Asymptotic stability of traveling waves for a dissipative nonlinear evolution system

Acta Mathematica Scientia 2015,35B(6):1325–1338 http://actams.wipm.ac.cn

ASYMPTOTIC STABILITY OF TRAVELING WAVES FOR A DISSIPATIVE NONLINEAR EVOLUTION SYSTEM∗

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Mina JIANG (

The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China E-mail : [email protected]

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Jianlin XIANG (

†

Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430079, China E-mail : [email protected] Abstract This paper is concerned with the existence and the nonlinear asymptotic stability of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations   ξt = −θx + βξxx , with initial data and end states

 θt = νξx + (ξθ)x + αθxx ,

(ξ, θ)(x, 0) = (ξ0 , θ0 )(x) → (ξ± , θ± ) as x → ±∞. We obtain the existence of traveling wave solutions by phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without restrictions on the coefficients α and ν by the method of energy estimates. Key words

dissipative evolution equations; traveling wave solutions; nonlinear stability; energy estimates

2010 MR Subject Classification

1

35K45; 35C07; 35B40

Introduction and Main Results

In physical and mechanical fields, many phenomena can be modeled by the systems of the nonlinear interaction between ellipticity and dissipation. Lorenz derived his famous equations ∗ Received September 3, 2014; revised November 24, 2014. Jiang’s research was supported by the Natural Science Foundation of China (11001095), the Ph.D. specialized grant of the Ministry of Education of China (20100144110001) and the Special Fund for Basic Scientific Research of Central Colleges (CCNU12C01001). Xiang’s research was supported by the Fundamental Research Funds for the Central Universities (2015IA009) and the Natural Science Foundation of China (61573012). † Corresponding author: Jianlin XIANG.

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in [1] from Rayleigh-Benard equations [2]. But the Rayleigh-Benard problem is a system of two highly nonlinear partial differential equations with three independent variables, except for the linearized system, it is difficult to analyze those equations in any way besides the numerical computations. Therefor it is useful to construct a manageable partial differential equation and its reduced system. Hsieh [3] proposed the following alternative system to try to yield more and new insight to Rayleigh-Benard equations   ξ = −(σ − β)ξ − σθ + βξ , t x xx (1.1)  θt = −(1 − α)θ + νξx + (ξθ)x + αθxx ,

where (t, x) ∈ [0, +∞) × R, σ, α, β and ν are all positive constants satisfying the relation β < σ and α < 1. ξ(x, t) and θ(x, t) denote a stream function and a temperature function respectively. As a preliminary work, Jian and Chen [4] first established the global existence of solutions of the system (1.1) with the initial condition (ξ0 , θ0 ) ∈ H 1 (R, R2 ) ∩ L1 (R, R2 ), and the optimal decay rate and optimal decay order were obtained by Wang [5]. Hsiao and Jian [6] obtained the global existence of classical solutions for the initial boundary value problem of the system (1.1) with initial condition (ξ0 , θ0 ) ∈ C 2,δ ([0, 1]) ∗ C 2,δ ([0, 1]) (0 < δ < 1) and periodic boundary condition

(ξ0 , θ0 )(0) = (ξ0 , θ0 )(1), (ξ)x , (θ)x (0, t) = (ξ)x , (θ)x (1, t), 0 ≤ t ≤ T. Moreover, Tang and Zhao [7] studied the following modified system   ξ = −(σ − β)ξ − σθ + βξ , t x xx  θt = −(1 − α)θ + νξx + 2ξθx + αθxx .

(1.2)

When the initial data (ξ0 , θ0 ) ∈ L2 (R, R2 ), they established the global existence, nonlinear stability and optimal decay rate of the solution to (1.2) with suitable restrictions on coefficients σ, α, β and ν. Furthermore, if (ξ0 , θ0 ) ∈ L1 (R, R2 ), they obtained the optimal decay rates of solutions for system (1.2). Zhu and Wang [8] extended the above results to more general case in which initial data satisfy (ξ0 (x), θ0 (x)) → (ξ± , θ± ), as x → ±∞, where (ξ+ − ξ− , θ+ − θ− ) 6= (0, 0). Duan and Zhu [9] investigated the asymptotics of diffusion wave toward the solution of system (1.2). Zhu et al. [10] obtained the global existence, nonlinear stability and decay rates of the solution to the diffusion wave for system (1.2). The optimal convergence rates of solutions to diffusion wave for the Cauchy problem of (1.1) and (1.2) were derived in [11] and [12], respectively. Hsieh [3] briefly discussed the linear stability of possible periodic traveling wave solution to (1.2) in cases of 0 ≤ α ≤ 1, ν > α as well as σ > 0, β > 0. But there has not been any rigorous results of traveling wave solutions to (1.1) or (1.2). The aim of this paper is to study the existence and the nonlinear stability of traveling wave solutions to the system (1.1). Considering that (1.1) is a system of second-order parabolic equations, it is very challenging to consider the traveling wave solutions due to the high dimensionally of the wave system. In this paper, we shall consider the following system   ξ = −θ + βξ , t x xx (1.3)  θt = νξx + (ξθ)x + αθxx ,

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with initial data and end states (ξ, θ)(x, 0) = (ξ0 , θ0 )(x) → (ξ± , θ± ) as x → ±∞,

(1.4)

where (t, x) ∈ [0, +∞) × R, α, ν and β are positive constants. In fact, (1.3) is a simplified version of the following system proposed by [13] with ε1 = ε2 = 0, λ = 0 and f (θ) = −θ:   ξ = ε ξ + λξξ + f (θ) + βξ , t 1 x x xx  θt = ε2 θ + νξx + (ξθ)x + αθxx .

