Entire solutions originating from three fronts to a two-dimensional nonlocal periodic lattice dynamical system

TL;DR

This paper investigates entire solutions of a two-dimensional nonlocal periodic lattice dynamical system, introducing new solutions originating from three fronts under specific conditions, expanding understanding of complex wave interactions.

Contribution

It introduces two novel types of entire solutions originating from three fronts, constructed using auxiliary functions and super- and sub-solutions, extending previous work on merging-front solutions.

Findings

Established existence of new entire solutions from three fronts

Used auxiliary rational functions to construct solutions

Extended prior results on merging-front solutions

Abstract

This paper is concerned with the entire solutions of a two-dimensional nonlocal periodic lattice dynamical system. With bistable assumption, it is well known that the system has three different types of traveling fronts. The existence of merging-front entire solutions originating from two fronts for the system have been established by Dong, Li \& Zhang [{\it Comm. Pur Appl. Anal.}, {\bf17}(2018), 2517-2545]. Under certain conditions on the wave speeds, and by some auxiliary rational functions with certain properties to construct appropriate super- and sub solutions of the system, we establish two new types of entire solutions which originating from three fronts.

Equations

$$u_{i,j}^{\prime }(t)=\sum _{k_1\in \mathbb{Z}}\sum _{k_2\in \mathbb{Z}}J(k_1,k_2)u_{i-k_1,j-k_2}(t)-u_{i,j}(t)+f_{i,j}(u_{i,j}(t)),i,j\in \mathbb{Z},t\in \mathbb{R},$$

$$u_{i,j}^{\prime }(t)=D[u_{i,j}](t)-d_{i,j}{u}_{i,j}(t)+f_{i,j}(u_{i,j}(t)),$$

$$D[u_{i,j}](t)$$

$$d_{i,j}$$

$$u_{i,j}(t)={\varphi }_{i,j}(icos\theta +jsin\theta +kt),\text{ or }{u}_{i,j}(t)={\varphi }_{i,j}(-icos\theta -jsin\theta +kt),$$

$${\varphi }_{i,j}(\cdot )={\varphi }_{i+N_1,j}(\cdot )={\varphi }_{i,j+N_2}(\cdot ),{\varphi }_{i,j}(-\infty )=e_1\text{ and }{\varphi }_{i,j}(\infty )=e_2,\forall i,j\in \mathbb{Z}.$$

$${\tilde{\phi }}_{i,j;1}(-\infty )={\phi }_{i,j;1}(-\infty )=0\text{ and }{\tilde{\phi }}_{i,j;1}(+\infty )={\phi }_{i,j;1}(+\infty )=1,\forall i,j\in \mathbb{Z},$$

$${\hat{\phi }}_{i,j;1}(-\infty )=1\text{ and }{\hat{\phi }}_{i,j;1}(+\infty )=0,\forall i,j\in \mathbb{Z}.$$

$$v_{i,j}^{\prime }(t)=\sum _{k_1\in \mathbb{Z}}\sum _{k_2\in \mathbb{Z}}J(k_1,k_2)v_{i-k_1,j-k_2}(t)-v_{i,j}(t)+g_{i,j}(v_{i,j}(t))\text{mbox}foranyi,j\in \mathbb{Z},t\in \mathbb{R},$$

$$\displaystyle\ \lim\limits_{t\rightarrow-\infty}\Big{\{}\sum_{1\leq i\leq 3}\sup_{p_{i-1}(t)<icos\theta+jsin\theta<p_{i}(t)}|W_{i,j}(t)-\varphi_{i,j}(t)(icos\theta+jsin\theta+v_{i}t+\theta_{i})|\Big{\}}=0,$$

$$\underset{t\to +\infty }{\lim }\underset{icos\theta +jsin\theta \in \mathbb{R}}{\sup }|U_{i,j}(t)-1|=0$$

$$\displaystyle\lim_{t\rightarrow-\infty}\big{\{}\sup_{icos\theta+jsin\theta\leq m_{1}}|U_{i,j}(t)-\hat{\varphi}_{i,j;1}(icos\theta+jsin\theta+\hat{v}t-\omega)|$$

$$+\underset{m_1\le icos\theta +jsin\theta \le m_2}{\sup }|U_{i,j}(t)-{\tilde{\phi }}_{i,j;2}(icos\theta +jsin\theta +{\tilde{v}}_{1}t+\omega )|$$

$$\displaystyle\qquad\qquad\qquad+\sup_{icos\theta+jsin\theta\geq m_{2}}|U_{i,j}(t)-\tilde{\varphi}_{i,j;3}(icos\theta+jsin\theta+v_{2}t+\omega)|\big{\}}=0,$$

$$\underset{t\to +\infty }{\lim }\underset{icos\theta +jsin\theta \in \mathbb{R}}{\sup }|V_{i,j}(t)-{\hat{\phi }}_{i,j;1}(icos\theta +jsin\theta +\hat{v}t+{\omega }_1)|=0$$

