Planar Waves of the Buffered Bistable System

Abstract

This paper is concerned with the large time behavior of disturbed planar fronts in the buffered bistable system in ℝ^𝑛 (𝑛≥2). We first show that the large time behavior of the disturbed fronts can be approximated by that of the mean curvature flow with a drift term for all large time up to 𝑡=+∞. And then we prove that the planar front is asymptotically stable in 𝐿^∞(ℝ^𝑛) under ergodic perturbations, which include quasiperiodic and almost periodic ones as special cases.

1. Introduction

Traveling waves in excitable systems have been the subject of a vast number of mathematical studies for the past50years. The basic mathematical theory can be used to describe wave propagation in a wide array of biological and chemical systems. Recently, Tsai and Sneyd [1, 2] studied the following buffered system: 𝜕𝑢𝜕𝜕𝑡=𝐷²𝑢𝜕𝑦²+𝑓(𝑢)+^𝑁_𝑖=1𝑘^𝑖₋𝑏^𝑖₀−𝑣_𝑖−𝑘^𝑖₊𝑢𝑣_𝑖,𝑦,𝑡∈ℝ×(0,∞),𝜕𝑣_𝑖𝜕𝑡=𝐷_𝑖𝜕²𝑣_𝑖𝜕𝑦²+𝑘^𝑖₋𝑏^𝑖₀−𝑣_𝑖−𝑘^𝑖₊𝑢𝑣_𝑖,𝑖=1,…,𝑁, (1) where 𝐷, 𝐷_𝑖, 𝑘^𝑖₋, 𝑘^𝑖₊ and 𝑏^𝑖₀ are positive constants. We call system (1) the full buffering model. When𝐷_𝑖=0 (𝑖=1,…,𝑁), Tsai and Sneyd [1] proved the existence, uniqueness, and stability of traveling wave fronts of system (1). The existence, uniqueness, and stability of traveling wave fronts of system (1) with𝐷_𝑖≠0, were obtained by Tsai and Sneyd [2]. About the buffered system (3), see also [3, 4], for more details.

In this paper, we consider the large time behavior of the Cauchy problem to buffering model inℝ^𝑛: 𝑢_𝑡=𝐷Δ𝑢+𝑓(𝑢)+^𝑁_𝑖=1𝑘^𝑖₋𝑏^𝑖₀−𝑣_𝑖−𝑘^𝑖₊𝑢𝑣_𝑖,𝑥∈ℝ^𝑛−1𝑣,𝑦∈ℝ,𝑡>0,_𝑖𝑡=𝐷_𝑖Δ𝑣_𝑖+𝑘^𝑖₋𝑏^𝑖₀−𝑣_𝑖−𝑘^𝑖₊𝑢𝑣_𝑖,𝑖=1,…,𝑁, (2) with initial value 𝑢𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑥∈ℝ^𝑛−1𝑣,𝑦∈ℝ,_𝑖𝑥,𝑦,0=𝑣_𝑖0𝑥,𝑦,𝑥∈ℝ^𝑛−1,𝑦∈ℝ, (3) where𝑛≥2,Δ=𝜕²/𝜕𝑥²₁+⋯+𝜕²/𝜕𝑥²_𝑛−1+𝜕²/𝜕𝑦². Throughout this paper, we always assume that𝐷=𝐷_𝑖=1. In other words, we consider a special case in the present paper and will study the general case in the further paper.

One of the most interesting and natural questions is the behavior of solutions(𝑢(𝑥,𝑦,𝑡),𝑣_𝑖(𝑥,𝑦,𝑡))as𝑡→+∞, in particular, the question about the stability of traveling wave fronts. Lv [5] studied the system (2) with𝑁=1 and obtained the asymptotic stability of planar waves onℝ^𝑛, where𝑛≥2. Under initial perturbation that decays at space infinity, they also proved that the perturbed solution converges to planar waves as 𝑡→∞. We remark that one can obtain the similar results to Theorems 1 and 3 when 𝑁≥2. For convenience, we only consider the case that𝑁=1.

It is well known that (2) onℝhas a traveling wave solution with the form (𝑢(𝑦,𝑡),𝑣_𝑖(𝑦,𝑡))=(𝜙(𝑦+𝑐𝑡),𝜓_𝑖(𝑦+𝑐𝑡)) for some constant𝑐∈ℝ, connecting two equilibrium (0,𝑏^𝑖₀) and (1,𝑘^𝑖₋𝑏^𝑖₀/(𝑘^𝑖₋+𝑘^𝑖₊)), under the condition that𝑓is of bistable. Here “𝑓is bistable” is meant that 𝑓^′(0)<0 and 𝑓^′(1)<0, where 𝑓(0)=𝑓(1)=0. A typical example is that 𝑓(𝑢)=𝑢(𝑢−𝑎)(1−𝑢), where 0<𝑎<1.

A traveling wave front of (2) is a monotone solution with the form (𝑢(𝑦,𝑡),𝑣_𝑖(𝑦,𝑡))=(𝜙(𝑦+𝑐𝑡),𝜓_𝑖(𝑦+𝑐𝑡)) (𝑐 is the wave speed), satisfying 𝐷𝜙^′′(𝜉)−𝑐𝜙^′(𝜉)𝜙(𝜉)++𝑓_𝑛𝑖=1𝑘^𝑖₋𝑏^𝑖₀−𝜓_𝑖−𝑘^𝑖₊𝜙𝜓_𝑖𝐷=0,𝜉∈ℝ,_𝑖𝜓_𝑖^′′(𝜉)−𝑐𝜓^′_𝑖(𝜉)+𝑘^𝑖₋𝑏^𝑖₀−𝜓_𝑖−𝑘^𝑖₊𝜙𝜓_𝑖𝜙=0,𝜉∈ℝ,_′(𝜉)>0,𝜓^′_𝑖(𝜉)<0,lim_{𝜉→−∞}𝜙(𝜉),𝜓_𝑖(𝜉)=0,𝑏^𝑖₀,lim_𝜉→∞𝜙(𝜉),𝜓_𝑖(𝜉)=1,𝑏^𝑖₂, (4) where𝜉=𝑦+𝑐𝑡,^′=𝑑/𝑑𝜉 and 𝑏^𝑖₂=𝑘^𝑖₋𝑏^𝑖₀/(𝑘^𝑖₋+𝑘^𝑖₊).

The function(𝜙(𝑦+𝑐𝑡),𝜓_𝑖(𝑦+𝑐𝑡))is also a traveling wave front for system (2) with𝑛≥2. We call it planar front. Its translation(𝜙(𝑦+𝑐𝑡+𝜉),𝜓_𝑖(𝑦+𝑐𝑡+𝜉)), where𝜉is any constant, is also called a planar front. Note that the equations in (2) from second to the last have the same properties; without loss of generality, we only consider the case that𝑁=1, and we write𝑣_𝑖as𝑣, that is, 𝑢_𝑡=Δ𝑢+𝑓(𝑢)+𝑘₁𝑏₀−𝑣−𝑘₂𝑢𝑣,𝑥∈ℝ^𝑛−1𝑣,𝑦∈ℝ,𝑡>0,_𝑡=Δ𝑣+𝑘₁𝑏₀−𝑣−𝑘₂𝑢𝑣,𝑥∈ℝ^𝑛−1,𝑦∈ℝ,𝑡>0, (5) with initial value 𝑢𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑥∈ℝ^𝑛−1𝑣,𝑦∈ℝ,𝑥,𝑦,0=𝑣₀𝑥,𝑦,𝑥∈ℝ^𝑛−1,𝑦∈ℝ. (6) Let𝑢^∗=𝑢 and 𝑣^∗=𝑏₀−𝑣, then (5) becomes (for simplicity, we omit the symbol∗) 𝑢_𝑡=Δ𝑢+𝑓(𝑢)+𝑘₁𝑣−𝑘₂𝑢𝑏₀−𝑣,𝑥∈ℝ^𝑛−1𝑣,𝑦∈ℝ,𝑡>0,_𝑡=Δ𝑣−𝑘₁𝑣+𝑘₂𝑢𝑏₀−𝑣,𝑥∈ℝ^𝑛−1,𝑦∈ℝ,𝑡>0. (7)

Obviously, system (7) is a cooperative model. The problem corresponding to traveling wave front (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡))=(𝜙(𝜉),𝜓(𝜉)) of system (7), connecting (0,0) and (1,𝑏₀−𝑏₂), is 𝜙^′′−𝑐𝜙^′+𝑓𝜙+𝑘₁𝜓−𝑘₂𝜙𝑏₀𝜓−𝜓=0,_′′−𝑐𝜓^′−𝑘₁𝜓+𝑘₂𝜙𝑏₀𝜙−𝜓=0,_′𝜓>0,_′>0,lim_{𝜉→−∞}𝜙,𝜓=(0,0),lim_𝜉→∞𝜙,𝜓=1,𝑏₀−𝑏₂. (8)

Recently, Matano et al. [6] considered the following Allen-Cahn equation: 𝑢_𝑡=Δ𝑢+𝑓(𝑢),𝑥∈ℝ^𝑛−1𝑢,𝑦∈ℝ,𝑡>0,𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑥∈ℝ^𝑛−1,𝑦∈ℝ. (9) Under the condition that𝑓is of the bistable type, they obtained the stability of planar fronts under any-possibly large-initial perturbations and almost periodic perturbations. When𝑓is monostable type, the stability of planar waves for (9) was obtained by Lv and Wang [7]. About the stability of planar wave, also see [8–10]. Just recently, Matano and Nara [11] reconsidered the Cauchy problem (9) under the condition that𝑓is bistable type. They proved that the planar front is asymptotically stable in𝐿^∞(ℝ^𝑛)under spatially ergodic perturbation and that the large time behavior of the disturbed planar front can be approximated by that of the mean curvature flow with a drift term for all large time up to+∞. Lv [12] studied the Cauchy problem (9) under that𝑓is monostable type and obtained that the planar front is asymptotically stable in𝐿^∞(ℝ^𝑛)under spatially ergodic perturbation.

Our work has been inspired by [11]; in this paper, we will consider the stability of planar fronts under spatially ergodic perturbation. Let us now state our main results.