The aim here is to establish the existence and asymptotic stability of traveling wave solutions of (1.3)–(1.4). For the simplicity of analysis, we introduce the following change of variables: u = −(θ + ν),

v = ξ.

Substituting (1.5) into (1.3), we obtain the following viscous conservation laws   u − (uv) = αu , t x xx  vt − ux = βvxx ,

(1.5)

(1.6)

with initial data and end states

(u, v)(x, 0) = (u0 , v0 )(x) → (u± , v± ) as x → ±∞,

(1.7)

where u± = −(θ± + ν), v± = ξ± . We consider the solutions of the system (1.6)-(1.7) in the following region D = {(u, v)|u ≥ 0, v ≤ 0, u± ≥ 0, v± ≤ 0}. (1.8) Obviously, the existence and stability of solutions to the Cauchy problem (1.3)–(1.4) are equivalent to that of the Cauchy problem (1.6)–(1.7). Thus inspired by the idea of [14], we shall first prove the existence and nonlinear stability of traveling wave solutions to (1.6)–(1.7) in the region D, and then transfer the results back to (1.3)–(1.4) with the transformations (1.5). We remark here that it is unclear if there are traveling wave solutions outside the region D. Since (1.6) is a system of conservation laws, it would be helpful to discuss the hyperbolicity to gain some insight into the properties of traveling wave solutions (i.e., viscous shock waves). In the absence of the viscous terms, (1.6) becomes   u − (uv) = 0, t x (1.9)  vt − ux = 0. The Jacobian matrix of (1.9) is J(u, v) =



−v −u −1

0



and its eigenvalues satisfy

λ2 + λv − u = 0. Since u ≥ 0, (1.10) has two real roots √ √ −v − v 2 + 4u −v + v 2 + 4u λ1 (u, v) = , λ2 (u, v) = , 2 2 with corresponding eigenvectors     −λ1 λ2 r1 (u, v) = , r2 (u, v) = . 1 −1

(1.10)

(1.11)

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Since u ≥ 0, hence λ1 ≤ 0 ≤ λ2 and then the system (1.9) is hyperbolic. Traveling wave solutions of (1.6)–(1.7) are solutions such that (u, v)(x, t) = (U, V )(x − st) =: (U, V )(z), z = x − st, where s denotes the wave speed. Substitute the above ansatz into (1.6)–(1.7), we have   −sU − (U V ) = αU , z z zz (1.12)  −sVz − Uz = βVzz , with boundary conditions

(U, V )(z) → (u± , v± ) as z → ±∞. From conservation laws (1.6), we integrate (1.12) once to obtain   αU = −sU − U V + ρ =: F (U, V ), z 1  βVz = −sV − U + ρ2 =: G(U, V ),

(1.13)

(1.14)

where ρ1 and ρ2 are constants satisfying

ρ1 = su+ + u+ v+ = su− + u− v− , ρ2 = sv+ + u+ = sv− + u− .

(1.15)

From (1.15), we deduce that the wave speed s satisfies the following quadratic equation s2 + v− s − u+ = 0.

(1.16)

Obviously, if u+ > 0, the discriminant of the quadratic (1.16) is positive so that (1.16) has two real roots with distinct signs. In this paper, we only consider the case of s > 0, the analysis can be directly extended to the case of s < 0. To ensure the uniqueness of traveling wave solutions, we assume that s satisfies the condition λ2 (u+ , v+ ) < s < λ2 (u− , v− ),

(1.17)

where λ2 (u, v) is defined in (1.11). We derive from the condition (1.17) that 0 ≤ u+ < u− ,

v− < v+ ≤ 0.

The positive wave speed s is given from (1.16) by q 2 + 4u v− + v− s=− + . 2 2 Since u+ ≥ 0, v− < 0, then s + v− > 0.

