Entire Solutions in Lattice Delayed Differential Equations with Nonlocal Interaction: Bistable Cases

Abstract

Abstract. This paper is concerned with entire solutions of a class of bistable delayed lattice differential equations with nonlocal interaction. Here an entire solution is meant by a solution defined for all (n, t) ∈ Z × R. Assuming that the equation has an increasing traveling wave front with nonzero wave speed and using a comparison argument, we obtain a two-dimensional manifold of entire solutions. In particular, it is shown that the traveling wave fronts are on the boundary of the manifold. Furthermore, uniqueness and stability of such entire solutions are studied.

Entire solutions in nonlocal dispersal equations with bistable nonlinearity

“…These entire solutions behave as two traveling wave solutions coming from both sides of the x-axis and annihilating in a finite time. Morita and Ninomiya [38] and Guo [27] have established entire solutions different from those obtained in [16,17,24,[28][29][30]35,37,45,47,48,51]. However, the issue of the existence of entire solutions for nonlocal dispersal equations (1.2) is still open.…”

“…However, the global attractors are rather complicated. Recently, many new types of entire solutions for reaction-diffusion equations with (without) nonlocal delayed nonlinearity have been obtained by several authors and these entire solutions provide essential information about the global attractors, for example, Chen and Guo [16], Chen et al [17], Fukao et al [24], Guo and Morita [28], Hamel and Nadirashvili [29,30], Li et al [35,37], Wang et al [45,47,48] and Yagisita [51]. These entire solutions behave as two traveling wave solutions coming from both sides of the x-axis and annihilating in a finite time.…”

Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case

“…These solutions can not only describe the interaction of traveling waves but also characterize new dynamics of diffusion equations. For the study of such entire solutions, we refer to [4,[14][15][16]19,23,27] for reaction-diffusion equations without delay, [22,35,37,42] for reaction-diffusion equations with nonlocal delay, [38,39] for delayed lattice differential equations with nonlocal interaction, [21,34] for nonlocal dispersal equations without delay ((1.9) below), [28,40,44] for reaction-diffusion systems, and [43] for periodic lattice dynamical systems. However, to the best of our knowledge, the issues on entire solutions for nonlocal dispersal equations with spatio-temporal delay have not been addressed, especially for infinite delay equations.…”

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

“…Such specific birth function has been widely used in mathematical biology literature, see e.g. Ma and Zou [18] and Wang et al [29]. Throughout this paper, we always assume that (G) and (B1) hold and (1.2) has a traveling wave front U (x + ct) connecting 0 and K with speed c = 0.…”

“…Recently, many types of front-like entire solutions have been observed for various evolution equations by mixing the traveling wave solutions and some spatially independent solutions, see [11][12][13][15][16][17]20,21,25,[27][28][29][30][31][32][33]. For examples, Hamel and Nadirashvili [12] established three-, four-and five-dimensional manifolds of entire solutions for the Fisher-KPP equation.…”