# Uniqueness of Traveling Waves for a Two-Dimensional Bistable Periodic Lattice Dynamical System

Authors: Chin-Chin Wu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study traveling waves for a two-dimensional lattice dynamical system with
bistable nonlinearity in periodic media. The existence and the monotonicity in time of
traveling waves can be derived in the same way as the one-dimensional lattice case. In this
paper, we derive the uniqueness of nonzero speed traveling waves by using the comparison
principle and the sliding method.


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## Body

## 1. Introduction

In this paper, we study the following two-dimensional (2D) lattice dynamical system:(1.1)u̇i,j(t):=ddtui,j(t)=D2[ui,j](t)+f(ui,j(t)),i,j∈Z,t>0,
where f is a C2 function in ℝ and(1.2)D2[ui,j]:=pi+1,jui+1,j+pi,jui-1,j+qi,j+1ui,j+1+qi,jui,j-1-di,jui,j,di,j:=pi+1,j+pi,j+qi,j+1+qi,j.
We assume that the coefficients pi,j,qi,j are positive and bounded such that(1.3)pi+N,j=pi,j=pi,j+N,qi+N,j=qi,j=qi,j+N,∀i,j∈Z,
for some positive integer N. Furthermore, we consider the case of bistable nonlinearity, namely,(1.4)f(0)=f(1)=f(a)=0,f′(0)<0<f′(a),f′(1)<0,
for some constant a∈(0,1). For simplicity, we only consider the case when a∈(0,1/2].We are interested in (planar) traveling wave solutions of (1.1) such that(1.5)ui,j(t+kNr+mNsc)=ui-kN,j-mN(t)∀k,m∈Z,t∈R,
for some (speed) c≠0 and(1.6)lim⁡ri+sj→-∞ui,j(t)=0,lim⁡ri+sj→+∞ui,j(t)=1,
for any t∈ℝ in the direction (r,s):=(cosθ,sinθ) for some θ∈(0,π/2).The study of lattice dynamical systems has attracted a lot of attention for past years. In particular, traveling wave solutions are important due to the wide applications of these special solutions. For example, the invading of one species to another can be described by traveling wave solutions (see, e.g., [1, 2]). The lattice dynamical system arises, for example, when the habitat is divided into discrete niches in certain biology models. We refer the reader to, for example, [3–9] for monostable nonlinearity and [10–16] for bistable nonlinearity in a one-dimensional lattice. In particular, in [16] the authors studied a very general model with bistable nonlinearity in a 1D lattice. Our purpose of this paper is to extend the result of [16] to the case of multidimensional lattice. For the study of multidimensional lattice, we refer to [17–19]. For the simplicity of presentation, we will only consider the 2D lattice dynamical system (1.1). Our results can be easily extended to the more general case with a convection term or spatially dependent nonlinearity as in [16].In a similar manner as that in [16] for 1D lattice case, we can prove the existence of traveling wave solutions of (1.1)–(1.6) with profile {Ui,j}i,j∈ℤ and speed c∈ℝ, by transforming the problem (1.1)–(1.6) into an integral formulation. Moreover, if the speed c>0, then we can obtain U̇i,j(t)<0 for all i,j∈ℤ and t∈ℝ. We will not repeat the proof here and focus on the study of the uniqueness of nonzero speed traveling waves. The uniqueness is in the sense that if there exist two traveling waves with nonzero speeds, then these two speeds are the same, and two wave profiles are the same except a translation. Due to that the nonlinearity is independent of spatial variable, our proof of the uniqueness is simpler and more transparent than that in [16]. In fact, motivated by the work of Fife and McLeod [20], Lemma 3.1 (below) provides some estimations in terms of a given traveling wave solution for the solution to the initial value problem for (1.1) with certain initial condition. Moreover, with Lemma 3.1, we employ the idea of moving coordinate and a sliding method to complete the proof of uniqueness (see Theorem 3.3).This paper is organized as follows. In Section2, we give some preliminaries including a comparison principle. Then we prove uniqueness of traveling wave with nonzero speed in Section 3.