(1.18)

(1.19) (1.20)

The first result of this paper considering the existence of traveling wave solutions of the transformed system (1.6)–(1.7) is stated as follows. Theorem 1.1 Suppose u+ > 0 and (1.17) hold. If the wave speed s is given by (1.19), then there exists a monotone traveling wave solution (U, V )(x − st) to the Cauchy problem (1.6)–(1.7), which is unique up to a translation and satisfies Uz < 0 and Vz > 0. Theorem 1.1 can be proved by using phase plane analysis, since the proof is the same as that of Theorem 2.1 in [14], we omit it here. Remark 1.1 Since Uz < 0 and Vz > 0, 0 ≤ u+ < U < u− and v− < V < v+ ≤ 0, then from (1.12) and (1.14), we can easily deduce that Uz , Vz , Uzz , Vzz are all bounded. Furthermore,

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from (1.11) we see that the system (1.9) is strictly hyperbolic, i.e., λ1 6= λ2 , when 0 < u+ < U < u− and v− < V < v+ ≤ 0. In the second part of this paper, we consider the nonlinear asymptotic stability of traveling wave solutions obtained in Theorem 1.1 under the small initial perturbation of the form    Z +∞  u0 (x) − U (x) u+ − u− dx = x0 , (1.21) v0 (x) − V (x) v+ − v− −∞ the coefficient x0 is uniquely determined by the initial data (u0 (x), v0 (x)). Decomposing the solution (u, v) of the system (1.6) and (1.7) by (u, v)(x, t) = (U, V )(x − st + x0 ) + (ϕx , ψx )(x, t),

(1.22)

where (ϕ, ψ)(x, t) =

Z

x

−∞


u(y, t) − U (y + x0 − st), v(y, t) − V (y + x0 − st) dy,

for all x ∈ R and t ≥ 0. By conservation laws (1.6) and the system (1.12), we obtain  Z +∞  u(x, t) − U (x + x0 − st) dx v(x, t) − V (x + x0 − st) −∞  Z +∞  u0 (x) − U (x + x0 ) = dx v0 (x) − V (x + x0 ) −∞    Z +∞ Z +∞  u0 (x) − U (x) U (x) − U (x + x0 ) = dx + dx v0 (x) − V (x) V (x) − V (x + x0 ) −∞ −∞    Z +∞  u0 (x) − U (x) u+ − u− = dx − x0 = ~0, v0 (x) − V (x) v+ − v− −∞ which implies that ϕ(±∞, t) = 0, ψ(±∞, t) = 0 for all t ≥ 0. Without loss of generality, we further assume that the translation x0 = 0, then (1.21) becomes    Z +∞  u0 (x) − U (x) 0 dx = . (1.23) v (x) − V (x) 0 0 −∞ Hence the initial conditions of the perturbation (ϕ, ψ) are given by Z x
(ϕ0 , ψ0 )(x) = u0 − U, v0 − V (y)dy.

(1.24)

−∞

Before giving the results about the stability of traveling wave solutions, we introduce some notations. Hereafter, Lp (R) denotes the usual Lebesgue space with norms Z  p1 kf kLp(R) = |f (x)|p dx (1 ≤ p < +∞), kf kL∞ (R) = ess sup |f (x)|, R

R

p

and when p = 2, we write k · kLp (R) = k · k. H (R)(p ≥ 1) denotes the usual p-th Sobolev space with equivalent norm X  12 p 2 k . kf kp = kf kH p (R) = k∂x f k k=0

The main result on the asymptotic stability of the traveling wave solutions obtained in Theorem 1.1 for the transformed system is stated as follows.

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Theorem 1.2 Assume that the hypotheses in Theorem 1.1 hold and let (U, V )(x − st) be a traveling wave solution of (1.6) obtained in Theorem 1.1. If the perturbation (1.23) holds, then there exists a constant δ0 > 0, such that if ku0 − U k1 + kv0 − V k1 + k(ϕ0 , ψ0 )k ≤ δ0 , the Cauchy problem (1.6)–(1.7) has a unique global solution (u, v)(x, t) satisfying u(x, t) ≥ γ0 > 0 for some γ0 > 0 for all x ∈ R, t ≥ 0, and
2 (u − U, v − V ) ∈ C([0, +∞); H 1 ) ∩ L2 ([0, +∞); H 1 ) . Furthermore, the solution (u, v)(x, t) has the following nonlinear asymptotic stability sup |(u, v)(x, t) − (U, V )(x − st)| → 0 as t → +∞.

(1.25)

x∈R

Finally, we shall transfer the results in Theorem 1.1 and Theorem 1.2 back to the original problem (1.3) and (1.4). In fact, from (1.5), we see that the traveling wave solutions of (1.3) has the form of (ξ, θ)(x, t) = (Ξ, Θ)(z) = (V (z), −U (z) − ν),

z = x − st,

(1.26)

which satisfies (Ξ, Θ)(z) → (ξ± , θ± ) = (v± , −u± − ν) as z → ±∞.

(1.27)

The condition (1.17) is equivalent to ˜2 (ξ+ , θ+ ) < s < λ ˜2 (ξ− , θ− ), λ

(1.28)

˜ 2 (ξ+ , θ+ ) = λ2 (u+ , v+ ), λ ˜2 (ξ− , θ− ) = λ2 (u− , v− ). (1.19) can be rewritten as where λ q 2 − 4(θ + ν) ξ− + ξ− s=− + . (1.29) 2 2 Now we transfer the results in Theorem 1.1 and Theorem 1.2 back to the system (1.3). From Theorem 1.1, we directly obtain the existence of traveling wave solutions for the original system (1.3) satisfying the initial value (1.4) as follows. Theorem 1.3 Suppose θ+ < −ν and (1.28) holds. If the wave speed s is given by (1.29), then there exists a monotone traveling wave solution (Ξ, Θ)(x− st) to the Cauchy problem (1.3) and (1.4), which is unique up to a translation and satisfies Ξz > 0 and Θz > 0. Furthermore, since ξ − Ξ = v − V,