$$\displaystyle\lim_{t\rightarrow-\infty}\big{\{}\sup_{icos\theta+jsin\theta\leq n_{1}}|V_{i,j}(t)-\hat{\varphi}_{i,j;1}(icos\theta+jsin\theta+\hat{v}t+\omega_{1})|$$

$$+\underset{n_1\le icos\theta +jsin\theta \le n_2}{\sup }|V_{i,j}(t)-{\tilde{\phi }}_{i,j;2}(icos\theta +jsin\theta +{\tilde{v}}_{1}t+{\omega }_1)|$$

$$\displaystyle\qquad\qquad\qquad+\sup_{icos\theta+jsin\theta\geq n_{2}}|V_{i,j}(t)-\hat{\varphi}_{i,j;2}(icos\theta+jsin\theta+\hat{v}_{1}t-\omega_{2})|\big{\}}=0,$$

$$u_{i,j}^{\prime }(t)\ge (or\le )\sum _{k_1\in \mathbb{Z}}\sum _{k_2\in \mathbb{Z}}J(k_1,k_2)u_{i-k_1,j-k_2}(t)-u_{i,j}(t)+f_{i,j}(u_{i,j}(t)),$$

$$\displaystyle\big{|}u_{i,j}^{\prime}(t;\varphi)\big{|}\leq M\text{ and }\big{|}u_{i,j}^{\prime\prime}(t;\varphi)\big{|}\leq M,\ \mbox{for }{i,j}\in\mathbb{Z}\ \mbox{and }t>t_{0}+1.$$

$$\left\{\begin{array}[]{ll}{c_{k}}\phi_{i,j;k}^{\prime}(\xi)=\sum_{k_{1}}\sum_{k_{2}}J(k_{1},k_{2})\phi_{i-k_{1},j-k_{2};k}(\xi)-\phi_{i,j;k}(\xi)+f_{i,j}(\phi_{i,j;k}(\xi)),\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \phi_{i+N_{1},j;k}(\xi)=\phi_{i,j;k}(\xi)=\phi_{i,j+N_{2};k}(\xi),i,j\in\mathbb{Z},~{}\xi\in\mathbb{R},\\ \phi_{i,j;k}(-\infty)=\alpha_{i},\phi_{i,j;k}(\infty)=\beta_{i}\end{array}\right.$$

$${\varphi }_{i,j;1}(0)=\frac{a}{2},{\varphi }_{i,j;2}(0)=\frac{a}{2},{\varphi }_{i,j;3}(0)=\frac{1+a}{2}.$$

$$0\le {\varphi }_{i,j;1}(t{)}^{\prime }(\xi ),{\varphi }_{i,j;2}(t{)}^{\prime }(\xi ),{\varphi }_{i,j;3}(t{)}^{\prime }(\xi )\le C_0{e}^{{\eta }_1\xi },\text{vskip}6.0ptplus2.0ptminus2.0pt$$

$$C_1{e}^{{\eta }_1\xi }$$

$$\frac{{\varphi }_{i,j;1}(t{)}^{\prime }(\xi )}{1-{\varphi }_{i,j;1}(t)(\xi )},\frac{{\varphi }_{i,j;2}(t{)}^{\prime }(\xi )}{{\varphi }_{i,j;2}(t)(\xi )},\frac{{\varphi }_{i,j;3}(t{)}^{\prime }(\xi )}{{\varphi }_{i,j;3}(t)(\xi )-a}\ge \rho ;$$

$$0\le {\varphi }_{i,j;1}(t{)}^{\prime }(\xi ),{\varphi }_{i,j;2}(t{)}^{\prime }(\xi ),{\varphi }_{i,j;3}(t{)}^{\prime }(\xi )\le C_0{e}^{-{\eta }_2\xi },\text{vskip}6.0ptplus2.0ptminus2.0pt$$

$$C_1{e}^{-{\eta }_2\xi }$$

$$\frac{{\varphi }_{i,j;1}(t{)}^{\prime }(\xi )}{{\varphi }_{i,j;1}(t)(\xi )},\frac{{\varphi }_{i,j;2}(t{)}^{\prime }(\xi )}{a-{\varphi }_{i,j;2}(t)(\xi )},\frac{{\varphi }_{i,j;3}(t{)}^{\prime }(\xi )}{1-{\varphi }_{i,j;3}(t)(\xi )}\ge \rho .$$

$$Q(y,z,w):=$$

$$Q(y,z,w)=\left\{{\begin{array}[]{ll}\displaystyle y+(1-y)z\frac{(1-a)(w-y)}{(1-y)z(1-a)+(a-z)(1-w)},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \displaystyle w+(a-z)(1-w)\frac{y-w}{(1-y)z(1-a)+(a-z)(1-w)}.\\ \end{array}}\right.$$

$$Q_y(y,z,w)=(a-z)(1-w)Q_1(y,z,w),$$

$$Q_z(y,z,w)=(1-y)(1-w)Q_2(y,z,w),$$

$$Q_w(y,z,w)=(1-y)z{Q}_3(y,z,w).$$

$$Q_{yy}(y,z,w)=$$

$$Q_{zz}(y,z,w)=$$

$$=$$

$$Q_{ww}(y,z,w)=$$

Full text

Keywords: Entire solution, Pulsating (periodic) traveling front, nonlocal periodic lattice dynamical system, super-sub solution

