Stability of traveling wavefronts for a discrete diffusive Lotka–Volterra competition system

J. Math. Anal. Appl. 447 (2017) 222–242

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Stability of traveling wavefronts for a discrete diffusive Lotka–Volterra competition system Ge Tian, Guo-Bao Zhang ∗ College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of China

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Article history: Received 15 July 2016 Available online 12 October 2016 Submitted by J. Shi

In this paper, we study a discrete diffusive Lotka–Volterra competition system. It is known that this system has traveling wavefronts. We prove that the traveling wavefronts are exponentially stable, when the initial perturbation around the traveling wavefronts decays exponentially as x → −∞, but can be arbitrarily large in other locations. The approach we use here is the comparison principle and the weighted energy method. © 2016 Elsevier Inc. All rights reserved.

Keywords: Traveling wavefronts Stability Discrete diffusive Lotka–Volterra competition system Weighted energy method

1. Introduction In this paper, we study the following discrete diffusive Lotka–Volterra competition system 

∂v1 ∂t (t, x) ∂v2 ∂t (t, x)

= D[v1 ](t, x) + r1 v1 (t, x)[1 − v1 (t, x) − b1 v2 (t, x)], = D[v2 ](t, x) + r2 v2 (t, x)[1 − v2 (t, x) − b2 v1 (t, x)],

(1.1)

where t > 0, x ∈ R, ri , bi are all positive constants, i = 1, 2, and D[vi ](t, x) = vi (t, x + 1) − 2vi (t, x) + vi (t, x − 1). This model is often used to describe the competing interaction of two species. Here v1 (t, x) and v2 (t, x) stand for the populations of two species at time t and location x, respectively. The parameter bi is the competition coefficient and ri is the net birth rate of species i, i = 1, 2. * Corresponding author. E-mail address: [email protected] (G.-B. Zhang). http://dx.doi.org/10.1016/j.jmaa.2016.10.012 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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The system (1.1) is the continuum version of the following lattice dynamical system:  ∂v

1,j (t)

∂t ∂v2,j (t) ∂t

= (v1,j+1 (t) − 2v1,j (t) + v1,j−1 (t)) + r1 v1,j (t)(1 − v1,j (t) − b1 v2,j (t)), = (v2,j+1 (t) − 2v2,j (t) + v2,j−1 (t)) + r2 v2,j (t)(1 − v2,j (t) − b2 v1,j (t)),

(1.2)

where t > 0 and j ∈ Z. Meanwhile, the system (1.1) can be regarded as a spatial discrete version of the following reaction–diffusion system: 

∂v1 ∂t (t, x) ∂v1 ∂t (t, x)

∂ 2 v1 ∂x2 (t, x) ∂ 2 v2 ∂x2 (t, x)

= =

+ r1 v1 (t, x)[1 − v1 (t, x) − b1 v2 (t, x)], + r2 v2 (t, x)[1 − v2 (t, x) − b2 v1 (t, x)],

(1.3)

where t > 0 and x ∈ R. It is easy to see that the corresponding diffusionless system of (1.1)–(1.3) is 

v1 (t) = r1 v1 (t)[1 − v1 (t) − b1 v2 (t)], v2 (t) = r2 v2 (t)[1 − v2 (t) − b2 v1 (t)].

(1.4)

b2 −1 The system (1.4) has four constant equilibria: (0, 0), (0, 1), (1, 0) and coexistence equilibrium ( b1b1b2−1 −1 , b1 b2 −1 ) provided that b1 b2 = 1. By a phase plane analysis, we have the following asymptotic behaviors as t → +∞ (see [6]):

(i) (ii) (iii) (iv)

(v1 , v2 ) → (1, 0) if 0 < b1 < 1 < b2 . (v1 , v2 ) → (0, 1) if 0 < b2 < 1 < b1 . (v1 , v2 ) → one of (0, 1), (1, 0) (depending on the initial condition) if b1 , b2 > 1. b2 −1 (v1 , v2 ) → ( b1b1b2−1 −1 , b1 b2 −1 ) (u and v coexist) if 0 < b1 , b2 < 1.

We need to point out that case (ii) can be reduced to the case (i) by exchanging the positions of v1 and v2 . The competition systems (1.2) and (1.3) have been studied quite extensively for past years, see, for example, [2–7,12,13,21] and the references cited therein. The traveling wave solution is among the central problems, since it can describe the propagation or invasion of species in population dynamics [5,6]. In mathematics, traveling wave solution of (1.2) (or (1.3)) is a special solution (v1,j (t), v2,j (t)) = (ϕ1 (ξ), ϕ2 (ξ)) with c > 0, ξ := j + ct, (or (v1 (t, x), v2 (t, x)) = (ϕ1 (ξ), ϕ2 (ξ)) with c > 0, ξ := x + ct), where c is the wave speed, ϕ1 and ϕ2 are called wave profiles. If ϕ1 and ϕ2 are monotone, then (ϕ1 , ϕ2 ) is called a traveling wavefront. For the continuum problem (1.3), we refer the readers to the work of Hosono [9,10], Kan-on [12], Kan-on and Fang [13] and Volpert et al. [23]. For the lattice system (1.2), Guo and Wu [5] proved that there is a positive constant (the minimal wave speed) such that a traveling wavefront connecting (0, 1) and (1, 0) of (1.2) exists if and only if its speed is above this minimal wave speed. They also showed that any wave profile of (1.2) is strictly monotone and unique up to translations. It is not hard to see that (1.1) and (1.2) take the same wave profile system, i.e., 

cϕ1 (ξ) = D[ϕ1 ](ξ) + r1 ϕ1 (ξ)[1 − ϕ1 (ξ) − b1 ϕ2 (ξ)], cϕ2 (ξ) = D[ϕ2 ](ξ) + r2 ϕ2 (ξ)[1 − ϕ2 (ξ) − b2 ϕ1 (ξ)],

where D[ϕ](ξ) := ϕ(ξ + 1) − 2ϕ(ξ) + ϕ(ξ − 1). Henceforth, the existence of wavefront of (1.1) connecting (0, 1) and (1, 0) with positive speed is assured. The purpose of this article is to establish the stability of traveling wavefronts of (1.1). The stability of traveling wavefronts for reaction–diffusion equations with monostable nonlinearity has been extensively studied, see [1,15,17–20,24–27] and reference therein. To our knowledge, there are main

