# Planar Waves of the Buffered Bistable System

Authors: Xiaohuan Wang; Guangying Lv
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

This paper is concerned with the large time behavior of disturbed planar fronts in the buffered bistable system inℝn
(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∞(ℝn) under ergodic perturbations, which include quasiperiodic and almost periodic ones as special cases.


---

## Body

## 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:
(1)∂u∂t=D∂2u∂y2+f(u)+∑i=1N‍[k-i(b0i-vi)-k+iuvi],ddddddddddddddddddddd(y,t)∈ℝ×(0,∞),∂vi∂t=Di∂2vi∂y2+k-i(b0i-vi)-k+iuvi,dddddddddddddddddddi=1,…,N,
where D, Di, k-i, k+i and b0i are positive constants. We call system (1) the full buffering model. WhenDi=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) withDi≠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 tobuffering model inℝn:
(2)ut=DΔu+f(u)+∑i=1N‍[k-i(b0i-vi)-k+iuvi],ddddddddddddddddx∈ℝn-1,y∈ℝ,t>0,vit=DiΔvi+k-i(b0i-vi)-k+iuvi,dddddddddddddddddi=1,…,N,
with initial value
(3)u(x,y,0)=u0(x,y),x∈ℝn-1,y∈ℝ,vi(x,y,0)=vi0(x,y),x∈ℝn-1,y∈ℝ,
wheren≥2,Δ=∂2/∂x12+⋯+∂2/∂xn-12+∂2/∂y2. Throughout this paper, we always assume thatD=Di=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),vi(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 onℝn, 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),vi(y,t))=(ϕ(y+ct),ψi(y+ct)) for some constantc∈ℝ, connecting two equilibrium (0,b0i) and (1,k-ib0i/(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),vi(y,t))=(ϕ(y+ct),ψi(y+ct)) (c is the wave speed), satisfying
(4)Dϕ′′(ξ)-cϕ′(ξ)+f(ϕ(ξ))+∑i=1n‍[k-i(b0i-ψi)-k+iϕψi]=0,ξ∈ℝ,Diψi′′(ξ)-cψi′(ξ)+k-i(b0i-ψi)-k+iϕψi=0,dddddddddddddddddddddddddddddξ∈ℝ,ϕ′(ξ)>0,ψi′(ξ)<0,limξ→-∞(ϕ(ξ),ψi(ξ))=(0,b0i),limξ→∞(ϕ(ξ),ψi(ξ))=(1,b2i),
whereξ=y+ct,′=d/dξ and b2i=k-ib0i/(k-i+k+i).The function(ϕ(y+ct),ψi(y+ct))is also a traveling wave front for system (2) withn≥2. We call it planar front. Its translation(ϕ(y+ct+ξ),ψi(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 writeviasv, that is,
(5)ut=Δu+f(u)+k1(b0-v)-k2uv,dddddddddx∈ℝn-1,y∈ℝ,t>0,vt=Δv+k1(b0-v)-k2uv,ddddx∈ℝn-1,y∈ℝ,t>0,
with initial value
(6)u(x,y,0)=u0(x,y),x∈ℝn-1,y∈ℝ,v(x,y,0)=v0(x,y),x∈ℝn-1,y∈ℝ.
Letu*=u and v*=b0-v, then (5) becomes (for simplicity, we omit the symbol*)
(7)ut=Δu+f(u)+k1v-k2u(b0-v),dddddddddx∈ℝn-1,y∈ℝ,t>0,vt=Δv-k1v+k2u(b0-v),x∈ℝn-1,y∈ℝ,t>0.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
(8)ϕ′′-cϕ′+f(ϕ)+k1ψ-k2ϕ(b0-ψ)=0,ψ′′-cψ′-k1ψ+k2ϕ(b0-ψ)=0,ϕ′>0,ψ′>0,limξ→-∞(ϕ,ψ)=(0,0),limξ→∞(ϕ,ψ)=(1,b0-b2).Recently, Matano et al. [6] considered the following Allen-Cahn equation:
(9)ut=Δu+f(u),x∈ℝn-1,y∈ℝ,t>0,u(x,y,0)=u0(x,y),x∈ℝn-1,y∈ℝ.