**AMS Subjective Classifications (2000): ** 34K05; 34A34; 34E05

1 Introduction

In this paper, we are interested in the entire solutions of the following nonlocal periodic lattice dynamical system

[TABLE]

where $u_{i,j}(t)$ means the density of a certain species in a periodic patchy environment; $\sum\limits_{k_{1}\in\mathbb{Z}}\sum\limits_{k_{2}\in\mathbb{Z}}J(k_{1},k_{2})u_{i-k_{1},j-k_{2}}(t)-u_{i,j}(t)$ is the nonlocal dispersal and represents transportation due to long range dispersion mechanism. The kernel function $J:\mathbb{Z}\times\mathbb{Z}\longrightarrow[0,\infty)$ is a probability function formulating the dispersal of individuals and satisfies $J(\cdot,\cdot)\geq 0$ is even $\sum\limits_{k_{1}\in\mathbb{Z}}\sum\limits_{k_{2}\in\mathbb{Z}}J(k_{1},k_{2})=1$, $J(k_{1},k_{2})=0$ if $|k_{1}|>k_{0}$ or $|k_{2}|>k_{0}$, where $k_{0}$ is a positive constant. Here the reaction term $f_{i,j}\in C^{2}(\mathbb{R})$ satisfies the bistable condition

1. $\rm(A1)$

$f_{i,j}(0)=f_{i,j}(a)=f_{i,j}(1)=0$, $i,j\in\mathbb{Z}$, $f_{i,j}^{\prime}(0)<0$, $f_{i,j}^{\prime}(1)<0$, $f_{i,j}^{\prime}(a)>0$, $f_{i,j}(u)<0$  for  $u\in(0,a)$  and   $f_{i,j}(u)>0$  for   $u\in(a,1)$

and the periodic condition

1. $\rm(A2)$

$f_{i,j}(\cdot)=f_{i+N_{1},j}(\cdot)=f_{i,j+N_{2}}(\cdot)$ for all $N_{1}$ and $N_{2}$ are two positive integers.

The system (1.1) can be also regarded as a nonlocal version of the following local diffusion system

[TABLE]

where

[TABLE]

where $p_{i,j},q_{i,j}>0$ and $p_{i+N_{1},j}=p_{i,j}=p_{i,j+N_{2}}$, $q_{i+N_{1},j}=q_{i,j}=q_{i,j+N_{2}}$. The reaction function satisfies $f_{i,j}(\cdot)=f_{i+N_{1},j}(\cdot)=f_{i,j+N_{2}}(\cdot)$ for all $i,j\in{\mathbb{Z}}$.

In the past decades, there are a lot of works devoted to front propagation for lattice dynamical systems in spatially periodic or homogeneous media, e.g.[20, 21, 4, 8, 22, 23, 24, 25, 26, 27, 28, 17, 18, 5, 6, 29, 19] and some references cited therein. Notice that the traveling front solution is a special type of entire solutions, but it is not enough to understand the whole dynamics. From the viewpoint of dynamics, for entire solutions which behave as two traveling wave fronts moving towards each other from both sides of the $x$ (or $j$) axis, this type of entire solutions are called “annihilating-front” entire solutions. Beyond that, there are also two common types of entire solutions. The first type behaves as 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 approaching each other from both sides of the $x$ (or $j$) axis and merging and converging to a single bistable front, see [13, 12, 11, 2, 14]. Such two types of entire solutions are often called as merging-front entire solutions. In [9], the authors have established the existence of merging-front entire solutions originating from two fronts for the system of (1.1) with both monostable and bistable nonlinearities.

Recently, there are other new types of entire solutions merging three fronts, which were addressed in [1, 3]. Motivated by these works, it is natural and interesting to study new entire solutions merging three fronts of the system (1.1). Now, there has been no results on the entire solutions merging three fronts for the nonlocal periodic lattice dynamical system. Therefore, in this paper, we will study the existence of the entire solutions originating from three fronts of (1.1).

Before to state our main results, we first give definitions of the pulsating traveling fronts and entire solutions for (1.1).

Definition 1.1** (Pulsating traveling fronts and entire solutions).**

1. (1)

A solution $u(t)=\{u_{i,j}(t)\}_{i,j\in\mathbb{Z}}$, $t\in\mathbb{R}$ of (1.1) is called a pulsating (or periodic) traveling front connecting $\{e_{1},e_{2}\}\subset\{0,a,1\}$ in the direction $(cos\theta$, $sin\theta)$ with the wave speed $k$, if

[TABLE]

for all $i,j\in\mathbb{Z},{\ }t\in\mathbb{R}$ and some function $\phi(\cdot)=\{\phi_{i,j}(\cdot)\}_{i,j\in\mathbb{Z}}$ which satisfies

[TABLE] 2. (2)

A function $u(t)=\{u_{i,j}(t)\}_{i,j\in{\mathbb{Z}}}$, $t\in\mathbb{R}$ is called an entire solution of (1.1) if for any $i,j\in{\mathbb{Z}}$, $u_{i,j}(t)$ is differentiable for all $t\in{\mathbb{R}}$ and satisfies (1.1) for $i,j\in{\mathbb{Z}}$ and $t\in{\mathbb{R}}$.