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three methods: the (technical) weighted energy method [1,15,20], the sub-supersolutions method and squeezing technique [17,24,25] and the combination of the comparison principle and the weighted energy method [18,19]. Recently, Yang, Li and Wu [26,27] extended the first and the third methods to partially degenerate monostable delayed reaction–diffusion system in epidemiology. Lv and Wang [16] used the third method to a delayed Lotka–Volterra cooperative system and obtained the nonlinear stability of traveling wavefronts. More recently, Yu, Xu and Zhang [29] investigated the stability of invasion traveling waves for a competition system with nonlocal dispersals by the third method. For the discrete diffusion equations with monostable nonlinearity, only a few articles considered the stability of traveling wave solutions [8,11,17,22, 28]. In particular, Ma and Zou [17] establish the asymptotic stability of the unique traveling wave of a discrete reaction–diffusion equation with delay by using the squeezing technique. Guo and Zimmer [8] proved the global stability of traveling wavefronts with speed c ≥ c∗ (c∗ is the minimal speed) of a discrete diffusion equation with nonlocal delay effects by using a combination of the weighted energy method and the Green function technique. More recently, Yang, Zhang and Tian [28] studied a non-monotone spatially discrete reaction–diffusion equation with time delay, and proved that all non-critical traveling waves φ(x + ct) with the wave speed c > c∗ , including monotone or nonmonotone ones, are time-exponentially stable, by the technical weighted-energy method. Meanwhile, Tian, Zhang and Yang [22] further showed the stability of critical traveling waves φ(x + c∗ t) by the technical weighted-energy method and some new flavors. Motivated by the works [16,22,26–29], in this paper, we study the stability of the traveling wavefronts connecting (0, 1) to (1, 0) for (1.1) by using the comparison principle and the weighted energy method. The key step is to establish the L2 -energy estimates for the solutions of the perturbed system (see, for example, (3.3) and (3.4)) of the system (1.1) in a suitable weighted Sobolev space Hw1 . We should point out that the energy estimates in our system are more complicated than that in single equations in [8,22,28]. In addition, compared with Yang, Li and Wu [26,27], Lv and Wang [16] and Yu, Xu and Zhang [29], due to the occurrence of the discrete dispersal operator D(v) in (1.1), the technical details for obtaining a priori estimate are very different. The rest of this paper is organized as follows. In section 2, we introduce some necessary notations and present the main results on the existence and stability of traveling wavefronts. In section 3, we are mainly devoted to the proof of the stability theorem. 2. Preliminaries and main result In this section, we first introduce the existence of traveling wavefronts of (1.1), then state our main theorem. In what follows, we always assume that 0 < b1 < 1 < b2 . Throughout this paper, we assume that (1.1) satisfies the initial conditions 

v1 (0, x) = v10 (0, x),

v2 (0, x) = v20 (0, x).

(2.1)

Let v1∗ = v1 , v2∗ = 1 − v2 . For the sake of convenience, we drop the star. Then (1.1) with the initial condition (2.1) can be reduced to the following cooperation system 

∂v1 ∂t (t, x) ∂v2 ∂t (t, x)

= D[v1 ](t, x) + r1 v1 (t, x)[1 − b1 − v1 (t, x) + b1 v2 (t, x)], = D[v2 ](t, x) + r2 (1 − v2 (t, x))[b2 v1 (t, x) − v2 (t, x)],

(2.2)

with the initial condition 

∗ (0, x), v1 (0, x) = v10 (0, x) =: v10 ∗ (0, x). v2 (0, x) = 1 − v20 (0, x) =: v20

(2.3)

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Substituting (v1 (t, x), v2 (t, x)) = (ϕ1 (ξ), ϕ2 (ξ)), ξ := x + ct, into (2.2), we have the following wave profile system  cϕ1 (ξ) = D[ϕ1 ](ξ) + r1 ϕ1 (ξ)[1 − b1 − ϕ1 (ξ) + b1 ϕ2 (ξ)], cϕ2 (ξ) = D[ϕ2 ](ξ) + r2 (1 − ϕ2 (ξ))[b2 ϕ1 (ξ) − ϕ2 (ξ)].

(2.4)

We require that (ϕ1 (ξ), ϕ2 (ξ)) satisfies the following asymptotic boundary conditions lim (ϕ1 (ξ), ϕ2 (ξ)) = (0, 0) and

ξ→−∞

lim (ϕ1 (ξ), ϕ2 (ξ)) = (1, 1).

ξ→+∞

The existence of monotone solutions for the system (2.4) connecting (0, 0) to (1, 1) has been proved by Guo and Wu [5]. Proposition 2.1. There exists c∗ > 0 such that for any c ≥ c∗ , there is an increasing traveling wave solution (ϕ1 (ξ), ϕ2 (ξ)) of (2.4) connecting (0, 0) and (1, 1), and for any c < c∗ , there is no such traveling wave. Before stating our main result, let us make the following notation. Notation: Throughout the paper, C > 0 denotes a generic constant, while Ci > 0 (i = 0, 1, 2 · · · ) represents a special constant. Letting I be an interval, especially I = R, L2 (I) is the space of the square integrable function on I, and H k (I) (k ≥ 0) is the Sobolev space of the L2 -function f (x) defined on I whose di 2 2 2 derivatives dx i f , i = 1, · · ·, k, also belong to L (I). Lw (I) represents the weight L -space with the weighted w(x) > 0 and its norm is defined by ⎛



||f (x)||L2w = ⎝

⎞ 12 w(x)f 2 (x)dx⎠ .

I

Hwk (I) is the weighted Sobolev space with the norm

||f (x)||Hwk

⎛ ⎞ 12 i 2 k   d =⎝ w(x) i f (x) dx⎠ . dx i=0 I

Letting T > 0 and B be a Banach space, we denote by C 0 ([0, T ]; B) the space of the B-valued continuous functions on [0, T ], and L2 ([0, T ]; B) as the space of B-valued L2 -function on [0, T ]. The corresponding spaces of the B-valued function on [0, ∞) are defined similarly. To obtain the stability of traveling wavefronts of (1.1), we need the following technical assumption: (H) 6r2 + 2r1 b1 + 1 < r2 b2 < 2r1 − 4r1 b1 − 1. Define two functions on σ as follows: M1 (σ) = 2r1 − 4r1 b1 − r2 b2 − (eσ + e−σ − 1) and M2 (σ) = r2 b2 − 6r2 − 2r1 b1 − (eσ + e−σ − 1). By the assumption (H), we can see that

226

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M1 (0) = 2r1 − 4r1 b1 − r2 b2 − 1 > 0 and M2 (0) = r2 b2 − 6r2 − 2r1 b1 − 1 > 0. Then by the continuity of M1 (σ) and M2 (σ) with respect to σ, there exists σ0 > 0 such that M1 (σ0 ) > 0 and M2 (σ0 ) > 0. Furthermore, define N1 (ξ) = −2r1 − 4r1 b1 − 2r2 b2 + 4r1 ϕ1 (ξ) + r2 b2 ϕ2 (ξ) − (eσ0 + e−σ0 − 1) and N2 (ξ) = −2r2 b2 − 6r2 − 2r1 b1 + 2r2 b2 ϕ1 (ξ) + r2 b2 ϕ2 (ξ) − (eσ0 + e−σ0 − 1), where (ϕ1 (ξ), ϕ2 (ξ)) is a traveling wavefront given in Proposition 2.1. It is easy to see that lim N1 (ξ) = M1 (σ0 ) > 0 and

ξ→+∞

lim N2 (ξ) = M2 (σ0 ) > 0,

ξ→+∞

which imply that there exists a number ξ0 > 0 large enough such that ⎧ ⎪ N1 (ξ0 ) = − 2r1 − 4r1 b1 − 2r2 b2 + 4r1 ϕ1 (ξ0 ) + r2 b2 ϕ2 (ξ0 ) ⎪ ⎪ ⎪ ⎪ ⎨ − (eσ0 + e−σ0 − 1) > 0, ⎪ N2 (ξ0 ) = − 2r2 b2 − 6r2 − 2r1 b1 + 2r2 b2 ϕ1 (ξ0 ) + r2 b2 ϕ2 (ξ0 ) ⎪ ⎪ ⎪ ⎪ ⎩ − (eσ0 + e−σ0 − 1) > 0.

