# Stability of Traveling Wave Fronts for a Three Species Predator-Prey Model with Nonlocal Dispersals

Authors: Dongmei Yuan; Yuzhen Bai
Journal: Complexity
 (2019)

---

## Abstract

In this paper, we consider a predator-prey model with nonlocal dispersals of two cooperative preys and one predator. We prove that the traveling wave fronts with the relatively large wave speed are exponentially stable as perturbation in some exponentially weighted spaces, when the difference between initial data and traveling wave fronts decay exponentially at negative infinity, but in other locations, the initial data can be very large. The adopted method is to use the weighted energy method and the squeezing technique with some new flavors to handle the nonlocal dispersals.


---

## Body

## 1. Introduction

In this paper, we investigate the stability of traveling wave fronts for a three species predator-prey model:(1)∂ux,t∂t=∫ℝJ1x−yuy,t−ux,tdy+r1u1−u+kv−a1z,x∈ℝ,t>0,∂vx,t∂t=∫ℝJ2x−yvy,t−vx,tdy+r2v1−v+hu−a2z,x∈ℝ,t>0,∂zx,t∂t=∫ℝJ3x−yzy,t−zx,tdy+r3z−1+a1u+a2v−z,x∈ℝ,t>0,with the initial condition(2)ux,0=u0x,0,vx,0=v0x,0,zx,0=z0x,0,where ri,i=1,2,3,a1,a2,h,andk are positive constants, ux,t and vx,t denote the densities of two cooperative preys at time t and location x, respectively, zx,t denotes the density of the predator at time t and location x, r1 and r2 are the intrinsic growth rates of ux,t and vx,t, respectively, r3 is the death rate of zx,t, h and k are interspecific cooperative coefficients between two preys, r1a1 and r2a2 are the predation rates, and r3a1 and r3a2 are the conversion rates. Ji:ℝ⟶ℝi=1,2,3 are probability functions of the random dispersal of individuals and satisfy the following assumptions:H1Jix=Ji−x≥0,x∈ℝ,∫ℝJixdx=1and∫ℝJiyeλydy<∞,λ>0 is the eigenvalue of the characteristic equation of model (1)H20<hk<1Wu [1] investigated the spreading speed for a predator-prey model with one predator and two preys:(3)∂ux,t∂t=d1uxx+r1u1−u−kv−a1z,x∈ℝ,t>0,∂vx,t∂t=d2vxx+r2v1−hu−v−a2z,x∈ℝ,t>0,∂zx,t∂t=d3zxx+r3z−1+a1u+a2v−z,x∈ℝ,t>0,with the initial condition(4)ux,0=u∗,vx,0=v∗,zx,0=z0x,where ux,t and vx,t denote the densities of two competitive preys at time t and location x, respectively and zx,t denotes the density of the predator at time t and location x. The parameter di,i=1,2,3, is the diffusion coefficients of u,v,andz, respectively, the remaining parameters are the same as model (1). Under certain conditions, the author characterized the asymptotic spreading speed by the parameters of model (3). The interaction of three species eological systems has been studied before (see [2–4]).Yu and Pei [5] studied the stability of traveling wave fronts for the cooperative system with nonlocal dispersals:(5)∂u1x,t∂t=∫ℝJ1x−yu1y,t−u1x,tdy+r1u1x,t1−a1u1x,t+b1u2x,t,∂u2x,t∂t=∫ℝJ2x−yu2y,t−u2x,tdy+r2u2x,t1−a2u2x,t+b2u1x,t,with the initial condition(6)u1x,0=u10x,0,u2x,0=u20x,0.The authors adopt the weighted energy method and the squeezing technique to prove the stability of the traveling wave fronts. The weighted energy method for treating time-delayed reaction diffusion equations was firstly introduced by Mei et al. [6]. Then, by combining the squeezing argument, it was developed for proving the global stability of wavefronts by [7–9]. For the nonlocal model using a single integrodifferential equation, the existence, uniqueness, and stability of traveling waves have been widely studied in [10–14]. For the multicomponent nonlocal systems, the existence of traveling waves was also investigated in [15, 16], while the stability of traveling waves is less investigated and can only be found in [5, 17]. Many researchers are widely focused on the complex dynamics of biological systems such as stochastic delay population system [18] and many researchers have studied the Lotka–Volterra time delay models with two competitive preys and one predator [19]. Note that the composite population systems with stochastic effects and time delays present some complex dynamics; thus, this causes widespread researchers concern [20, 21].Inspired by works [1, 5], we will study the stability of traveling wave fronts for problems (1) and (2). We consider the case when both preys are cooperative. In this case, there is the constant state:(7)u,v,z=1+k1−hk,1+h1−hk,0=k1,k2,k3.The rest of this paper is organized as follows. In Section2, we present some preliminaries and summarize our main results. Section 3 is concerned with the proof of the main result by the technical weighted energy method and establishes the desired priori estimate, and then we get it by the squeezing technique. Finally, we make a simple summary in Section 4.Before stating our main result, we introduce some notations.Notations.
Throughout this paper,C>0 denotes a generic constant, while Ci>0i=0,1,2,… represents a specific constant. Let I be an interval, typically I=ℝ. L2I is the space of square integrable functions defined on I, and HkIk≥0 is the Sobolev space of the L2 functions fx defined on the interval I whose derivatives di/dxifi=1,…,k also belong to L2I. Further, Lw2I be the weighted L2 space with a weight function wx>0, with the norm defined as(8)fLw2I=∫Iwxfx2dx1/2.
H
w
k
I be the weighted Sobolev space with the norm given by(9)fHwkI=∑i=0k∫Iwxdifxdxi2dx1/2.
LetT>0 be a number and B be a Banach space. We denote by C00,T;B the space of the B-valued continuous functions on 0,T, and L20,T;B as the space of the B-valued L2 functions on 0,T. The corresponding spaces of the B-valued functions on 0,∞ are defined similarly.