Theorem 1 (large time behavior). Let𝑛≥2. Let(𝑢(𝑥,𝑦,𝑡),𝑣(𝑥,𝑦,𝑡))be a solution problem (7) with (6) whose initial value(𝑢₀(𝑥,𝑦),𝑣₀(𝑥,𝑦))satisfies (0,0)≤𝑢₀𝑥,𝑦,𝑣₀𝑥,𝑦≤1,𝑏₀−𝑏₂,limsup_{𝑦→−∞}sup_{𝑥∈ℝ𝑛−1}𝑢₀𝑥,𝑦<𝛿,liminf_𝑦→+∞inf_{𝑥∈ℝ𝑛−1}𝑢₀𝑥,𝑦>1−𝛿,limsup_{𝑦→−∞}sup_{𝑥∈ℝ𝑛−1}𝑣₀𝑥,𝑦<𝑘₀𝛿,liminf_𝑦→+∞inf_{𝑥∈ℝ𝑛−1}𝑣₀𝑥,𝑦>𝑏₀−𝑏₂−𝑘₀𝛿, (10) where𝛿∈(0,𝛿₀] and𝑘₀>0are defined as in Lemma 4. Then there exist a constant𝑇>0 and a smooth function𝛾(𝑥,𝑡)such that(i)for each𝑡∈[𝑇,+∞) and 𝑥∈ℝ^𝑛−1, one has(𝑢(𝑥,𝑦,𝑡),𝑣(𝑥,𝑦,𝑡))=(𝜙(0),𝜓(0))if and only if𝑦=𝛾(𝑥,𝑡);(ii)it holds that lim_𝑡→∞sup_{(𝑥,𝑦)∈ℝ𝑛}𝑢𝑥,𝑦,𝑡−𝜙𝑦−𝛾(𝑥,𝑡)=0,lim_𝑡→∞sup_{(𝑥,𝑦)∈ℝ𝑛}𝑣𝑥,𝑦,𝑡−𝜓𝑦−𝛾(𝑥,𝑡)=0; (11)(iii)for any𝜀>0, there exists𝜏_𝜀∈[𝑇,∞)such that the solution𝑈(𝑥,𝑡)of the problem 𝑈_𝑡1+∇_𝑥𝑈²∇=div^𝑥𝑈1+∇_𝑥𝑈²−𝑐,𝑥∈ℝ^𝑛−1𝑈(),𝑡>0,𝑥,0=𝛾𝑥,𝜏_𝜀,𝑥∈ℝ^𝑛−1 (12)satisfies sup_{𝑥∈ℝ𝑛−1,𝑡≥𝜏𝜀}𝛾(𝑥,𝑡)−𝑈𝑥,𝑡−𝜏_𝜀≤𝜀, (13) where∇_𝑥denotes the(𝑛−1)-dimensional gradient.

The statement (ii) of Theorem 1 implies that the solution (𝑢(𝑥,𝑦,𝑡),𝑣(𝑥,𝑦,𝑡)) behaves like the function (𝜙(𝑦−𝛾(𝑥,𝑡)),𝜓(𝑦−𝛾(𝑥,𝑡))) for large𝑡; thus the large time behavior of the solution (𝑢(𝑥,𝑦,𝑡),𝑣(𝑥,𝑦,𝑡)) is basically determined by the position of the(𝜙(0),𝜓(0))-level set (𝛾(𝑥,𝑡),𝛾(𝑥,𝑡)). The last statement shows that the behavior of𝛾(𝑥,𝑡)can be approximated by the solution𝑈(𝑥,𝑡)of the mean curvature flow onℝ^𝑛−1with a drift term𝑐. Comparing the above theorem with Theorem  1.1 in [5], it is to see that we delete the assumptions that initial perturbations decay to zero as|𝑥|+|𝑦|→∞, and thus the result in this paper is better than that of [5].

In order to obtain the stability of planar wave, we need the following definition.

Definition 2 (unique ergodicity in the 𝑥-direction). A bounded uniformly continuous function 𝑝(𝑥,𝑦): ℝ^𝑛−1×ℝ→ℝ is called uniquely ergodic in the𝑥-direction if there exists a unique probability measure on ℍ_𝑝, that is, 𝜎_𝑎-invariant for any𝑎∈ℝ^𝑛−1, ℍ_𝑝∶=𝜎_𝑎𝑔∣𝑎∈ℝ^{𝑛−1𝐿∞loc(ℝ𝑛−1)},𝜎_𝑎𝑝(𝑥)=𝑝(𝑥+𝑎), (14) and𝐴^𝑋stands for the closure of a set𝐴in the𝑋-topology.

Theorem 3. In addition to the assumptions of Theorem 1, assume further that𝑢₀(𝑥,𝑦) and 𝑣₀(𝑥,𝑦)are uniquely ergodic in the𝑥-direction. Then there exist constants𝜇₁,𝜇₂∈ℝsuch that lim_𝑡→∞sup_{(𝑥,𝑦)∈ℝ𝑛}𝑢𝑥,𝑦,𝑡−𝜙𝑦+𝑐𝑡+𝜇₁=0,lim_𝑡→∞sup_{(𝑥,𝑦)∈ℝ𝑛}𝑣𝑥,𝑦,𝑡−𝜓𝑦+𝑐𝑡+𝜇₂=0. (15)

From [11], we see that ℙ⊂ℚℙ⊂𝔸ℙ⊂𝕊𝔼⊂𝕌𝔼, (16) whereℙ,ℚℙ,𝔸ℙ,𝕊𝔼,𝕌𝔼denote, respectively, the sets of periodic functions, quasiperiodic functions, almost periodic functions, strictly ergodic functions, and uniquely ergodic functions. Hence, the above Theorem 3 is a general result.

The rest of this paper is organized as follows. In Section 2, some known results and related problem are considered. Section 3 is concerned with the proofs of Theorems 1 and 3.

2. Some Known Results and Related Problem

In this section, we first study the one-dimensional problem (7) in a moving frame, that is, 𝑢_𝑡=𝑢_𝑧𝑧−𝑐𝑢_𝑧+𝑓(𝑢)+𝑘₁𝑣−𝑘₂𝑢𝑏₀,𝑣−𝑣𝑧∈ℝ,𝑡>0,_𝑡=𝑣_𝑧𝑧−𝑐𝑣_𝑧−𝑘₁𝑣+𝑘₂𝑢𝑏₀−𝑣,𝑧∈ℝ,𝑡>0, (17) where𝑧=𝑦+𝑐𝑡.

The traveling wave solutions of (17) have been studied by many authors; see [1, 2], for more details. Tsai and Sneyd [1, 2] obtained the existence, uniqueness, and stability of traveling wave fronts of system (17). They obtained the following lemmas; see Lemma  6.1 in [2] and Lemma  3.5 in [5].

Lemma 4. Assume that𝑓is bistable and(𝜙(𝑧),𝜓(𝑧))is a traveling wave solution of (17). Then there exist 𝛿₀, 𝛽, 𝑘₀>0 and 𝜎≥1 such that, for any 𝛿∈(0,𝛿₀], the function defined by 𝑢⁺(𝑥,𝑧,𝑡)=𝜙𝑧+𝜎𝛿1−e^−𝛽𝑡+𝛿e^−𝛽𝑡,𝑣⁺(𝑥,𝑧,𝑡)=𝜓𝑧+𝜎𝛿1−e^−𝛽𝑡+𝑘₀𝛿e^−𝛽𝑡 (18) is a subsolution. More precisely, it satisfies the following inequality: 𝑁𝑢⁺∶=𝑢⁺_𝑡−Δ𝑢⁺+𝑐𝑢⁺_𝑧−𝑓𝑢⁺−𝑘₁𝑣⁺+𝑘₂𝑢⁺𝑏₀−𝑣⁺𝑁𝑣≥0,₊∶=𝑣⁺_𝑡−Δ𝑣⁺+𝑐𝑣⁺_𝑧+𝑘₁𝑣⁺−𝑘₂𝑢⁺𝑏₀−𝑣⁺≥0. (19)

Lemma 5. Assume that𝑓is bistable and (𝜙(𝑧),𝜓(𝑧))is a traveling wave solution of (17). Then there exist 𝛿₀, 𝛽, 𝑘₀>0 and 𝜎≥1 such that, for any 𝛿∈(0,𝛿₀], the function defined by 𝑢⁻(𝑥,𝑧,𝑡)=𝜙𝑧−𝜎𝛿1−e^−𝛽𝑡−𝛿e^−𝛽𝑡,𝑣⁻(𝑥,𝑧,𝑡)=𝜓𝑧−𝜎𝛿1−e^−𝛽𝑡−𝑘₀𝛿e^−𝛽𝑡 (20) is a subsolution. More precisely, it satisfies the following inequality: 𝑁[𝑢⁻]∶=𝑢⁻_𝑡−Δ𝑢⁻+𝑐𝑢⁻_𝑧−𝑓(𝑢⁻)−𝑘₁𝑣⁻+𝑘₂𝑢⁻𝑏₀−𝑣⁻𝑁[𝑣≤0,₋]∶=𝑣⁻_𝑡−Δ𝑣⁻+𝑐𝑣⁻_𝑧+𝑘₁𝑣⁻−𝑘₂𝑢⁻𝑏₀−𝑣⁻≤0. (21)

The proof of the above two lemmas is similar to that of Lemmas 3.5 and 3.6 in [5], and we omit them here. Now we consider the following problem: 𝑈_𝑡1+∇_𝑥𝑈²∇=div^𝑥𝑈1+∇_𝑥𝑈²−𝑐,𝑥∈ℝ^𝑚𝑈(),𝑡>0,𝑥,0=𝑈₀(𝑥),𝑥∈ℝ^𝑚. (22)

Lemma 6 (see [11]). Let𝑚≥1. Let𝑈(𝑥,𝑡)be a solution to the problem (22) whose initial value𝑈₀(𝑥)is bounded, Lipschitz continuous, and uniquely ergodic onℝ^𝑚. Then there exists a constant𝜇∈ℝsuch that lim_𝑡→∞sup_{𝑥∈ℝ𝑚}𝑈(𝑥,𝑡)+𝑐𝑡+𝜇=0. (23)

Lemma 7 (see [11]). Let𝑈(𝑥,𝑡;𝜑) and 𝑉(𝑥,𝑡;𝜑)denote the solutions of the following equations: 𝑈_𝑡1+∇_𝑥𝑈²∇=div^𝑥𝑈1+∇_𝑥𝑈²−𝑐,𝑥∈ℝ^𝑚𝑉,𝑡>0,_𝑡=Δ_𝑥𝑐𝑉−2∇_𝑥𝑉²−𝑐,𝑥∈ℝ^𝑚,𝑡>0, (24) under the initial conditions𝑈(⋅,0)=𝑉(⋅,0)=𝜑∈𝑊^2,∞(ℝ^𝑚). Then, for any constant𝜀>0, there exists a constant𝛿>0such that if‖∇_𝑥𝜑‖_𝑊1,∞≤𝛿, it holds that sup_{𝑥∈ℝ𝑚}𝑈𝑥,𝑡;𝜑−𝑉𝑥,𝑡;𝜑≤𝜀∀𝑡>0. (25)

The following lemma is a little different from Lemma  3.10 in [11], but the proof is similar.