(2.5)

For above σ0 and ξ0 , we define a weight function w(ξ) by  w(ξ) =

e−σ0 (ξ−ξ0 ) , 1,

ξ ≤ ξ0 , ξ > ξ0 .

(2.6)

For convenience, we still denote (2.3) by v1 (0, x) = v10 (0, x) and v2 (0, x) = v20 (0, x).

(2.7)

Now, we state our main theorem: Theorem 2.2 (Stability). Assume that (H) holds and 0 < b1 < 1 < b2 . For any given traveling wavefront (ϕ1 (x + ct), ϕ2 (x + ct)) with the wave speed c > max{c∗ , c˜}, where c˜ =

max{c1 , c2 } , σ0

where c1 = 2r1 + 4r1 b1 + 2r2 b2 + (eσ0 + e−σ0 − 1),

(2.8)

c2 = 6r2 + 2r1 b1 + 2r2 b2 + (eσ0 + e−σ0 − 1),

(2.9)

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if the initial data satisfy (0, 0) ≤ (v10 (0, x), v20 (0, x)) ≤ (1, 1),

x ∈ R,

and the initial perturbations satisfy  v10 (0, x) − ϕ1 (x) ∈ Hw1 (R), v20 (0, x) − ϕ2 (x) ∈ Hw1 (R), then the nonnegative solution of the Cauchy problem (2.2) and (2.7) uniquely exists and satisfies (0, 0) ≤ (v1 (t, x), v2 (t, x)) ≤ (1, 1),

t > 0, x ∈ R,

and v1 (t, x) − ϕ1 (x + ct) ∈ C([0, +∞); Hw1 (R)) ∩ L2 ([0, +∞); Hw1 (R)), v2 (t, x) − ϕ2 (x + ct) ∈ C([0, +∞); Hw1 (R)) ∩ L2 ([0, +∞); Hw1 (R)), where w(x) was defined by (2.6). Moreover, (v1 (t, x), v2 (t, x)) converges to the traveling wavefront (ϕ1 (x + ct), ϕ2 (x + ct)) exponentially in time t, i.e., sup |v1 (t, x) − ϕ1 (x + ct)| ≤ Ce−μt x∈R

and sup |v2 (t, x) − ϕ2 (x + ct)| ≤ Ce−μt x∈R

for all t > 0, where C and μ are some positive constants. 3. Stability In this section, we are going to prove the stability of the traveling wavefronts by the weighted energy method with the comparison principle together. The global existence and uniqueness of the solution, and the comparison principle to the initial value problem (2.2) and (2.7) can be proved by an argument similar to [14, Lemma 3.2]. Lemma 3.1 (Boundedness). Assume that (H) holds and 0 < b1 < 1 < b2 , the initial data satisfy (0, 0) ≤ (v10 (0, x), v20 (0, x)) ≤ (1, 1) for x ∈ R. Then the nonnegative solution (v1 (t, x), v2 (t, x)) of (2.2) and (2.7) satisfies (0, 0) ≤ (v1 (t, x), v2 (t, x)) ≤ (1, 1) for (t, x) ∈ R+ × R. Lemma 3.2 (Comparison principle). Assume that (H) holds and 0 < b1 < 1 < b2 . Let (v1− (t, x), v2− (t, x)) and − − + + (v1+ (t, x), v2+ (t, x)) be the solution of (2.2) with the initial data (v10 (0, x), v20 (0, x)) and (v10 (0, x), v20 (0, x)), respectively. If

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− − + + (0, 0) ≤ (v10 (0, x), v20 (0, x)) ≤ (v10 (0, x), v20 (0, x)) ≤ (1, 1)

for x ∈ R, then (0, 0) ≤ (v1− (t, x), v2− (t, x)) ≤ (v1+ (t, x), v2+ (t, x)) ≤ (1, 1) for (t, x) ∈ R+ × R. Next, we mainly discuss the stability of the solution to (2.2) and (2.7). Define ⎧ − ⎪ v10 (0, x) = min{v10 (0, x), ϕ1 (x)}, x ∈ R, ⎪ ⎪ ⎪ ⎨v + (0, x) = max{v (0, x), ϕ (x)}, x ∈ R, 10 1 10 − ⎪ v20 (0, x) = min{v20 (0, x), ϕ2 (x)}, x ∈ R, ⎪ ⎪ ⎪ ⎩ + v20 (0, x) = max{v20 (0, x), ϕ2 (x)}, x ∈ R, which implies ⎧ − + ⎪ (0, x) ≤ v10 (0, x) ≤ v10 (0, x) ≤ 1, x ∈ R, 0 ≤ v10 ⎪ ⎪ ⎪ ⎨0 ≤ v − (0, x) ≤ ϕ (x) ≤ v + (0, x) ≤ 1, x ∈ R, 1 10 10 − + ⎪ (0, x) ≤ v (0, x) ≤ v20 (0, x) ≤ 1, x ∈ R, 0 ≤ v ⎪ 20 20 ⎪ ⎪ ⎩ − + 0 ≤ v20 (0, x) ≤ ϕ2 (x) ≤ v20 (0, x) ≤ 1, x ∈ R.

(3.1)

Then by the comparison principle, we have ⎧ ⎪ 0 ≤ v1− (t, x) ≤ v1 (t, x) ≤ v1+ (t, x) ≤ 1, (t, x) ∈ R+ × R, ⎪ ⎪ ⎪ ⎨0 ≤ v − (t, x) ≤ ϕ (x + ct) ≤ v + (t, x) ≤ 1, (t, x) ∈ R+ × R, 1 1 1 − + ⎪ 0 ≤ v2 (t, x) ≤ v2 (t, x) ≤ v2 (t, x) ≤ 1, (t, x) ∈ R+ × R, ⎪ ⎪ ⎪ ⎩ 0 ≤ v2− (t, x) ≤ ϕ2 (x + ct) ≤ v2+ (t, x) ≤ 1, (t, x) ∈ R+ × R.