θ − Θ = −(u − U ),

the perturbation (1.23) is equivalent to    Z +∞  θ0 (x) − Θ(x) 0 dx = . ξ0 (x) − Ξ(x) 0 −∞

(1.30)

(1.31)

Thus Theorem 1.2 leads to the nonlinear asymptotic stability of traveling wave solutions of the original problem (1.3) and (1.4). Theorem 1.4 Suppose the hypotheses in Theorem 1.3 hold and let (Ξ, Θ)(x − st) be a traveling wave solution of (1.3) obtained in Theorem 1.3. If the perturbation (1.31) holds, then there exists a constant δ1 > 0, such that if kξ0 − Ξk1 + kθ0 − Θk1 + k(ϕ0 , ψ0 )k ≤ δ1 , the Cauchy problem (1.3) and (1.4) has a unique global solution (ξ, θ)(x, t) satisfying θ(x, t) ≤ θ1 < 0 for some θ1 < 0 for all x ∈ R, t ≥ 0, and
2 (ξ − Ξ, θ − Θ) ∈ C([0, +∞); H 1 ) ∩ L2 ([0, +∞); H 1 ) .

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Furthermore, the solution (ξ, θ)(x, t) has the following nonlinear asymptotic stability sup |(ξ, θ)(x, t) − (Ξ, Θ)(x − st)| → 0 as t → +∞.

(1.32)

x∈R

The rest of this paper is organized as follows. In Section 2, The nonlinear asymptotic stability of the traveling wave solutions of the transformed system is proved. In Section 3, we transfer the results of the transformed system (1.6) back to the original system (1.3) and prove Theorems 1.3 and 1.4.

2

Stability of Traveling Wave Solutions for the Transformed System

In this section, we apply the method of energy estimates to prove the nonlinear asymptotic stability of the traveling wave solution obtained in Theorem 1.1. Here, we have to mention the significant work about the traveling wave solutions investigated by [16] as well as [17] and [18], they provided a standard method of energy estimates. Hereafter, we use C to denote a generic constant changing from one line to another, and an integral lacking limits of integration to denote an integral over the whole real line R. By virtue of (1.22), the solution of (1.6) is decomposed by (u, v)(x, t) = (U, V )(x − st) + (ϕx , ψx )(x, t) = (U, V )(z) + (ϕ¯z , ψ¯z )(z, t),

(2.1)

¯ in some functional space which will be defined below. For simplicity of notation, with (ϕ, ¯ ψ) ¯ will be denoted by (ϕ, ψ) in what follows. (ϕ, ¯ ψ) Substituting (2.1) into (1.6), applying (1.12) and integrating the resultant equations with respect to z, we obtain the equations for the perturbation (ϕ, ψ)   ϕ = αϕ + (s + V )ϕ + U ψ + ϕ ψ , t zz z z z z (2.2)  ψt = βψzz + sψz + ϕz , with initial data

(ϕ, ψ)(z, 0) = (ϕ0 , ψ0 )(z), z ∈ R,

(2.3)

where (ϕ0 , ψ0 ) is defined in (1.24). We look for solutions of the reformulated problem (2.2)–(2.3) in the following space



X(0, T ) = { ϕ(z, t), ψ(z, t) : ϕ, ψ ∈ C [0, T ); H 2 ∩ C 1 (0, T ); H 1 , ϕz , ψz ∈ L2 (0, T ); H 2 } with 0 ≤ T ≤ +∞. Define

N (t) = sup {kϕ(·, τ )k2 + kψ(·, τ )k2 }.

(2.4)

τ ∈[0,t]

By the Sobolev embedding theorem, it holds that sup{|ϕ|, |ϕz |, |ψ|, |ψz |} ≤ N (t), for t ≥ 0.

(2.5)

z∈R

Now we state the following theorem on stability of travelling waves. Theorem 2.1 Assume (ϕ0 , ψ0 ) ∈ H 2 (R)2 . If u+ > 0, then there exists a constant δ0 > 0 such that if N (0) ≤ δ0 , (2.6)

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the Cauchy problem (2.2)–(2.3) has a unique global solution (ϕ, ψ) ∈ X(0, +∞) satisfying Z tZ Z t 2 2 k(ϕ(·, t), ψ(·, t))k2 + |Uz (z)|ϕ (z, τ )dzdτ + k(ϕz (·, τ ), ψz (·, τ ))k22 dτ 0

0

≤ CN 2 (0)

(2.7)

for all t ∈ [0, +∞), where C > 0 is a constant. Moreover the following asymptotic behavior holds sup |(ϕz , ψz )(z, t)| → 0 as t → +∞.