According to [16, Theorem 6], it is easy to see that (1.1) has two increasing pulsating traveling fronts $\tilde{\varphi}_{i,j;1}(icos\theta+jsin\theta+\tilde{v}t)$ and $\varphi_{i,j;1}(-icos\theta-jsin\theta+\bar{v}t)$ satisfying

[TABLE]

where $\tilde{v}$ and $\bar{v}>0$. Let $\hat{v}=-\bar{v}$, then $\hat{\varphi}_{i,j;1}(icos\theta+jsin\theta+\hat{v}t)=\varphi_{i,j;1}(-icos\theta-jsin\theta+\bar{v}t)=\varphi_{i,j;1}(-(icos\theta+jsin\theta+\hat{v}t))$, which satisfies

[TABLE]

Moreover, if we restrict $f_{i,j}(u)$ in the intervals $[0,a]$ and $[a,1]$, respectively, then (1.1) can be regarded as two monostable equations. Thus, we make the following assumption:

(A3) f_{i,j}(u)\Big{\{}\begin{array}[]{ll}\geq f^{\prime}_{i,j}(a)(u-a),&\mbox{if }u\in[0,a],\\ \leq f^{\prime}_{i,j}(a)(u-a),&\mbox{if }u\in[a,1],\end{array}\mbox{for all }i,j\in\mathbb{Z}.

For the case $u_{i,j}(t)\in[0,a]$, by considering $v_{i,j}(t)=a-u_{i,j}(t)$, (1.1) can be reduce to

[TABLE]

where $g_{i,j}(v)=-f_{i,j}(a-v)$. Then it follows from [2, 9] that there exists a $v^{*}(\theta)>0$ such that for every $\check{v}_{1},\hat{v}_{1}>v^{*}$, (1.5) has two increasing pulsating traveling fronts $\check{W}_{i,j}(icos\theta+jsin\theta+\check{v}_{1}t)$ and $\hat{W}_{i,j}(icos\theta+jsin\theta+\hat{v}_{1}t)$ which satisfy

$\check{W}_{i,j}(-\infty)=\hat{W}_{i,j}(-\infty)=0\text{ and }\check{W}_{i,j}(+\infty)=\hat{W}_{i,j}(+\infty)=a,$ for $i,j\in\mathbb{Z}$.

Let’s define $v^{*}_{1}=-v^{*}$, $\tilde{v}_{1}=-\check{v}_{1}<-v^{*}=v^{*}_{1}$, $\tilde{\varphi}_{i,j;2}(icos\theta+jsin\theta+\tilde{v}_{1}t)=a-\check{W}_{i,j}(-icos\theta-jsin\theta+\check{v}_{1}t)=a-\check{W}_{i,j}(-(icos\theta+jsin\theta)+\tilde{v}_{1}t)$ for any $\check{v}_{1}<v^{*}_{1}$

and $\hat{\varphi}_{i,j;2}(icos\theta+jsin\theta+\hat{v}_{1}t)=a-\check{W}_{i,j}(icos\theta+jsin\theta+\hat{v}t)$ for any $\hat{v}_{1}>v^{*}$ then $\tilde{\varphi}_{i,j;2}(icos\theta+jsin\theta+\tilde{v}_{1}t)$ and $\hat{\varphi}_{i,j;2}(icos\theta+jsin\theta+\hat{v}_{1}t)$ are pulsating traveling fronts of (1.1) with the wave speeds $\tilde{v}_{1}<v^{*}_{1}$ and $\hat{v}_{1}>v^{*}$, respectively,

$\tilde{\varphi}_{i,j;2}(-\infty)=0$, $\tilde{\varphi}_{i,j;2}(+\infty)=a$, and $\hat{\varphi}_{i,j;2}(-\infty)=a$, $\hat{\varphi}_{i,j;2}(+\infty)=0$

Similarly, for the case $u_{i,j}(t)\in[a,1]$, we can also obtain that (letting $v_{i,j}(t)=u_{i,j}(t)-a)$ there exists a ${v_{2}}^{*}(\theta)>0$ such that for every $v_{2}>{v_{2}}^{*}$, (1.1) admits an increasing pulsating traveling front $\tilde{\varphi}_{i,j;3}(j+v_{2}t)$ with

$\tilde{\varphi}_{i,j;3}(-\infty)=a$ and $\tilde{\varphi}_{i,j;3}(+\infty)=1$ for any $icos\theta+jsin\theta\in\mathbb{Z}$.