(3.2)

In order to prove the stability of the traveling wavefronts presented in Theorem 2.2, we need the following three steps: Step 1. The convergence of vi+ (t, x) to ϕi (x + ct), i = 1, 2. Let ξ := x + ct and V1 (t, ξ) = v1+ (t, x) − ϕ1 (x + ct),

+ V10 (0, ξ) = v10 (0, x) − ϕ1 (x)

V2 (t, ξ) = v2+ (t, x) − ϕ2 (x + ct),

+ V20 (0, ξ) = v20 (0, x) − ϕ2 (x).

and

Then by (3.1) and (3.2), we have (0, 0) ≤ (V10 (0, ξ), V20 (0, ξ)) ≤ (1, 1) and (0, 0) ≤ (V1 (t, ξ), V2 (t, ξ)) ≤ (1, 1). From (2.2) and (2.4), we can see that (V1 (t, ξ), V2 (t, ξ)) satisfies

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V1t + cV1ξ = [V1 (t, ξ + 1) + V1 (t, ξ − 1) − 2V1 (t, ξ)] + V1 (r1 − r1 b1 − 2r1 ϕ1 + r1 b1 V2 + r1 b1 ϕ2 ) − r1 V12 + r1 b1 ϕ1 V2 ,

(3.3)

V2t + cV2ξ = [V2 (t, ξ + 1) + V2 (t, ξ − 1) − 2V2 (t, ξ)] + V2 (2r2 ϕ2 − r2 − r2 b2 V1 − r2 b2 ϕ1 ) + r2 V22 + r2 b2 (1 − ϕ2 )V1 ,

(3.4)

with the initial data V1 (0, ξ) = V10 (0, ξ), V2 (0, ξ) = V20 (0, ξ), ξ ∈ R. It is easy to see that Vi0 (0, ξ) ∈ Hw1 (R), i = 1, 2. Then we can obtain that Vi (t, ξ) ∈ C([0, +∞), Hw1 (R)), i = 1, 2. In order to establish the energy estimates, sufficient regularity of the solution to (3.3) and (3.4) is required. We thus mollify the initial data as follows Vi0ε (0, ξ) = (Jε ∗ Vi0 )(0, ξ)  Jε (ξ − y)Vi0 (0, y) dy ∈ Hw2 (R), i = 1, 2, = R

where Jε (ξ) is the mollifier. Let Vi (t, ξ), i = 1, 2, be the solution to (3.3) and (3.4) with this mollified initial data. Then we have Viε (t, ξ) ∈ C([0, +∞), Hw2 (R)), i = 1, 2. By taking the limit ε → 0, we can obtain the corresponding energy estimate for the original solution Vi (t, ξ). For the sake of simplicity, below we formally use Vi (t, ξ) to establish the desired energy estimates. Multiplying (3.3) and (3.4) by e2μt w(ξ)V1 (t, ξ) and e2μt w(ξ)V2 (t, ξ), respectively, where μ > 0 will be specified later in Lemma 3.4, we obtain 

 c  1 2μt e wV12 + e2μt wV12 − e2μt wV1 (V1 (t, ξ + 1) + V1 (t, ξ − 1)) 2 2 ξ t    cw − μ − r1 + r1 b1 + 2r1 ϕ1 − r1 b1 V2 − r1 b1 ϕ2 e2μt wV12 + 2− 2w

= − r1 e2μt wV13 + r1 b1 ϕ1 e2μt wV1 V2

(3.5)

and 

 c  1 2μt 2 e wV2 e2μt wV22 − e2μt wV2 (V2 (t, ξ + 1) + V2 (t, ξ − 1)) + 2 2 ξ t   c w − μ − 2r2 ϕ2 + r2 + r2 b2 V1 + r2 b2 ϕ1 e2μt wV22 + 2− 2w

= r2 e2μt wV23 + r2 b2 (1 − ϕ2 )e2μt wV1 V2 .

(3.6)

Integrating (3.5) and (3.6) over R × [0, t] with respect to ξ and t, and noting the vanishing term at far fields, since V1 ∈ Hw1 and V2 ∈ Hw1 ,  ∞ e2μt wV12 2

c

ξ=−∞

we can see

= 0 and

 ∞ e2μt wV22 2

c

ξ=−∞

= 0,

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230

t  e

2μt

||V1 (t)||2L2w

−2

e2μs wV1 (V1 (s, ξ + 1) + V1 (s, ξ − 1))dξds R

0

t

 

 c w − μ − r1 + r1 b1 + 2r1 ϕ1 − r1 b1 V2 − r1 b1 ϕ2 e2μs wV12 dξds 2w

2−

+2 R

0

t  ≤

||V10 (0)||2L2w

ϕ1 e2μs wV1 (s, ξ)V2 (s, ξ)dξds

+ 2r1 b1 0

(3.7)

R

and t  e

2μt

||V2 (t)||2L2w

−2

e2μs wV2 (V2 (s, ξ + 1) + V2 (s, ξ − 1))dξds 0

t

  2−

+2 R

0

R

 c w − μ − 2r2 ϕ2 + r2 + r2 b2 V1 + r2 b2 ϕ1 e2μs wV22 dξds 2w t 

≤

||V20 (0)||2L2w

+2

t  r2 e

0

2μs

wV23 dξds

r2 b2 (1 − ϕ2 )e2μs wV1 V2 dξds.

+2

R

0

(3.8)

R

By the Cauchy–Schwarz inequality 2xy ≤ x2 + y 2 , we have t  e2μs w(ξ)Vi (s, ξ)Vi (s, ξ + 1)dξds

2 0

R

t ≤

 e2μs

  w(ξ) Vi2 (s, ξ) + Vi2 (s, ξ + 1) dξds

R

0

t =

⎛ ⎞   w(ξ − 1) e2μs ⎝ w(ξ)Vi2 (s, ξ)dξ + w(ξ)Vi2 (s, ξ)dξ ⎠ ds w(ξ) R

0

(3.9)

R

and t  e2μs w(ξ)Vi (s, ξ)Vi (s, ξ − 1)dξds

2 0

R

t ≤

 e2μs

  w(ξ) Vi2 (s, ξ) + Vi2 (s, ξ − 1) dξds

R

0

t =

⎛ ⎞   w(ξ + 1) w(ξ)Vi2 (s, ξ)dξ ⎠ ds, e2μs ⎝ w(ξ)Vi2 (s, ξ)dξ + w(ξ) R

0

R

where i = 1, 2. Similarly, we obtain t  ϕ1 e2μs w(ξ)V1 (s, ξ)V2 (s, ξ)dξds

2 0

R

(3.10)

G. Tian, G.-B. Zhang / J. Math. Anal. Appl. 447 (2017) 222–242

t 

t 

≤

ϕ1 e

2μs

w(ξ)V12 (s, ξ)dξds

ϕ1 e2μs w(ξ)V22 (s, ξ)dξds

+

R

0

231

(3.11)

R

0

and t  r2 b2 (1 − ϕ2 )e2μs w(ξ)V1 (s, ξ)V2 (s, ξ)dξds

2 R

0

t  ≤

r2 b2 (1 − ϕ2 )e2μs w(ξ)V12 (s, ξ)dξds + 0

(3.12)

t 

R

r2 b2 (1 − ϕ2 )e2μs w(ξ)V22 (s, ξ)dξds. 0

R

Substituting (3.9)–(3.12) into (3.7) and (3.8), we have

e

2μt

||V1 (t)||2L2w

t   w 4 − c − 2μ − 2r1 + 2r1 b1 + 4r1 ϕ1 − 2r1 b1 V2 − 2r1 b1 ϕ2 + w R

0



w(ξ − 1) w(ξ + 1) + − r1 b1 ϕ1 − 2 + w(ξ) w(ξ)

 e2μs w(ξ)V12 dξds

t  ≤

||V10 (0)||2L2w

ϕ1 e2μs w(ξ)V22 (s, ξ)dξds

+ r1 b1

(3.13)

R

0

and

e

2μt

||V2 (t)||2L2w

t   w 4 − c − 2μ − 4r2 ϕ2 + 2r2 (1 − V2 ) + 2r2 b2 V1 + 2r2 b2 ϕ1 + w 0



R

w(ξ − 1) w(ξ + 1) + − r2 b2 (1 − ϕ2 ) − 2 + w(ξ) w(ξ)

 e2μs w(ξ)V22 dξds

(3.14)

t  ≤

||V20 (0)||2L2w

r2 b2 (1 − ϕ2 )e2μs w(ξ)V12 dξds.