(2.8)

z∈R

The proof of Theorem 2.1 is based on the following local existence theorem and a priori estimates. Proposition 2.2 (Local existence) For any ǫ > 0, there exists a positive constant T depending on ǫ such that if (ϕ0 , ψ0 ) ∈ H 2 (R)2 with N (0) ≤ 2ǫ , the problem (2.2)–(2.3) has a unique solution (ϕ, ψ) ∈ X(0, T ) which satisfies N (t) ≤ 2N (0)

(2.9)

for any 0 ≤ t ≤ T . The local existence in Proposition 2.2 can be obtained by a standard way (cf. [19]) and so we omit the proof here. Proposition 2.3 (A priori estimates) Suppose that (ϕ, ψ) ∈ X(0, T ) is a solution obtained in Proposition 2.2 for a positive constant T . Then there exists a positive constant δ1 > 0, independent of T , such that if N (t) < δ1 for any 0 ≤ t ≤ T , the solution (ϕ, ψ) of (2.2)–(2.3) satisfies Z tZ Z t k(ϕ(·, t), ψ(·, t))k22 + |Uz (z)|ϕ2 (z, τ )dzdτ + k(ϕz (·, τ ), ψz (·, τ ))k22 dτ 0

0

≤ CN 2 (0)

(2.10)

for all t ∈ [0, T ], where C > 0 is a constant. The proof of Proposition 2.3 follows from a series of lemmas as follows. Lemma 2.4 (L2 -estimates) Assume that (ϕ0 , ψ0 ) ∈ H 2 (R)2 and (ϕ, ψ) be a solution of (2.2)–(2.3). If u+ > 0, then there exist constants µ0 > 0 and C > 0 such that the solution (ϕ, ψ) of (2.2)–(2.3) satisfies Z tZ kϕ(·, t)k2 + kψ(·, t)k2 + µ0 |Uz (z)||ϕ(z, τ )|2 dzdτ 0 Z t Z t +α k(ϕz (·, τ )k2 dτ + β kψz (·, τ ))k2 dτ 0 0   Z tZ ≤ C kϕ0 k2 + kψ0 k2 + |ϕϕz ψz |dzdτ . (2.11) 0

Proof Multiplying the first equation of (2.2) by ϕ/U and the second equation by ψ, and adding the resulting equations, we obtain ϕϕt α s+V ϕϕz ψz + ψψt = ϕϕzz + ϕϕz + + ψz ϕ + βψzz ψ + sψz ψ + ϕz ψ. U U U U

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We then integrate the equation with respect to z and have Z Z  2 Z Z  α i 1 d α 2 1 h s + V  ϕ + ψ 2 dz + ϕz dz + − ϕ2 dz + β ψz2 dz 2 dt U U 2 U U zz z Z ϕϕz ψz = dz. (2.12) U

Setting

H(z) =

s + V 

−

α

, U U zz z then by (1.12), (1.14), (1.15) and the fact that Uz < 0, we derive that

(2.13)

2ρ1 |Uz |. (2.14) U3 Since ρ1 = su− + u− v− = u− (s + v− ) > 0 by (1.18) and (1.20) and 0 < u+ < U < u− , it follows that 2ρ1 2u− (s + v− ) > =: µ0 > 0. (2.15) U3 u3− H(z) =

Hence substituting (2.14) and (2.15) into (2.12), we have Z  2 Z Z Z Z  ϕ 1 d α 2 µ0 ϕϕz ψz 2 2 2 + ψ dz + ϕ dz + |Uz |ϕ dz + β ψz dz ≤ dz. 2 dt U U z 2 U

Therefore, integrating the inequality with respect to t, due to the boundedness of U , we obtain Z tZ kϕ(·, t)k2 + kψ(·, t)k2 + µ0 |Uz (z)||ϕ(z, τ )|2 dzdτ 0 Z tZ Z tZ 2 +α |ϕz (z, τ )| dzdτ + β |ψz (z, τ )|2 dzdτ 0 0   Z tZ 2 2 ≤ C kϕ0 k + kψ0 k + |ϕϕz ψz |dzdτ , (2.16) 0

where C is a positive constant. The proof of Lemma 2.4 is thus completed.



Next we deduce the estimates for the first order derivatives of (ϕ, ψ). Lemma 2.5 (H 1 -estimates) Assume that the hypotheses in Lemma 2.4 hold. Then the solution (ϕ, ψ) of (2.2)–(2.3) satisfies Z t Z t kϕz (·, t)k2 + kψz (·, t)k2 + α kϕzz (·, τ )k2 dτ + β kψzz (·, τ )k2 dτ 0 0   Z tZ
(2.17) ≤ C kϕ0 k21 + kψ0 k21 + |ϕ| + |ϕz | + |ϕzz | |ϕz ψz |dzdτ . 0

Proof Multiplying the first equation of (2.2) by −ϕzz /U and the second equation by −ψzz , and adding the resulting equations, we have  ψ2 
ϕzz α s + V  ϕ2z  ϕz ϕzz ψz 2 − ϕt − ψzz ψt = − ϕ2zz − βψzz − − ϕz ψz z − −s z . U U U 2 z U 2 z We then integrate the equation with respect to z to get Z  2 Z 2 Z  1 d ϕz ϕzz 2 2 + ψz dz + α dz + β ψzz dz 2 dt U U Z h   Z 1 1  s + V  2i ϕz ϕzz ψz = − dz. (2.18) ϕz ϕt + ϕz dz − U z 2 U U z