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∞(ℝn)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∞(ℝn)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
(10)(0,0)≤(u0(x,y),v0(x,y))≤(1,b0-b2),limsupy→-∞supx∈ℝn-1u0(x,y)<δ,liminfy→+∞infx∈ℝn-1u0(x,y)>1-δ,limsupy→-∞supx∈ℝn-1v0(x,y)<k0δ,liminfy→+∞infx∈ℝn-1v0(x,y)>b0-b2-k0δ,
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∈ℝn-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(11)limt→∞sup(x,y)∈ℝn|u(x,y,t)-ϕ(y-γ(x,t))|=0,limt→∞sup(x,y)∈ℝn|v(x,y,t)-ψ(y-γ(x,t))|=0;(iii)
for anyε>0, there existsτε∈[T,∞)such that the solutionU(x,t)of the problem
(12)Ut1+|∇xU|2=div(∇xU1+|∇xU|2)-c,dddddddddddddddddddx∈ℝn-1,t>0,U(x,0)=γ(x,τε),x∈ℝn-1satisfies
(13)supx∈ℝn-1,t≥τε|γ(x,t)-U(x,t-τε)|≤ε,
where∇xdenotes the(n-1)-dimensional gradient.The statement (ii) of Theorem1 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 onℝn-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 thex-direction).
A bounded uniformly continuous functionp(x,y): ℝn-1×ℝ→ℝ is called uniquely ergodic in thex-direction if there exists a unique probability measure on ℍp, that is, σa-invariant for anya∈ℝn-1,
(14)ℍp∶={σag∣a∈ℝn-1}¯Lloc∞(ℝn-1),(σap)(x)=p(x+a),
andA¯Xstands for the closure of a setAin theX-topology.Theorem 3.
In addition to the assumptions of Theorem1, assume further thatu0(x,y) and v0(x,y)are uniquely ergodic in thex-direction. Then there exist constantsμ1,μ2∈ℝsuch that
(15)limt→∞sup(x,y)∈ℝn|u(x,y,t)-ϕ(y+ct+μ1)|=0,limt→∞sup(x,y)∈ℝn|v(x,y,t)-ψ(y+ct+μ2)|=0.From [11], we see that
(16)ℙ⊂ℚℙ⊂𝔸ℙ⊂𝕊𝔼⊂𝕌𝔼,
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 Section2, 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,
(17)ut=uzz-cuz+f(u)+k1v-k2u(b0-v),ddddddddddddddddddddz∈ℝ,t>0,vt=vzz-cvz-k1v+k2u(b0-v),ddddddddddddddddz∈ℝ,t>0,
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
(18)u+(x,z,t)=ϕ(z+σδ(1-e-βt))+δe-βt,v+(x,z,t)=ψ(z+σδ(1-e-βt))+k0δe-βt
is a subsolution. More precisely, it satisfies the following inequality:
(19)N[u+]∶=ut+-Δu++cuz+-f(u+)-k1v++k2u+(b0-v+)≥0,N[v+]∶=vt+-Δv++cvz++k1v+-k2u+(b0-v+)≥0.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
(20)u-(x,z,t)=ϕ(z-σδ(1-e-βt))-δe-βt,v-(x,z,t)=ψ(z-σδ(1-e-βt))-k0δe-βt
is a subsolution. More precisely, it satisfies the following inequality:
(21)N[u-]∶=ut--Δu-+cuz--f(u-)-k1v-+k2u-(b0-v-)≤0,N[v-]∶=vt--Δv-+cvz-+k1v--k2u-(b0-v-)≤0.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:
(22)Ut1+|∇xU|2=div(∇xU1+|∇xU|2)-c,dddddddddddddddddddx∈ℝm,t>0,U(x,0)=U0(x),x∈ℝm.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 onℝm. Then there exists a constantμ∈ℝsuch that
(23)limt→∞supx∈ℝm|U(x,t)+ct+μ|=0.Lemma 7 (see [11]).
LetU(x,t;φ) and V(x,t;φ)denote the solutions of the following equations:
(24)Ut1+|∇xU|2=div(∇xU1+|∇xU|2)-c,ddddddddddddddddddddx∈ℝm,t>0,Vt=ΔxV-c2|∇xV|2-c,x∈ℝm,t>0,
under the initial conditionsU(·,0)=V(·,0)=φ∈W2,∞(ℝm). Then, for any constantε>0, there exists a constantδ>0such that if∥∇xφ∥W1,∞≤δ, it holds that
(25)supx∈ℝm|U(x,t;φ)-V(x,t;φ)|≤ε∀t>0.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
(26)Vt=ΔxV-c2|∇xV|2-c,x∈ℝn-1,t>0,V(x,0)=V0(x),x∈ℝn-1.
Then the following estimates hold:
(27)supx∈ℝn-1|Vxi(x,t)|≤min{C0t-1/2,C1},supx∈ℝn-1|Vxixj(x,t)|≤min{C0t-1,C2},supx∈ℝn-1|Vxixjxk(x,t)|≤C3(1+t)-3/2,supx∈ℝn-1|Vxit(x,t)|≤C4(1+t)-3/2,
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
(28)C1⟶0as∥∇xV0∥L∞⟶0;(iii)
C2depends only onc,∥V0∥L∞, and ∥∇xV0∥W1,∞ and satisfies
(29)C2⟶0as∥∇xV0∥W1,∞⟶0;(iv)
C3 and C4depend only on c and ∥V0∥W3,∞.