## 2. Preliminaries

The following lemma can be easily deduced from (1.5) and (1.6).Lemma 2.1.
Let{ui,j}i,j∈ℤ be a solution of (1.1)–(1.6). If c>0(<0), then ui,j(t)→0(→1) as t→∞ and ui,j(t)→1(→0) as t→-∞ for each i,j.We can determine the sign of the speedc (when c≠0) as follows.Lemma 2.2.
Suppose thatc≠0, then c has the same sign as [-∫01f(s)ds].Proof.
ForK≥max{1,N/|c|}, an integration by parts gives
(2.1)∫-KK[u̇i,j(t)]2dt=∫-KKu̇i,j(t){D2[ui,j](t)+f(ui,j(t))}dt=∫-KKpi+1,ju̇i,jui+1,jdt-∫-KKpi,jui,ju̇i-1,jdt+pi,jui-1,jui,j|-KK+∫-KKqi,j+1u̇i,jui,j+1dt-∫-KKqi,jui,ju̇i,j-1dt+qi,jui,j-1ui,j|-KK-12di,jui,j2|-KK+∫-KKu̇i,jf(ui,j)dtfor1≤i,j≤N.
Then 1≤∀j≤N,
(2.2)∑i=1N∫-KK[u̇i,j(t)]2dt=∫-KKpN+1,ju̇N,juN+1,jdt-∫-KKp1,ju̇0,ju1,jdt+∑i=1Npi,jui-1,jui,j|-KK+∑i=1N∫-KKqi,j+1u̇i,jui,j+1dt+∑i=1Nqi,jui,j-1ui,j|-KK-∑i=1N∫-KKqi,jui,ju̇i,j-1dt-∑i=1N12di,jui,j2|-KK+∑i=1N∫-KKu̇i,jf(ui,j)dt.
Therefore,
(2.3)∑j=1N∑i=1N∫-KK[u̇i,j(t)]2dt=∑j=1Np1,j{∫-KKu̇N,juN+1,jdt-∫-KKu̇0,ju1,jdt}+∑i=1Nqi,1{∫-KKu̇i,Nui,N+1dt-∫-KKu̇i,0ui,1dt}+∑j=1N∑i=1N{pi,jui-1,jui,j|-KK+qi,jui,j-1ui,j|-KK-12di,jui,j2|-KK}+∑j=1N∑i=1N∫-KKu̇i,jf(ui,j)dt.
Sending K to ∞, using (1.5) and Lemma 2.1, it follows that
(2.4)∑j=1N∑i=1N∫-∞∞[u̇i,j(t)]2dt=-sgn⁡(c)N2∫01f(s)ds.
Hence, the lemma follows.As a simple consequence of Lemma2.2, we have c=0 if f is of balanced type, that is, ∫01f(s)ds=0. Notice that we cannot guarantee the speed c is zero or not by using the method developed in [16]. In fact, for the 1D lattice case, the classical work of Keener [21] indicates that the propagation failure (i.e., c=0) occurs when the diffusion coefficient is sufficiently small, even when f is of unbalanced type. A similar result for 1D periodic case can be found in [22]. For our model, the problem for the propagation failure is still open.Setu:={ui,j}i,j∈ℤ. Define 𝒩i,ju(t):=u̇i,j(t)-D2[ui,j](t)-f(ui,j(t)). Then we have the following comparison principle.Lemma 2.3.
Assume thatt0∈ℝ,i0,j0∈ℤ∪{∞} and c∈ℝ. Suppose that u:={ui,j(t)}i,j∈ℤand v:={vi,j(t)}i,j∈ℤ are bounded and continuous on
(2.5){(i,j,t)∈Z2×R∣t≥t0,ri+sj+ct≤ri0+sj0+1},
such that
(2.6)Ni,ju≥Ni,jv∀t>t0,ri+sj+ct≤ri0+sj0,ui,j(t0)≥vi,j(t0)∀ri+sj+ct0<ri0+sj0,ui,j(t)≥vi,j(t)∀t≥t0,ri0+sj0≤ri+sj+ct≤ri0+sj0+1,
then ui,j(t)≥vi,j(t) for all t>t0, ri+sj+ct≤ri0+sj0. Moreover, if there exists some (i1,j1) with ri1+sj1+ct0<ri0+sj0 such that ui1,j1(t0)>vi1,j1(t0), then ui,j(t)>vi,j(t) for all t>t0, ri+sj+ct<ri0+sj0.Since the proof is quite similar to the one given in [16, Lemma  1], we safely omit it here.