## 2. Preliminaries and Main Result

In this section, we consider the existence of traveling wave fronts for the Cauchy problems (1) and (2) by using the comparison principle and squeezing technique. Traveling waves solution of system (1) is a special solution u,v,z with the form ux,t=ϕ1x+ct=ϕ1ξ, vx,t=ϕ2x+ct=ϕ2ξ, and zx,t=ϕ3x+ct=ϕ3ξ with c>0 is the wave speed, where the wave profile ϕ1,ϕ2,ϕ3∈C1ℝ,ℝ,ℝ and satisfies(10)cϕ1′ξ=∫ℝJ1y−ξϕ1y−ϕ1ξdy+r1ϕ11−ϕ1+kϕ2−a1ϕ3,cϕ2′ξ=∫ℝJ2y−ξϕ2y−ϕ2ξdy+r2ϕ21−ϕ2+hϕ1−a2ϕ3,cϕ3′ξ=∫ℝJ3y−ξϕ3y−ϕ3ξdy+r3ϕ3−1+a1ϕ1+a2ϕ2−ϕ3.In order to state our stability result, we need to give the following two propositions in [5].Proposition 1.
Assume thatH1-H2 hold. Then there exists a positive number c∗ such that, for any c>c∗, (1) and (2) admits a traveling wavefront ϕ1ξ,ϕ2ξ,ϕ3ξ satisfying(11)limt⟶+∞ϕ1,ϕ2,ϕ3=k1,k2,k3≜K,limt⟶−∞ϕ1,ϕ2,ϕ3=0,0,0≜0.Proposition 2.
Assume thatH1-H2 hold. Let c0x,0=u0x,0,v0x,0,z0x,0∈0,KX, where X=B∪Cℝ,ℝ3 is the Banach space of all bounded and uniformly continuous functions from ℝ to ℝ3 with the usual supremum norm ⋅X. Then, (1) and (2) admit a unique, bounded, and nonnegative solution cx,t,c0=ux,t,c0,vx,t,c0,zx,t,c0 on x,t∈ℝ×0,+∞ with the initial data ux,0,c0,vx,0,c0,zx,0,c0, and 0≤cx,t,c0≤K for any x∈ℝ and t>0. Furthermore, letting u−x,t,v−x,t,z−x,t and u+x,t,v+x,t,z+x,t be the solutions of (1) with no diffusion term with the initial data u−x,0,v−x,0,z−x,0 and u+x,0,v+x,0,z+x,0, respectively, and if(12)0≤u−x,0,v−x,0,z−x,0≤u+x,0,v+x,0,z+x,0≤K,forx∈ℝ,then(13)0≤u−x,t,v−x,t,z−x,t≤u+x,t,v+x,t,z+x,t≤K,x∈ℝ,t≥0.Propositions1 and 2 are based on reference [5] and are generalized from two dimension to three dimension. Their proofs are similar to the ones of Theorem 3.5 in [22] and Lemma 3.2 in [23], respectively.In order to state our stability result, we need to add the following conditions:H30<r2h<min1/k2r1−r1kk1−1/41/k2r2−r1kk1−1/4,H40<r3h<min1/a2r3−1/8,H50<r3k<min1/a1r3−1/8.Define three functions onη as follows:(14)f1η=2r1−2r1kk1−2r2hk2−∫−∞0J1ye−ηydy,f2η=2r2−2r2hk2−2r1kk1−∫−∞0J2ye−ηydy,f3η=4r3−2a1r3k−2a2r3h−∫−∞0J3ye−ηydy.According toH1-H5, it is easily checked(15)f10=2r1−2r1kk1−2r2hk2−12>0,f20=2r2−2r2hk2−2r1kk1−12>0,f30=4r3+2a1r3k+2a2r3h−12>0.Therefore, by continuity, there existsη0>0 such that f1η0>0,f2η0>0 and f3η0>0.In addition, define three functions onξ as follows:(16)g1ξ=−2r1+4r1ϕ1−4r1kk2−2r1kk1−2r2hk2−∫−∞0J1ye−η0ydy,g2ξ=−2r2+4r2ϕ2−4r2hk1−2r2hk2−2r1kk1−∫−∞0J2ye−η0ydy,g3ξ=−4r3+4r3ϕ3−4r3k3−2a1kr3−2a2hr3−∫−∞0J3ye−η0ydy,where ϕ1,ϕ2,ϕ3 is a traveling wave front of (10). We can easily prove(17)limξ⟶+∞g1ξ=f1η0>0,limξ⟶+∞g2ξ=f2η0>0,limξ⟶+∞g3ξ=f3η0>0,which imply that(18)g1ξ0=−2r1+4r1ϕ1−4r1kk2−2r1kk1−2r2hk2−∫−∞0J1ye−η0ydy>0,g2ξ0=−2r2+4r2ϕ2−4r2hk1−2r2hk2−2r1kk1−∫−∞0J2ye−η0ydy>0,g3ξ0=−4r3+4r3ϕ3−4r3k3−2a1kr3−2a2hr3−∫−∞0J3ye−η0ydy>0,where ξ0 is chosen to be large enough.Define a weight function by(19)wξ=e−η0ξ−ξ0,ξ≤ξ0,1,ξ>ξ0.Now, we state the stability result.Theorem 1.
Assume thatH1-H5 hold. If any traveling wave front ϕ1x+ct,ϕ2x+ct,ϕ3x+ct of (10) with the wave speed c>maxc∗,1/η0maxc1,c2,c3, where(20)c1=2r1+4r1kk2+2r1kk1+2r2hk2−12−∫−∞0J1ye−η0ydy,c2=2r2+4r2hk1+2r2hk2+2r1kk1−12−∫−∞0J2ye−η0ydy,c3=4r3+4r3k3+2a1kr3+2a2hr3−12−∫−∞0J3ye−η0ydy,the initial data satisfy(21)0,0,0≤u0x,0,v0x,0,z0x,0≤k1,k2,k3,x∈ℝ,and the initial perturbation satisfies(22)u0x,0−ϕ1x∈Hw1ℝ,v0x,0−ϕ2x∈Hw1ℝ,z0x,0−ϕ3x∈Hw1ℝ.Then, the nonnegative solution of problems (1) and (2) exists uniquely and satisfies(23)0,0,0≤ux,t,vx,t,zx,t≤k1,k2,k3,x∈ℝ,t>0,ux,t−ϕ1x+ct,vx,t−ϕ2x+ct,zx,t−ϕ3x+ct∈C0,+∞;Hw1ℝ∩L20,+∞;Hw1ℝ,where wx is defined by (19). Moreover, ux,t,vx,t,zx,t converges to the traveling wave front ϕ1x+ct,ϕ2x+ct,ϕ3x+ct exponentially in time t, i.e.,(24)supx∈ℝux,t−ϕ1x+ct≤Ce−μt,supx∈ℝvx,t−ϕ2x+ct≤Ce−μt,supx∈ℝzx,t−ϕ3x+ct≤Ce−μt,for all t>0, where C and μ are some positive constants.


## 3. Stability

In this section, the existence of the solution to problems (1) and (2) can be proved via the upper and lower solutions method (see [24]). We first examine the case when two species cooperate while remaining in the same area without diffusive movement:(25)∂u1x,t∂t=r1u1x,t1−a1u1x,t+b1u2x,t,∂u2x,t∂t=r2u2x,t1−a2u2x,t+b2u1x,t,where all parameters are positive and have four constant equilibria 0,0,0,1/a2,and1/a1,0 and coexistence equilibrium a2+b1/a1a2−b1b2,a1+b2/a1a2−b1b2=k1,k2 provided that a1a2>b1b2.Li and Lin [22] investigated the existence of traveling wave fronts connecting 0,0 to the constant equilibrium k1,k2 for system (5) by using the known theory [25]. Integrodifferential system (5) is related to the classic Laplacian diffusion system, for example,(26)Jx=δx+δ″x,where δ is the Dirac delta (see, Medlock et al. [26]), and then (5) reduces to(27)∂u1x,t∂t=∂2u1x,t∂x2+r1u1x,t1−a1u1x,t+b1u2x,t,∂u2x,t∂t=∂2u2x,t∂x2+r2u2x,t1−a2u2x,t+b2u1x,t.Because time delay often affects the evolutionary process, the system was incorporated into discrete and nonlocal delays while the existence of traveling wave fronts was obtained (see, [27–29]). The existence of traveling wave solutions of system (1) is similar to them. That is, the initial data satisfies(28)0,0,0≤u0x,0,v0x,0,z0x,0≤k1,k2,k3,x∈ℝ,and the initial perturbation satisfies(29)u0x,0−ϕ1x∈Hw1ℝ,v0x,0−ϕ2x∈Hw1ℝ,z0x,0−ϕ3x∈Hw1ℝ,and then the nonnegative solution of problems (1) and (2) exists uniquely and satisfies(30)0,0,0≤ux,t,vx,t,zx,t≤k1,k2,k3,x∈ℝ,t>0,ux,t−ϕ1x+ct,vx,t−ϕ2x+ct,zx,t−ϕ3x+ct∈C0,+∞;Hw1ℝ∩L20,+∞;Hw1ℝ.Here, we omit the details.Define(31)u0−x,0=minu0x,0,ϕ1x,x∈ℝ,u0+x,0=maxu0x,0,ϕ1x,x∈ℝ,v0−x,0=minv0x,0,ϕ2x,x∈ℝ,v0+x,0=maxv0x,0,ϕ2x,x∈ℝ,z0−x,0=minz0x,0,ϕ3x,x∈ℝ,z0+x,0=maxz0x,0,ϕ3x,x∈ℝ,and then we obtain(32)0≤u0−x,0≤u0x,0≤u0+x,0≤k1,x∈ℝ,0≤u0−x,0≤ϕ1x≤u0+x,0≤k1,x∈ℝ,0≤v0−x,0≤v0x,0≤v0+x,0≤k2,x∈ℝ,0≤v0−x,0≤ϕ2x≤v0+x,0≤k2,x∈ℝ,0≤z0−x,0≤z0x,0≤z0+x,0≤k3,x∈ℝ,0≤z0−x,0≤ϕ3x≤z0+x,0≤k3,x∈ℝ.Therefore, it follows from the comparison principle that(33)0≤u−x,t≤ux,t≤u+x,t≤k1,x,t∈ℝ×ℝ+,0≤u−x,t≤ϕ1x+ct≤u+x,t≤k1,x,t∈ℝ×ℝ+,0≤v−x,t≤vx,t≤v+x,t≤k2,x,t∈ℝ×ℝ+,0≤v−x,t≤ϕ2x+ct≤v+x,t≤k2,x,t∈ℝ×ℝ+,0≤z−x,t≤zx,t≤z+x,t≤k3,x,t∈ℝ×ℝ+,0≤z−x,t≤ϕ3x+ct≤z+x,t≤k3,x,t∈ℝ×ℝ+.Letting(34)U1ξ,t=u+x,t−ϕ1ξ,U10ξ,0=u0+−ϕ1ξ,x,t∈ℝ×ℝ+,U2ξ,t=v+x,t−ϕ2ξ,U20ξ,0=v0+−ϕ2ξ,x,t∈ℝ×ℝ+,U3ξ,t=z+x,t−ϕ3ξ,U30ξ,0=z0+−ϕ3ξ,x,t∈ℝ×ℝ+,where ξ=x+ct, and it follows from (32) and (33) that(35)0,0,0≤U10ξ,0,U20ξ,0,U30ξ,0≤k1,k2,k3,0,0,0≤U1ξ,t,U2ξ,t,U3ξ,t≤k1,k2,k3.