## 3. Proofs of Theorems1 and 3

In this section, we consider problem (7) inℝn 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
(30)w(x,z,t)=u(x,y,t),ω(x,z,t)=v(x,y,t),z=y+ct,
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)
(31)ut=Δu-cuz+f(u)+k1v-k2u(b0-v),dddddddddddddddx∈ℝn-1,z∈ℝ,t>0,vt=Δv-cvz-k1v+k2u(b0-v),ddddddx∈ℝn-1,z∈ℝ,t>0,u(x,z,0)=u0(x,z),x∈ℝn-1,z∈ℝ,v(x,z,0)=v0(x,z),x∈ℝn-1,z∈ℝ,
whereΔ=∂2/∂x12+⋯+∂2/∂xn-12+∂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
(32)limsupz→-∞supx∈ℝn-1u0(x,z)<δ,liminfz→+∞infx∈ℝn-1u0(x,z)>1-δ,limsupz→-∞supx∈ℝn-1v0(x,z)<k0δ,liminfz→+∞infx∈ℝn-1v0(x,z)>b0-b2-k0δ,
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
(33)limt→∞infx∈ℝn-1(u(x,z,t),v(x,z,t))≥(ϕ(z-z*),ψ(z-z*)),dddddddddddddddddddddddddduniformlyinz∈ℝ,limt→∞supx∈ℝn-1(u(x,z,t),v(x,z,t))≤(ϕ(z-z*),ψ(z-z*)),ddddddddddddddddddddddddduniformlyinz∈ℝ.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
(34)u(x,z,T)≤ϕ(z-z0)+δ1=u+(z-z0,0),(35)v(x,z,T)≤ψ(z-z0)+δ2=v+(z-z0,0).
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
(36)(0,0)≤(u(x,z,t),v(x,z,t))≤(1,b0-b2),dddddddddddddddddd(x,z)∈ℝn,t>0.
Note thatlimz→∞(ϕ(z),ψ(z))=(1,b0-b2), there exists a constantz1∈ℝsuch that, for eachT>0andz≥z1,
(37)u(x,z,t)≤ϕ(z)+δ1=u+(z,0),v(x,z,t)≤ψ(z)+δ2=v+(z,0).
Next, we show that(38)(limsupz→-∞supx∈ℝn-1u(x,z,T),limsupz→-∞supx∈ℝn-1v(x,z,T))<(δ1,δ2),
for eachT>0. For this purpose, choose positive constants β1, β2, and M such that
(39)limsupz→-∞supx∈ℝn-1u0(x,z)<β1<δ,limsupz→-∞supx∈ℝn-1v0(x,z)<β2<k0δ
and that
(40)(u0(x,z),v0(x,z))≤(β1+Mecz,β2+Mecz),ddddddddddddddddddddddddd(x,z)∈ℝn.
Then the functions
(41)(w(x,z,t),ω(x,z,t))=(min{β1+Mec(z+at),1},min{β2+Mec(z+at),b0-b2})
are a supersolution of (31) ifa>0is chosen sufficiently large. Hence,
(42)(u(x,z,T),v(x,z,T))≤(w(x,z,T),ω(x,z,T)).
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 onℝn-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 {(xi,ti)} such that0<t1<t2<⋯→∞and that
(43)(u(x+xi,z,t+ti),v(x+xi,z,t+ti))⟶(w(x,z,t),ω(x,z,t))asi⟶∞inCloc2,1(ℝn×ℝ).
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{(xi,ti)}with0<t1<t2<⋯→∞, there exist a subsequence {(xi′,ti′)} and a ω-limit point W(x,z,t)∶=(w(x,z,t),ω(x,z,t)) of U(x,z,t)such that
(44)U(x+xi′,z,t+ti′)⟶W(x,z,t),dddddasi⟶∞inCloc2,1(ℝn×ℝ).
Indeed, by the assumption in Theorem 1, it is easy to see thatU(x,z,t)is bounded. Then byLp-estimates and Schauder’s estimate (see [13, 14]), the solutionU(x,z,t)belongs toC2+α,1+α/2(ℝn×[δ,T])for any0<δ<T. Furthermore,
(45)∥U(x,z,t)∥C2+α,1+α/2(ℝn×[δ,T])≤C,
whereC>0is a constant independent ofT>0. Let{Qk}k=1,2,…be a sequence of compact subsets of ℝn×ℝ satisfying
(46)Q1⊂Q2⊂⋯,limk→∞Qk=ℝn×ℝ.