+ 0

R

Combining (3.13) and (3.14), we get   e2μt ||V1 (t)||2L2w + ||V2 (t)||2L2w t  + 0

  1 2 e2μs Bμ,w V12 (s, ξ) + Bμ,w V22 (s, ξ) w(ξ)dξds

(3.15)

R

≤ ||V10 (0)||2L2w + ||V20 (0)||2L2w , where 1 Bμ,w (t, ξ) = A1w (t, ξ) − 2μ,

with

2 Bμ,w (t, ξ) = A2w (t, ξ) − 2μ,

(3.16)

G. Tian, G.-B. Zhang / J. Math. Anal. Appl. 447 (2017) 222–242

232

w (ξ) − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A1w (t, ξ) = 4 − c

(3.17)

and w (ξ) − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + . + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A2w (t, ξ) = 4 − c

(3.18)

i We shall prove Bμ,w (t, ξ) > 0, i = 1, 2, which is one of the crucial steps in the proof of the stability of traveling wavefronts of (1.1). For this we need the following key Lemma.

Lemma 3.3. Assume that (H) holds and 0 < b1 < 1 < b2 . For any c > max{c∗ , c˜}, there exist some positive constants Ci such that Aiw (t, ξ) ≥ Ci ,

i = 1, 2,

for all ξ ∈ R, t ≥ 0. Proof. Since c > max{c∗ , c˜}, we obtain cσ0 > c1 and cσ0 > c2 . By (2.8) and (2.9), we get cσ0 − 2r1 − 4r1 b1 − 2r2 b2 − (eσ0 + e−σ0 − 1) > 0 and cσ0 − 6r2 − 2r1 b1 − 2r2 b2 − (eσ0 + e−σ0 − 1) > 0. We first show that A1w (t, ξ) ≥ C1 for some positive constant C1 . Case 1: ξ < ξ0 − 1. In this case, ξ < ξ0 , ξ + 1 < ξ0 and ξ − 1 < ξ0 . Hence, w(ξ) = e−σ0 (ξ−ξ0 ) , w(ξ −  (ξ) = −σ0 , w(ξ−1) = eσ0 and w(ξ+1) = e−σ0 . Note that 1) = e−σ0 (ξ−1−ξ0 ) , w(ξ + 1) = e−σ0 (ξ+1−ξ0 ) , ww(ξ) w(ξ) w(ξ) (0, 0) ≤ (ϕ1 , ϕ2 ) ≤ (1, 1) and (0, 0) ≤ (V1 , V2 ) ≤ (1, 1). Then it can be verified that w (ξ) − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A1w (t, ξ) = 4 − c

= 4 + cσ0 − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2   + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + eσ0 + e−σ0 ≥ cσ0 − 2r1 − 3r1 b1 − r2 b2 − (eσ0 + e−σ0 − 1) + 1 > r1 b1 + r2 b2 > 0. Case 2: ξ0 − 1 ≤ ξ ≤ ξ0 . In this case, ξ ≤ ξ0 , ξ + 1 ≥ ξ0 and ξ − 1 < ξ0 , which means that w(ξ) = e−σ0 (ξ−ξ0 ) ,  (ξ) σ0 σ0 (ξ−ξ0 ) w(ξ − 1) = e−σ0 (ξ−1−ξ0 ) , w(ξ + 1) = 1, ww(ξ) = −σ0 , w(ξ−1) and w(ξ+1) . Thus we get w(ξ) = e w(ξ) = e

G. Tian, G.-B. Zhang / J. Math. Anal. Appl. 447 (2017) 222–242

233

w (ξ) − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A1w (t, ξ) = 4 − c

= 4 + cσ0 − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2   + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + eσ0 + eσ0 (ξ−ξ0 ) ≥ 4 + cσ0 − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2 + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − (3 + eσ0 ) ≥ 4 + cσ0 − 2r1 − 3r1 b1 − r2 b2 − (eσ0 + e−σ0 + 3 − e−σ0 ) = cσ0 − 2r1 − 3r1 b1 − r2 b2 − (eσ0 + e−σ0 − 1) + e−σ0 > r1 b1 + r2 b2 > 0. Case 3: ξ0 < ξ ≤ ξ0 +1. In this case, ξ > ξ0 , ξ+1 > ξ0 and ξ−1 ≤ ξ0 . Thus, w(ξ) = 1, w(ξ−1) = e−σ0 (ξ−1−ξ0 ) ,  (ξ) −σ0 (ξ−1−ξ0 ) w(ξ + 1) = 1, ww(ξ) = 0, w(ξ−1) and w(ξ+1) w(ξ) = e w(ξ) = 1. Then by (2.5), we obtain w (ξ) − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A1w (t, ξ) = 4 − c

= 4 − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2   + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 3 + e−σ0 (ξ−ξ0 −1) ≥ 4 − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2 + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − (3 + eσ0 ) ≥ 4 − 2r1 − 3r1 b1 − r2 b2 + 4r1 ϕ1 (ξ0 ) + r2 b2 ϕ2 (ξ0 ) − (eσ0 + e−σ0 + 3 − e−σ0 ) = − 2r1 − 3r1 b1 − r2 b2 + 4r1 ϕ1 (ξ0 ) + r2 b2 ϕ2 (ξ0 ) − (eσ0 + e−σ0 − 1) + e−σ0 = N1 (ξ0 ) + r1 b1 + r2 b2 + e−σ0 > r1 b1 + r2 b2 > 0. Case 4: ξ > ξ0 +1. In this case, ξ > ξ0 , ξ +1 > ξ0 and ξ −1 > ξ0 . Hence, w(ξ) = 1, w(ξ −1) = 1, w(ξ +1) = 1, w (ξ) w(ξ−1) w(ξ+1) σ0 + e−σ0 ≥ 2, we have w(ξ) = 0, w(ξ) = 1 and w(ξ) = 1. By (2.5) and considering e w (ξ) − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A1w (t, ξ) = 4 − c

= −2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) − r2 b2 + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) ≥ −2r1 − 3r1 b1 − r2 b2 + 4r1 ϕ1 (ξ0 ) + r2 b2 ϕ2 (ξ0 ) − (eσ0 + e−σ0 − 1) + 1 = N1 (ξ0 ) + r1 b1 + r2 b2 + 1

234

G. Tian, G.-B. Zhang / J. Math. Anal. Appl. 447 (2017) 222–242

> r1 b1 + r2 b2 > 0. Let C1 := r1 b1 + r2 b2 . Then we obtain A1w (t, ξ) ≥ C1 . Next, we prove A2w (t, ξ) ≥ C2 for some positive constant C2 . Case 1: ξ < ξ0 − 1. In this case, ξ < ξ0 , ξ + 1 ≤ ξ0 and ξ − 1 < ξ0 . Then we obtain w (ξ) − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A2w (t, ξ) = 4 − c