### 3.1. A Prior Estimate

This section is devoted to establishing a prior estimate, which is the core of this paper. The approach is the weighted energy method.Define(36)Bμ,wiξ,t=Awiξ,t−2μ,i=1,2,3,4,5,6,where(37)Aw1ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2a1r1U3+2a1r1ϕ3−kr1ϕ1+a1r1ϕ1−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy,(38)Aw2ξ,t=−cw′w−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+2a2r2U3+2a2r2ϕ3−hr2ϕ2+a2r2ϕ2−kr1ϕ1+r1a1ϕ1+1−∫ℝJ2ywξ+ywξdy,(39)Aw3ξ,t=−cw′w+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2−a1r3ϕ1−a2r3ϕ3−kr1ϕ1−hr2ϕ2+1−∫ℝJ3ywξ+ywξdy,(40)Aw4ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+4r1U1+2a1r1U3+2a1r1ϕ3−kr1U1−kr1ϕ1+a1r1U1+a1r1ϕ1+hr2U2−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy,(41)Aw5ξ,t=−cw′w−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+4r2U2+2a2r2U3+2a2r2ϕ3−kr1U1−kr1ϕ1+a2r2U2+a2r2ϕ2−r2hU2−r2hϕ2+r1a1ϕ1+1−∫ℝJ2ywξ+ywξdy,(42)Aw6ξ,t=−cw′w+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2+4r3U3−a1r3U3−a1r3ϕ3−a2r3U3−a2r3ϕ3−kr1ϕ1−hr2ϕ2−r3ϕ3+1−∫ℝJ3ywξ+ywξdy.In order to get the basic estimate, we must prove thatAwiξ,t≥C>0,i=1,2,3,4,5,6 and Bwiξ,t≥C>0,i=1,2,3,4,5,6, for some constant C. We need the following key lemma.Lemma 1.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, there exists some positive constant C such that(43)Awiξ,t≥C,i=1,2,3,for allξ∈ℝ,t≥0.Proof.
We proveAw1ξ,t≥C1 for some positive constant C1. Since c>maxc∗,1/η0maxc1,c2,c3, we have cη0≥c1,cη0≥c2 and cη0≥c3, that is,(44)cη0−2r1−6r1k−2r2h+12−∫ℝJ1yeη0ydy>0,cη0−2r2−6r2h−2r1k+12−∫ℝJ2yeη0ydy>0,cη0−4r3−2a1r3−2a2r3+12−∫ℝJ3yeη0ydy>0.Case 1.
ξ
≤
ξ
0. Since, wξ=e−η0ξ−ξ0 and wξ is nonincreasing, it follows from (37) that(45)Aw1ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2a1r1U3+2a1r1ϕ3−kr1ϕ1+a1r1ϕ1−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy≥cη0−2r1−4r1kk2−r1kk1−hr2k2+1−∫−∞ξ0−ξJ1ye−η0ydy−∫​ξ0−ξ+∞J1yeη0ξ−ξ0dy≥cη0−2r1−4r1kk2−r1kk1−hr2k2+1−∫ℝJ1yeη0ydy−∫0+∞J1yeη0ξ−ξ0dy≥cη0−2r1−4r1kk2−r1kk1−hr2k2+12−∫ℝJ1yeη0ydy>r1k+r2h>0.Case 2ξ>ξ0.
In this case,wξ=1 and w′ξ/wξ=0. Thus, it follows from (37) that(46)Aw1ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2a1r1U3+2a1r1ϕ3−kr1ϕ1+a1r1ϕ1−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy≥−2r1+4r1ϕ1ξ0−4r1kk2−r1kk1−hr2k2−∫−∞0J1ye−η0ydy>r1k+r2h>0.
Next, settingC1=r1k+r2h>0, we have Aw1ξ,t≥C1. We can prove Aw2ξ,t≥C2andAw3ξ,t≥C3 in the same way.Lemma 2.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, there exists some positive constant C such that(47)Awiξ,t≥C,i=4,5,6,for allξ∈ℝ,t≥0.Proof.
We proveAw4ξ,t≥C7 for some positive constant C7.Case 3.
ξ
≤
ξ
0. It follows from wξ=e−η0ξ−ξ0 and the monotonicity of wξ that(48)Aw4ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+4r1U1+2a1r1U3+2a1r1ϕ3−kr1U1−kr1ϕ1+a1r1U1+a1r1ϕ1+hr2U2−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy≥cη0−2r1−4r1kk2−2r1kk1−2hr2k2+1−∫ℝJ1ywξ+ywξdy≥cη0−2r1−4r1kk2−2r1kk1−2hr2k2+1−∫−∞ξ0−ξJ1ye−η0ydy−∫ξ0−ξ+∞J1yeη0ξ−ξ0dy≥cη0−2r1−4r1kk2−2r1kk1−2hr2k2+1−∫ℝJ1yeη0ydy−∫0+∞J1yeη0ξ−ξ0dy≥cη0−2r1−4r1kk2−2r1kk1−2hr2k2+12−∫ℝJ1yeη0ydy=C7>0.Case 4.
ξ
>
ξ
0. According to wξ=1 and w′ξ/wξ=0, we know(49)Aw4ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+4r1U1+2a1r1U3+2a1r1ϕ3−kr1U1−kr1ϕ1+a1r1U1+a1r1ϕ1+hr2U2−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy≥−2r1+4r1ϕ1−4r1kk2−2r1kk1−2hr2k2+1−∫−∞ξ0−ξJ1ye−η0ξ+y−ξ0dy−∫ξ0−ξ+∞J1ydy≥−2r1+4r1ϕ1−4r1kk2−2r1kk1−2hr2k2−∫−∞0J1ye−η0ydy=C8>0.
In view of the above argument and lettingC4≔minC7,C8, we have Aw4ξ,t≥C4. We can prove Aw5ξ,t≥C5andAw6ξ,t≥C6 in the same way. This completes the proof.
We easily check the following result.Lemma 3.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, there exists some positive constant C such that(50)Bμ,wiξ,t≥C,i=1,2,3,4,5,6,hold for all ξ∈ℝ, and 0<μ<1/2mini=1,2,3,4,5,6Ci.
Now we are going to derive the energy estimates forU1t,ξ,U2t,ξ, and U3t,ξ in the weighted Sobolev space Hw1ℝ.Lemma 4.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, it holds(51)U1tLw22+U2tLw22+U3tLw22+∫0te−2μt−sU1sLw22+U2sLw22+U3sLw22ds≤Ce−2μtU100Lw22+U200Lw22+U300Lw22,for some positive constant C.Proof.
From (1) and (10), U1ξ,t,U2ξ,t,U3ξ,t satisfies(52)U1t+cU1ξ=∫ℝJ1yU1ξ−y,t−U1ξ,tdy+r1kϕ1U2−r1a1ϕ1U3+U1ξ,tr1−2r1ϕ1+r1kU2+r1kϕ2−r1a1U3−r1a1ϕ3−r1U12,U2t+cU2ξ=∫ℝJ2yU2ξ−y,t−U2ξ,tdy+r2hϕ2U1−r2a2ϕ2U3+U2ξ,tr2−2r2ϕ2+r2hU1+r2hϕ1−r2a2U3−r2a2ϕ3−r2U22,U3t+cU3ξ=∫ℝJ3yU3ξ−y,t−U3ξ,tdy+a1r3ϕ3U1+a2r3ϕ3U2+U3ξ,t−r3−2r3ϕ3+a1r3U1+a1r3ϕ1+a2r3U2+a2r3ϕ2−r3U32,with the initial data U1ξ,0=U10ξ,0,U2ξ,0=U20ξ,0,andU3ξ,0=U30ξ,0,ξ∈ℝ. Multiplying the three equations of (52) by e2μtwξU1ξ,t,e2μtwξU2ξ,t, and e2μtwξU3ξ,t, respectively, where μ>0 is defined like that, we have(53)12e2μtwU12t+c2e2μtwU12ξ−e2μtwU1∫ℝJ1yU1ξ−y,t−U1ξ,tdy+−c2w′w−μ−r1+2r1ϕ1−r1kU2−r1kϕ2+r1a1U3+r1a1ϕ3e2μtwU12=−r1e2μtwU13+kr1ϕ1e2μtwU1U2−r1a1ϕ1e2μtwU1U3,(54)12e2μtwU22t+c2e2μtwU22ξ−e2μtwU2∫ℝJ2yU2ξ−y,t−U2ξ,tdy+−c2w′w−μ−r2+2r2ϕ2−r2hU1−r2hϕ1+r2a2U3+r2a2ϕ3e2μtwU22=−r2e2μtwU23+hr2ϕ2e2μtwU1U2−r2a2ϕ2e2μtwU2U3,(55)12e2μtwU32t+c2e2μtwU32ξ−e2μtwU3∫ℝJ3yU3ξ−y,t−U3ξ,tdy+−c2w′w−μ+r3+2r3ϕ3−a1r3U1−a1r3ϕ1−r3a2U2−r3a2ϕ2e2μtwU32=−r3e2μtwU33+a1r3ϕ3e2μtwU1U3+a2r3ϕ3e2μtwU2U3.
Integrating (53)–(55) over ℝ×0,t with respect to ξ and t and noting the vanishing term at far field(56)c2e2μtwU12ξ⟶−∞∞=0,c2e2μtwU22ξ⟶−∞∞=0,c2e2μtwU32ξ⟶−∞∞=0,we obtain(57)e2μtU1tLw22−2∫0t∫ℝe2μswξU1ξ,s∫ℝJ1yU1ξ−y,t−U1ξ,tdydξds+∫0t∫ℝe2μs−cw′w−2μ−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2r1a1U3+2r1a1ϕ3wξU12dξds≤U100Lw22+2kr1∫0t∫ℝϕ1e2μswξU1ξ,sU2ξ,sdξds−2r1a1∫0t∫ℝϕ1e2μswξU1ξ,sU3ξ,sdξds,(58)e2μtU2tLw22−2∫0t∫ℝe2μswξU2ξ,s∫ℝJ2yU2ξ−y,t−U2ξ,tdydξds+∫0t∫ℝe2μs−cw′w−2μ−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+2r2a2U3+2r2a2ϕ3wξU22dξds≤U200Lw22+2hr2∫0t∫ℝϕ2e2μswξU1ξ,sU2ξ,sdξds−2r2a2∫0t∫ℝϕ2e2μswξU2ξ,sU3ξ,sdξds,(59)e2μtU3tLw22−2∫0t∫ℝe2μswξU3ξ,s∫ℝJ3yU3ξ−y,t−U3ξ,tdydξds+∫0t∫ℝe2μs−cw′w−2μ+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2wξU32dξds≤U300Lw22+2a1r3∫0t∫ℝϕ3e2μswξU1ξ,sU3ξ,sdξds+2a2r3∫0t∫ℝϕ3e2μswξU2ξ,sU3ξ,sdξds.
