Entire Solutions Of A Nonlocal Dispersal Equation With Spatio-temporal Delays:Bistable Case

It is well-known that spatio-temporal delays and nonlocal dispersal exist inevitablly in nature. Recently, many researchers considered the effect of the spatio-temporal delays and nonlocal dispersal to the differential equations. They constructed the nonlocal dispersal equations with spatio-temporal delays which are more real to illustrate the realistic problems. However, the study of the nonlocal dispersal equations with spatio-temporal delays should be considered not only that the nonlocal dispersal term will make the maximum principle fail, but also the spatio-temporal delayed term will make the mathematical research more difficult.In this paper, we mainly consider the entire solutions of the following non-local diffusion equation with spatio-temporal delays where f is bistable type. Here the entire solutions are defined in the whole space and for all t ∈ R. In this paper, we study existence, uniqueness and the large-time behavior of the traveling waves at first and establish a new comparison principle. By using two traveling waves which are coming from opposite direction, we establish the Cauchy problem with initial time t=-n and construct the entire solutions by the uniformly convergence of the solutions of the Cauchy problem when n→∞. Moreover, we also obtain the asymptotic behavior and some qualitative properties of the entire solutions by constructing sup-solutions and sub-solutions.