For (1.1), the traveling front $u_{i,j}(t)$ with the speed $v$ exists if $u_{i,j}(t)=\varphi(icos\theta+jsin\theta+vt)$ for some function and it connects two different two different constant states. Now we set $z:=icos\theta+jsin\theta+vt$ and substitute into (1.1), then let $(\varphi_{i,j},v_{i}),i=1,2,3$ be the traveling fronts of (1.1). If the entire solution $\varphi_{i,j}(t)$ of (1.1) satisfies

[TABLE]

where $v_{1}<v_{2}<v_{3},$ $\theta_{1},\theta_{2},\theta_{3}$ are some constants, $p_{i}(t):=-(v_{i}+v_{i+1})t/2$, $p_{0}=-\infty$ and $p_{3}=+\infty$, it is called the entire solution originating from three fronts of (1.1).

Theorem 1.2**.**

Assume that (A1), (A2) and (A3) hold. Let $(\hat{v},\hat{\varphi}_{i,j;1}),\,\,(\tilde{v}_{1},\tilde{\varphi}_{i,j;2})$ and $(v_{2},\tilde{\varphi}_{i,j;3})$ be pulsating traveling fronts described as above such that $\hat{v}<\tilde{v}_{1}<{v_{2}}$. Then there exists an entire solution $U_{i,j}(t):\mathbb{Z}\times\mathbb{R}\rightarrow[0,1]$ originating from three fronts of (1.1) and $\omega\in\mathbb{R}$ which satisfy

[TABLE]

and

[TABLE]

where $m_{1}=-\frac{(\hat{v}+\tilde{v}_{1})t}{2}$ and $m_{2}=-\frac{(\tilde{v}_{1}+v_{2})t}{2}$.

Theorem 1.3**.**

Assume that (A1), (A2) and (A3) hold. Let $(\hat{v},\hat{\varphi}_{i,j;1}),$ $(\tilde{v}_{1},\tilde{\varphi}_{i,j;2})$ and $(\hat{v}_{1},\hat{\varphi}_{i,j;2})$ be pulsating traveling fronts with $\hat{v}<\tilde{v}_{1}<\hat{v}_{1}$. Then there exists an entire solution $V_{i,j}(t):\mathbb{Z}\times\mathbb{R}\rightarrow[0,1]$ originating from three fronts of (1.1) and $\omega_{1},\omega_{2}\in\mathbb{R}$ which satisfy

[TABLE]

and

[TABLE]

where $n_{1}=-\frac{(\hat{v}+\tilde{v}_{1})t}{2}$ and $n_{2}=-\frac{(\tilde{v}_{1}+\hat{v}_{1})t}{2}$.

In order to verify Theorems 1.2-1.3, we will adopt the elementary method of super- and subsolutions and a comparison principle. Since the comparison principle can be well applied, we only need to construct a suitable pair of super- and subsolutions by some auxiliary rational functions with certain properties which were developed by Morita and Ninomiya in [14]. This technique had been used to prove some types of entire solutions originating from two fronts of (1.1). Therefore, we would apply the technique to establish some new types of entire solutions originating from three fronts of (1.1), i.e. Theorems 1.2-1.3.

Now we recall the definitions of super- and subsolutions of (1.1) and some known results on the existence, the comparison principle and the prior estimates for solutions of (1.1).

Definition 1.4** (Super- and subsolutions).**

1. (1)

Let $t_{0}<T$ be any two real constants. A sequence of continuous functions $\{u_{i,j}(t)\}_{i,j\in{\mathbb{Z}}}$ is called a supersolution (or subsolution) of (1.1) on $[t_{0},T)$ if

[TABLE]

for $t\in[t_{0},T)$. 2. (2)

Let $T$ be a real constant. A sequence of continuous functions $\{u_{i,j}(t)\}_{i,j\in{\mathbb{Z}}}$ is called a supersolution (or subsolution) of (1.1) on $(-\infty,T)$ if for any $t_{0}<T$, $\{u_{i,j}(t)\}_{i,j\in{\mathbb{Z}}}$ is a supersolution (or subsolution) of (1.1) on $[t_{0},T)$.

Lemma 1.5**.**

[9, Existence and Comparison principle]**

1. (1)

For any $\varphi=\{\varphi_{i,j}\}_{i,j\in{\mathbb{Z}}}$ with $\varphi_{i,j}\in[0,1]$, equation (1.1) admits a unique solution $u(t;\varphi)=\{u_{j}(t;\varphi)\}_{i,j\in{\mathbb{Z}}}$ on $[t_{0},+\infty)$ which satisfies $u_{i,j}(t_{0};\varphi)=\varphi_{i,j}$ and $u_{i,j}\in C^{1}([t_{0},\infty),[0,1])$ for all $i,j\in{\mathbb{Z}}$. 2. (2)

Suppose \big{\{}u^{+}_{i,j}(t)\big{\}}_{i,j\in{\mathbb{Z}}} and \big{\{}u^{-}_{i,j}(t)\big{\}}_{i,j\in{\mathbb{Z}}} are bounded supersolution and subsolution of (1.1) on $[t_{0},+\infty)$, respectively, such that $u^{+}_{i,j}(t),u^{-}_{i,j}(t)\geq 0$ and $u^{+}_{i,j}(t_{0})\geq u^{-}_{i,j}(t_{0})$ for $i,j\in{\mathbb{Z}}$ and $t\geq t_{0}$, then $u^{+}_{i,j}(t)\geq u^{-}_{i,j}(t)\geq 0$ for all $i,j\in{\mathbb{Z}}$ and $t\geq t_{0}$.