By using the Cauchy–Schwarz inequality2xy≤x2+y2, we get(60)2∫0t∫ℝe2μswξUiξ,s∫ℝJiyUiξ−y,s−Uiξ,sdydξds≤∫0t∫ℝe2μswξUi2ξ,t∫ℝJiwξ+ywξdy−1dξds.
Similarly, we obtain(61)2∫0t∫ℝϕie2μswξU1U2dξds≤∫0t∫ℝϕie2μswξU12+U22dξds,i=1,2,(62)2∫0t∫ℝϕie2μswξU2U3dξds≤∫0t∫ℝϕie2μswξU22+U32dξds,i=2,3,(63)2∫0t∫ℝϕie2μswξU1U3dξds≤∫0t∫ℝϕie2μswξU12+U32dξds,i=1,3.
Substituting (60)–(63) into (57)–(59), respectively, we have(64)e2μtU1tLw22+∫0t∫ℝe2μs−cw′w−2μ−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2r1a1U3+2r1a1ϕ3−kr1ϕ1+a1r1ϕ1+1−∫ℝJ1ywξ+ywξdywξU12ξ,sdξds≤U100Lw22+kr1∫0t∫ℝϕ1e2μswξU22ξ,sdξds−r1a1∫0t∫ℝϕ1e2μswξU32ξ,sdξds,(65)e2μtU2tLw22+∫0t∫ℝe2μs−cw′w−2μ−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+2r2a2U3+2r2a2ϕ3−r2hϕ2+a2r2ϕ2+1−∫ℝJ2ywξ+ywξdywξU22dξds≤U200Lw22+hr2∫0t∫ℝϕ2e2μswξU12ξ,sdξds−r2a2∫0t∫ℝϕ2e2μswξU32ξ,sdξds,(66)e2μtU3tLw22+∫0t∫ℝe2μs−cw′w−2μ+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2−a1r3ϕ3−a2r3ϕ3+1−∫ℝJ3ywξ+ywξdywξU32dξds≤U300Lw22+a1r3∫0t∫ℝϕ3e2μswξU12ξ,sdξds+a2r3∫0t∫ℝϕ3e2μswξU22ξ,sdξds.
Adding the three inequalities (64), (65), and (66), we obtain(67)e2μtU1tLw22+U2tLw22+U3tLw22+∫0t∫ℝe2μsBμ,w1wξU12ξ,s+Bμ,w2wξU22ξ,s+Bμ,w3wξU32ξ,sdξds≤U100Lw22+U200Lw22+U300Lw22,where Bμ,w1,Bμ,w2, and Bμ,w3 are defined in (36). According to (36), we can verify(68)U1tLw22+U2tLw22+U3tLw22+∫0te−2μt−sU1sLw22+U2sLw22+U3sLw22dξds≤Ce−2μtU100Lw22+U200Lw22+U300Lw22,for some positive constant C. This completes the proof.Lemma 5.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, it holds(69)U1ξtLw22+U2ξtLw22+U3ξtLw22+∫0te−2μt−sU1ξsLw22+U2ξsLw22+U3ξsLw22ds≤Ce−2μtU100Hw12+U200Hw12+U300Hw12,for some positive constant C.Proof.
Differentiating (52) with respect to ξ, we have(70)U1tξ+cU1ξξ=∫ℝJ1yU1ξξ−y,t−U1ξξ,tdy+U1ξξ,tr1−2r1ϕ1+r1kU2+r1kϕ2−r1a1U3−r1a1ϕ3−2r1U1U1ξ+r1kϕ1′U2+kr1ϕ1U2ξ−r1a1ϕ1′U3−r1a1ϕ1U3ξ+U1−2r1ϕ1′+r1kU2ξ+r1kϕ2′−r1a1U3ξ−r1a1ϕ3′,U2tξ+cU2ξξ=∫ℝJ2yU2ξξ−y,t−U2ξξ,tdy+U2ξξ,tr2−2r2ϕ2+r2hU1+r2hϕ1−r2a2U3−r2a2ϕ3−2r2U2U2ξ+r2hϕ2′U1+r2hϕ2U1ξ−r2a2ϕ2′U3−r2a2ϕ2U3ξ+U2−2r2ϕ2′+r2hU1ξ+r2hϕ1′−r2a2U3ξ−r2a2ϕ3′,U3tξ+cU3ξξ=∫ℝJ3yU3ξξ−y,t−U3ξξ,tdy+U3ξξ,t−r3−2r3ϕ3+a1r3U1+a1r3ϕ1+a2r3U2+a2r3ϕ2−2r3U3U3ξ+a1r3ϕ3′U1+a1r3ϕ3U1ξ+a2r3ϕ3′U2+a2r3ϕ3U2ξ+U3−2r3ϕ3′+a1r3U1ξ+a1r3ϕ1′+a2r3U2ξ+a2r3ϕ2′.
Multiplying the three equations of (70) by e2μtwξU1ξξ,t,e2μtwξU2ξξ,t, and e2μtwξU3ξξ,t, respectively, it holds(71)12e2μtwU1ξ2t+c2e2μtwU1ξ2ξ−e2μtwU1ξ∫ℝJ1yU1ξξ−y,t−U1ξξ,tdy+−c2w′w−μ−r1+2r1ϕ1−r1kU2−r1kϕ2+r1a1U3+r1a1ϕ3+2r1U1e2μtwU1ξ2=−2r1ϕ1′+r1kU2ξ+r1kϕ2′−r1a1U3ξ−r1a1ϕ3′e2μtwU1U1ξ+kr1ϕ1′e2μtwU2U1ξ+kr1ϕ1e2μtwU2ξU1ξ−r1a1ϕ1′e2μtwU3U1ξ−r1a1ϕ1e2μtwU1ξU3ξ,(72)12e2μtwU2ξ2t+c2e2μtwU2ξ2ξ−e2μtwU2ξ∫ℝJ2yU2ξξ−y,t−U2ξξ,tdy+−c2w′w−μ−r2+2r2ϕ2−r2hU1−r2hϕ1+r2a2U3+r2a2ϕ3+2r2U2e2μtwU2ξ2=−2r2ϕ2′+r2hU1ξ+r2hϕ1′−r2a2U3ξ−r2a2ϕ3′e2μtwU2U2ξ+r2hϕ2′e2μtwU1U2ξ+hr2ϕ2e2μtwU1ξU2ξ−r2a2ϕ2′e2μtwU3U2ξ−a2r2ϕ2e2μtwU2ξU3ξ,(73)12e2μtwU3ξ2t+c2e2μtwU3ξ2ξ−e2μtwU3ξ∫ℝJ3yU3ξξ−y,t−U3ξξ,tdy+−c2w′w−μ+r3+2r3ϕ3−a1r3U1−a1r3ϕ1−r3a2U2−r3a2ϕ2+2r3U3e2μtwU3ξ2=−2r3ϕ3′+a1r3U1ξ+a1r3ϕ1′+a2r3U2ξ+a2r3ϕ2′e2μtwU3U3ξ+a1r3ϕ3′e2μtwU1U3ξa1r3ϕ3e2μtwU1ξU3ξ+a2r3ϕ3′e2μtwU2U3ξ+a2r3ϕ3e2μtwU2ξU3ξ.
According toU1∈Hw1,U2∈Hw1, and U3∈Hw1, we have(74)c2e2μtwU1ξ2ξ⟶−∞∞=0,c2e2μtwU2ξ2ξ⟶−∞∞=0,c2e2μtwU3ξ2ξ⟶−∞∞=0.
Integrating (71)–(73) over ℝ×0,t with respect to ξ and t and according to the Cauchy–Schwarz inequality, we obtain(75)e2μtU1ξtLw22+∫0t∫ℝe2μs−cw′w−2μ−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2r1a1U3+2r1a1ϕ3+4r1U1−∫ℝJ1ywξ+ywξdy+1−r1kU1+a1r1U1−kr1ϕ1+a1r1ϕwξU1ξ2dξds≤U10ξtLwa22+kr1∫0t∫ℝU1+ϕ1e2μswξU2ξ2dξds−a1r1∫0t∫ℝU1+ϕ1e2μswξU3ξ2dξds+2kr1∫0t∫ℝe2μsϕ1′wξU2U1ξdξds−2a1r1∫0t∫ℝe2μsϕ1′wξU3U1ξdξds+2∫0t∫ℝe2μs−2r1ϕ1′+kr1ϕ2′−a1r1ϕ3′wξU1U1ξdξds,(76)e2μtU2ξtLw22+∫0t∫ℝe2μs−cw′w−2μ−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+2r2a2U3+2r2a2ϕ3+4r2U2−∫ℝJ2ywξ+ywξdy+1−r2hU2+a2r2U2−r2hϕ2+a2r2ϕ2wξU2ξ2dξds≤U20ξ0Lw22+hr2∫0t∫ℝU2+ϕ2e2μswξU1ξ2dξds−a2r2∫0t∫ℝU2+ϕ2e2μswξU3ξ2dξds+2r2h∫0t∫ℝe2μsϕ2′wξU1U2ξdξds−2a2r2∫0t∫ℝe2μsϕ2′wξU3U2ξdξds+2∫0t∫ℝe2μs−2r2ϕ2′+r2hϕ1′−r2a2ϕ3′wξU2U2ξdξds,(77)e2μtU3ξtLw22+∫0t∫ℝe2μs−cw′w−2μ+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2+4r3U3−∫ℝJ2ywξ+ywξdy+1−a1r3U3−a2r3U3−a1r3ϕ3−a2r3ϕ3wξU3ξ2dξds≤U30ξ0Lw22+a1r3∫0t∫ℝU3+ϕ3e2μswξU1ξ2dξds+a2r3∫0t∫ℝU3+ϕ3e2μswξU2ξ2dξds+2a1r3∫0t∫ℝe2μsϕ3′wξU1U3ξdξds+2a2r3∫0t∫ℝe2μsϕ3′wξU2U3ξdξds+2∫0t∫ℝe2μs−2r3ϕ3′+a1r3ϕ3′+a2r3ϕ2′wξU3U3ξdξds.
Adding the three inequalities (75), (76), and (77), we obtain(78)e2μtU1ξtLw22+U2ξtLw22+U3ξtLw22+∫0t∫ℝe2μsBμ,w4wξU1ξ2ξ,s+Bμ,w5wξU2ξ2ξ,s+Bμ,w6wξU3ξ2ξ,sdξds≤U10ξ0Lw22+U20ξ0Lw22+U30ξ0Lw22+2∫0t∫ℝQξ,swξe2μsdξds,where Bμ,w4,Bμ,w5, and Bμ,w6 are given by (36), and(79)Qξ,t=kr1ϕ1′U2U1ξ−r1a1ϕ1′U3U1ξ+−2r1ϕ1′+kr1ϕ2′−a1r1ϕ3′wξU1U1ξ+r2hϕ2′U1U2ξ−a2r2ϕ2′U3U2ξ+−2r2ϕ2′+hr2ϕ1′−a2r2ϕ3′wξU2U2ξ+a1r3ϕ3′U1U3ξ+a2r3ϕ3′U2U3ξ+−2r3ϕ3′+a1r3ϕ1′+a2r3ϕ2′wξU3U3ξ.
Note that0,0,0≤U1ξ,t,U2ξ,t,U3ξ,t≤k1,k2,k3, and ϕ1′,ϕ2′,ϕ3′ are bounded on ℝ. There exists a positive constant C0 such that(80)kr1ϕ1′≤C0,r1a1ϕ1′≤C0,−2r1ϕ1′+kr1ϕ2′−a1r1ϕ3′≤C0,r2hϕ2′≤C0,a2r2ϕ2′≤C0,−2r2ϕ2′+hr2ϕ1′−a2r2ϕ3′≤C0,a1r3ϕ3′≤C0,a2r3ϕ3′≤C0,−2r3ϕ3′+a1r3ϕ1′+a2r3ϕ2′≤C0.
According to Lemma4, we obtain the following inequality(81)∫0te2μsU1tLw22+U2tLw22+U3tLw22ds≤CU100Lw22+U200Lw22+U300Lw22.
By using the Young inequality2xy≤ηx2+1/ηy2 and (51) we have(82)2∫0t∫ℝe2μsQξ,swξdξds≤C0∫0t∫ℝe2μswξ1ηU12ξ,s+U22ξ,s+U32ξ,s+ηU1ξ2ξ,s+U2ξ2ξ,s+U3ξ2ξ,sdξds≤C0η∫0te2μsU1sLw22+U2sLw22+U3sLw22ds+ηC0∫0te2μsU1ξsLw22+U2ξsLw22+U3ξsLw22ds≤C0CηU100Lw22+U200Lw22+U300Lw22+ηC0∫0te2μsU1ξsLw22+U2ξsLw22+U3ξsLw22ds.
Choosingη>0 such that ηC0<1/2minc4,c5,c6 and combining with (81), we have(83)U1ξtLw22+U2ξtLw22+U3ξtLw22+∫0te−2μt−sU1ξsLw22+U2ξsLw22+U3ξsLw22ds≤Ce−2μtU100Hw12+U200Hw12+U300Hw12,for some positive constant C. This completes the proof.
Combining Lemma4 and Lemma 5 and noting that wξ≥1 on R, we obtain the following result.Lemma 6.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, it holds(84)U1tHw1≤Ce−μtU100Hw12+U200Hw12+U300Hw121/2,U2tHw1≤Ce−μtU100Hw12+U200Hw12+U300Hw121/2,U3tHw1≤Ce−μtU100Hw12+U200Hw12+U300Hw121/2,for some positive constant C and all t>0.