Then, for eachk, the sequence of functions{U(x,z,t)(x+xi,z,t+ti)}i=1,2,…is defined onQkfor all largeiand the restrictions of these functions ontoQkare relatively compact inC2,1(Qk)by virtue of the estimate (45). By using diagonal argument, we can choose a subsequence{(xi′,ti′)}and a functionw(x,z,t)defined onℝn×ℝsuch that, for anyk≥1, it holds that
(47)limi→∞∥U(x+xi′,z,t+ti′)-W(x,z,t)∥C2,1(Qk)=0,
which meansU(x+xi′,z,t+ti′)→W(x,z,t)asi→∞inCloc2,1(Qk).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 onℝn-1×ℝ×ℝand satisfies
(48)ut=Δu-cuz+f(u)+k1v-k2u(b0-v),dddddddddddddddddd(x,z)∈ℝn,t∈ℝ,vt=Δv-cvz-k1v+k2u(b0-v),dddddddddddd(x,z)∈ℝn,t∈ℝ.
Assume further that there exist two constantsz*,z*∈ℝ such that
(49)(ϕ(z-z*),ψ(z-z*))≤(u(x,z,t),v(x,z,t))≤(ϕ(z-z*),ψ(z-z*)),(x,z)∈ℝn,t∈ℝ.
Then there exists a constantz0∈[z*,z*]such that
(50)u(x,z,t)=ϕ(z-z0),v(x,z,t)=ψ(z-z0),ddd(x,z)∈ℝn,t∈ℝ.Proof.
Define(51)us(x,z,t)=u(x+y,z+s,t+T),ddddddddddy∈ℝn-1,s,T∈ℝ,vs(x,z,t)=v(x+y,z+s,t+T),ddddddddddy∈ℝn-1,s,T∈ℝ.
Then,
(52)(ϕ(z+s-z*),ψ(z+s-z*))≤(us(x,z,t),vs(x,z,t))≤(ϕ(z+s-z*),ψ(z+s-z*)).
It follows from the monotonicity ofϕandψthat there existss∈ℝsatisfying
(53)us(x,z,t)≥u(x,z,t),vs(x,z,t)≥v(x,z,t).
Indeed, lets=z*-z*, then(us(x,z,t),vs(x,z,t))≥(u(x,z,t),v(x,z,t)). Now, let
(54)s*=inf{s∈ℝ,(us′,vs′)≥(u,v),inℝn×ℝ∀s′≥s}.
Sincelimz→-∞(ϕ(z),ψ(z))=(0,0), limz→∞(ϕ(z),ψ(z))=(1,b0-b2) and by using (49), there exists constantA1>0such that
(55)(u(x,z,t),v(x,z,t))≥(1-γ,b0-b2-γ)ddddddddddd∀z≥A1,(x,z,t)∈ℝn×ℝ,(u(x,z,t),v(x,z,t))≤(γ,γ)∀z≤-A1,(x,z,t)∈ℝn×ℝ.
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)∈ℝn×ℝ,-A≤z≤A}. If infS(us*-u)>0 and infS(vs*-v)>0, then there exists η0∈(0,s*) such that (us*-η0,vs*-η0)≥(u,v) in S for all η∈[0,η0]. Denote E={(x,z,t)∈ℝn×ℝ,z>A}, w=us*-η-u and ω=vs*-η-v, we see that (w,ω)≥(0,0) on ∂E and satisfies
(56)wt-Δw+cwz=f(us*-η)-f(u)+(k1+k2u)ω-k2(b0-vs1*-η)w≥(k1+k2u)ω-k2(b0-vs*-η)w-Bwωt-Δω+cωz=-(k1+k2u)ω+k2(b0-vs*-η)w
for some constantB(remember thatf∈C1(ℝ) andus*-η,u,vs*-ηandvare 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,(us*-η,vs*-η)≥(u,v)inℝn×ℝfor allη∈[0,η0]. This contradicts the minimality ofs*. It follows then that
(57)infS(us*-u)=0orinfS(vs*-v)=0.
We first consider the case thatinfS(us*-u)=0andinfS(vs*-v)>0. As a consequence, there existz∞∈[-A,A]and a sequence(xn,zn,tn)n∈Nsuch that
(58)zn⟶z∞,us*(xn,zn,tn)-u(xn,zn,tn)⟶0asn⟶∞.
Callun(x,z,t)=u(x+xn,z+zn,t+tn)andvn(x,z,t)=v(x+xn,z+zn,t+tn). Note thatvn(x,z,t)is a bounded sequence. Up to extraction of a subsequence, the functions(un,vn)converge locally uniformly to a solution(u∞,v∞)of the following system:
(59)ut=Δu-cuz+f(u)+k1v-k2u(b0-v),vt=Δv-cvz-k1v+k2u(b0-v).