Lemma 8. Let𝑉(𝑥,𝑡)be a solution to the problem 𝑉_𝑡=Δ_𝑥𝑐𝑉−2∇_𝑥𝑉²−𝑐,𝑥∈ℝ^𝑛−1𝑉(),𝑡>0,𝑥,0=𝑉₀(𝑥),𝑥∈ℝ^𝑛−1. (26) Then the following estimates hold: sup_{𝑥∈ℝ𝑛−1}𝑉_𝑥𝑖(𝑥,𝑡)≤min𝐶₀𝑡^−1/2,𝐶₁,sup_{𝑥∈ℝ𝑛−1}𝑉_{𝑥𝑖𝑥𝑗}(𝑥,𝑡)≤min𝐶₀𝑡⁻¹,𝐶₂,sup_{𝑥∈ℝ𝑛−1}𝑉_{𝑥𝑖𝑥𝑗𝑥𝑘}(𝑥,𝑡)≤𝐶₃(1+𝑡)^−3/2,sup_{𝑥∈ℝ𝑛−1}𝑉_𝑥𝑖𝑡(𝑥,𝑡)≤𝐶₄(1+𝑡)^−3/2, (27) for each 1≤𝑖,𝑗,𝑘≤𝑛−1, where 𝐶₀, 𝐶₁, 𝐶₂, 𝐶₃, and 𝐶₄ are positive constants such that(i)𝐶₀depends only on𝑐 and ‖𝑉₀‖_𝐿∞;(ii)𝐶₁depends only on𝑐,‖𝑉₀‖_𝐿∞, and ‖∇_𝑥𝑉₀‖_𝐿∞ and satisfies 𝐶₁⟶0𝑎𝑠∇_𝑥𝑉_0𝐿∞⟶0; (28)(iii)𝐶₂depends only on𝑐,‖𝑉₀‖_𝐿∞, and ‖∇_𝑥𝑉₀‖_𝑊1,∞ and satisfies 𝐶₂⟶0𝑎𝑠∇_𝑥𝑉_0𝑊1,∞⟶0; (29)(iv)𝐶₃ and 𝐶₄depend only on 𝑐 and ‖𝑉₀‖_𝑊3,∞.

3. Proofs of Theorems 1 and 3

In this section, we consider problem (7) inℝ^𝑛 and prove our main results. Firstly, we give rough upper and lower bounds for the solution at large time, then introduce the notion of𝜔-limit points of the solution and study basic properties of(𝜙(0),𝜓(0))-level surface of the solution. Lastly, we construct a fine set of supersolutions and subsolutions and give the proofs of the main theorems.

We will express solutions(𝑢(𝑥,𝑦,𝑡),𝑣(𝑥,𝑦,𝑡))of (1) in a moving frame, and thus the planar waves can be viewed as stationary states. Letting 𝑤(𝑥,𝑧,𝑡)=𝑢𝑥,𝑦,𝑡,𝜔(𝑥,𝑧,𝑡)=𝑣𝑥,𝑦,𝑡,𝑧=𝑦+𝑐𝑡, (30) problem (7) with (6) is rewritten as (we write(𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡))as(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))for simplicity) 𝑢_𝑡=Δ𝑢−𝑐𝑢_𝑧+𝑓(𝑢)+𝑘₁𝑣−𝑘₂𝑢𝑏₀−𝑣,𝑥∈ℝ^𝑛−1𝑣,𝑧∈ℝ,𝑡>0,_𝑡=Δ𝑣−𝑐𝑣_𝑧−𝑘₁𝑣+𝑘₂𝑢𝑏₀−𝑣,𝑥∈ℝ^𝑛−1𝑢(),𝑧∈ℝ,𝑡>0,𝑥,𝑧,0=𝑢₀(𝑥,𝑧),𝑥∈ℝ^𝑛−1𝑣(),𝑧∈ℝ,𝑥,𝑧,0=𝑣₀(𝑥,𝑧),𝑥∈ℝ^𝑛−1,𝑧∈ℝ, (31) whereΔ=𝜕²/𝜕𝑥²₁+⋯+𝜕²/𝜕𝑥²_𝑛−1+𝜕²/𝜕𝑧². Throughout this section, we always assume that the initial value(𝑢₀,𝑣₀)satisfies(0,0)≤(𝑢₀(𝑥,𝑧),𝑣₀(𝑥,𝑧))≤(1,𝑏₀−𝑏₂)and limsup_{𝑧→−∞}sup_{𝑥∈ℝ𝑛−1}𝑢₀(𝑥,𝑧)<𝛿,liminf_𝑧→+∞inf_{𝑥∈ℝ𝑛−1}𝑢₀(𝑥,𝑧)>1−𝛿,limsup_{𝑧→−∞}sup_{𝑥∈ℝ𝑛−1}𝑣₀(𝑥,𝑧)<𝑘₀𝛿,liminf_𝑧→+∞inf_{𝑥∈ℝ𝑛−1}𝑣₀(𝑥,𝑧)>𝑏₀−𝑏₂−𝑘₀𝛿, (32) where𝛿and𝑘₀are defined in Theorem 1. We first consider upper and lower bounds of the solution of (31) at large time.

Lemma 9. Let(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))be a solution of (31). Then there exists constant 𝑧_∗, 𝑧^∗∈ℝsuch that lim_𝑡→∞inf_{𝑥∈ℝ𝑛−1}(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))≥𝜙𝑧−𝑧^∗,𝜓𝑧−𝑧^∗,𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑧∈ℝ,lim_𝑡→∞sup_{𝑥∈ℝ𝑛−1}(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))≤𝜙𝑧−𝑧_∗,𝜓𝑧−𝑧_∗,𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑧∈ℝ. (33)

Proof. We only show the upper bound of(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡)), since the other is similar. Let(𝑢⁺,𝑣⁺)be as in Lemma 4. Then it suffices to show that there exist constants𝑇>0and𝑧₀∈ℝsuch that 𝑢(𝑥,𝑧,𝑇)≤𝜙𝑧−𝑧₀+𝛿₁=𝑢⁺𝑧−𝑧₀,𝑣(),0𝑥,𝑧,𝑇≤𝜓𝑧−𝑧₀+𝛿₂=𝑣⁺𝑧−𝑧₀,0. (34)(35) Indeed, the comparison principle and (34)-(35) give (𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))≤(𝑢⁺(𝑧−𝑧₀,𝑡−𝑇),𝑣⁺(𝑧−𝑧₀,𝑡−𝑇)) for𝑡≥𝑇, which yield upper bound by letting𝑡→∞.
Since(0,0)≤(𝑢₀(𝑥,𝑧),𝑣₀(𝑥,𝑧))≤(1,𝑏₀−𝑏₂), we see from the comparison principle that (0,0)≤(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))≤1,𝑏₀−𝑏₂,(𝑥,𝑧)∈ℝ^𝑛,𝑡>0. (36) Note thatlim_𝑧→∞(𝜙(𝑧),𝜓(𝑧))=(1,𝑏₀−𝑏₂), there exists a constant𝑧₁∈ℝsuch that, for each𝑇>0and𝑧≥𝑧₁, 𝑢(𝑥,𝑧,𝑡)≤𝜙(𝑧)+𝛿₁=𝑢⁺(),𝑣()(𝑧)𝑧,0𝑥,𝑧,𝑡≤𝜓+𝛿₂=𝑣⁺(𝑧,0). (37)
Next, we show that limsup_{𝑧→−∞}sup_{𝑥∈ℝ𝑛−1}𝑢(𝑥,𝑧,𝑇),limsup_{𝑧→−∞}sup_{𝑥∈ℝ𝑛−1}𝑣(<𝛿𝑥,𝑧,𝑇)₁,𝛿₂, (38) for each𝑇>0. For this purpose, choose positive constants 𝛽₁, 𝛽₂, and 𝑀 such that limsup_{𝑧→−∞}sup_{𝑥∈ℝ𝑛−1}𝑢₀(𝑥,𝑧)<𝛽₁<𝛿,limsup_{𝑧→−∞}sup_{𝑥∈ℝ𝑛−1}𝑣₀(𝑥,𝑧)<𝛽₂<𝑘₀𝛿 (39) and that 𝑢₀(𝑥,𝑧),𝑣₀(𝑥,𝑧)≤𝛽₁+𝑀𝑒^𝑐𝑧,𝛽₂+𝑀𝑒^𝑐𝑧,(𝑥,𝑧)∈ℝ^𝑛. (40) Then the functions (𝑤()(=𝛽𝑥,𝑧,𝑡,𝜔𝑥,𝑧,𝑡))min₁+𝑀𝑒^{𝑐(𝑧+𝑎𝑡)},1,min𝛽₂+𝑀𝑒^{𝑐(𝑧+𝑎𝑡)},𝑏₀−𝑏₂ (41) are a supersolution of (31) if𝑎>0is chosen sufficiently large. Hence, (𝑢(𝑥,𝑧,𝑇),𝑣(𝑥,𝑧,𝑇))≤(𝑤(𝑥,𝑧,𝑇),𝜔(𝑥,𝑧,𝑇)). (42) This proves (38). Combining (37) and (38), it is easy to see that (34)-(35) hold. This completes the proof.

Now, we introduce the notion of𝜔-limit points of the solution(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))of (31), where we consider a sequence both in𝑥and𝑡. Then we show that any𝜔-limit point is a planar wave under the assumption (34).

Definition 10. A vector-valued function(𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡))defined onℝ^𝑛−1×ℝ×ℝis called a𝜔-limit point of the solution(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))of (31) if there exists a sequence {(𝑥_𝑖,𝑡_𝑖)} such that0<𝑡₁<𝑡₂<⋯→∞and that 𝑢𝑥+𝑥_𝑖,𝑧,𝑡+𝑡_𝑖,𝑣𝑥+𝑥_𝑖,𝑧,𝑡+𝑡_𝑖⟶(𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡))as𝑖⟶∞in𝐶^2,1_locℝ^𝑛×ℝ. (43)
The following remark tells us how to construct𝜔-limit point, which is similar to Remark  4.4 in [11].