### 3.2. Proof of Theorem 1

Using Sobolev embedding inequalityH1ℝ↪Cℝ and wξ≥1, we have(85)supx∈ℝU1ξ,t≤CU1tH1≤CU1tHw1≤Ce−μt,supx∈ℝU2ξ,t≤CU2tH1≤CU2tHw1≤Ce−μt,supx∈ℝU3ξ,t≤CU3tH1≤CU3tHw1≤Ce−μt.In view of Lemma6, we obtain the following results(86)supx∈ℝu+−ϕ1x+ct=supξ∈ℝU1ξ,t≤Ce−μt,supx∈ℝv+−ϕ2x+ct=supξ∈ℝU2ξ,t≤Ce−μt,supx∈ℝz+−ϕ3x+ct=supξ∈ℝU3ξ,t≤Ce−μt,for t>0. Similarly, for t>0, we can also verify(87)supx∈ℝu−−ϕ1x+ct=supξ∈ℝU1ξ,t≤Ce−μt,supx∈ℝv−−ϕ2x+ct=supξ∈ℝU2ξ,t≤Ce−μt,supx∈ℝz−−ϕ3x+ct=supξ∈ℝU3ξ,t≤Ce−μt.Hence, by the squeezing technique, we obtain(88)supx∈ℝux,t−ϕ1x+ct≤Ce−μt,t>0,supx∈ℝvx,t−ϕ2x+ct≤Ce−μt,t>0,supx∈ℝzx,t−ϕ3x+ct≤Ce−μt,t>0.This completes the proof of Theorem1.