It is easy to see that
(60)w(x,z,t)=u∞(x+y,z+s*,t+T)-u(x,z,t)≥0,ω(x,z,t)=v∞(x+y,z+s*,t+T)-v(x,z,t)>0inℝn×ℝ
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
(61)wt(0,z∞,0)=0,Δw(0,z∞,0)≥0,wz(0,z∞,0)=0,ω(0,z∞,0)>0.
So we get a contradiction. Similarly, one can deal with the other case:infS(vs*-v)=0 and infS(us*-u)>0.
Now we consider the last case:infS(us*-u)=0andinfS(vs*-v)=0. As the case thatinfS(us*-u)=0andinfS(vs*-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)inℝn×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,
(62)u0(x,z,t)=u(x+y,z,t+T)≥u(x,z,t),v0(x,z,t)=v(x+y,z,t+T)≥v(x,z,t).
SinceT∈ℝandy∈ℝn-1are arbitrary, we conclude thatuandvdepend onzonly, namely,(u(x,z,t),v(x,z,t))=(ϕ(z-z1),ψ(z-z2))for somezi∈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 Lemma9, anyω-limit point of(u,v)satisfies
(63)ϕ(z-z*)≤w(x,z,t)≤ϕ(z-z*),ψ(z-z*)≤ω(x,z,t)≤ψ(z-z*),(x,z)∈ℝn,t∈ℝ,
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
(64)(w(x,z,t),ω(x,z,t))=(ϕ(z-z0),ψ(z-z0)),(x,z)∈ℝn,t∈ℝ.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
(65)infx∈ℝn-1,|z|≤R,t≥T(uz(x,z,t),vz(x,z,t))>(0,0).Proof.
We prove this lemma by contradiction. If the above claim does not hold, then there exists a sequence{(xk,zk,tk)}such that{zk}⊂[-R,R],0<t1<t2<⋯→∞ and that
(66)liminfk→∞(uz(xk,zk,tk),vz(xk,zk,tk))≤(0,0).
Replace{(xk,zk,tk)}by its subsequence if necessary; we may assume without loss of generality thatzkconverges to some limitz∞∈[-R,R]and that
(67)(uz,vz)(x+xk,z+zk,t+tk)⟶(w,ω)(x,z,t)ddddddddddddddask⟶∞inCloc2,1(ℝn×ℝ),
where(w,ω)is a limit point of(u,v). Hence,
(68)(wz(0,z∞,0),ωz(0,z∞,0))=limk→∞(uz(xk,zk,tk),vz(xk,zk,tk))≤(0,0).
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 Lemmas14 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
(69)inf(x,z,t)∈D(uz(x,z,t),vz(x,z,t))>0,
where
(70)D={(x,z,t)∈ℝn×[T,∞),(ϕ(0)2,ψ(0)2)≤(u(x,z,t),v(x,z,t))≤(1+ϕ(0)2,b0-b2+ψ(0)2)}.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
(71)limt→∞supx∈ℝn-1,|z|≤R|uxi(x,z,t)|=0,limt→∞supx∈ℝn-1,|z|≤R|uxixj(x,z,t)|=0,limt→∞supx∈ℝn-1,|z|≤R|vxi(x,z,t)|=0,limt→∞supx∈ℝn-1,|z|≤R|vxixj(x,z,t)|=0,
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
(72)(u(x,z,t),v(x,z,t))=(ϕ(0),ψ(0)),dddddddifandonlyifz=Γ(x,t)
for any(x,t)∈ℝn-1×[T,∞). Furthermore, the following estimates hold:(i)
for each1≤i,j≤n-1,
(73)limt→∞supx∈ℝn-1|Γxi(x,z,t)|=0,limt→∞supx∈ℝn-1|Γxixj(x,z,t)|=0;(ii)
there exists a constantM>0such that, for each1≤i,j,k≤n-1,
(74)supx∈ℝn-1|Γxixjxk(x,z,t)|≤Mfort≥T.Proof.
Since(75)D={(x,z,t)∈ℝn×[T,∞),(ϕ(0)2,ψ(0)2)≤(u(x,z,t),v(x,z,t))≤(1+ϕ(0)2,b0-b2+ψ(0)2)}
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
(76)limt→∞sup(x,z)∈ℝn|u(x,z,t)-ϕ(z-Γ(x,t))|=0,limt→∞sup(x,z)∈ℝn|v(x,z,t)-ψ(z-Γ(x,t))|=0.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{(xk,zk,tk)}such that 0<t1<t2<⋯→∞ and that
(77)|u(xk,zk,tk)-ϕ(zk-Γ(xk,tk))|≥δ.