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Denoting

Z h

1

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1  s + V  2i ϕz dz, U z 2 U z and substituting the first equation of (2.2) into I, we obtain Z Z Z   Z   s + V  i 1 h α  Vz 2 1 1 − ϕ2z dz + ϕz dz − U ϕz ψz dz − ϕ2 ψz dz. I= 2 U zz U U U z U z z z By (2.13) and (2.14), we have Z Z   Z   Z Vz 2 1 1 ρ1 2 |U |ϕ dz + ϕ dz − U ϕ ψ dz − ϕ2 ψz dz. (2.19) I =− z z z z z 3 U U U z U z z We then use Cauchy-Schwarz inequality and the boundedness of U, Uz and Uzz to derive that 1 h 1 i
α ϕ2zz −U (2.20) ϕz ψz ≤ − U ϕz ψ + + C |Uz |ψ 2 + ϕ2z . U z U z 2 U z (2.19) and (2.20) lead to Z Z Z
α ϕ2zz 2 2 I≤ dz + C |Uz |ψ + ϕz dz + C ϕ2z |ψz |dz, (2.21) 2 U with the help of the boundedness of U , Uz as well as Vz . Therefore, substituting (2.21) into (2.18), we have Z  2 Z 2 Z  1 d ϕz ϕzz α 2 + ψz2 dz + dz + β ψzz dz 2 dt U 2 U Z Z Z
≤C ψ 2 + ϕ2z dz + C ϕ2z |ψz |dz + C |ϕz ϕzz ψz |dz I=

−

ϕz ϕt +

for some positive constant C. Integrating the result with respect to t, we obtain Z Z tZ

2 ϕ2z + ψz2 dz + αϕ2zz + βψzz dzdτ 0   Z tZ   ≤ C kϕ0 k21 + kψ0 k21 + |ϕ| + |ϕz | + |ϕzz | |ϕz ψz |dzdτ ,

(2.22)

0

here we use the boundedness of U, Uz , Uzz as well as Vz again and inequality (2.11). The proof of Lemma 2.5 is finished.  Finally, we estimate the second order derivatives of (ϕ, ψ). Lemma 2.6 Assume that the hypotheses in Lemma 2.4 hold. Then there exists a positive constant C such that the solution (ϕ, ψ) of (2.2)–(2.3) satisfies Z t Z t 2 2 2 kϕzz (·, t)k + kψzz (·, t)k + α kϕzzz (·, τ )k dτ + β kψzzz (·, τ )k2 dτ 0 0   Z tZ
2 2 ≤ C kϕ0 k2 + kψ0 k2 + |ϕ| + |ϕz | + |ϕzz | |ϕz ψz |dzdτ 0

+C

Z tZ 0


|ϕzz | + |ϕzzz | |(ϕz ψz )z |dzdτ.

(2.23)

Proof Multiplying the first equation of (2.2) by 1/U , differentiating the resultant equation with respect to z twice, and differentiating the second equation of (2.2) with respect to z twice, we obtain  α  s + V  ϕ ψ  1  z z   ϕt = ϕzz + ϕz + ψzzz + , U U U U zz zz zz zz (2.24)   ψtzz = βψzzzz + sψzzz + ϕzzz .

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Now we multiply the first equation of (2.24) by ϕzz and the second equation by ψzz , integrate the results with respect to z and then add them to obtain     Z  2 Z   1 d ϕzz 1 1 2 + ψzz dz + ϕt + 2 ϕtz ϕzz dz 2 dt U U zz U z    Z  Z  Z  Z α s+V ϕz ψz = ϕzz ϕzz dz + ϕz ϕzz dz + ϕzz dz + β ψzzzz ψzz dz U U U zz zz zz =: T1 + T2 + T3 + T4 . (2.25) Since (ϕ, ψ) ∈ X(0, T ), by the integration by parts, we calculate that Z 2 Z   1 ϕzzz α T1 = −α dz + ϕ2 dz, U 2 U zz zz   Z  Z  3 1 s+V s+V 2 ϕz dz + ϕ2zz dz, T2 = − 2 U 2 U zzz z Z 2 T4 = −β ψzzz dz.

Substituting (2.26) into (2.25) gives rise to  Z  2 Z Z 2 1 d ϕzz ϕzzz 2 2 + ψzz dz + α dz + β ψzzz dz 2 dt U U    Z   1 1 + ϕt + 2 ϕtz ϕzz dz U zz U z      Z  Z   1 s+V s+V α 1 2 =− ϕz dz + 3 + ϕ2zz dz 2 U 2 U U zzz z zz  Z  ϕz ψz ϕzz dz. + U zz

(2.26)

(2.27)