Remark 11. Let𝐔(𝑥,𝑧,𝑡)∶=(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡)) be a solution of (31). Then for any sequence{(𝑥_𝑖,𝑡_𝑖)}with0<𝑡₁<𝑡₂<⋯→∞, there exist a subsequence {(𝑥^′_𝑖,𝑡^′_𝑖)} and a 𝜔-limit point 𝐖(𝑥,𝑧,𝑡)∶=(𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡)) of 𝐔(𝑥,𝑧,𝑡)such that 𝐔𝑥+𝑥^′_𝑖,𝑧,𝑡+𝑡^′_𝑖⟶𝐖(𝑥,𝑧,𝑡),as𝑖⟶∞in𝐶^2,1_locℝ^𝑛×ℝ. (44) Indeed, by the assumption in Theorem 1, it is easy to see that𝐔(𝑥,𝑧,𝑡)is bounded. Then by𝐿^𝑝-estimates and Schauder’s estimate (see [13, 14]), the solution𝐔(𝑥,𝑧,𝑡)belongs to𝐶^{2+𝛼,1+𝛼/2}(ℝ^𝑛×[𝛿,𝑇])for any0<𝛿<𝑇. Furthermore, ‖𝐔(𝑥,𝑧,𝑡)‖_𝐶^{2+𝛼,1+𝛼/2}_(ℝ^𝑛_{×[𝛿,𝑇])}≤𝐶, (45) where𝐶>0is a constant independent of𝑇>0. Let{𝑄_𝑘}_𝑘=1,2,…be a sequence of compact subsets of ℝ^𝑛×ℝ satisfying 𝑄₁⊂𝑄₂⊂⋯,lim_𝑘→∞𝑄_𝑘=ℝ^𝑛×ℝ. (46) Then, for each𝑘, the sequence of functions{𝐔(𝑥,𝑧,𝑡)(𝑥+𝑥_𝑖,𝑧,𝑡+𝑡_𝑖)}_𝑖=1,2,…is defined on𝑄_𝑘for all large𝑖and the restrictions of these functions onto𝑄_𝑘are relatively compact in𝐶^2,1(𝑄_𝑘)by virtue of the estimate (45). By using diagonal argument, we can choose a subsequence{(𝑥^′_𝑖,𝑡^′_𝑖)}and a function𝑤(𝑥,𝑧,𝑡)defined onℝ^𝑛×ℝsuch that, for any𝑘≥1, it holds that lim_𝑖→∞𝐔𝑥+𝑥^′_𝑖,𝑧,𝑡+𝑡^′_𝑖−𝐖(𝑥,𝑧,𝑡)_{𝐶2,1(𝑄𝑘)}=0, (47) which means𝐔(𝑥+𝑥^′_𝑖,𝑧,𝑡+𝑡^′_𝑖)→𝐖(𝑥,𝑧,𝑡)as𝑖→∞in𝐶^2,1_loc(𝑄_𝑘).

In order to prove that any𝜔-limit point is a planar wave, the following Lemma is needed.

Lemma 12. Let(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))be a vector-valued function that is defined onℝ^𝑛−1×ℝ×ℝand satisfies 𝑢_𝑡=Δ𝑢−𝑐𝑢_𝑧+𝑓(𝑢)+𝑘₁𝑣−𝑘₂𝑢𝑏₀,()−𝑣𝑥,𝑧∈ℝ^𝑛𝑣,𝑡∈ℝ,_𝑡=Δ𝑣−𝑐𝑣_𝑧−𝑘₁𝑣+𝑘₂𝑢𝑏₀,()−𝑣𝑥,𝑧∈ℝ^𝑛,𝑡∈ℝ. (48)
Assume further that there exist two constants 𝑧_∗,𝑧^∗∈ℝ such that 𝜙𝑧−𝑧^∗,𝜓𝑧−𝑧^∗≤(𝑢()(≤𝜙𝑥,𝑧,𝑡,𝑣𝑥,𝑧,𝑡))𝑧−𝑧_∗,𝜓𝑧−𝑧_∗,(𝑥,𝑧)∈ℝ^𝑛,𝑡∈ℝ. (49) Then there exists a constant𝑧₀∈[𝑧_∗,𝑧^∗]such that 𝑢(𝑥,𝑧,𝑡)=𝜙𝑧−𝑧₀,𝑣(𝑥,𝑧,𝑡)=𝜓𝑧−𝑧₀,(𝑥,𝑧)∈ℝ^𝑛,𝑡∈ℝ. (50)

Proof. Define 𝑢^𝑠(𝑥,𝑧,𝑡)=𝑢𝑥+𝑦,𝑧+𝑠,𝑡+𝑇,𝑦∈ℝ^𝑛−1𝑣,𝑠,𝑇∈ℝ,_𝑠(𝑥,𝑧,𝑡)=𝑣𝑥+𝑦,𝑧+𝑠,𝑡+𝑇,𝑦∈ℝ^𝑛−1,𝑠,𝑇∈ℝ. (51) Then, 𝜙𝑧+𝑠−𝑧^∗,𝜓𝑧+𝑠−𝑧^∗≤𝑢^𝑠(𝑥,𝑧,𝑡),𝑣^𝑠()≤𝜙𝑥,𝑧,𝑡𝑧+𝑠−𝑧_∗,𝜓𝑧+𝑠−𝑧_∗. (52) It follows from the monotonicity of𝜙and𝜓that there exists𝑠∈ℝsatisfying 𝑢^𝑠()(),𝑣𝑥,𝑧,𝑡≥𝑢𝑥,𝑧,𝑡_𝑠(𝑥,𝑧,𝑡)≥𝑣(𝑥,𝑧,𝑡). (53) Indeed, let𝑠=𝑧^∗−𝑧_∗, then(𝑢^𝑠(𝑥,𝑧,𝑡),𝑣^𝑠(𝑥,𝑧,𝑡))≥(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡)). Now, let 𝑠_∗=inf𝑠∈ℝ,𝑢^𝑠′,𝑣^𝑠′≥(𝑢,𝑣),inℝ^𝑛×ℝ∀𝑠^′≥𝑠. (54) Sincelim_{𝑧→−∞}(𝜙(𝑧),𝜓(𝑧))=(0,0), lim_𝑧→∞(𝜙(𝑧),𝜓(𝑧))=(1,𝑏₀−𝑏₂) and by using (49), there exists constant𝐴₁>0such that (𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))≥1−𝛾,𝑏₀−𝑏₂−𝛾∀𝑧≥𝐴₁,(𝑥,𝑧,𝑡)∈ℝ^𝑛(𝑢()(≤×ℝ,𝑥,𝑧,𝑡,𝑣𝑥,𝑧,𝑡))𝛾,𝛾∀𝑧≤−𝐴₁,(𝑥,𝑧,𝑡)∈ℝ^𝑛×ℝ. (55) Note that one can assume that𝛾∈(0,min{1/2,(𝑏₀−𝑏₂−𝛾)/2}). Let𝐴=max{𝐴₁,((𝑧^∗−𝑧_∗)/2)}, then (𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))≥(1−𝛾,𝑏₀−𝑏₂−𝛾) for 𝑧≥𝐴 and (𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))≤(𝛾,𝛾) for 𝑧≤−𝐴. It is easy to see that 𝑠_∗≤2𝐴 and 𝑠_∗>−∞. Assume that 𝑠_∗>0 and call 𝑆={(𝑥,𝑧,𝑡)∈ℝ^𝑛×ℝ,−𝐴≤𝑧≤𝐴}. If inf_𝑆(𝑢^𝑠∗−𝑢)>0 and inf_𝑆(𝑣^𝑠∗−𝑣)>0, then there exists 𝜂₀∈(0,𝑠_∗) such that (𝑢^{𝑠∗−𝜂0},𝑣^{𝑠∗−𝜂0})≥(𝑢,𝑣) in 𝑆 for all 𝜂∈[0,𝜂₀]. Denote 𝐸={(𝑥,𝑧,𝑡)∈ℝ^𝑛×ℝ,𝑧>𝐴}, 𝑤=𝑢^{𝑠∗−𝜂}−𝑢 and 𝜔=𝑣^{𝑠∗−𝜂}−𝑣, we see that (𝑤,𝜔)≥(0,0) on 𝜕𝐸 and satisfies 𝑤_𝑡−Δ𝑤+𝑐𝑤_𝑧=𝑓𝑢^{𝑠∗−𝜂}−𝑓(𝑢)+𝑘₁+𝑘₂𝑢𝜔−𝑘₂𝑏₀−𝑣^{𝑠1∗−𝜂}𝑤≥𝑘₁+𝑘₂𝑢𝜔−𝑘₂𝑏₀−𝑣^{𝑠∗−𝜂}𝜔𝑤−𝐵𝑤_𝑡−Δ𝜔+𝑐𝜔_𝑧=−𝑘₁+𝑘₂𝑢𝜔+𝑘₂𝑏₀−𝑣^{𝑠∗−𝜂}𝑤 (56) for some constant𝐵(remember that𝑓∈𝐶¹(ℝ) and𝑢^{𝑠∗−𝜂},𝑢,𝑣^{𝑠∗−𝜂}and𝑣are bounded). The parabolic maximum principle implies that(𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡))≥(0,0)in𝐸. Similarly, we can prove that(𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡))≥(0,0)for𝑧≤−𝐴. Therefore,(𝑢^{𝑠∗−𝜂},𝑣^{𝑠∗−𝜂})≥(𝑢,𝑣)inℝ^𝑛×ℝfor all𝜂∈[0,𝜂₀]. This contradicts the minimality of𝑠_∗. It follows then that inf_𝑆𝑢^𝑠∗−𝑢=0orinf_𝑆𝑣^𝑠∗−𝑣=0. (57) We first consider the case thatinf_𝑆(𝑢^𝑠∗−𝑢)=0andinf_𝑆(𝑣^𝑠∗−𝑣)>0. As a consequence, there exist𝑧_∞∈[−𝐴,𝐴]and a sequence(𝑥_𝑛,𝑧_𝑛,𝑡_𝑛)_𝑛∈𝑁such that 𝑧_𝑛⟶𝑧_∞,𝑢^𝑠∗𝑥_𝑛,𝑧_𝑛,𝑡_𝑛−𝑢𝑥_𝑛,𝑧_𝑛,𝑡_𝑛⟶0as𝑛⟶∞. (58) Call𝑢_𝑛(𝑥,𝑧,𝑡)=𝑢(𝑥+𝑥_𝑛,𝑧+𝑧_𝑛,𝑡+𝑡_𝑛)and𝑣_𝑛(𝑥,𝑧,𝑡)=𝑣(𝑥+𝑥_𝑛,𝑧+𝑧_𝑛,𝑡+𝑡_𝑛). Note that𝑣_𝑛(𝑥,𝑧,𝑡)is a bounded sequence. Up to extraction of a subsequence, the functions(𝑢_𝑛,𝑣_𝑛)converge locally uniformly to a solution(𝑢_∞,𝑣_∞)of the following system: 𝑢_𝑡=Δ𝑢−𝑐𝑢_𝑧+𝑓(𝑢)+𝑘₁𝑣−𝑘₂𝑢𝑏₀,𝑣−𝑣_𝑡=Δ𝑣−𝑐𝑣_𝑧−𝑘₁𝑣+𝑘₂𝑢𝑏₀−𝑣. (59) It is easy to see that 𝑤(𝑥,𝑧,𝑡)=𝑢_∞𝑥+𝑦,𝑧+𝑠_∗()𝜔(),𝑡+𝑇−𝑢𝑥,𝑧,𝑡≥0,𝑥,𝑧,𝑡=𝑣_∞𝑥+𝑦,𝑧+𝑠_∗(),𝑡+𝑇−𝑣𝑥,𝑧,𝑡>0inℝ^𝑛×ℝ (60) and 𝑤(0,𝑧_∞,0)=0; that is, 𝑤(𝑥,𝑧,𝑡) attains the local minimum at (0,𝑧_∞,0). We only consider the equation of 𝑢. It is easy to verify that𝑤cannot attain the local minimum because 𝑤_𝑡0,𝑧_∞,0=0,Δ𝑤0,𝑧_∞𝑤,0≥0,_𝑧0,𝑧_∞𝜔,0=0,0,𝑧_∞,0>0. (61) So we get a contradiction. Similarly, one can deal with the other case:inf_𝑆(𝑣^𝑠∗−𝑣)=0 and inf_𝑆(𝑢^𝑠∗−𝑢)>0.
Now we consider the last case:inf_𝑆(𝑢^𝑠∗−𝑢)=0andinf_𝑆(𝑣^𝑠∗−𝑣)=0. As the case thatinf_𝑆(𝑢^𝑠∗−𝑢)=0andinf_𝑆(𝑣^𝑠∗−𝑣)>0, we obtain a solution(𝑢_∞,𝑣_∞)of system (59) and(𝑤(0,𝑧_∞,0),𝜔(0,𝑧_∞,0))=0. It follows from the strong parabolic maximum principle that(𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡))=(0,0)for all𝑡≤0and then(𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡))≡(0,0)inℝ^𝑛×ℝby uniqueness of the solution of the Cauchy problem (31). Thus, 𝑢_∞(0,0,0)=𝑢_∞(𝑘𝑦,𝑘𝑠_∗,𝑘𝑇)for all𝑘∈ℤ. But𝑢_∞(𝑘𝑦,𝑘𝑠_∗,𝑘𝑇)→1as𝑘→∞since𝑠_∗>0. This is a contradiction.
Thus,𝑠_∗≤0, 𝑢⁰()(),𝑣𝑥,𝑧,𝑡=𝑢𝑥+𝑦,𝑧,𝑡+𝑇≥𝑢𝑥,𝑧,𝑡₀(𝑥,𝑧,𝑡)=𝑣𝑥+𝑦,𝑧,𝑡+𝑇≥𝑣(𝑥,𝑧,𝑡). (62) Since𝑇∈ℝand𝑦∈ℝ^𝑛−1are arbitrary, we conclude that𝑢and𝑣depend on𝑧only, namely,(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))=(𝜙(𝑧−𝑧₁),𝜓(𝑧−𝑧₂))for some𝑧_𝑖∈ℝ(𝑖=1,2). Note that(𝜙(𝑧),𝜓(𝑧))is a solution of (59), we conclude that𝑧₁=𝑧₂=𝑧₀. This completes the proof.