Lemma 1.6**.**

[9, A Prior estimate]**  Let u(t;\varphi)=\big{\{}u_{i,j}(t;\varphi)\big{\}}_{i,j\in\mathbb{Z}} be a solution of (1.1) with initial value $\varphi=\{\varphi_{i,j}\}_{i,j\in\mathbb{Z}}$ and $\varphi_{i,j}\in[0,1]$. For any fixed $t_{0}\in\mathbb{R}$, there exists a positive constant $M$ (independent of $t_{0}$ and $\varphi$) such that

[TABLE]

2 Proof of Theorem 1.2

Set $c_{1}:=\hat{v}$, $c_{2}:=\tilde{v}_{1}$ and $c_{3}:=v_{2}$. let $\phi_{i,j;1}=\hat{\varphi}_{i,j;1}$, $\phi_{i,j;2}=\tilde{\varphi}_{i,j;2}$, $\phi_{i,j;3}=\tilde{\varphi}_{i,j;3}$, $i,j\in\mathbb{Z},k=1,2,3,$ be traveling fronts of (1.1) that satisfy

[TABLE]

where $(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2},\alpha_{3},\beta_{3})=(1,0,0,a,a,1)$.

Without loss of generality, we assume that

[TABLE]

Lemma 2.1**.**

According to the arguments in [9, Lemma 2.5], there exist positive numbers $C_{0},\,C_{1},$ $C_{2}$, $\eta_{1},\,\eta_{2}$ and $\rho$, which depend on $\hat{v}$, $\tilde{v}_{1}$, $v_{2}$, $\hat{v}$, $\tilde{v}_{1}$, $\hat{v}_{1}$ and $\theta$, such that for all $i,j\in\mathbb{Z}$ and $\xi\leq 1$,

[TABLE]

For all $i,j\in\mathbb{Z}$ and $\xi\geq-1$,

[TABLE]

2.1 Construction of sub- and supersolutions

To construct a pair of super-solution and sub-solution of (1.1), we introduce the following auxiliary function (see [1])

[TABLE]

where $D_{1}:=\{[0,1]\times[0,a]\times[a,1]\}\backslash(\{(1,a,w)|a\leq w\leq 1\}\cup\{(1,z,1)|0\leq z\leq a\}\cup\{(y,0,1)|0\leq y\leq 1\})$.

Lemma 2.2**.**

[1, Lemma 3.1]** The function $Q(y,z,w)$ has the following properties.

• (i)

[TABLE]

• (ii)

There exist functions $Q_{i},i=1,2,3$, such that

[TABLE]

• (iii)

There exist functions $R_{j},j=1,\cdot\cdot\cdot,16,$ such that

[TABLE]

Next, we consider the following four ordinary differential equations with the initial conditions (see [1, 15, 14]):

[TABLE]

where $L,s_{2},s_{1},p_{0},r_{0}$ and $\kappa$ are positive constants. (2.11)-(2.14) have the following unique solutions for $t\leq 0$, respectively,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For a sufficiently large constant $\delta>0$, we choose $p_{0}$ and $r_{0}$ satisfying

[TABLE]

Moreover,

[TABLE]

Also, there exist a positive constant $N$ such that for all $t\leq 0$

[TABLE]

and $p_{1}(t),p_{2}(t),r_{1}(t),r_{2}(t)\leq-\delta$.

Therefore, by the choice of $k$, there exists a $t_{0}<0$ such that

[TABLE]

[TABLE]

Lemma 2.3**.**

Let $p_{1}(t)$ and $p_{2}(t)$ be solution of (2.15) and (2.16). For any $p_{2}(t)\leq p_{1}(t)\leq 0$. There exist positive constants $\epsilon_{1}$, $\epsilon_{2}$ and $\epsilon_{3}$ such that

[TABLE]

From (2.9) and Lemma (2.10), we see that

[TABLE]

Which implies that there is a smooth function $\mathcal{F}_{1}$ satisfying

[TABLE]

Since $Q(0,z,a)=z$ and $Q_{2}^{\prime}(0,z,a)=1$, we have

[TABLE]

which implies $\mathcal{F}_{1}(0,z,a)=0$. Applying the mean value theorem to $\mathcal{F}_{1}$ yields

[TABLE]

Thus we obtain

[TABLE]

From the above discussion, we can easily obtain that there exists a positive constant $C$ such that

[TABLE]

for $z\in R$, $p_{1}(t)$, $p_{2}(t)<-\delta$, $l=1,\cdot\cdot\cdot,16$, $m,n=1,2.$

Set $\bar{c}:=(c_{1}+c_{2})/2$, $s_{1}:=(c_{2}-c_{1})/2$ and $s_{2}:=c_{3}-\bar{c}$. Define the functions $\overline{U}_{i,j}(t)$ and $\underline{U}_{i,j}(t)$ by

[TABLE]

constitute a pair of super- and subsolution of (1.1) for $t\leq 0$. Moreover, there exists a positive constant $C$ such that

[TABLE]

Proof.