## 3. Uniqueness

In this section, we will study the uniqueness of traveling waves of (1.1)–(1.6). Firstly, applying a method of Fife-McLeod [20], we can derive the following result.Lemma 3.1.
Suppose thatu:={ui,j(t)}i,j∈ℤ is a solution of (1.1) for t>0, such that
(3.1)limsup⁡ri+sj→-∞ui,j(0)<a<lim⁡inf⁡ri+sj→∞ui,j(0)
and 0<ui,j(t)<1 for all t≥0. If  Ui,jc(t) is a traveling wave solution with speed c≠0, then there exist a sufficiently large positive integer i0 and j0, depending on the initial value, and positive numbers β0 (depending on the value of f′(s) near s=0 and s=1), δ0=δ0(β0,l,f,Uc)>0, such that for all i,j∈ℤ, t>0,
(3.2)Ui-i0N,j-j0Nc(t-δe-βt)-le-βt≤ui,j(t)≤Ui+i0N,j+j0Nc(t+δe-βt)+le-βt
if β∈(0,β0], δ≥δ0,
(3.3)l∈[max⁡{limsup⁡ri+sj→-∞ui,j(0),1-lim⁡inf⁡ri+sj→∞ui,j(0)},min⁡{a-ϵ,1-a-ϵ}],
for some small ϵ>0.Proof.
We will only consider the case whenc>0. In this case, we have U̇i,jc(t)<0 for all t∈ℝ.
First, we let(3.4)Φ(0,p)={[f(0)-f(p)]/p,p>0,-f′(0),p=0.
Clearly,Φ(0,p) is continuous. Fixing ϵ with
(3.5)0<ϵ<min⁡⁡{a-limsup⁡ri+sj→-∞ui,j(0),lim⁡inf⁡ri+sj→∞ui,j(0)-a},
there exists μ1>0, such that [f(0)-f(p)]/p≥2μ1 for all 0<p≤a-ϵ. Since f(s) is continuous, we would find Δ1>0, such that (f(u)-f(u+p))/p≥μ1 for all 0≤u≤Δ1, 0<p≤a-ϵ. By the same reasoning, there exist μ2>0 and Δ2>0, such that (f(u)-f(u+p))/p≥μ2 for all p≤a-ϵ, 1-Δ2≤u≤1.
Defineũ:={ũi,j}i,j∈ℤ, where
(3.6)ũi,j(t):=min⁡{1,Ui+ĩ0N,j+j̃0Nc(t+δ̃e-β̃t)+l̃e-β̃t}.
Choosel̃ satisfying limsupri+sj→-∞ui,j(0)<l̃<a-ϵ<a, and let i0, j0 be determined later. We claim that
(3.7)ui,j(0)≤Ui+ĩ0N,j+j̃0Nc(δ̃)+l̃,∀i,j∈Z,
for some ĩ0 and j̃0. Since limsupri+sj→-∞ui,j(0)<l̃, there exists m, such that
(3.8)ui,j(0)<l̃,∀ri+sj≤m.
Moreover, since limri+sj→∞Ui,j(0)=1, there exist ĩ0 and j̃0, such that
(3.9)ui,j(0)≤Ui+ĩ0N,j+j̃0Nc(δ̃)+l̃,∀ri+sj≥m.
Combining (3.8) and (3.9), we have proved the claim (3.7).
Now, we prove𝒩i,jũ≥0. If ũi,j(t)=Ui+ĩ0N,j+j̃0Nc(t+δ̃e-β̃t)+l̃e-β̃t, then
