Abstract

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 traveling 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 $t → ±∞$.