## 3.1. A Prior Estimate

This section is devoted to establishing a prior estimate, which is the core of this paper. The approach is the weighted energy method.Define(36)Bμ,wiξ,t=Awiξ,t−2μ,i=1,2,3,4,5,6,where(37)Aw1ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2a1r1U3+2a1r1ϕ3−kr1ϕ1+a1r1ϕ1−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy,(38)Aw2ξ,t=−cw′w−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+2a2r2U3+2a2r2ϕ3−hr2ϕ2+a2r2ϕ2−kr1ϕ1+r1a1ϕ1+1−∫ℝJ2ywξ+ywξdy,(39)Aw3ξ,t=−cw′w+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2−a1r3ϕ1−a2r3ϕ3−kr1ϕ1−hr2ϕ2+1−∫ℝJ3ywξ+ywξdy,(40)Aw4ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+4r1U1+2a1r1U3+2a1r1ϕ3−kr1U1−kr1ϕ1+a1r1U1+a1r1ϕ1+hr2U2−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy,(41)Aw5ξ,t=−cw′w−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+4r2U2+2a2r2U3+2a2r2ϕ3−kr1U1−kr1ϕ1+a2r2U2+a2r2ϕ2−r2hU2−r2hϕ2+r1a1ϕ1+1−∫ℝJ2ywξ+ywξdy,(42)Aw6ξ,t=−cw′w+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2+4r3U3−a1r3U3−a1r3ϕ3−a2r3U3−a2r3ϕ3−kr1ϕ1−hr2ϕ2−r3ϕ3+1−∫ℝJ3ywξ+ywξdy.In order to get the basic estimate, we must prove thatAwiξ,t≥C>0,i=1,2,3,4,5,6 and Bwiξ,t≥C>0,i=1,2,3,4,5,6, for some constant C. We need the following key lemma.Lemma 1.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, there exists some positive constant C such that(43)Awiξ,t≥C,i=1,2,3,for allξ∈ℝ,t≥0.Proof.
We proveAw1ξ,t≥C1 for some positive constant C1. Since c>maxc∗,1/η0maxc1,c2,c3, we have cη0≥c1,cη0≥c2 and cη0≥c3, that is,(44)cη0−2r1−6r1k−2r2h+12−∫ℝJ1yeη0ydy>0,cη0−2r2−6r2h−2r1k+12−∫ℝJ2yeη0ydy>0,cη0−4r3−2a1r3−2a2r3+12−∫ℝJ3yeη0ydy>0.Case 1.
ξ
≤
ξ
0. Since, wξ=e−η0ξ−ξ0 and wξ is nonincreasing, it follows from (37) that(45)Aw1ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2a1r1U3+2a1r1ϕ3−kr1ϕ1+a1r1ϕ1−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy≥cη0−2r1−4r1kk2−r1kk1−hr2k2+1−∫−∞ξ0−ξJ1ye−η0ydy−∫​ξ0−ξ+∞J1yeη0ξ−ξ0dy≥cη0−2r1−4r1kk2−r1kk1−hr2k2+1−∫ℝJ1yeη0ydy−∫0+∞J1yeη0ξ−ξ0dy≥cη0−2r1−4r1kk2−r1kk1−hr2k2+12−∫ℝJ1yeη0ydy>r1k+r2h>0.Case 2ξ>ξ0.
In this case,wξ=1 and w′ξ/wξ=0. Thus, it follows from (37) that(46)Aw1ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2a1r1U3+2a1r1ϕ3−kr1ϕ1+a1r1ϕ1−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy≥−2r1+4r1ϕ1ξ0−4r1kk2−r1kk1−hr2k2−∫−∞0J1ye−η0ydy>r1k+r2h>0.
Next, settingC1=r1k+r2h>0, we have Aw1ξ,t≥C1. We can prove Aw2ξ,t≥C2andAw3ξ,t≥C3 in the same way.Lemma 2.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, there exists some positive constant C such that(47)Awiξ,t≥C,i=4,5,6,for allξ∈ℝ,t≥0.Proof.
We proveAw4ξ,t≥C7 for some positive constant C7.Case 3.
ξ
≤
ξ
0. It follows from wξ=e−η0ξ−ξ0 and the monotonicity of wξ that(48)Aw4ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+4r1U1+2a1r1U3+2a1r1ϕ3−kr1U1−kr1ϕ1+a1r1U1+a1r1ϕ1+hr2U2−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy≥cη0−2r1−4r1kk2−2r1kk1−2hr2k2+1−∫ℝJ1ywξ+ywξdy≥cη0−2r1−4r1kk2−2r1kk1−2hr2k2+1−∫−∞ξ0−ξJ1ye−η0ydy−∫ξ0−ξ+∞J1yeη0ξ−ξ0dy≥cη0−2r1−4r1kk2−2r1kk1−2hr2k2+1−∫ℝJ1yeη0ydy−∫0+∞J1yeη0ξ−ξ0dy≥cη0−2r1−4r1kk2−2r1kk1−2hr2k2+12−∫ℝJ1yeη0ydy=C7>0.Case 4.
ξ
>
ξ
0. According to wξ=1 and w′ξ/wξ=0, we know(49)Aw4ξ,t=−cw′w−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+4r1U1+2a1r1U3+2a1r1ϕ3−kr1U1−kr1ϕ1+a1r1U1+a1r1ϕ1+hr2U2−hr2ϕ2+r2a2ϕ2+1−∫ℝJ1ywξ+ywξdy≥−2r1+4r1ϕ1−4r1kk2−2r1kk1−2hr2k2+1−∫−∞ξ0−ξJ1ye−η0ξ+y−ξ0dy−∫ξ0−ξ+∞J1ydy≥−2r1+4r1ϕ1−4r1kk2−2r1kk1−2hr2k2−∫−∞0J1ye−η0ydy=C8>0.
In view of the above argument and lettingC4≔minC7,C8, we have Aw4ξ,t≥C4. We can prove Aw5ξ,t≥C5andAw6ξ,t≥C6 in the same way. This completes the proof.
We easily check the following result.Lemma 3.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, there exists some positive constant C such that(50)Bμ,wiξ,t≥C,i=1,2,3,4,5,6,hold for all ξ∈ℝ, and 0<μ<1/2mini=1,2,3,4,5,6Ci.
Now we are going to derive the energy estimates forU1t,ξ,U2t,ξ, and U3t,ξ in the weighted Sobolev space Hw1ℝ.Lemma 4.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, it holds(51)U1tLw22+U2tLw22+U3tLw22+∫0te−2μt−sU1sLw22+U2sLw22+U3sLw22ds≤Ce−2μtU100Lw22+U200Lw22+U300Lw22,for some positive constant C.Proof.
From (1) and (10), U1ξ,t,U2ξ,t,U3ξ,t satisfies(52)U1t+cU1ξ=∫ℝJ1yU1ξ−y,t−U1ξ,tdy+r1kϕ1U2−r1a1ϕ1U3+U1ξ,tr1−2r1ϕ1+r1kU2+r1kϕ2−r1a1U3−r1a1ϕ3−r1U12,U2t+cU2ξ=∫ℝJ2yU2ξ−y,t−U2ξ,tdy+r2hϕ2U1−r2a2ϕ2U3+U2ξ,tr2−2r2ϕ2+r2hU1+r2hϕ1−r2a2U3−r2a2ϕ3−r2U22,U3t+cU3ξ=∫ℝJ3yU3ξ−y,t−U3ξ,tdy+a1r3ϕ3U1+a2r3ϕ3U2+U3ξ,t−r3−2r3ϕ3+a1r3U1+a1r3ϕ1+a2r3U2+a2r3ϕ2−r3U32,with the initial data U1ξ,0=U10ξ,0,U2ξ,0=U20ξ,0,andU3ξ,0=U30ξ,0,ξ∈ℝ. Multiplying the three equations of (52) by e2μtwξU1ξ,t,e2μtwξU2ξ,t, and e2μtwξU3ξ,t, respectively, where μ>0 is defined like that, we have(53)12e2μtwU12t+c2e2μtwU12ξ−e2μtwU1∫ℝJ1yU1ξ−y,t−U1ξ,tdy+−c2w′w−μ−r1+2r1ϕ1−r1kU2−r1kϕ2+r1a1U3+r1a1ϕ3e2μtwU12=−r1e2μtwU13+kr1ϕ1e2μtwU1U2−r1a1ϕ1e2μtwU1U3,(54)12e2μtwU22t+c2e2μtwU22ξ−e2μtwU2∫ℝJ2yU2ξ−y,t−U2ξ,tdy+−c2w′w−μ−r2+2r2ϕ2−r2hU1−r2hϕ1+r2a2U3+r2a2ϕ3e2μtwU22=−r2e2μtwU23+hr2ϕ2e2μtwU1U2−r2a2ϕ2e2μtwU2U3,(55)12e2μtwU32t+c2e2μtwU32ξ−e2μtwU3∫ℝJ3yU3ξ−y,t−U3ξ,tdy+−c2w′w−μ+r3+2r3ϕ3−a1r3U1−a1r3ϕ1−r3a2U2−r3a2ϕ2e2μtwU32=−r3e2μtwU33+a1r3ϕ3e2μtwU1U3+a2r3ϕ3e2μtwU2U3.
Integrating (53)–(55) over ℝ×0,t with respect to ξ and t and noting the vanishing term at far field(56)c2e2μtwU12ξ⟶−∞∞=0,c2e2μtwU22ξ⟶−∞∞=0,c2e2μtwU32ξ⟶−∞∞=0,we obtain(57)e2μtU1tLw22−2∫0t∫ℝe2μswξU1ξ,s∫ℝJ1yU1ξ−y,t−U1ξ,tdydξds+∫0t∫ℝe2μs−cw′w−2μ−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2r1a1U3+2r1a1ϕ3wξU12dξds≤U100Lw22+2kr1∫0t∫ℝϕ1e2μswξU1ξ,sU2ξ,sdξds−2r1a1∫0t∫ℝϕ1e2μswξU1ξ,sU3ξ,sdξds,(58)e2μtU2tLw22−2∫0t∫ℝe2μswξU2ξ,s∫ℝJ2yU2ξ−y,t−U2ξ,tdydξds+∫0t∫ℝe2μs−cw′w−2μ−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+2r2a2U3+2r2a2ϕ3wξU22dξds≤U200Lw22+2hr2∫0t∫ℝϕ2e2μswξU1ξ,sU2ξ,sdξds−2r2a2∫0t∫ℝϕ2e2μswξU2ξ,sU3ξ,sdξds,(59)e2μtU3tLw22−2∫0t∫ℝe2μswξU3ξ,s∫ℝJ3yU3ξ−y,t−U3ξ,tdydξds+∫0t∫ℝe2μs−cw′w−2μ+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2wξU32dξds≤U300Lw22+2a1r3∫0t∫ℝϕ3e2μswξU1ξ,sU3ξ,sdξds+2a2r3∫0t∫ℝϕ3e2μswξU2ξ,sU3ξ,sdξds.