= 4 + cσ0 − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ)   − r2 b2 + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + eσ0 + e−σ0 ≥ cσ0 − 4r2 − r2 b2 − r1 b1 − (eσ0 + e−σ0 − 1) + 1 > 2r2 + r2 b2 + r1 b1 > 0. Case 2: ξ0 − 1 ≤ ξ ≤ ξ0 . In this case, ξ ≤ ξ0 , ξ + 1 ≥ ξ0 and ξ − 1 < ξ0 . Then we get w (ξ) − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A2w (t, ξ) = 4 − c

= 4 + cσ0 − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ)   − r2 b2 + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + eσ0 + eσ0 (ξ−ξ0 ) ≥ 4 + cσ0 − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) − r2 b2 + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − (3 + eσ0 ) ≥ 4 + cσ0 − 4r2 − r2 b2 − r1 b1 − (eσ0 + e−σ0 + 3 − e−σ0 ) ≥ cσ0 − 4r2 − r2 b2 − r1 b1 − (eσ0 + e−σ0 − 1) + e−σ0 > 2r2 + r2 b2 + r1 b1 > 0. Case 3: ξ0 < ξ ≤ ξ0 + 1. In this case, ξ > ξ0 , ξ + 1 > ξ0 and ξ − 1 ≤ ξ0 . Then by (2.5), one has w (ξ) − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A2w (t, ξ) = 4 − c

= 4 − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) − r2 b2   + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 3 + e−σ0 (ξ−ξ0 −1) ≥ 4 − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) − r2 b2 + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − (3 + eσ0 ) ≥ 4 − 4r2 + 2r2 b2 ϕ1 (ξ0 ) − r2 b2 − r1 b1 + r2 b2 ϕ2 (ξ0 ) − (eσ0 + e−σ0 + 3 − e−σ0 ) = − 4r2 + 2r2 b2 ϕ1 (ξ0 ) − r2 b2 − r1 b1 + r2 b2 ϕ2 (ξ0 ) − (eσ0 + e−σ0 − 1) + e−σ0

G. Tian, G.-B. Zhang / J. Math. Anal. Appl. 447 (2017) 222–242

235

= N2 (ξ0 ) + 2r2 + r2 b2 + r1 b1 + e−σ0 > 2r2 + r2 b2 + r1 b1 > 0. Case 4: ξ > ξ0 + 1. In this case, ξ > ξ0 , ξ + 1 > ξ0 and ξ − 1 > ξ0 . By (2.5) and considering eσ0 + e−σ0 ≥ 2, we obtain w (ξ) − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) − r2 b2 w(ξ)   w(ξ − 1) w(ξ + 1) + + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) − 2 + w(ξ) w(ξ)

A2w (t, ξ) = 4 − c

= − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) − r2 b2 + r2 b2 ϕ2 (ξ) − r1 b1 ϕ1 (ξ) ≥ − 4r2 + 2r2 b2 ϕ1 (ξ0 ) − r2 b2 − r1 b1 + r2 b2 ϕ2 (ξ0 ) − (eσ0 + e−σ0 − 1) + 1 = N2 (ξ0 ) + 2r2 + r2 b2 + r1 b1 + 1 > 2r2 + r2 b2 + r1 b1 > 0. Thus, we get A2w (t, ξ) ≥ C2 , where C2 := 2r2 + r2 b2 + r1 b1 > 0. This completes the proof. 2 Lemma 3.4. Assume that (H) holds and 0 < b1 < 1 < b2 . For any c > max{c∗ , c˜}, there exist some positive constants Ci such that i Bμ,w (t, ξ) ≥ Ci ,

for all ξ ∈ R and 0 < μ <

i = 1, 2,

mini=1,2 {Ci } . 2

Lemma 3.5. Assume that (H) holds and 0 < b1 < 1 < b2 . For any c > max{c∗ , c˜}, it holds t ||V1 (t)||2L2w

+

||V2 (t)||2L2w

+

  e−2μ(t−s) ||V1 (s)||2L2w + ||V2 (s)||2L2w ds

0

  ≤ Ce−2μt ||V10 (0)||2L2w + ||V20 (0)||2L2w ,

(3.19)

for some positive constant C. Similarly, differentiating the (3.3) with respect to ξ, we can obtain ⎧ ⎪ V1tξ + cV1ξξ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ V2tξ + cV2ξξ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

[V1ξ (t, ξ + 1) + V1ξ (t, ξ − 1) − 2V1ξ (t, ξ)] + V1ξ (r1 − r1 b1 − 2r1 ϕ1 + r1 b1 V2 + r1 b1 ϕ2 ) + r1 b1 ϕ1 V2 − 2r1 V1 V1ξ + (−2r1 ϕ1 + r1 b1 V2ξ + r1 b1 ϕ2 )V1 + r1 b1 ϕ1 V2ξ , [V2ξ (t, ξ + 1) + V2ξ (t, ξ − 1) − 2V2ξ (t, ξ)] + V2ξ (2r2 ϕ2 − r2 − r2 b2 V1 − r2 b2 ϕ1 ) + 2r2 V2 V2ξ − r2 b2 V1 ϕ2 + (2r2 ϕ2 − r2 b2 V1ξ − r2 b2 ϕ1 )V2 + r2 b2 (1 − ϕ2 )V1ξ .

Multiplying (3.20) by e2μt w(ξ)V1ξ (t, ξ) and e2μt w(ξ)V2ξ (t, ξ), respectively, it holds

(3.20)

G. Tian, G.-B. Zhang / J. Math. Anal. Appl. 447 (2017) 222–242

236



 c  1 2μt 2 2 e w(ξ)V1ξ + e2μt w(ξ)V1ξ − e2μt w(ξ)V1ξ (V1ξ (t, ξ + 1) + V1ξ (t, ξ − 1)) 2 2 ξ t   c w 2 − μ − r1 + r1 b1 + 2r1 ϕ1 − r1 b1 V2 − r1 b1 ϕ2 e2μt w(ξ)V1ξ + 2− 2w

= (−2r1 ϕ1 + r1 b1 V2ξ + r1 b1 ϕ2 )V1 e2μt w(ξ)V1ξ (t, ξ) 2 2μt − 2r1 V1 V1ξ e w(ξ) + (r1 b1 ϕ1 V2 + r1 b1 ϕ1 V2ξ )e2μt w(ξ)V1ξ (t, ξ)

(3.21)

and 

 c  1 2μt 2 2 e w(ξ)V2ξ + e2μt w(ξ)V2ξ − e2μt w(ξ)V2ξ (V2ξ (t, ξ + 1) + V2ξ (t, ξ − 1)) 2 2 ξ t   c w 2 − μ − 2r2 ϕ2 + r2 + r2 b2 V1 + r2 b2 ϕ1 e2μt w(ξ)V2ξ + 2− 2w

2 2μt = (2r2 ϕ2 − r2 b2 V1ξ − r2 b2 ϕ1 )V2 e2μt w(ξ)V2ξ (t, ξ) + 2r2 V2 V2ξ e w(ξ)

− r2 b2 ϕ2 V1 e2μt w(ξ)V2ξ (t, ξ) + r2 b2 (1 − ϕ2 )V1ξ e2μt w(ξ)V2ξ (t, ξ).