We only prove that $\overline{U}_{i,j}(t)$ is a super-solution, since the other case can be discussed similarly. For convenience, we denote

[TABLE]

[TABLE]

Direct computations show that

[TABLE]

where $Q:=Q(\phi_{i,j;1}(z_{1}),\phi_{i,j;2}(z_{2}),\phi_{i,j;3}(z_{3}))$, $\phi_{i,j;1}:=\phi_{i,j;1}(z_{1})$, $\phi_{i,j;2}:=\phi_{i,j;2}(z_{2})$, $\phi_{i,j;3}:=\phi_{i,j;3}(z_{3})$, $\phi_{i-k_{1},j-k_{2};1}=\phi_{i-k_{1},j-k_{2};1}(z_{1}-k_{1}cos\theta-k_{2}sin\theta)$, $\phi_{i-k_{1},j-k_{2};2}=\phi_{i-k_{1},j-k_{2};2}(z_{2}-k_{1}cos\theta-k_{2}sin\theta)$, $\phi_{i-k_{1},j-k_{2};3}=\phi_{i-k_{1},j-k_{2};3}(z_{3}-k_{1}cos\theta-k_{2}sin\theta)$ and

[TABLE]

Denote

[TABLE]

Lemma 2.4**.**

There exists a large $\sigma>0$. The following statements hold that

[TABLE]

Moreover, if $p_{1},p_{2}\leq-\sigma$, then the following assertions hold:

[TABLE]

It follows from that

[TABLE]

By direct computations, we obtain that

[TABLE]

[TABLE]

where

[TABLE]

There is a positive constant $M$ such that

[TABLE]

When $z\leq p_{1}(t)$, based on Lemma 2.1, Lemma 2.2, Lemma 2.3 and (2.29), we have $z_{1}=z-p_{1}(t)\leq 0$, $z_{2}=z+p_{1}(t)\leq 0$ and $z_{3}=z+p_{2}(t)\leq 0$. Then it follows that

[TABLE]

For $p_{1}(t)\leq z\leq 0$, based on Lemma 2.1, Lemma 2.2, Lemma 2.3 and (2.30), we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\leq 0$ and $z_{3}=z+p_{2}(t)\leq 0$. Then it follows that

[TABLE]

Then, we can obtain that

[TABLE]

where $L_{1}=max\{\tilde{L}_{1},\tilde{L}_{2}\}$.

For $0\leq z\leq-p_{1}(t)$, based on Lemma 2.1, Lemma 2.2, Lemma 2.3 and (2.30), we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\leq 0$ and $z_{3}=z+p_{2}(t)\leq 0$. Then it follows that

[TABLE]

For $-p_{1}(t)\leq z\leq(-p_{2}-p_{1}(t))/2$, based on Lemma 2.1, Lemma 2.2, Lemma 2.3 and (2.31), we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\geq 0$ and $z_{3}=z+p_{2}(t)\leq 0$. Then it follows that

[TABLE]

Then, we can obtain that

[TABLE]

where $L_{2}=max\{\tilde{L}_{3},\tilde{L}_{4}\}$.

For $(-p_{2}-p_{1}(t))/2\leq z\leq-p_{2}$, based on Lemma2.1, Lemma 2.2, Lemma 2.3 and (2.31), we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\geq 0$ and $z_{3}=z+p_{2}(t)\leq 0$. Then it follows that

[TABLE]

For $z\geq-p_{2}$, based on Lemma 2.1, Lemma 2.2, Lemma 2.3 and (2.32), we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\geq 0$ and $z_{3}=z+p_{2}(t)\geq 0$. Then it follows that

[TABLE]

Then, we can obtain that

[TABLE]

where $L_{3}=max\{\tilde{L}_{5},\tilde{L}_{6}\}$.

Let us show that

[TABLE]

where $L_{1}^{\prime},L_{2}^{\prime},L_{3}^{\prime}$ are positive constants.

Next, we estimate $|\mathcal{F}/\mathcal{A}|$. For $z\leq p_{1}(t)$, we have $z_{1}=z-p_{1}(t)\leq 0$, $z_{2}=z+p_{1}(t)\leq 0$ and $z_{3}=z+p_{2}(t)\leq 0$, based on Lemma 2.1, Lemma 2.2(ii), Lemma 2.3 and (2.29). Then it follows that

[TABLE]

For $p_{1}(t)\leq z\leq 0$,we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\leq 0$ and $z_{3}=z+p_{2}(t)\leq 0$, based on Lemma 2.1, Lemma 2.2(ii), Lemma 2.3, and (2.30). Then it follows that

[TABLE]

We can obtain that

[TABLE]

where $L_{1}^{\prime}=max\{\bar{L}_{1},\bar{L}_{2}\}$.

