Entire solutions of non-quasi-monotone delayed reaction—diffusion equations with applications

Abstract

We are interested in finding the entire solutions of non-quasi-monotone delayed non-local reaction-diffusion equations. It is well known that the comparison principle is not applicable for such equations. To overcome this difficulty, we introduce two auxiliary quasi-monotone equations and establish some comparison arguments for the three systems. Some new types of entire solutions are then constructed using the comparison argument, the travelling wavefronts and a spatially independent solution of the auxiliary equations. We also extend our arguments to a delayed cellular neural network with non-monotonic output functions and a delayed non-local lattice differential equation with non-monotonic birth functions.

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.…”

Traveling Wave Solutions of a Delayed Cooperative System

“…Very recently, Meng et al [19] investigated the monotone traveling wave solutions of Equation (4) if the intraspecific delay is small, which leads to the quasimonotonicity in the sense of exponential ordering [5]. Besides the traveling wave solutions, there are also some other features of entire solutions formulating by wave type solutions (see [20][21][22] for some examples of nonmonotone equations).…”

Traveling wave solutions for a neutral reaction–diffusion equation with non-monotone reaction

“…which has been wildly investigated in the literature (see [22,26,27,31] and references therein). As a prototype of such equations, we mention the evolution model of the adult population of a single species with two age classes and moving around in an unbounded 1dimensional spatial domain as follows:…”