Entire solutions in a two-dimensional nonlocal lattice dynamical system

Fang-Di Dong,Li Zhang,Wan-Tong Li

doi:10.3934/cpaa.2018120

Abstract

This paper is concerned with entire solutions for a two-dimensional periodic lattice dynamical system with nonlocal dispersal. In the bistable case, by applying comparison principle and constructing appropriate upper- and lowersolutions, two different types of entire solutions are constructed. The first type behaves like a monostable front merges with a bistable front and one chases another from the same side; while the other type can be represented by two monostable fronts merge and converge to a single bistable front. In the monostable case, we first establish the existence and properties of spatially periodic solutions which connect two steady states. Then new types of entire solutions are constructed by mixing a heteroclinic orbit of the spatially averaged ordinary differential equations with traveling wave fronts with different speeds. Further, for a class of special heterogeneous reaction function, we establish the uniqueness and continuous dependence of the entire solution on parameters, such as wave speeds and shifted variables.

Highlights

In this paper, we are interested in entire solutions of the following nonlocal periodic lattice dynamical system ui,j (t) =J (k1, k2)ui−k1,j−k2 (t) − ui,j (t) + fi,j (ui,j (t)) (1)k1 k2 where i, j, k1, k2 ∈ Z, t ∈ R
We provide the definition of pulsating traveling front
We give the uniqueness and continuous dependence on parameters of the entire solution established in Theorem 1.4 when the reaction function fi,j satisfies some special condition

Summary

Introduction

We are interested in entire solutions of the following nonlocal periodic lattice dynamical system ui,j (t) =J (k1, k2)ui−k1,j−k2 (t) − ui,j (t) + fi,j (ui,j (t)) (1)k1 k2 where i, j, k1, k2 ∈ Z, t ∈ R. Let θ ∈ [0, 2π) be a constant, a solution u(t) = {ui,j(t)}i,j∈Z, t ∈ R of (1) is called a pulsating (periodic) traveling front connecting e1 and e2 in the direction (cos θ, sin θ) with speed c, if ui,j(t) = φi,j(i cos θ + j sin θ + ct) or ui,j(t) = φi,j(−i cos θ − j sin θ + ct) for all i, j ∈ Z, t ∈ R and some function φ(·) = {φi,j(·)}i,j∈Z which satisfies φi,j (·) = φi,j+N2 (·) = φi+N1,j (·), φi,j (−∞) = e1 and φi,j (∞) = e2, where {e1, e2} ∈ {0, a, 1}.

Results

Conclusion