Entire Solutions with Annihilating Fronts to a Nonlocal Dispersal Equation with Bistable Nonlinearity and Spatio-Temporal Delay

Abstract

This paper deals with the entire solutions to a nonlocal dispersal bistable equation with spatio-temporal delay. Assuming that the equation has a traveling wave front with non-zero wave speed, we establish the existence of entire solutions with annihilating-fronts by using the comparison principle combined with explicit constructions of sub-and supersolutions. These entire solutions constitute a two-dimensional manifold and the traveling wave fronts belong to the boundary of the manifold. We also prove the uniqueness, Liapunov stability and continuous dependence on the shift parameters of the entire solutions.

Spatial dynamics of a nonlocal bistable reaction diffusion equation

“…Alfaro et al [3] researched the case of the nonlinearities with the form of u(u−θ)(1−φ * u). For more results about bistable reaction-diffusion equation can be referred to [8,25,26,30]. It should be pointed out that the difficulties caused by different nonlinear term (the integral located at different place) are different.…”

Bistable traveling waves in a delayed competitive system

“…In literature, the traveling wave solutions of bistable models have been widely studied, which includes the existence, stability and uniqueness of traveling wave solutions. For delayed reaction-diffusion equations with bistable nonlinearities, we may refer to [1,10,14,16,19,20,22,27] for the investigation of traveling wave solutions. When delayed systems are concerned, several works studied the existence and properties of bistable traveling wave solutions, see [9,18] for delayed Lotka-Volterra systems and [7] for epidemic models.…”