On the other hand, by virtue of Lemma9 and boundedness ofΓ(x,t), we can choose constants R>0 and T>0 such that
(78)supx∈ℝn-1,|z|≥R,t≥T|u(x,z,t)-ϕ(z-Γ(x,t))|<δ,
which means that{zk}is bounded. We can choose subsequence of{(xk,zk,tk)}, which we denote again by{(xk,zk,tk)}such that
(79)z∞≔limk→∞zk,γ∞∶=limk→∞Γ(xk,tk)u(x+xk,z,t+tk)⟶w(x,z,t)dddask⟶∞inCloc2,1(ℝn×ℝ),
wherewis someω-limit point ofu. This and (77) show that
(80)|w(0,γ∞,0)-ϕ(z∞-γ∞)|=limk→∞|u(xk,zk,tk)-ϕ(zk-γ(xk,tk))|≥δ.
On the other hand, we have
(81)w(0,γ∞,0)=limk→∞u(xk,Γ(xk,tk),tk)=ϕ(0).
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
(82)Vt=ΔxV-c2|∇xV|2,x∈ℝn-1,t>0,v(x,0)=v0(x),x∈ℝn-1.Lemma 19 (supersolution).
For any constant M>0, there exist positive constant δ, ε0, k0 and smooth functionsp(t)andq(t)satisfying
(83)p(0)>0,q(0)=0,0≤p(t),q(t)≤εfort≥0,
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
(84)u+(x,z,t)=ϕ(z-V(x,t)1+|∇xV|2+q(t))+p(t),v+(x,z,t)=ψ(z-V(x,t)1+|∇xV|2+q(t))+k0p(t)
satisfies
(85)L[u+]∶=ut+-Δu++cuz+-f(u+)-k1v++k2u+(b0-v+)≥0,(x,z)∈ℝn,t>0,L[v+]∶=vt+-Δv++cvz++k1v+-k2u+(b0-v+)≥0,(x,z)∈ℝn,t>0.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
(86)η(x,z,t)=z-V(x,t)1+|∇xV|2.
Then, by using (8), we have
(87)L[u+]=(ηt-Δxη+cηz-c)ϕ′+(1-|∇xη|2-ηz2)ϕ′′+q′(t)ϕ′+p′(t)-f(ϕ+p(t))+f(ϕ)-(k1+k2ϕ)k0p(t)+k2p(t)(b0-ψ-k0p(t))L[v+]=(ηt-Δxη+cηz-c)ψ′+(1-|∇xη|2-ηz2)ψ′′+q′(t)ψ′+k0p′(t)+(k1+k2ϕ)k0p(t)-k2p(t)(b0-ψ-k0p(t)).
By rewriting the above expression in terms ofV, we obtain
(88)L[u+]=(I0-I2)ϕ′+(I1-3I3)ηϕ′-2I2ηϕ′′-I3η2ϕ′′+J1,L[v+]=(I0-I2)ψ′+(I1-3I3)ηψ′-2I2ηψ′′-I3η2ψ′′+J2,
whereIi, Jj (i=0,1,2,3; j=1,2) are functions given by
(89)I0=-Vt-c1+|∇xV|2+div(∇x1+|∇xV|2)-c,I1=-∑i=1n-1‍VxiVxit1+|∇xV|2+∑i,j=1n-1‍Vxixj2+VxjVxixixj1+|∇xV|2,I2=∑i,j=1n-1‍VxiVxj+VxjVxixj(1+|∇xV|2)3/2,I3=∑i=1n-1‍(∑i=1n-1VxiVxij1+|∇xV|2)2,J1=(ϕ′q′(t)p(t)+p′(t)p(t)-∫01‍f′(ϕ+τp(t))dτ-(k1+k2ϕ)k0+k2(b0-ψ-k0p(t))q′(t)p(t)+p′(t)p(t))·p(t),J2=(ψ′q′(t)k0p(t)+p′(t)p(t)+(k1+k2ϕ)+k2(b0-ψ-k0p(t))1k0q′(t)k0p(t))·k0p(t).
Step
2. Now we estimateIi (i=1,2,3). Define Iu(x,z,t) and Iv(x,z,t) by
(90)Iu(x,z,t)=(I0-I2)ϕ′+(I1-3I3)ηϕ′-2I2ηϕ′′-I3η2ϕ′′,Iv(x,z,t)=(I0-I2)ψ′+(I1-3I3)ηψ′-2I2ηψ′′-I3η2ψ′′.
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:
(91)ηϕ′(η+q(t)),ηϕ′′(η+q(t)),η2ϕ′′(η+q(t)),ηψ′(η+q(t)),ηψ′′(η+q(t)),η2ψ′′(η+q(t)).
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
(92)|Iu(x,z,t)|≤P(t),|Iv(x,z,t)|≤P(t)whereP(t)=min{C2t-2,C1}.