(3.10)Ni,jũ=(1-δ̃β̃e-β̃t)(Ui+ĩ0N,j+j̃0Nc)′(t+δ̃e-β̃t)-l̃β̃e-β̃t-D2[Ui+ĩ0N,j+j̃0Nc(t+δ̃e-β̃t)+l̃e-β̃t]-f(Ui+ĩ0N,j+j̃0Nc(t+δ̃e-β̃t)+l̃e-β̃t)=-δ̃β̃e-β̃t(Ui+ĩ0N,j+j̃0Nc)′(t+δ̃e-β̃t)-l̃β̃e-β̃t+f(Ui+ĩ0N,j+j̃0Nc(t+δ̃e-β̃t))-f(Ui+ĩ0N,j+j̃0Nc(t+δ̃e-β̃t)+l̃e-β̃t).
Divide into three cases.
Case 1.
0≤Ui,jc≤Δ:=min{Δ1,Δ2}. From the above discussion, since l̃e-β̃t≤l̃<a-ϵ, we have
(3.11)f(Ui,jc)-f(Ui,jc+l̃e-β̃t)≥μ1l̃e-β̃t.
Since U̇i,jc<0,𝒩i,jũ≥-l̃β̃e-β̃t+μ1l̃e-β̃t=(μ1-β̃)l̃e-β̃t. Choosing β̃≤μ1, we have 𝒩i,jũ≥0 in this case.Case 2.
1-Δ≤Ui,jc≤1. As in Case 1, we have μ2>0 such that 𝒩i,jũ≥-l̃β̃e-β̃t+μ2l̃e-β̃t=(μ2-β̃)l̃e-β̃t≥0 if β̃≤μ2.Case 3.
Δ≤Ui,jc≤1-Δ. If σ:=minΔ≤Ui,jc≤1-Δ|U̇i,jc|>0 and k:=maxs∈[0,1]|f′(s)|>0, then 𝒩i,jũ≥δ̃β̃σe-β̃t-l̃β̃e-β̃t-kl̃e-β̃t=(δ̃β̃σ-l̃β̃-kl̃)e-β̃t. Choosing δ̃≥l̃(β̃+k)/β̃σ, we have 𝒩i,jũ≥0 in this case.Then 𝒩i,jũ≥0 for all cases. Hence, the second inequality of (3.2) follows from a comparison principle. By the same way, we have the first inequality of (3.2). This proves the lemma.Note that we have the following different type of super- and sub-solutions which can be verified by a similar way as that of Lemma3.1.Lemma 3.2.
Suppose thatw:={wi,j(t)}i,j∈ℤ is a solution of (1.1)–(1.6). For any t0∈R, l̃∈(0,l0), l0:=min{a-ϵ,1-a-ϵ}, ϵ is a small number, and σ1≥σ0=σ0(β,f,wi,jc)>0, and let w±={wi,j±(t)}i,j∈ℤ, where
(3.12)wi,j±(t):=wi,j(t+t0∓σ1l̃(1-e-βt))±l̃e-βt,
then ±𝒩i,jw±≥0.We now prove the following uniqueness result.Theorem 3.3.
Suppose that{Ui,jc}, {U¯i,jc¯} are two traveling wave solutions of (1.1)–(1.6) with c, c¯≠0, then one has c=c¯ and Ui,jc(ξ)=U¯i,jc¯(ξ+ξ*) for some ξ*∈ℝ.Proof.
As before, we only consider the case when bothc and c¯ are positive. Since
(3.13)lim⁡ri+sj→-∞Ui,j(0)=0<a<lim⁡ri+sj→∞Ui,j(0)=1,
by Lemma 3.1, we have
(3.14)Ui-i0N,j-j0Nc(t-δe-βt)-le-βt≤U¯i,jc¯(t)≤Ui+i0N,j+j0Nc(t+δe-βt)+le-βt,
for all t>0, i,j∈ℤ with the constants i0,j0,l,β,δdefined in Lemma 3.1.
LetIk=I+kN, Jm=J+mN, and I,J∈{1,…,N}, k,m∈ℤ. Take (i,j)=(Ik,Jm). We get for all t>0, k,m∈ℤ,