For $0\leq z\leq-p_{1}(t)$, we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\leq 0$ and $z_{3}=z+p_{2}(t)\leq 0$, based on Lemma 2.1, Lemma 2.3, (2.26) and (2.30). Then it follows that

[TABLE]

For $-p_{1}(t)\leq z\leq(-p_{2}-p_{1}(t))/2$, we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\geq 0$ and $z_{3}=z+p_{2}(t)\leq 0$, based on Lemma 2.1, Lemma 2.2(ii), Lemma 2.3, and (2.31). Then it follows that

[TABLE]

We can obtain that

[TABLE]

where $L_{2}^{\prime}=max\{\bar{L}_{3},\bar{L}_{4}\}$.

For $(-p_{2}-p_{1}(t))/2\leq z\leq-p_{2}$, we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\geq 0$ and $z_{3}=z+p_{2}(t)\leq 0$, based on Lemma 2.1, Lemma 2.2(ii), Lemma 2.3, and (2.31). Then it follows that

[TABLE]

For $z\geq-p_{2}$, we have $z_{1}=z-p_{1}(t)\geq 0$, $z_{2}=z+p_{1}(t)\geq 0$ and $z_{3}=z+p_{2}(t)\geq 0$, based on Lemma 2.1, Lemma 2.2(ii), Lemma 2.3, and (2.32). Then it follows that

[TABLE]

We can obtain that

[TABLE]

where $L_{3}^{\prime}=max\{\bar{L}_{5},\bar{L}_{6}\}$. For all $t\leq t_{0}$, where $M_{1}=max\{L_{1},L_{2},L_{3}\}$ and $M_{2}=max\{L_{1}^{\prime},L_{2}^{\prime},L_{3}^{\prime}\}$. Then choosing $L>(M_{1},M_{2})$, it follows that

[TABLE]

By (2.11), (2.12), Lemma 2.4, hence $\overline{U}_{i,j}(t)$ is a super-solution of (1.1) for $t<t_{0}$. Similarly, we can prove that $\underline{U}_{i,j}(t)$ is a subsolution of (1.1) for $t<t_{0}$. Then we obtain

[TABLE]

We have

[TABLE]

hence the lemma is proved.

From [9], existence and uniqueness of entire solution of (1.1) can be shown. There exists a unique entire solution $u_{i,j}(t)$ of (1.1) such that $\underline{U}_{i,j}(t)\leq u_{i,j}(t)\leq\overline{U}_{i,j}(t)$ for $t<T$. Then we Define

[TABLE]

Furthermore, on the premise that the supersolution and subsolution are established. By [2, Theorem 3.3], we can get the asymptotic behavior of (1.1).

3 Proof of theorem 1.3

Since the proof of theorem 1.3 is quite similar to that of theorem 1.2, we only point out the main differences in this section. Similar to that of theorem 1.2. Set $c_{1}:=\hat{v}$, $c_{2}:=\tilde{v_{1}}$ and $c_{3}:=\hat{v_{1}}$. Let $\phi_{i,j;1}=\hat{\varphi}_{i,j;1}$, $\phi_{i,j;2}=\tilde{\varphi}_{i,j;2}$, $\phi_{i,j;3}=\hat{\varphi}_{i,j;2}$, $j\in\mathbb{Z},k=1,2,3,$ be traveling fronts of (1.1) that satisfy

[TABLE]

where $(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2},\alpha_{3},\beta_{3})=(1,0,0,a,a,0)$.

First, we consider the auxiliary rational function $\tilde{Q}(y,z,w)$

[TABLE]

Then we take the function $\tilde{p}_{i}^{\prime}(t)$ and $\tilde{r}_{i}^{\prime}(t)$, $i=1,2$ are the solutions of the following initial value problems:

[TABLE]

where $L,s_{1},s_{2},\tilde{p}_{0},\tilde{r}_{0}$ and $\kappa$ are the same as in Section 2.

Define the function $\overline{U}^{*}_{i,j}(t)$ and $\underline{U}^{*}_{i,j}(t)$ as follows.

[TABLE]

By the similar argument as in the function $\overline{U}^{*}_{i,j}(t)$ and $\underline{U}^{*}_{i,j}(t)$ are a pair of super-sub-solution of (1.1) for $t\leq t_{0}$ with some constant $t_{0}<0$. Hence, the existence and uniqueness of the entire solution $u_{i,j}(t)$ of (1.1) can be shown and $u_{i,j}(t)$ satisfies

[TABLE]

for all $i,j\in Z$ and $t\leq t_{0}$, Moreover, it is not difficult to check the entire solution $u_{i,j}(t)$ satisfies by using the similar argument as in and taking

[TABLE]

Similar to the proof argument as Section 2, we can also obtain that $\overline{U}^{*}_{i,j}(t)$ and $\underline{U}^{*}_{i,j}(t)$ are a pair of supersolution and subsolution of (1.1). Since the proof of Theorem 1.3 is quite similar to that of theorem 1.2, we omit the detail of the proof.