By using the Cauchy–Schwarz inequality2xy≤x2+y2, we get(60)2∫0t∫ℝe2μswξUiξ,s∫ℝJiyUiξ−y,s−Uiξ,sdydξds≤∫0t∫ℝe2μswξUi2ξ,t∫ℝJiwξ+ywξdy−1dξds.
Similarly, we obtain(61)2∫0t∫ℝϕie2μswξU1U2dξds≤∫0t∫ℝϕie2μswξU12+U22dξds,i=1,2,(62)2∫0t∫ℝϕie2μswξU2U3dξds≤∫0t∫ℝϕie2μswξU22+U32dξds,i=2,3,(63)2∫0t∫ℝϕie2μswξU1U3dξds≤∫0t∫ℝϕie2μswξU12+U32dξds,i=1,3.
Substituting (60)–(63) into (57)–(59), respectively, we have(64)e2μtU1tLw22+∫0t∫ℝe2μs−cw′w−2μ−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2r1a1U3+2r1a1ϕ3−kr1ϕ1+a1r1ϕ1+1−∫ℝJ1ywξ+ywξdywξU12ξ,sdξds≤U100Lw22+kr1∫0t∫ℝϕ1e2μswξU22ξ,sdξds−r1a1∫0t∫ℝϕ1e2μswξU32ξ,sdξds,(65)e2μtU2tLw22+∫0t∫ℝe2μs−cw′w−2μ−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+2r2a2U3+2r2a2ϕ3−r2hϕ2+a2r2ϕ2+1−∫ℝJ2ywξ+ywξdywξU22dξds≤U200Lw22+hr2∫0t∫ℝϕ2e2μswξU12ξ,sdξds−r2a2∫0t∫ℝϕ2e2μswξU32ξ,sdξds,(66)e2μtU3tLw22+∫0t∫ℝe2μs−cw′w−2μ+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2−a1r3ϕ3−a2r3ϕ3+1−∫ℝJ3ywξ+ywξdywξU32dξds≤U300Lw22+a1r3∫0t∫ℝϕ3e2μswξU12ξ,sdξds+a2r3∫0t∫ℝϕ3e2μswξU22ξ,sdξds.
Adding the three inequalities (64), (65), and (66), we obtain(67)e2μtU1tLw22+U2tLw22+U3tLw22+∫0t∫ℝe2μsBμ,w1wξU12ξ,s+Bμ,w2wξU22ξ,s+Bμ,w3wξU32ξ,sdξds≤U100Lw22+U200Lw22+U300Lw22,where Bμ,w1,Bμ,w2, and Bμ,w3 are defined in (36). According to (36), we can verify(68)U1tLw22+U2tLw22+U3tLw22+∫0te−2μt−sU1sLw22+U2sLw22+U3sLw22dξds≤Ce−2μtU100Lw22+U200Lw22+U300Lw22,for some positive constant C. This completes the proof.Lemma 5.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, it holds(69)U1ξtLw22+U2ξtLw22+U3ξtLw22+∫0te−2μt−sU1ξsLw22+U2ξsLw22+U3ξsLw22ds≤Ce−2μtU100Hw12+U200Hw12+U300Hw12,for some positive constant C.Proof.
Differentiating (52) with respect to ξ, we have(70)U1tξ+cU1ξξ=∫ℝJ1yU1ξξ−y,t−U1ξξ,tdy+U1ξξ,tr1−2r1ϕ1+r1kU2+r1kϕ2−r1a1U3−r1a1ϕ3−2r1U1U1ξ+r1kϕ1′U2+kr1ϕ1U2ξ−r1a1ϕ1′U3−r1a1ϕ1U3ξ+U1−2r1ϕ1′+r1kU2ξ+r1kϕ2′−r1a1U3ξ−r1a1ϕ3′,U2tξ+cU2ξξ=∫ℝJ2yU2ξξ−y,t−U2ξξ,tdy+U2ξξ,tr2−2r2ϕ2+r2hU1+r2hϕ1−r2a2U3−r2a2ϕ3−2r2U2U2ξ+r2hϕ2′U1+r2hϕ2U1ξ−r2a2ϕ2′U3−r2a2ϕ2U3ξ+U2−2r2ϕ2′+r2hU1ξ+r2hϕ1′−r2a2U3ξ−r2a2ϕ3′,U3tξ+cU3ξξ=∫ℝJ3yU3ξξ−y,t−U3ξξ,tdy+U3ξξ,t−r3−2r3ϕ3+a1r3U1+a1r3ϕ1+a2r3U2+a2r3ϕ2−2r3U3U3ξ+a1r3ϕ3′U1+a1r3ϕ3U1ξ+a2r3ϕ3′U2+a2r3ϕ3U2ξ+U3−2r3ϕ3′+a1r3U1ξ+a1r3ϕ1′+a2r3U2ξ+a2r3ϕ2′.
Multiplying the three equations of (70) by e2μtwξU1ξξ,t,e2μtwξU2ξξ,t, and e2μtwξU3ξξ,t, respectively, it holds(71)12e2μtwU1ξ2t+c2e2μtwU1ξ2ξ−e2μtwU1ξ∫ℝJ1yU1ξξ−y,t−U1ξξ,tdy+−c2w′w−μ−r1+2r1ϕ1−r1kU2−r1kϕ2+r1a1U3+r1a1ϕ3+2r1U1e2μtwU1ξ2=−2r1ϕ1′+r1kU2ξ+r1kϕ2′−r1a1U3ξ−r1a1ϕ3′e2μtwU1U1ξ+kr1ϕ1′e2μtwU2U1ξ+kr1ϕ1e2μtwU2ξU1ξ−r1a1ϕ1′e2μtwU3U1ξ−r1a1ϕ1e2μtwU1ξU3ξ,(72)12e2μtwU2ξ2t+c2e2μtwU2ξ2ξ−e2μtwU2ξ∫ℝJ2yU2ξξ−y,t−U2ξξ,tdy+−c2w′w−μ−r2+2r2ϕ2−r2hU1−r2hϕ1+r2a2U3+r2a2ϕ3+2r2U2e2μtwU2ξ2=−2r2ϕ2′+r2hU1ξ+r2hϕ1′−r2a2U3ξ−r2a2ϕ3′e2μtwU2U2ξ+r2hϕ2′e2μtwU1U2ξ+hr2ϕ2e2μtwU1ξU2ξ−r2a2ϕ2′e2μtwU3U2ξ−a2r2ϕ2e2μtwU2ξU3ξ,(73)12e2μtwU3ξ2t+c2e2μtwU3ξ2ξ−e2μtwU3ξ∫ℝJ3yU3ξξ−y,t−U3ξξ,tdy+−c2w′w−μ+r3+2r3ϕ3−a1r3U1−a1r3ϕ1−r3a2U2−r3a2ϕ2+2r3U3e2μtwU3ξ2=−2r3ϕ3′+a1r3U1ξ+a1r3ϕ1′+a2r3U2ξ+a2r3ϕ2′e2μtwU3U3ξ+a1r3ϕ3′e2μtwU1U3ξa1r3ϕ3e2μtwU1ξU3ξ+a2r3ϕ3′e2μtwU2U3ξ+a2r3ϕ3e2μtwU2ξU3ξ.
According toU1∈Hw1,U2∈Hw1, and U3∈Hw1, we have(74)c2e2μtwU1ξ2ξ⟶−∞∞=0,c2e2μtwU2ξ2ξ⟶−∞∞=0,c2e2μtwU3ξ2ξ⟶−∞∞=0.
Integrating (71)–(73) over ℝ×0,t with respect to ξ and t and according to the Cauchy–Schwarz inequality, we obtain(75)e2μtU1ξtLw22+∫0t∫ℝe2μs−cw′w−2μ−2r1+4r1ϕ1−2r1kU2−2r1kϕ2+2r1a1U3+2r1a1ϕ3+4r1U1−∫ℝJ1ywξ+ywξdy+1−r1kU1+a1r1U1−kr1ϕ1+a1r1ϕwξU1ξ2dξds≤U10ξtLwa22+kr1∫0t∫ℝU1+ϕ1e2μswξU2ξ2dξds−a1r1∫0t∫ℝU1+ϕ1e2μswξU3ξ2dξds+2kr1∫0t∫ℝe2μsϕ1′wξU2U1ξdξds−2a1r1∫0t∫ℝe2μsϕ1′wξU3U1ξdξds+2∫0t∫ℝe2μs−2r1ϕ1′+kr1ϕ2′−a1r1ϕ3′wξU1U1ξdξds,(76)e2μtU2ξtLw22+∫0t∫ℝe2μs−cw′w−2μ−2r2+4r2ϕ2−2r2hU1−2r2hϕ1+2r2a2U3+2r2a2ϕ3+4r2U2−∫ℝJ2ywξ+ywξdy+1−r2hU2+a2r2U2−r2hϕ2+a2r2ϕ2wξU2ξ2dξds≤U20ξ0Lw22+hr2∫0t∫ℝU2+ϕ2e2μswξU1ξ2dξds−a2r2∫0t∫ℝU2+ϕ2e2μswξU3ξ2dξds+2r2h∫0t∫ℝe2μsϕ2′wξU1U2ξdξds−2a2r2∫0t∫ℝe2μsϕ2′wξU3U2ξdξds+2∫0t∫ℝe2μs−2r2ϕ2′+r2hϕ1′−r2a2ϕ3′wξU2U2ξdξds,(77)e2μtU3ξtLw22+∫0t∫ℝe2μs−cw′w−2μ+2r3+4r3ϕ3−2a1r3U1−2a1r3ϕ1−2a2r3U2−2a2r3ϕ2+4r3U3−∫ℝJ2ywξ+ywξdy+1−a1r3U3−a2r3U3−a1r3ϕ3−a2r3ϕ3wξU3ξ2dξds≤U30ξ0Lw22+a1r3∫0t∫ℝU3+ϕ3e2μswξU1ξ2dξds+a2r3∫0t∫ℝU3+ϕ3e2μswξU2ξ2dξds+2a1r3∫0t∫ℝe2μsϕ3′wξU1U3ξdξds+2a2r3∫0t∫ℝe2μsϕ3′wξU2U3ξdξds+2∫0t∫ℝe2μs−2r3ϕ3′+a1r3ϕ3′+a2r3ϕ2′wξU3U3ξdξds.
Adding the three inequalities (75), (76), and (77), we obtain(78)e2μtU1ξtLw22+U2ξtLw22+U3ξtLw22+∫0t∫ℝe2μsBμ,w4wξU1ξ2ξ,s+Bμ,w5wξU2ξ2ξ,s+Bμ,w6wξU3ξ2ξ,sdξds≤U10ξ0Lw22+U20ξ0Lw22+U30ξ0Lw22+2∫0t∫ℝQξ,swξe2μsdξds,where Bμ,w4,Bμ,w5, and Bμ,w6 are given by (36), and(79)Qξ,t=kr1ϕ1′U2U1ξ−r1a1ϕ1′U3U1ξ+−2r1ϕ1′+kr1ϕ2′−a1r1ϕ3′wξU1U1ξ+r2hϕ2′U1U2ξ−a2r2ϕ2′U3U2ξ+−2r2ϕ2′+hr2ϕ1′−a2r2ϕ3′wξU2U2ξ+a1r3ϕ3′U1U3ξ+a2r3ϕ3′U2U3ξ+−2r3ϕ3′+a1r3ϕ1′+a2r3ϕ2′wξU3U3ξ.
Note that0,0,0≤U1ξ,t,U2ξ,t,U3ξ,t≤k1,k2,k3, and ϕ1′,ϕ2′,ϕ3′ are bounded on ℝ. There exists a positive constant C0 such that(80)kr1ϕ1′≤C0,r1a1ϕ1′≤C0,−2r1ϕ1′+kr1ϕ2′−a1r1ϕ3′≤C0,r2hϕ2′≤C0,a2r2ϕ2′≤C0,−2r2ϕ2′+hr2ϕ1′−a2r2ϕ3′≤C0,a1r3ϕ3′≤C0,a2r3ϕ3′≤C0,−2r3ϕ3′+a1r3ϕ1′+a2r3ϕ2′≤C0.
According to Lemma4, we obtain the following inequality(81)∫0te2μsU1tLw22+U2tLw22+U3tLw22ds≤CU100Lw22+U200Lw22+U300Lw22.
By using the Young inequality2xy≤ηx2+1/ηy2 and (51) we have(82)2∫0t∫ℝe2μsQξ,swξdξds≤C0∫0t∫ℝe2μswξ1ηU12ξ,s+U22ξ,s+U32ξ,s+ηU1ξ2ξ,s+U2ξ2ξ,s+U3ξ2ξ,sdξds≤C0η∫0te2μsU1sLw22+U2sLw22+U3sLw22ds+ηC0∫0te2μsU1ξsLw22+U2ξsLw22+U3ξsLw22ds≤C0CηU100Lw22+U200Lw22+U300Lw22+ηC0∫0te2μsU1ξsLw22+U2ξsLw22+U3ξsLw22ds.
Choosingη>0 such that ηC0<1/2minc4,c5,c6 and combining with (81), we have(83)U1ξtLw22+U2ξtLw22+U3ξtLw22+∫0te−2μt−sU1ξsLw22+U2ξsLw22+U3ξsLw22ds≤Ce−2μtU100Hw12+U200Hw12+U300Hw12,for some positive constant C. This completes the proof.
Combining Lemma4 and Lemma 5 and noting that wξ≥1 on R, we obtain the following result.Lemma 6.
Assume thatH1-H5 hold. For any c>maxc∗,1/η0maxc1,c2,c3, it holds(84)U1tHw1≤Ce−μtU100Hw12+U200Hw12+U300Hw121/2,U2tHw1≤Ce−μtU100Hw12+U200Hw12+U300Hw121/2,U3tHw1≤Ce−μtU100Hw12+U200Hw12+U300Hw121/2,for some positive constant C and all t>0.