Step
3. Now we determine the constantC1,ε0and the smooth functionspi(t)and q(t) (i=1,2). Set
(93)F(u,v)=f(u)+k1v-k2u(b0-v),G(u,v)=-k1v+k2u(b0-v).
Then, direct calculation shows that
(94)Fu=f′(u)-k2(b0-v),Fv=k1+k2u,Gu=k2(b0-v),Gv=-(k1+k2u),Fu(0,0)Gv(0,0)-Fv(0,0)Gu(0,0)=-k1f′(0)>0,Fu(1,b0-b2)Gv(1,b0-b2)-Fv(1,b0-b2)Gu(1,b0-b2)=-(k1+k2)f′(1)>0.
Hence, there exist constantsε0>0,α11,α22<0andα12,α21>0such that
(95)α11<Fu<0,α22<Gv<0,0<Fv<α12,0<Gv<α21,onAεi,i=1,2,
whereε∈(0,ε0]and
(96)Aε0={(u,v)∣|u|≤ε,|v|≤ε},Aε1={(u,v)∣|u-1|≤ε,|v-b0+b2|≤ε}.
It follows that there exist positive constantsK0andk0such that for allK∈(0,K0), we have
(97)2K+α11<0,2K+α22<0,(2K+α11)(2K+α22)>α12α21,-α212K+α22<k0<-2K+α11α12.
This implies that
(98)-2K-α11-k0α12>0,-2K-α22-1k0α21>0.
Denoteξ=(z-V(x,t))/1+|∇xV|2+q(t). Using the above estimates and the facts thatlimz→-∞ϕ(z)=0, limz→-∞ψ(z)=0, limz→∞ϕ(z)=1andlimz→∞ψ(z)=b0-b2, there exist constantsM1,M2>0such that
(99)-2K-f′(ϕ+θδ1e-βt)-k0(k1+k2ϕ)+k2(b0-ψ)>0dddddddddddddddddddddddddforξ≤-M2orξ≥M1,-2K+k1+k2ϕ-k21k0(b0-ψ)>0ddddddddddforξ≤-M2orξ≥M1.
Set(100)δ0=min{-2K-α11-k0α122k2k0,-2K-α22-α21/k02k2k0,1,1k0}.
We define the constantC1>0andC0≥1by
(101)C1=K2ε2k0216(1+k0)2C2C02,C0=max{1,2K+∥f′∥L∞([0,1])+(k1+k2)k0+δ0k2inf-M2≤ξ≤M1ϕ′(ξ),2Kk0-k1k0+k2b0inf-M2≤ξ≤M1ψ′(ξ)2K+∥f′∥L∞([0,1])+(k1+k2)k0+δ0k2inf-M2≤ξ≤M1ϕ′(ξ)}.
We choose functionsp(t),q(t)∈C∞[0,∞)satisfying
(102)P(t)≤min{1,k0}Kp(t),Kp(t)≤2(1+1k0)P(t),K|p′(t)|≤2|P′(t)|,q(t)=∫0t‍p(s)ds.
Then (107) holds, since we have
(103)p(0)≥Kε2k016(1+k0)2C2C02>0,0<p(t)≤Kε2k08C2C02(1+k0)≤ε,0≤∫0∞‍p(s)ds≤ε.
Step
4. Now we complete the proof. Since|Iu(x,z,t)|≤P(t)and|Iv(x,z,t)|≤P(t)by Step 2, it suffices to show the inequalityJi(x,z,t)≥P(t) (i=1,2). Letting ξ=(z-V(x,t))/1+|∇xV|2+q(t), whenξ∈(-∞,-M2]∪[M1,∞), we have
(104)J1≥(p′(t)p(t)-∫01‍f′(ϕ+τp(t))dτ-(k1+k2)k0+k2(b0-ψ-p(t))p′(t)p(t)-∫01‍)·p(t)≥Kp(t)≥P(t),J2≥(p′(t)p(t)+(k1+k2ϕ)+k2(b0-ψ-p(t))1k0)·k0p(t)≥k0Kp(t)≥P(t),
since we have
(105)supt≥0|p′(t)|p(t)≤supt≥02|P′(t)|Kp(t)≤supt≥02|P′(t)|P(t)=KεC0<K.
For any-M2≤ξ≤M1, we have
(106)L[u+]≥(C0inf-M2≤ξ≤M1ϕ′(ξ)-2K-∥f′∥L∞([0,1])-(k1+k2)k0-δ0k2C0inf-M2≤ξ≤M1)p(t)≥0,L[v+]≥(C0k0inf-M2≤ξ≤M1ψ′(ξ)-2K+k1-k2b0k0)≥0.