From Lemma 9, any𝜔-limit point of(𝑢,𝑣)satisfies 𝜙𝑧−𝑧^∗≤𝑤(𝑥,𝑧,𝑡)≤𝜙𝑧−𝑧_∗,𝜓𝑧−𝑧^∗≤𝜔(𝑥,𝑧,𝑡)≤𝜓𝑧−𝑧_∗,(𝑥,𝑧)∈ℝ^𝑛,𝑡∈ℝ, (63) for some constant 𝑧^∗,𝑧_∗∈ℝ. Combining the above lemma, we immediately have the following result.

Lemma 13. Let (𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡)) be a solution of (31). Then any𝜔-limit point (𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡)) of (𝑢,𝑣) is a planar wave; that is, there exists a constant 𝑧₀ℝ such that (𝑤()(=𝜙𝑥,𝑧,𝑡,𝜔𝑥,𝑧,𝑡))𝑧−𝑧₀,𝜓𝑧−𝑧₀,(𝑥,𝑧)∈ℝ^𝑛,𝑡∈ℝ. (64)

Now, we derive estimate for the derivatives of the solution of (31).

Lemma 14 (monotonicity in𝑧). Let(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))be a solution of (31). Then for any constant 𝑅>0, there exists a constant𝑇>0such that inf_{𝑥∈ℝ𝑛−1,|𝑧|≤𝑅,𝑡≥𝑇}𝑢_𝑧(𝑥,𝑧,𝑡),𝑣_𝑧(𝑥,𝑧,𝑡)>(0,0). (65)

Proof. We prove this lemma by contradiction. If the above claim does not hold, then there exists a sequence{(𝑥_𝑘,𝑧_𝑘,𝑡_𝑘)}such that{𝑧_𝑘}⊂[−𝑅,𝑅],0<𝑡₁<𝑡₂<⋯→∞ and that liminf_𝑘→∞𝑢_𝑧𝑥_𝑘,𝑧_𝑘,𝑡_𝑘,𝑣_𝑧𝑥_𝑘,𝑧_𝑘,𝑡_𝑘≤(0,0). (66) Replace{(𝑥_𝑘,𝑧_𝑘,𝑡_𝑘)}by its subsequence if necessary; we may assume without loss of generality that𝑧_𝑘converges to some limit𝑧_∞∈[−𝑅,𝑅]and that 𝑢_𝑧,𝑣_𝑧𝑥+𝑥_𝑘,𝑧+𝑧_𝑘,𝑡+𝑡_𝑘⟶(𝑤,𝜔)(𝑥,𝑧,𝑡)as𝑘⟶∞in𝐶^2,1_locℝ^𝑛×ℝ, (67) where(𝑤,𝜔)is a limit point of(𝑢,𝑣). Hence, 𝑤_𝑧0,𝑧_∞,0,𝜔_𝑧0,𝑧_∞,0=lim_𝑘→∞𝑢_𝑧𝑥_𝑘,𝑧_𝑘,𝑡_𝑘,𝑣_𝑧𝑥_𝑘,𝑧_𝑘,𝑡_𝑘≤(0,0). (68) On the other hand, Lemma 13 shows that(𝑤(𝑥,𝑧,𝑡),𝜔(𝑥,𝑧,𝑡))=(𝜙(𝑧−𝑧₀),𝜓(𝑧−𝑧₀))for some𝑧₀∈ℝ. This gets a contradiction because𝜙^′,𝜓^′>0. The proof of the lemma is completed.

By using Lemmas 14 and 9, the following corollary is obtained.

Corollary 15. Let(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))be a solution of (31). Then there exists a constant𝑇>0such that inf_{(𝑥,𝑧,𝑡)∈𝐷}𝑢_𝑧(𝑥,𝑧,𝑡),𝑣_𝑧(𝑥,𝑧,𝑡)>0, (69) where 𝐷=(𝑥,𝑧,𝑡)∈ℝ^𝑛×[),𝜙(0)𝑇,∞2,𝜓(0)2≤(𝑢()(≤(0)𝑥,𝑧,𝑡,𝑣𝑥,𝑧,𝑡))1+𝜙2,𝑏₀−𝑏₂+𝜓(0)2. (70)

Lemma 16 (decay of𝑥-derivatives). Let(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))be a solution of (31). Then for any constant𝑅>0, it holds that lim_𝑡→∞sup_{𝑥∈ℝ𝑛−1,|𝑧|≤𝑅}𝑢_𝑥𝑖(𝑥,𝑧,𝑡)=0,lim_𝑡→∞sup_{𝑥∈ℝ𝑛−1,|𝑧|≤𝑅}𝑢_{𝑥𝑖𝑥𝑗}(𝑥,𝑧,𝑡)=0,lim_𝑡→∞sup_{𝑥∈ℝ𝑛−1,|𝑧|≤𝑅}𝑣_𝑥𝑖(𝑥,𝑧,𝑡)=0,lim_𝑡→∞sup_{𝑥∈ℝ𝑛−1,|𝑧|≤𝑅}𝑣_{𝑥𝑖𝑥𝑗}(𝑥,𝑧,𝑡)=0, (71) for each1≤𝑖,𝑗≤𝑛−1.

The proof of this lemma is similar to Lemma  4.9 in [11], and we omit it here.

Next we study the(𝜙(0),𝜓(0))-level surface of the solution of (31). From Corollary 15 and Lemma 16, we can derive the following lemma that the(𝜙(0),𝜓(0))-level surface of the solution(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))has a graphical representation𝑧=Γ(𝑥,𝑡)for all𝑡.