Now we substitute the first equation of (2.2) into the last term on the left-hand side of (2.27) to get      1 1 ϕt + 2 ϕtz ϕzz U U z   zz           1 1 1 1 = α ϕ2zz ϕ2zz + (s + V ) +2 Vz ϕz ϕzz + 2(s + V ) U U z U zz U z z  z       1 1 1 −U ψz ϕzz + ϕz ψz ϕzz + 2 U ψz ϕzz U zz U zz U z z     1 1 −2U ψz ϕzzz + 2 ϕz ψz )z ϕzz . (2.28) U z U z Substituting (2.28) into (2.27) yields  Z  2 Z 2 Z 1 d ϕzz ϕzzz 2 2 + ψzz dz + α dz + β ψzzz dz 2 dt U U           Z Z 1 s+V 1 s+V α 1 2 =− ϕz dz + 3 + − 4(s + V ) ϕ2 dz 2 U 2 U U zz U z zz zzz z      Z  Z   1 1 1 − (s + V ) +2 Vz ϕz ϕzz dz + U ψz ϕzz dz U zz U z U zz Z   Z   Z   1 1 1 +2 U ψz ϕzzz dz − ϕz ψz ϕzz dz − 2 ϕz ψz )z ϕzz dz U z U zz U z

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Z 

ϕz ψz U



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ϕzz dz.

(2.29)

zz

Since Z 

 ϕz ψz ϕzz dz U zz Z   Z   Z 1 1 1 = ϕz ψz ϕzz dz + (ϕz ψz )z ϕzz dz − (ϕz ψz )z ϕzzz dz, U zz U z U

(2.30)

hence substituting (2.30) into (2.29) and integrating the result with respect to t, by the CauchySchwarz inequality, we deduce that Z Z Z tZ
2 2 ϕ2zz dz + ψzz dz + αϕ2zzz + βψzzz dzdτ 0 Z tZ Z

2 dz + C ϕ2z + ϕ2zz + ϕ2zzz + ψz2 dzdτ ≤C ϕ20zz + ψ0zz 0 Z tZ
+C |ϕzz | + |ϕzzz | |(ϕz ψz )z |dzdτ (2.31) 0

for some constant C > 0, where the boundedenss of U, Uz , Uzz as well as V and Vz is used. Therefore, the combination of (2.31) with (2.11) and (2.17) just gives (2.23), which completes the proof of Lemma 2.6.  Proof of Proposition 2.3 From Proposition 2.2, we see that it is only need to verify the a priori estimates (2.10) holds. In fact, by Lemmas 2.4–2.6, the solution of (2.2)–(2.3) satisfies Z t Z t kϕ(·, t)k22 + kψ(·, t)k22 + α kϕz (·, τ )k22 dτ + β kψz (·, τ )k22 dτ 0 0 Z tZ 2 +µ0 |Uz (z)|ϕ (z, τ )dzdτ 0   Z tZ
2 2 ≤ C kϕ0 k2 + kψ0 k2 + |ϕ| + |ϕz | + |ϕzz | |ϕz ψz |dzdτ 0

+C

Z tZ 0


|ϕzz | + |ϕzzz | |(ϕz ψz )z |dzdτ.

(2.32)

By the Sobolev embedding theorem, all of the nonlinear terms on the right-hand side of (2.32) Rt
can be bounded by CN (t) 0 kϕz (·, τ )k22 + kψz (·, τ )k22 dτ for some positive constant C. Hence Z t Z tZ
N 2 (t) + kϕz (·, τ )k22 + kψz (·, τ )k22 dτ + |Uz (z)|ϕ2 (z, τ )dzdτ 0 0 Z t
≤ CN 2 (0) + CN (t) kϕz (·, τ )k22 + kψz (·, τ )k22 dτ 0

1 for t ∈ [0, T ] and some positive constant C. Thus, let N (t) ≤ 2C , we obtain Z t Z tZ
N 2 (t) + kϕz (·, τ )k22 + kψz (·, τ )k22 dτ + |Uz (z)|ϕ2 (z, τ )dzdτ ≤ CN 2 (0) 0

0

for any t ∈ [0, T ], which completes the proof of Proposition 2.3.



Now we can use Proposition 2.2 and Proposition 2.3 to prove Theorem 2.1. Proof of Theorem 2.1 From Proposition 2.2 and Proposition 2.3, we see that if u+ > 0, there exists a constant δ0 > 0, such that if N (0) ≤ δ0 , the Cauchy problem (2.2)–(2.3) has

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a unique global solution (ϕ, ψ) ∈ X(0, +∞) satisfying (2.7) by the continuation argument. Furthermore, by the standard argument (see [20, 21]), we derive from the global estimate (2.7) that k(ϕz (·, t), ψz (·, t)k1 → 0 as t → +∞. Then for all z ∈ R, it follows that Z z 2 ϕz (z, t) = 2 ϕz ϕzz (y, t)dy −∞

≤2

Z

+∞

−∞

ϕ2z (y, t)dy

 21  Z

+∞

−∞

ϕ2zz (y, t)dy

 12

→ 0 as t → +∞.

(2.33)

Similarly, for all z ∈ R, ψz2 (z, t) → 0 as t → +∞.

(2.34)

Hence (2.8) holds from (2.33) and (2.34). The proof of Theorem 2.1 is finished.