## 3.2. Proof of Theorem 1

Using Sobolev embedding inequalityH1ℝ↪Cℝ and wξ≥1, we have(85)supx∈ℝU1ξ,t≤CU1tH1≤CU1tHw1≤Ce−μt,supx∈ℝU2ξ,t≤CU2tH1≤CU2tHw1≤Ce−μt,supx∈ℝU3ξ,t≤CU3tH1≤CU3tHw1≤Ce−μt.In view of Lemma6, we obtain the following results(86)supx∈ℝu+−ϕ1x+ct=supξ∈ℝU1ξ,t≤Ce−μt,supx∈ℝv+−ϕ2x+ct=supξ∈ℝU2ξ,t≤Ce−μt,supx∈ℝz+−ϕ3x+ct=supξ∈ℝU3ξ,t≤Ce−μt,for t>0. Similarly, for t>0, we can also verify(87)supx∈ℝu−−ϕ1x+ct=supξ∈ℝU1ξ,t≤Ce−μt,supx∈ℝv−−ϕ2x+ct=supξ∈ℝU2ξ,t≤Ce−μt,supx∈ℝz−−ϕ3x+ct=supξ∈ℝU3ξ,t≤Ce−μt.Hence, by the squeezing technique, we obtain(88)supx∈ℝux,t−ϕ1x+ct≤Ce−μt,t>0,supx∈ℝvx,t−ϕ2x+ct≤Ce−μt,t>0,supx∈ℝzx,t−ϕ3x+ct≤Ce−μt,t>0.This completes the proof of Theorem1.


## 4. Conclusions

Inspired by the works of [1, 5], in this paper, we consider a predator-prey model with nonlocal dispersals of two cooperative preys and one predator. Since the predation models of the three species add to the computational difficulty, it takes a lot of effort to the priority estimation, and the key point in proving the stability of the traveling front is devoted to establishing a prior estimate by using the weighted energy method. By the standard Sobolev embedding inequality and the squeezing technique, we prove the stability of the traveling wave solution.In recent years, there has been great progress in modeling and analysis dynamical behavior of predator-prey population involving both time delay and spatial diffusion. In a pioneer work, some researchers have studied a scalar reaction diffusion equation with a single discrete delay by using the phase-plan technique; we can take more attention and initiate the study of traveling wave solutions to delayed reaction diffusion systems on the basis of this paper. Futhermore, we will consider the existence and stability of traveling wave fronts for three-dimensional diffusion systems with convolution delay by the comparison principle and squeezing technique in the future. Another interesting and difficulty problem is the stability of the traveling wave solution under quasi-monotone or nonquasi-monotone assumptions. We leave these issues for future research.



---

*Source: 8742958-2019-12-14.xml*