Lemma 17. Let(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))be a solution of (31) and let𝑇>0be as defined in Corollary 15. Then there exists a smooth bounded functionΓ(𝑥,𝑡)such that (𝑢()(=𝜙(0)(0),()𝑥,𝑧,𝑡,𝑣𝑥,𝑧,𝑡)),𝜓𝑖𝑓𝑎𝑛𝑑𝑜𝑛𝑙𝑦𝑖𝑓𝑧=Γ𝑥,𝑡 (72) for any(𝑥,𝑡)∈ℝ^𝑛−1×[𝑇,∞). Furthermore, the following estimates hold:(i)for each1≤𝑖,𝑗≤𝑛−1, lim_𝑡→∞sup_{𝑥∈ℝ𝑛−1}Γ_𝑥𝑖(𝑥,𝑧,𝑡)=0,lim_𝑡→∞sup_{𝑥∈ℝ𝑛−1}Γ_{𝑥𝑖𝑥𝑗}(𝑥,𝑧,𝑡)=0; (73)(ii)there exists a constant𝑀>0such that, for each1≤𝑖,𝑗,𝑘≤𝑛−1, sup_{𝑥∈ℝ𝑛−1}Γ_{𝑥𝑖𝑥𝑗𝑥𝑘}(𝑥,𝑧,𝑡)≤𝑀𝑓𝑜𝑟𝑡≥𝑇. (74)

Proof. Since 𝐷=(𝑥,𝑧,𝑡)∈ℝ^𝑛×[),𝜙(0)𝑇,∞2,𝜓(0)2≤(𝑢()(≤(0)𝑥,𝑧,𝑡,𝑣𝑥,𝑧,𝑡))1+𝜙2,𝑏₀−𝑏₂+𝜓(0)2 (75) is bounded in the𝑧-direction by virtue of Lemma 9 and the facts(𝜙,𝜓)(−∞)=(0,0)and(𝜙,𝜓)(+∞)=(1,𝑏₀−𝑏₂), we can define a bounded functionΓ(𝑥,𝑡)satisfying (72) thanks to Corollary 15. HereΓ(𝑥,𝑡)is smooth by the implicit function theorem, since𝑢(𝑥,𝑧,𝑡)is smooth for𝑡>0. The other estimates follow from Lemma 16, and we omit it here. This completes the proof.

The following lemma shows that the large time behavior of the solution can be essentially determined by the(𝜙(0),𝜓(0))-level surfaceΓ(𝑥,𝑡).

Lemma 18. Let(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))be a solution of (31) and letΓ(𝑥,𝑡)be as defined in Lemma 17. Then, it holds that lim_𝑡→∞sup_{(𝑥,𝑧)∈ℝ𝑛}𝑢(𝑥,𝑧,𝑡)−𝜙(𝑧−Γ(𝑥,𝑡))=0,lim_𝑡→∞sup_{(𝑥,𝑧)∈ℝ𝑛}𝑣(𝑥,𝑧,𝑡)−𝜓(𝑧−Γ(𝑥,𝑡))=0. (76)

Proof. We prove this lemma by contradiction, and we only consider𝑢. If the above claim does not hold, there exist a constant𝛿>0and a sequence{(𝑥_𝑘,𝑧_𝑘,𝑡_𝑘)}such that 0<𝑡₁<𝑡₂<⋯→∞ and that 𝑢𝑥_𝑘,𝑧_𝑘,𝑡_𝑘−𝜙𝑧_𝑘−Γ𝑥_𝑘,𝑡_𝑘≥𝛿. (77)
On the other hand, by virtue of Lemma 9 and boundedness ofΓ(𝑥,𝑡), we can choose constants 𝑅>0 and 𝑇>0 such that sup_{𝑥∈ℝ𝑛−1,|𝑧|≥𝑅,𝑡≥𝑇}𝑢(𝑥,𝑧,𝑡)−𝜙(𝑧−Γ(𝑥,𝑡))<𝛿, (78) which means that{𝑧_𝑘}is bounded. We can choose subsequence of{(𝑥_𝑘,𝑧_𝑘,𝑡_𝑘)}, which we denote again by{(𝑥_𝑘,𝑧_𝑘,𝑡_𝑘)}such that 𝑧_∞∶=lim_𝑘→∞𝑧_𝑘,𝛾_∞∶=lim_𝑘→∞Γ𝑥_𝑘,𝑡_𝑘𝑢𝑥+𝑥_𝑘,𝑧,𝑡+𝑡_𝑘⟶𝑤(𝑥,𝑧,𝑡)as𝑘⟶∞in𝐶^2,1_locℝ^𝑛×ℝ, (79) where𝑤is some𝜔-limit point of𝑢. This and (77) show that 𝑤0,𝛾_∞,0−𝜙𝑧_∞−𝛾_∞=lim_𝑘→∞𝑢𝑥_𝑘,𝑧_𝑘,𝑡_𝑘−𝜙𝑧_𝑘−𝛾𝑥_𝑘,𝑡_𝑘≥𝛿. (80) On the other hand, we have 𝑤0,𝛾_∞,0=lim_𝑘→∞𝑢𝑥_𝑘,Γ𝑥_𝑘,𝑡_𝑘,𝑡_𝑘=𝜙(0). (81) Lemma 13 implies that𝑤(𝑥,𝑧,𝑡)≡𝜙(𝑧−𝛾_∞). This contradicts (80). This completes the proof of this lemma.

By setting𝑦=𝑧−𝑐𝑡and𝛾(𝑥,𝑡)=Γ(𝑥,𝑡)−𝑐𝑡, we obtain the statement (i), (ii) of Theorem 1 from Lemmas 17 and 18. Thus, it remains to prove the statement (iii). This will be done at the end of this section.

In the following, we construct supersolutions of (31). For this purpose, we consider the problem of the form 𝑉_𝑡=Δ_𝑥𝑐𝑉−2∇_𝑥𝑉²,𝑥∈ℝ^𝑛−1𝑣(),𝑡>0,𝑥,0=𝑣₀(𝑥),𝑥∈ℝ^𝑛−1. (82)

Lemma 19 (supersolution). For any constant 𝑀>0, there exist positive constant 𝛿, 𝜀₀, 𝑘₀ and smooth functions𝑝(𝑡)and𝑞(𝑡)satisfying 𝑝(0)𝑞(0)(𝑡)(𝑡)>0,=0,0≤𝑝,𝑞≤𝜀𝑓𝑜𝑟𝑡≥0, (83) where𝜀∈(0,𝜀₀], such that if𝑉(𝑥,𝑡)is any solution of (82) with‖𝑉₀‖_𝑊3,∞≤𝑀and‖∇_𝑥𝑉₀‖_𝑊1,∞≤𝛿, then the function defined by 𝑢⁺()()𝑥,𝑧,𝑡=𝜙𝑧−𝑉𝑥,𝑡1+∇_𝑥𝑉²(𝑡)(𝑡),𝑣+𝑞+𝑝₊()()𝑥,𝑧,𝑡=𝜓𝑧−𝑉𝑥,𝑡1+∇_𝑥𝑉²+𝑞(𝑡)+𝑘₀𝑝(𝑡) (84) satisfies 𝐿𝑢⁺∶=𝑢⁺_𝑡−Δ𝑢⁺+𝑐𝑢⁺_𝑧−𝑓𝑢⁺−𝑘₁𝑣⁺+𝑘₂𝑢⁺𝑏₀−𝑣⁺≥0,(𝑥,𝑧)∈ℝ^𝑛𝐿𝑣,𝑡>0,₊∶=𝑣⁺_𝑡−Δ𝑣⁺+𝑐𝑣⁺_𝑧+𝑘₁𝑣⁺−𝑘₂𝑢⁺𝑏₀−𝑣⁺≥0,(𝑥,𝑧)∈ℝ^𝑛,𝑡>0. (85)