(3.22)

Integrating (3.21) and (3.22) over R × [0, t] with respect to ξ and t, and noting the vanishing term at far fields, since V1 ∈ Hw2 and V2 ∈ Hw2 , c 2

e

2μt

2 w(ξ)V1ξ

 ∞

= 0 and

c

ξ=−∞

2

e

2μt

2 w(ξ)V2ξ

 ∞

= 0,

ξ=−∞

we can obtain t  e

2μt

||V1ξ (t)||2L2w

−2

e2μs w(ξ)V1ξ (V1ξ (s, ξ + 1) + V1ξ (s, ξ − 1))dξds 0

R

t  

 c w 2 2− − μ − r1 + r1 b1 + 2r1 ϕ1 − r1 b1 V2 − r1 b1 ϕ2 e2μs w(ξ)V1ξ dξds 2w

+2 0

R

t  ≤

||V1ξ,0 (0)||2L2w

ϕ1 e2μs w(ξ)V1ξ (s, ξ)V2ξ (s, ξ)dξds

+ 2r1 b1 R

0

t 

(−2r1 ϕ1 + r1 b1 V2ξ + r1 b1 ϕ2 )V1 e2μt w(ξ)V1ξ (t, ξ)dξds

+2 0

R

t  + 2r1 b1 0

ϕ1 e2μs w(ξ)V2 (s, ξ)V1ξ (s, ξ)dξds

R

and t  e

2μt

||V2ξ (t)||2L2w

−2

e2μs w(ξ)V2ξ (V2ξ (s, ξ + 1) + V2ξ (s, ξ − 1))dξds 0

t

  2−

+2 0

R

R

 c w 2 − μ − 2r2 ϕ2 + r2 + r2 b2 V1 + r2 b2 ϕ1 e2μs w(ξ)V2ξ dξds 2w

(3.23)

G. Tian, G.-B. Zhang / J. Math. Anal. Appl. 447 (2017) 222–242

237

t  ≤

||V2ξ,0 (0)||2L2w

2 e2μs w(ξ)V2 (s, ξ)V2ξ (s, ξ)dξds

+ 4r2 0

t

R

 (1 − ϕ2 )e2μs w(ξ)V1ξ (s, ξ)V2ξ (s, ξ)dξds

+ 2r2 b2 0

R

t



0

R

ϕ2 e2μs w(ξ)V1 (s, ξ)V2ξ (s, ξ)dξds

− 2r2 b2 t  +2

(2r2 ϕ2 − r2 b2 V1ξ − r2 b2 ϕ1 )w(ξ)e2μs V2 (s, ξ)V2ξ (s, ξ)dξds.

(3.24)

R

0

By a similar argument as in Lemma 3.5, we can have e

2μt

||V1ξ (t)||2L2w

t   w 4 − c − 2μ − 2r1 + 2r1 b1 + 4r1 ϕ1 − 2r1 b1 V2 + w 0

R

− 2r1 b1 ϕ2 − r1 b1 (ϕ1 (ξ) + V1 (s, ξ))   w(ξ − 1) w(ξ + 1) 2 + e2μs w(ξ)V1ξ dξds − 2+ w(ξ) w(ξ) t  ≤

||V1ξ,0 (0)||2L2w

2 (ϕ1 (ξ) + V1 (s, ξ))e2μs w(ξ)V2ξ (s, ξ)dξds

+ r1 b1 0

t



0

R

R

(−2r1 ϕ1 + r1 b1 ϕ2 )V1 e2μs w(ξ)V1ξ (s, ξ)dξds

+2

t  + 2r1 b1 0

ϕ1 e2μs w(ξ)V2 (s, ξ)V1ξ (s, ξ)dξds

(3.25)

R

and e

2μt

||V2ξ (t)||2L2w

t   w 4 − c − 2μ − 4r2 ϕ2 + 2r2 (1 − 2V2 (s, ξ)) + w 0

R

+ 2r2 b2 V1 + 2r2 b2 ϕ1 − r2 b2 (1 − ϕ2 (ξ)) − r2 b2 V2   w(ξ − 1) w(ξ + 1) 2 + e2μs w(ξ)V2ξ − 2+ dξds w(ξ) w(ξ) t  ≤

||V2ξ,0 (0)||2L2w

2 (1 − ϕ2 (ξ) + V2 )e2μs w(ξ)V1ξ (s, ξ)dξds

+ r2 b2 0

t  − 2r2 b2 0

t



+2 0

R

R

ϕ2 (ξ)e2μs w(ξ)V1 (s, ξ)V2ξ (s, ξ)dξds

R

(2r2 ϕ2 (ξ) − r2 b2 ϕ1 (ξ))w(ξ)e2μs V2 (s, ξ)V2ξ (s, ξ)dξds.

(3.26)

G. Tian, G.-B. Zhang / J. Math. Anal. Appl. 447 (2017) 222–242

238

Adding the (3.25) and (3.26) together, we get   e2μt ||V1ξ (t)||2L2w + ||V2ξ (t)||2L2w t  3 2 4 2 e2μs (Bμ,w V1ξ (s, ξ) + Bμ,w V2ξ (s, ξ))w(ξ)dξds

+ 0

R

t  ≤

||V1ξ,0 (0)||2L2w

+

||V2ξ,0 (0)||2L2w

Q(s, ξ)w(ξ)e2μs dξds,

+2 0

(3.27)

R

where Q(t, ξ) = (−2r1 ϕ1 + r1 b1 ϕ2 )V1 (t, ξ)V1ξ (t, ξ) + r1 b1 ϕ1 V2 (t, ξ)V1ξ (t, ξ) + (2r2 ϕ2 − r2 b2 ϕ1 )V2 (t, ξ)V2ξ (t, ξ) − r2 b2 ϕ2 V1 (t, ξ)V2ξ (t, ξ) and 3 Bμ,w (t, ξ) = A3w (t, ξ) − 2μ,

4 Bμ,w (t, ξ) = A4w (t, ξ) − 2μ,

(3.28)

with A3w (t, ξ) = 4 − c

w (ξ) − 2r1 + 2r1 b1 + 4r1 ϕ1 (ξ) − 2r1 b1 V2 (t, ξ) − 2r1 b1 ϕ2 (ξ) w(ξ)

− r1 b1 ϕ1 (ξ) − r2 b2 (1 − ϕ2 (ξ) + V2 ) − r1 b1 V1 (t, ξ)   w(ξ − 1) w(ξ + 1) + − 2+ w(ξ) w(ξ)

(3.29)

and A4w (t, ξ) = 4 − c

w (ξ) − 4r2 ϕ2 (ξ) + 2r2 b2 V1 (t, ξ) + 2r2 (1 − 2V2 (t, ξ)) + 2r2 b2 ϕ1 (ξ) w(ξ)

− r2 b2 (1 − ϕ2 (ξ)) − r1 b1 (ϕ1 (ξ) + V1 (t, ξ)) − r2 b2 V2 (t, ξ)   w(ξ − 1) w(ξ + 1) + . − 2+ w(ξ) w(ξ)