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
(107)p(0)>0,q(0)=0,0≤p(t),q(t)≤εfort≥0,
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
(108)u-(x,z,t)=ϕ(z-V(x,t)1+|∇xV|2-q(t))-p(t),v-(x,z,t)=ψ(z-V(x,t)1+|∇xV|2-q(t))-k0p(t)
satisfies
(109)L[u-]∶=ut--Δu-+cuz--f(u-)-k1v-+k2u-(b0-v-)≤0,(x,z)∈ℝn,t>0,L[v-]∶=vt--Δv-+cvz-+k1v--k2u-(b0-v-)≤0,(x,z)∈ℝn,t>0.The proof of Lemma20 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 constantτε>0such that the functionV(x,t)defined by
(110)Vt=ΔxV-c2|∇xv1|2,x∈ℝn-1,t>0,v1(x,0)=Γ(x,τε),x∈ℝn-1
satisfies
(111)supx∈ℝn-1|Γ(x,t)≥V(x,t-τε)|≤ε,t≥τε.Proof.
From Corollary15, we can choose constantsT>0,M>0, andL>0such that
(112)supt≥T∥Γ(·,t)∥W3,∞≤M,inf(x,z,t)∈Duz(x,z,t)≥L,inf(x,z,t)∈Dvz(x,z,t)≥L,D={(x,z,t)∈ℝn×[T,∞),(ϕ(0)2,ψ(0)2)≤(u(x,z,t),v(x,z,t))≤(1+ϕ(0)2,b0-b2+ψ(0)2)}.
For the constantMand ε^∶=min{ε0,min{Lε,(1+ϕ(0))/[2(1+∥ϕ′∥L∞)]}}, we choose a constantδ>0and functionsp(t),q(t)satisfying
(113)p(0)>0,q(0)=0,0≤p(t),q(t)≤ε^fort≥0,
as in Lemma 19. Then it follows from Lemma 17 that there exists a constant τε∈[T,∞) such that ∥Γ(·,τε)∥W1,∞≤δ. Moreover, using Lemma 18, by choosing τε larger if necessary, we have
(114)u(x,z,τε)≤ϕ(z-Γ(x,τε))+p(0)2≤ϕ(z-Γ(x,τε)1+|∇xΓ(x,τε)|2)+p(0),v(x,z,τε)≤ψ(z-Γ(x,τε))+k0p(0)2≤ψ(z-Γ(x,τε)1+|∇xΓ(x,τε)|2)+k0p(0),
where we used the smallness of|∇xΓ|. For suchτε, we defineV(x,t)as a function satisfying (110). Then Lemma 19 shows that the function defined by
(115)u+(x,z,t)=ϕ(z-V(x,t-τε)1+|∇xV|2+q(t-τε))+p(t-τε),v+(x,z,t)=ψ(z-V(x,t-τε)1+|∇xV|2+q(t-τε))+k0p(t-τε)
is a supersolution of (31) fort≥τε. Since (114) implies that u(x,z,τε)≤u+(x,z,τε) and v(x,z,τε)≤v+(x,z,τε), the comparison principle gives that u(x,z,t)≤u+(x,z,t) andv(x,z,t)≤v+(x,z,t) for t≥τε. Then, due to p(t),q(t)≤ε^, we have
(116)u(x,V(x,t-τε),t)-ϕ(0)≤u+(x,V(x,t-τε),t)-ϕ(0)=ϕ(q(t-τε))+p(t)-ϕ(0)≤(1+∥ϕ′∥L∞)ε^≤min{Lε,(1+ϕ(0))2}.
Noting thatu(x,Γ(x,t),t)=ϕ(0), we get
(117)Lε≥u(x,V(x,t-τε),t)-u(x,Γ(x,t-τε),t)≥infu∈[ϕ(0),(1+ϕ(0))/2],t≥τεuz×(V(x,t)-Γ(x,t))≥L(V(x,t)-Γ(x,t)),
which implies thatV(x,t-T)-Γ(x,t)≤εfort>τε. Similarly, by using the subsolution(u-,v-)given in Lemma 20, we can show thatV(x,t-T)-Γ(x,t)≥-εfort>τε. This completes the proof.Now we are ready to complete the proof of Theorem1.Proof of Theorem1.
The statements (i) and (ii) of Theorem1 are directly from Lemmas 17 and 18, respectively. Thus, we only prove the statement (iii).
By Lemma21, 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
(118)Vt=ΔxV-c2|∇xV|2,x∈ℝn-1,t>0.
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
(119)V~t=ΔxV~-c2|∇xV~|2-c,x∈ℝn-1,t>0.
Hence, the statement (iii) of Theorem 1 follows from Lemma 21. This completes the proof of Theorem 1.Next, we prove Theorem3. 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 Theorem3.
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.



---

*Source: 936296-2013-11-20.xml*