Proof. We divide the proof into four steps. The first two steps are similar to that of Lemma  4.12 in [11], and we only give the last results.
Step 1. Set 𝜂()=()𝑥,𝑧,𝑡𝑧−𝑉𝑥,𝑡1+∇_𝑥𝑉². (86) Then, by using (8), we have 𝐿𝑢⁺=𝜂_𝑡−Δ_𝑥𝜂+𝑐𝜂_𝑧−𝑐𝜙^′+1−∇_𝑥𝜂²−𝜂²_𝑧𝜙^′′+𝑞^′(𝑡)𝜙^′+𝑝^′(𝑡)(𝑡)𝜙−𝑘−𝑓𝜙+𝑝+𝑓₁+𝑘₂𝜙𝑘₀𝑝(𝑡)+𝑘₂𝑝(𝑡)𝑏₀−𝜓−𝑘₀𝑝(𝑡)𝐿𝑣⁺=𝜂_𝑡−Δ_𝑥𝜂+𝑐𝜂_𝑧−𝑐𝜓^′+1−∇_𝑥𝜂²−𝜂²_𝑧𝜓^′′+𝑞^′(𝑡)𝜓^′+𝑘₀𝑝^′(𝑡)+𝑘₁+𝑘₂𝜙𝑘₀𝑝(𝑡)−𝑘₂𝑝(𝑡)𝑏₀−𝜓−𝑘₀𝑝(𝑡). (87) By rewriting the above expression in terms of𝑉, we obtain 𝐿𝑢⁺=𝐼₀−𝐼₂𝜙^′+𝐼₁−3𝐼₃𝜂𝜙^′−2𝐼₂𝜂𝜙^′′−𝐼₃𝜂²𝜙^′′+𝐽₁,𝐿𝑣⁺=𝐼₀−𝐼₂𝜓^′+𝐼₁−3𝐼₃𝜂𝜓^′−2𝐼₂𝜂𝜓^′′−𝐼₃𝜂²𝜓^′′+𝐽₂, (88) where𝐼_𝑖, 𝐽_𝑗 (𝑖=0,1,2,3; 𝑗=1,2) are functions given by 𝐼₀𝑉=−^𝑡−𝑐1+∇_𝑥𝑉²∇+div^𝑥1+∇_𝑥𝑉²𝐼−𝑐,₁=−^𝑛−1_𝑖=1𝑉_𝑥𝑖𝑉_𝑥𝑖𝑡1+∇_𝑥𝑉²+^𝑛−1_𝑖,𝑗=1𝑉²_{𝑥𝑖𝑥𝑗}+𝑉_𝑥𝑗𝑉_{𝑥𝑖𝑥𝑖𝑥𝑗}1+∇_𝑥𝑉²,𝐼₂=^𝑛−1_𝑖,𝑗=1𝑉_𝑥𝑖𝑉_𝑥𝑗+𝑉_𝑥𝑗𝑉_{𝑥𝑖𝑥𝑗}1+∇_𝑥𝑉^23/2,𝐼₃=^𝑛−1_𝑖=1∑^𝑛−1_𝑖=1𝑉_𝑥𝑖𝑉_𝑥𝑖𝑗1+∇_𝑥𝑉²²,𝐽₁=𝜙^′𝑞^′(𝑡)𝑝(𝑡)+𝑝^′(𝑡)𝑝(𝑡)−¹₀𝑓^′𝜙+𝜏𝑝(𝑡)𝑑𝜏−𝑘₁+𝑘₂𝜙𝑘₀+𝑘₂𝑏₀−𝜓−𝑘₀𝑝(𝑡)(𝑡),𝐽⋅𝑝₂=𝜓^′𝑞^′(𝑡)𝑘₀𝑝(𝑡)+𝑝^′(𝑡)𝑝(𝑡)+𝑘₁+𝑘₂𝜙+𝑘₂𝑏₀−𝜓−𝑘₀𝑝(𝑡1)𝑘₀⋅𝑘₀𝑝(𝑡). (89)
Step 2. Now we estimate𝐼_𝑖 (𝑖=1,2,3). Define 𝐼_𝑢(𝑥,𝑧,𝑡) and 𝐼_𝑣(𝑥,𝑧,𝑡) by 𝐼_𝑢(𝑥,𝑧,𝑡)=𝐼₀−𝐼₂𝜙^′+𝐼₁−3𝐼₃𝜂𝜙^′−2𝐼₂𝜂𝜙^′′−𝐼₃𝜂²𝜙^′′,𝐼_𝑣(𝑥,𝑧,𝑡)=𝐼₀−𝐼₂𝜓^′+𝐼₁−3𝐼₃𝜂𝜓^′−2𝐼₂𝜂𝜓^′′−𝐼₃𝜂²𝜓^′′. (90) It follows from Theorem  2.1 in [5] that 𝜙^′(𝑧), 𝜙^′′(𝑧), 𝜓^′′(𝑧) and 𝜓^′′(𝑧) decay to zero exponentially as |𝑧|→∞. Noting that 𝑞(𝑡) is assumed to be bounded, the following functions are all bounded: 𝜂𝜙^′𝜂+𝑞(𝑡),𝜂𝜙^′′(𝑡),𝜂𝜂+𝑞₂𝜙^′′𝜂+𝑞(𝑡),𝜂𝜓^′𝜂+𝑞(𝑡),𝜂𝜓^′′(𝑡),𝜂𝜂+𝑞₂𝜓^′′𝜂+𝑞(𝑡). (91) Now we choose a constant𝐶₁>0arbitrarily. Then from the above boundedness and Lemma 8, we can choose a constant𝐶₂≥1depending only on𝑐and𝑀, and a constant𝛿>0depending only on𝐶₁such that if‖∇_𝑥𝑉₀‖_𝑊1,∞≤𝛿, it holds that 𝐼_𝑢((𝑡),𝐼𝑥,𝑧,𝑡)≤𝑃_𝑣(𝑥,𝑧,𝑡)≤𝑃(𝑡)where𝑃(𝑡)=min𝐶₂𝑡⁻²,𝐶₁. (92)
Step 3. Now we determine the constant𝐶₁,𝜀₀and the smooth functions𝑝_𝑖(𝑡)and 𝑞(𝑡) (𝑖=1,2). Set 𝐹(𝑢,𝑣)=𝑓(𝑢)+𝑘₁𝑣−𝑘₂𝑢𝑏₀,𝐺()−𝑣𝑢,𝑣=−𝑘₁𝑣+𝑘₂𝑢𝑏₀−𝑣. (93) Then, direct calculation shows that 𝐹_𝑢=𝑓^′(𝑢)−𝑘₂𝑏₀,𝐹−𝑣_𝑣=𝑘₁+𝑘₂𝐺𝑢,_𝑢=𝑘₂𝑏₀,𝐺−𝑣_𝑣=−𝑘₁+𝑘₂𝑢,𝐹_𝑢(0,0)𝐺_𝑣(0,0)−𝐹_𝑣(0,0)𝐺_𝑢(0,0)=−𝑘₁𝑓^′(0)𝐹>0,_𝑢1,𝑏₀−𝑏₂𝐺_𝑣1,𝑏₀−𝑏₂−𝐹_𝑣1,𝑏₀−𝑏₂𝐺_𝑢1,𝑏₀−𝑏₂=−𝑘₁+𝑘₂𝑓^′(1)>0. (94) Hence, there exist constants𝜀₀>0,𝛼₁₁,𝛼₂₂<0and𝛼₁₂,𝛼₂₁>0such that 𝛼₁₁<𝐹_𝑢𝛼<0,₂₂<𝐺_𝑣<0,0<𝐹_𝑣<𝛼₁₂,0<𝐺_𝑣<𝛼₂₁,on𝐴^𝑖_𝜀,𝑖=1,2, (95) where𝜀∈(0,𝜀₀]and 𝐴⁰_𝜀=)∣|𝑢||𝑣|},𝐴{(𝑢,𝑣≤𝜀,≤𝜀_1𝜀=(𝑢,𝑣)∣|𝑢−1|≤𝜀,𝑣−𝑏₀+𝑏₂≤𝜀. (96) It follows that there exist positive constants𝐾₀and𝑘₀such that for all𝐾∈(0,𝐾₀), we have 2𝐾+𝛼₁₁<0,2𝐾+𝛼₂₂<0,2𝐾+𝛼₁₁2𝐾+𝛼₂₂>𝛼₁₂𝛼₂₁,−𝛼₂₁2𝐾+𝛼₂₂<𝑘₀<−2𝐾+𝛼₁₁𝛼₁₂. (97) This implies that −2𝐾−𝛼₁₁−𝑘₀𝛼₁₂>0,−2𝐾−𝛼₂₂−1𝑘₀𝛼₂₁>0. (98) Denote𝜉=(𝑧−𝑉(𝑥,𝑡))/1+|∇_𝑥𝑉|²+𝑞(𝑡). Using the above estimates and the facts thatlim_{𝑧→−∞}𝜙(𝑧)=0, lim_{𝑧→−∞}𝜓(𝑧)=0, lim_𝑧→∞𝜙(𝑧)=1andlim_𝑧→∞𝜓(𝑧)=𝑏₀−𝑏₂, there exist constants𝑀₁,𝑀₂>0such that −2𝐾−𝑓^′𝜙+𝜃𝛿₁𝑒^−𝛽𝑡−𝑘₀𝑘₁+𝑘₂𝜙+𝑘₂𝑏₀−𝜓>0for𝜉≤−𝑀₂or𝜉≥𝑀₁,−2𝐾+𝑘₁+𝑘₂𝜙−𝑘₂1𝑘₀𝑏₀−𝜓>0for𝜉≤−𝑀₂or𝜉≥𝑀₁. (99)
Set 𝛿₀=min−2𝐾−𝛼₁₁−𝑘₀𝛼₁₂2𝑘₂𝑘₀,−2𝐾−𝛼₂₂−𝛼₂₁/𝑘₀2𝑘₂𝑘₀1,1,𝑘₀. (100) We define the constant𝐶₁>0and𝐶₀≥1by 𝐶₁=𝐾²𝜀²𝑘²₀161+𝑘₀²𝐶₂𝐶²₀,𝐶₀𝑓=max1,2𝐾+^′_{𝐿∞([0,1])}+𝑘₁+𝑘₂𝑘₀+𝛿₀𝑘₂inf_{−𝑀2≤𝜉≤𝑀1}𝜙^′(𝜉),2𝐾𝑘₀−𝑘₁𝑘₀+𝑘₂𝑏₀inf_{−𝑀2≤𝜉≤𝑀1}𝜓^′(𝜉). (101) We choose functions𝑝(𝑡),𝑞(𝑡)∈𝐶^∞[0,∞)satisfying 𝑃(𝑡)≤min1,𝑘₀(𝑡),(𝑡)1𝐾𝑝𝐾𝑝≤21+𝑘₀𝑃(𝑡),𝐾𝑝^′(𝑡)≤2𝑃^′(𝑡,𝑞(𝑡)=_𝑡0)𝑝(𝑠)𝑑𝑠. (102) Then (107) holds, since we have 𝑝(0)≥𝐾𝜀²𝑘₀161+𝑘₀²𝐶₂𝐶²₀(𝑡)≤>0,0<𝑝𝐾𝜀²𝑘₀8𝐶₂𝐶²₀1+𝑘₀≤𝜀,0≤^∞₀𝑝(𝑠)𝑑𝑠≤𝜀. (103)
Step 4. Now we complete the proof. Since|𝐼_𝑢(𝑥,𝑧,𝑡)|≤𝑃(𝑡)and|𝐼_𝑣(𝑥,𝑧,𝑡)|≤𝑃(𝑡)by Step 2, it suffices to show the inequality𝐽_𝑖(𝑥,𝑧,𝑡)≥𝑃(𝑡) (𝑖=1,2). Letting 𝜉=(𝑧−𝑉(𝑥,𝑡))/1+|∇_𝑥𝑉|²+𝑞(𝑡), when𝜉∈(−∞,−𝑀₂]∪[𝑀₁,∞), we have 𝐽₁≥𝑝^′(𝑡)𝑝(𝑡)−¹₀𝑓^′(𝑡)−𝑘𝜙+𝜏𝑝𝑑𝜏₁+𝑘₂𝑘₀+𝑘₂𝑏₀(𝑡)(𝑡)(𝑡)(𝑡),𝐽−𝜓−𝑝⋅𝑝≥𝐾𝑝≥𝑃₂≥𝑝^′(𝑡)𝑝(𝑡)+𝑘₁+𝑘₂𝜙+𝑘₂𝑏₀(𝑡)1−𝜓−𝑝𝑘₀⋅𝑘₀𝑝(𝑡)≥𝑘₀𝐾𝑝(𝑡)≥𝑃(𝑡), (104) since we have sup_𝑡≥0𝑝^′(𝑡)𝑝(𝑡)≤sup_𝑡≥02𝑃^′(𝑡)𝐾𝑝(𝑡)≤sup_𝑡≥02𝑃^′(𝑡)𝑃(𝑡)=𝐾𝜀𝐶₀<𝐾. (105) For any−𝑀₂≤𝜉≤𝑀₁, we have 𝐿𝑢⁺≥𝐶₀inf_{−𝑀2≤𝜉≤𝑀1}𝜙^′(𝜉)−2𝐾−𝑓^′_{𝐿∞([0,1])}−𝑘₁+𝑘₂𝑘₀−𝛿₀𝑘₂𝑝(𝑡)𝐿𝑣≥0,₊≥𝐶₀𝑘₀inf_{−𝑀2≤𝜉≤𝑀1}𝜓^′(𝜉)−2𝐾+𝑘₁−𝑘₂𝑏₀𝑘₀≥0. (106) In summary, we have𝐿[𝑢⁺]≥0and𝐿[𝑢⁺]≥0. This completes the proof.