(3.15)UI+kN-i0N,J+mN-j0Nc(t-δe-βt)-le-βt≤U¯I+kN,J+mNc¯(t)≤UI+kN+i0N,J+mN+j0Nc(t+δe-βt)+le-βt,∀I,J.
By the property (1.5) for all t>0, k,m∈ℤ,
(3.16)UI-i0N,J-j0Nc(t-(rk+sm)Nc-δe-βt)-le-βt≤U¯I,Jc¯(t-(rk+sm)Nc¯)≤UI+i0N,J+j0Nc(t-(rk+sm)Nc+δe-βt)+le-βt,∀I,J.
Setting the moving coordinate(3.17)ξ∶=t-(rk+sm)Nc¯,
we have for any ξ∈ℝ, for all I,J,
(3.18)UI-i0N,J-j0Nc(ξ+(rk+sm)(Nc¯-Nc)-δe-βt)-le-βt≤U¯I,Jc¯(ξ)≤UI+i0N,J+j0Nc(ξ+(rk+sm)(Nc¯-Nc)+δe-βt)+le-βt.
Suppose that c≠c¯. We may assume that c<c¯. Fixing ξ and sending t→∞, this leads that either U¯I,Jc¯(ξ)≡0 or U¯I,Jc¯(ξ)≡1, which is a contradiction. Hence, c=c¯.
We now suppress the dependence ofc, and we obtain
(3.19)UI-i0N,J-j0N(ξ)≤U¯I,J(ξ)≤UI+i0N,J+j0N(ξ),∀ξ∈R,∀I,J.
Forξ0:=(ri0+sj0)N/c, we have
(3.20)UI,J(ξ+ξ0)≤U¯I,J(ξ)≤UI,J(ξ-ξ0),∀ξ∈R,∀I,J.
Define
(3.21)ξ*∶=inf⁡{ξ̃∣UI,J(ξ+ξ̃)≤U¯I,J(ξ),∀ξ∈R,∀I,J},ξ*∶=sup⁡{ξ̃∣U¯I,J(ξ)≤UI,J(ξ+ξ̃),∀ξ∈R,∀I,J}.
Since U̇i,j(t)<0 due to c>0, we have ξ*≤ξ*. Assume that ξ*<ξ*. By the strong comparison principle, we know that
(3.22)U¯I,J(ξ)<UI,J(ξ+ξ*),∀ξ∈R,∀I,J.
Since lim|ξ|→∞U̇I,J(ξ)=0, there exists M>0 such that
(3.23)2σ1|U̇I,J(ξ)|≤1,∀|ξ|≥M,∀I,J,
where σ1 is the number mentioned in Lemma 3.2. If |ξ+ξ*|≤M+1, by the continuity of UI,J, we would find h∈(0,η), η:=min{1/(2σ1),l0}, such that
(3.24)U¯I,J(ξ)<UI,J(ξ+ξ*+2σ1h),∀I,J.
If |ξ+ξ*|≥M+1, then
(3.25)UI,J(ξ+ξ*+2σ1h)-U¯I,J(ξ)>UI,J(ξ+ξ*+2σ1h)-UI,J(ξ+ξ*)=U̇I,J(ξ+ξ*+θ2σ1h)(2σ1h),θ∈(0,1)≥-h,∀I,J.
Combining (3.24) and (3.25), we get
(3.26)U¯I,J(ξ)≤UI,J(ξ+ξ*+2σ1h)+h,∀ξ∈R,∀I,J.
Hence,
(3.27)U¯I,J(0)≤UI,J(ξ*+2σ1h)+h,∀I,J.
By Lemma 3.2 and the comparison principle, we have
(3.28)U¯I,J(ξ)≤UI,J(ξ+ξ*+2σ1h-σ1h(1-e-βt))+he-βt,∀ξ∈R,∀I,J.
Fixing ξ and sending t→∞,
(3.29)U¯I,J(ξ)≤UI,J(ξ+ξ*+σ1h).
This contradicts with the definition of ξ*. Hence, Ui,j(ξ)=U¯i,j(ξ+ξ*).Hence, we obtain the uniqueness (up to translations) of the traveling wave solution with nonzero speed.



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