Front-like entire solutionsfor a delayed nonlocal dispersalequation with convolution typebistable nonlinearity

Abstract

This paper is concerned with front-like entire solutions of a delayed nonlocal dispersal equation with convolution type bistable nonlinearity. Here, a solution defined for all $(x, t)\in \mathbb {R}^2$ is an entire solution. It is known that the equation has an increasing traveling wavefront with nonzero wave speed under some reasonable conditions. We first give the asymptotic behavior of traveling wavefronts at infinity. Then, by the comparison argument and sub-super-solutions method, we construct new types of entire solutions other than traveling wavefronts and equilibrium solutions of the equation, which behave like two increasing traveling wavefronts propagating from both sides of the $x$-axis and annihilate at a finite time. Finally, various qualitative properties of the entire solutions are also investigated.