Lemma 20 (subsolution). For any constants𝑀>0, there exist positive constant 𝛿, 𝜀₀, 𝑘₀ and smooth functions 𝑝(𝑡) and 𝑞(𝑡) satisfying 𝑝(0)𝑞(0)(𝑡)(𝑡)>0,=0,0≤𝑝,𝑞≤𝜀𝑓𝑜𝑟𝑡≥0, (107) where𝜀∈(0,𝜀₀], such that if𝑉(𝑥,𝑡)is any solution of (82) with‖𝑉₀‖_𝑊3,∞≤𝑀and‖∇_𝑥𝑉₀‖_𝑊1,∞≤𝛿, then the function defined by 𝑢⁻()()𝑥,𝑧,𝑡=𝜙𝑧−𝑉𝑥,𝑡1+∇_𝑥𝑉²(𝑡)(𝑡),𝑣−𝑞−𝑝₋()()𝑥,𝑧,𝑡=𝜓𝑧−𝑉𝑥,𝑡1+∇_𝑥𝑉²−𝑞(𝑡)−𝑘₀𝑝(𝑡) (108) satisfies 𝐿[𝑢⁻]∶=𝑢⁻_𝑡−Δ𝑢⁻+𝑐𝑢⁻_𝑧−𝑓(𝑢⁻)−𝑘₁𝑣⁻+𝑘₂𝑢⁻𝑏₀−𝑣⁻≤0,(𝑥,𝑧)∈ℝ^𝑛𝐿[𝑣,𝑡>0,₋]∶=𝑣⁻_𝑡−Δ𝑣⁻+𝑐𝑣⁻_𝑧+𝑘₁𝑣⁻−𝑘₂𝑢⁻𝑏₀−𝑣⁻≤0,(𝑥,𝑧)∈ℝ^𝑛,𝑡>0. (109)

The proof of Lemma 20 is similar to that of Lemma 19, and we omit it here.

Lemma 21 (approximation ofΓ(𝑥,𝑡)). Let(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))be a solution of (31) and letΓ(𝑥,𝑡)be as defined in Lemma 17. Then for any𝜀>0, there exists a constant𝜏_𝜀>0such that the function𝑉(𝑥,𝑡)defined by 𝑉_𝑡=Δ_𝑥𝑐𝑉−2∇_𝑥𝑣₁²,𝑥∈ℝ^𝑛−1𝑣,𝑡>0,₁(𝑥,0)=Γ𝑥,𝜏_𝜀,𝑥∈ℝ^𝑛−1 (110) satisfies sup_{𝑥∈ℝ𝑛−1}Γ(𝑥,𝑡)≥𝑉𝑥,𝑡−𝜏_𝜀≤𝜀,𝑡≥𝜏_𝜀. (111)

Proof. From Corollary 15, we can choose constants𝑇>0,𝑀>0, and𝐿>0such that sup_𝑡≥𝑇‖Γ(⋅,𝑡)‖_𝑊^3,∞≤𝑀,inf_{()𝑥,𝑧,𝑡∈𝐷}𝑢_𝑧(𝑥,𝑧,𝑡)≥𝐿,inf_{(𝑥,𝑧,𝑡)∈𝐷}𝑣_𝑧())𝑥,𝑧,𝑡≥𝐿,𝐷=(𝑥,𝑧,𝑡∈ℝ^𝑛×[),𝜙(0)𝑇,∞2,𝜓(0)2≤(𝑢()(≤(0)𝑥,𝑧,𝑡,𝑣𝑥,𝑧,𝑡))1+𝜙2,𝑏₀−𝑏₂+𝜓(0)2. (112) For the constant𝑀and 𝜀̂∶=min{𝜀₀,min{𝐿𝜀,(1+𝜙(0))/[2(1+‖𝜙^′‖_𝐿∞)]}}, we choose a constant𝛿>0and functions𝑝(𝑡),𝑞(𝑡)satisfying 𝑝(0)𝑞(0)(𝑡)(𝑡)≤̂>0,=0,0≤𝑝,𝑞𝜀for𝑡≥0, (113) as in Lemma 19. Then it follows from Lemma 17 that there exists a constant 𝜏_𝜀∈[𝑇,∞) such that ‖Γ(⋅,𝜏_𝜀)‖_𝑊1,∞≤𝛿. Moreover, using Lemma 18, by choosing 𝜏_𝜀 larger if necessary, we have 𝑢𝑥,𝑧,𝜏_𝜀≤𝜙𝑧−Γ𝑥,𝜏_𝜀+𝑝(0)2≤𝜙𝑧−Γ𝑥,𝜏_𝜀1+∇_𝑥Γ𝑥,𝜏_𝜀²(0),𝑣+𝑝𝑥,𝑧,𝜏_𝜀≤𝜓𝑧−Γ𝑥,𝜏_𝜀+𝑘₀𝑝(0)2≤𝜓𝑧−Γ𝑥,𝜏_𝜀1+∇_𝑥Γ𝑥,𝜏_𝜀²+𝑘₀𝑝(0), (114) where we used the smallness of|∇_𝑥Γ|. For such𝜏_𝜀, we define𝑉(𝑥,𝑡)as a function satisfying (110). Then Lemma 19 shows that the function defined by 𝑢⁺(𝑥,𝑧,𝑡)=𝜙𝑧−𝑉𝑥,𝑡−𝜏_𝜀1+∇_𝑥𝑉²+𝑞𝑡−𝜏_𝜀+𝑝𝑡−𝜏_𝜀,𝑣⁺(𝑥,𝑧,𝑡)=𝜓𝑧−𝑉𝑥,𝑡−𝜏_𝜀1+∇_𝑥𝑉²+𝑞𝑡−𝜏_𝜀+𝑘₀𝑝𝑡−𝜏_𝜀 (115) is a supersolution of (31) for𝑡≥𝜏_𝜀. Since (114) implies that 𝑢(𝑥,𝑧,𝜏_𝜀)≤𝑢⁺(𝑥,𝑧,𝜏_𝜀) and 𝑣(𝑥,𝑧,𝜏_𝜀)≤𝑣⁺(𝑥,𝑧,𝜏_𝜀), the comparison principle gives that 𝑢(𝑥,𝑧,𝑡)≤𝑢⁺(𝑥,𝑧,𝑡) and𝑣(𝑥,𝑧,𝑡)≤𝑣⁺(𝑥,𝑧,𝑡) for 𝑡≥𝜏_𝜀. Then, due to 𝑝(𝑡),𝑞(𝑡)≤𝜀̂, we have 𝑢𝑥,𝑉𝑥,𝑡−𝜏_𝜀,𝑡−𝜙(0)≤𝑢⁺𝑥,𝑉𝑥,𝑡−𝜏_𝜀(0)𝑞,𝑡−𝜙=𝜙𝑡−𝜏_𝜀(𝑡)(0)≤𝜙+𝑝−𝜙1+^′_𝐿∞𝜀̂(0)≤min𝐿𝜀,1+𝜙2. (116) Noting that𝑢(𝑥,Γ(𝑥,𝑡),𝑡)=𝜙(0), we get 𝐿𝜀≥𝑢𝑥,𝑉𝑥,𝑡−𝜏_𝜀,𝑡−𝑢𝑥,Γ𝑥,𝑡−𝜏_𝜀,𝑡≥inf_{𝜙(0),(0𝑢∈1+𝜙)/2,𝑡≥𝜏𝜀}𝑢_𝑧×(𝑉()((𝑉()(,𝑥,𝑡−Γ𝑥,𝑡))≥𝐿𝑥,𝑡−Γ𝑥,𝑡)) (117) which implies that𝑉(𝑥,𝑡−𝑇)−Γ(𝑥,𝑡)≤𝜀for𝑡>𝜏_𝜀. Similarly, by using the subsolution(𝑢⁻,𝑣⁻)given in Lemma 20, we can show that𝑉(𝑥,𝑡−𝑇)−Γ(𝑥,𝑡)≥−𝜀for𝑡>𝜏_𝜀. This completes the proof.

Now we are ready to complete the proof of Theorem 1.

Proof of Theorem 1. The statements (i) and (ii) of Theorem 1 are directly from Lemmas 17 and 18, respectively. Thus, we only prove the statement (iii).
By Lemma 21, the large time behavior of the(𝜙(0),𝜓(0))-level surfaceΓ(𝑥,𝑡)of the solution𝑢(𝑥,𝑧,𝑡)of (31) can be approximated by function𝑉(𝑥,𝑡)of the equation 𝑉_𝑡=Δ_𝑥𝑐𝑉−2∇_𝑥𝑉²,𝑥∈ℝ^𝑛−1,𝑡>0. (118) This means that the(𝜙(0),𝜓(0))-level surface𝛾(𝑥,𝑡)=Γ(𝑥,𝑡)−𝑐𝑡of the solution(𝑢(𝑥,𝑦,𝑡),𝑣(𝑥,𝑦,𝑡))of (7) can be approximated by function𝑉(𝑥,𝑡)of the equation 𝑉_𝑡=Δ_𝑥𝑐𝑉−2∇_𝑥𝑉²−𝑐,𝑥∈ℝ^𝑛−1,𝑡>0. (119) Hence, the statement (iii) of Theorem 1 follows from Lemma 21. This completes the proof of Theorem 1.

Next, we prove Theorem 3. Firstly, similar to Lemma  4.15 in [11], one can prove that the(𝜙(0),𝜓(0))-level surfaceΓ(𝑥,𝑡)remains uniquely ergodic for all large𝑡≥0.

Lemma 22 (ergodicity of(𝜙(0),𝜓(0))-level surface). Let(𝑢(𝑥,𝑧,𝑡),𝑣(𝑥,𝑧,𝑡))be a solution of (31) and assume that(𝑢₀(𝑥,𝑧),𝑣₀(𝑥,𝑧))is uniquely ergodic in the𝑥-direction. Then the(𝜙(0),𝜓(0))-level surfaceΓ(𝑥,𝑡)defined in Lemma 17 is uniquely ergodic for each𝑡≥𝑇, where𝑇>0is the constant in Lemma 17.

Proof of Theorem 3. Let𝑇>0andΓ(𝑥,𝑡)be as in Lemma 17, thenΓ(𝑥,𝑡)is uniquely ergodic for each𝑡≥𝑇from Lemma 22. Combining Lemmas 6 and 7, one can easily prove Theorem 3.

Acknowledgments

The first author was supported in part by PRC Grants NSFC 11226168 and 11301146, and the second author was supported in part by PRC Grants NSFC 11171064, 11301146, and 11226168 and the Natural Science Foundation of Jiangsu province BK2011583. The authors are grateful to the referees for their valuable suggestions and comments on the original paper.