(3.30)

Lemma 3.6. Assume that (H) holds and 0 < b1 < 1 < b2 . For any c > max{c∗ , c˜}, there exist some positive constants Ci such that Aiw (t, ξ) ≥ Ci ,

i = 3, 4

for all ξ ∈ R, t ≥ 0. The proof is similar to that in Lemma 3.3, so we omit the proof here. Lemma 3.7. Assume that (H) holds and 0 < b1 < 1 < b2 . For any c > max{c∗ , c˜}, there exist some positive constants Ci such that i Bμ,w (t, ξ) ≥ Ci ,

for all ξ ∈ R and 0 < μ <

mini=3,4 {Ci } . 2

i = 3, 4,

G. Tian, G.-B. Zhang / J. Math. Anal. Appl. 447 (2017) 222–242

239

Now we estimate the last term on the right-hand side of (3.27). By the properties of traveling wave (ϕ1 (ξ), ϕ2 (ξ)), we can obtain that (ϕ1 (ξ), ϕ2 (ξ)) is bounded for all ξ ∈ R. Thus, there exists a positive constant C5 such that | − 2r1 ϕ1 + r1 b1 ϕ2 | ≤ C5 , |r1 b1 ϕ1 | ≤ C5 , |r2 ϕ2 − r2 b2 ϕ1 | ≤ C5 , |r2 b2 ϕ2 | ≤ C5 . According to Lemma 3.5, it is easy to get the following inequality t

  e2μs ||V1 (s)||2L2w + ||V2 (s)||2L2w ds ≤ C(||V10 (0)||2L2w + ||V20 (0)||2L2w ).

0

By using the Young-inequality 2xy ≤ ηx2 + η1 y 2 and (3.19), we have t  Q(s, ξ)w(ξ)e2μs dξds

2 R

0

t  ≤ C5

e 0

=

C5 η

t

2μs

R

     1 2 2 2 2 V (s, ξ) + V2 (s, ξ) + η V1ξ (s, ξ) + V2ξ (s, ξ) dξds w(ξ) η 1



t



e2μs ||V1 (s)||2L2w + ||V2 (s)||2L2w ds + C5 η 0

≤





(3.31)

e2μs ||V1ξ (s)||2L2w + ||V2ξ (s)||2L2w ds 0

  C5 C ||V10 (0)||2L2w + ||V20 (0)||2L2w + C5 η η

t

  ||V1ξ (s)||2L2w + ||V2ξ (s)||2L2w ds.

0

Choosing η > 0 such that C5 η <

min{C3 ,C4 } 2

and combining with (3.27), we have t

||V1ξ (t)||2L2w −2μt

≤ Ce



+

||V2ξ (t)||2L2w

+

  e−2μ(t−s) ||V1ξ (s)||2L2w + ||V2ξ (s)||2L2w ds

0

||V10 (0)||2Hw1

+ ||V20 (0)||2Hw1



for some positive constant C. Lemma 3.8. Assume that (H) holds and 0 < b1 < 1 < b2 . For any c > max{c∗ , c˜}, it holds t ||V1ξ (t)||2L2w 

+

||V2ξ (t)||2L2w

+

  e−2μ(t−s) ||V1ξ (s)||2L2w + ||V2ξ (s)||2L2w ds

0

≤ Ce−2μt ||V10 (0)||2Hw1 + ||V20 (0)||2Hw1



(3.32)

for some positive constant C. Combining Lemmas 3.5 and 3.8, it is easy to obtain a priori estimate as follows. Lemma 3.9 (A priori estimate). Assume that (H) holds and 0 < b1 < 1 < b2 . For any c > max{c∗ , c˜}, it holds

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  12 ||V1 (t)||Hw1 ≤ Ce−μt ||V10 (0)||2Hw1 + ||V20 (0)||2Hw1 , and   12 ||V2 (t)||Hw1 ≤ Ce−μt ||V10 (0)||2Hw1 + ||V20 (0)||2Hw1 for some positive constant C, 0 < μ <

mini=1,2,3,4 {Ci } 2

and all t > 0.

According to the standard Sobolev embedding inequality Hw1 (R) → H 1 (R) → C(R) since w(ξ) ≥ 1, one has sup |V1 (t, x)| ≤ C||V1 (t, x)||2H 1 ≤ C||V1 (t, x)||2Hw1 x∈R

and sup |V2 (t, x)| ≤ C||V2 (t, x)||2H 1 ≤ C||V2 (t, x)||2Hw1 . x∈R

By Lemma 3.9, we obtain the following results. Lemma 3.10. It holds that sup |v1+ (t, x) − ϕ1 (x + ct)| = sup |V1 (t, ξ)| ≤ Ce−μt , x∈R

ξ∈R

sup |v2+ (t, x) − ϕ2 (x + ct)| = sup |V2 (t, ξ)| ≤ Ce−μt x∈R

ξ∈R

for some positive constant C and t > 0. Step 2. The convergence of vi− (t, x) to ϕi (x + ct), i = 1, 2. Let ξ := x + ct and V1 (t, ξ) = ϕ1 (ξ) − v1− (t, x),

− V10 (0, ξ) = ϕ1 (x) − v10 (0, x)

V2 (t, ξ) = ϕ2 (ξ) − v2− (t, x),

+ V20 (0, ξ) = ϕ2 (x) − v20 (0, x).

and

As shown in the process of Step 1, we can similarly prove the convergence of vi−(t, x) to ϕi (x + ct), i = 1, 2, i.e., Lemma 3.11. It holds that sup |v1− (t, x) − ϕ1 (x + ct)| = sup |V1 (t, ξ)| ≤ Ce−μt , x∈R

sup |v2− (t, x) x∈R

ξ∈R

− ϕ2 (x + ct)| = sup |V2 (t, ξ)| ≤ Ce−μt ξ∈R

for some positive constant C and t > 0. Step 3. The convergence of vi (t, x) to ϕi (x + ct), i = 1, 2.

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241

Lemma 3.12. It holds that sup |v1 (t, x) − ϕ1 (x + ct)| ≤ Ce−μt x∈R

and sup |v2 (t, x) − ϕ2 (x + ct)| ≤ Ce−μt x∈R

for some positive constant C and t > 0. Proof. From (3.2), we can see that (v1− (t, x), v2− (t, x)) ≤ (v1 (t, x), v2 (t, x)) ≤ (v1+ (t, x), v2+ (t, x)).

(3.33)

Then we get |v1 (t, x) − ϕ1 (x + ct)| ≤ max{|v1+ (t, x) − ϕ1 (x + ct)|, |v1− (t, x) − ϕ1 (x + ct)|} and |v2 (t, x) − ϕ2 (x + ct)| ≤ max{|v2+ (t, x) − ϕ2 (x + ct)|, |v2− (t, x) − ϕ2 (x + ct)|}. In view of the convergence results in Lemmas 3.10 and 3.11, we have sup |v1 (t, x) − ϕ1 (x + ct)| ≤ Ce−μt ,

t>0

x∈R

and sup |v2 (t, x) − ϕ2 (x + ct)| ≤ Ce−μt ,

t > 0.

x∈R

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