Now we can prove Theorem 1.2 as follows. Proof of Theorem 1.2 By (2.1) and the definition of N (0) in (2.4), it holds that ku0 − U k1 + kv0 − V k1 + k(ϕ0 , ψ0 )k ≤ N (0) ≤ δ0 , thus by Theorem 2.1, the Cauchy problem (1.6)–(1.7) has a unique global solution (u, v)(x, t) satisfying
2 (u − U, v − V ) ∈ C([0, +∞); H 1 ) ∩ L2 ([0, +∞); H 1 ) ,

and the nonlinear asymptotic stability

sup |(u, v)(x, t) − (U, V )(x − st)| → 0 as t → +∞. x∈R

Furthermore, we know that u > 0 from Theorem 2.1. In fact, if the initial perturbation (2.3) satisfies (2.6), then by (2.7) there is a constant C > 0 such that √ |ϕz (z, t)| ≤ 2N (t) ≤ CN (0) ≤ Cδ0 . Then for all x ∈ R and t ≥ 0, it follows from (2.1) that
u(x, t) = u(x, t) − U (z) + U (z) = ϕz (z, t) + U (z) ≥ −CN (0) + u+ ≥ −Cδ0 + u+ > 0 provided that δ0 is small enough, which completes the proof of Theorem 1.2.

3



Proof of the Results of the Original System

In the final section, we transfer the results of the transformed problem (1.6) and (1.7) to the original problem (1.3) and (1.4). Proof of Theorem 1.3 By (1.26) and (1.27), Theorem 1.3 follows from Theorem 1.1 directly.  Proof of Theorem 1.4 can be decomposed by

From (1.5) and (2.1), we see that the solution of (1.3) and (1.4)

(ξ, θ)(x, t) = (Ξ, Θ)(x − st) + (ψx , −ϕx )(x, t),

(3.1)

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with initial conditions (ψ0 , −ϕ0 )(x) =

Z

x

−∞

Again by (1.30), we have


v0 − V, U − u0 (y)dy.

kθ − Θkp = ku − U kp , kξ − Ξkp = kv − V kp . Then it follows from (3.1)–(3.3) and Theorem 1.2 that the results of Theorem 1.4 hold.

(3.2)

(3.3) 

Acknowledgements The authors would like to express their thanks to Prof. Zhian Wang for his useful suggestions. References [1] Lorenz E N. Deterministic non-periodic flows. J Atmos Sci, 1963, 20: 130–141 [2] Curry J H, Herring J R, Loncaric J, Orszag S A. Order and disorder in two-and three-dimension B´ ernard convection. J Fluid Mech, 1984, 1: 1–38 [3] Hsieh D Y. On partial diffrential equations related to Lorenz system. J Math Phys, 1987, 28: 1589–1597 [4] Jian H Y, Chen D G. On the Cauchy problem for certain system of semilineaar parabolic equations. Acta Math Sinica English Ser, 1998, 14: 17–34 [5] Wang Z A. Large time profile of solutions for a dissipative nonlinear evolution system with conservational form. J Phys A: Math Gen, 2005, 38: 10955–10969 [6] Hsiao L, Jian H Y. Global smooth solutions to the spatically periodic Cauchy problem for dissipative non-linear evolution equations. J Math Anal Appl, 1997, 213: 262–274 [7] Tang S Q, Zhao H J. Nonlinear stability for dissipative nonlinear evolution equations with ellipticity. J Math Anal Appl, 1999, 233: 336–358 [8] Zhu C J, Wang Z A. Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity. Z Angew Math Phys, 2004, 55: 994–1014 [9] Duan R J, Zhu C J. Asymptotics of dissipative nonlinear evolution equations with ellipticity: Different and States. J Math Anal Appl, 2005, 303: 15–35 [10] Zhu C J, Zhang Z Y, Yin H. Convergence to diffusion waves for nonlinear evolution equations with ellipticity and damping, and with different end states. Acta Math Sin Engl Ser, 2006, 22: 1357–1370 [11] Wang Z A. Optimal decay rates to diffusion wave for nonlinear evolution equations with ellipticity. J Math Anal Appl, 2006, 319: 740–763 [12] Wang Z A. Optimal decay rates of solutions to dissipative nonlinear evolution equations with elliptitity. Z Angew Math Phys, 2006, 57: 399–418 [13] Jian H Y, Wang X P, Hsieh D H. The global attractor of a dissipative nonlinear evolution system. J Math Anal Appl, 1999, 238: 124–142 [14] Li T, Wang Z A. Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis. J Differential Equations, 2011, 250(3): 1310–1333 [15] Lax P. Hyperbolic systems of conservation laws ii. Comm Pure Appl Math, 1957, 10: 537–566 [16] Goodman J B. Nonlinear asymptotic stability of viscous shock profiles for conservation laws. ARMA, 1986, 95: 325–344 [17] Matsumura A, Nishihara K. On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J Appl Math, 1985, 2: 17–25 [18] Kawashima S, Matsumura A. Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Comm Math Phys, 1985, 101: 97–127 [19] Nishida T. Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics. Publ Math D’Orsay vol 78-02, D´ epartement de Math´ ematique, Universit´ e de Paris-Sud: Orsay, France, 1978 [20] Li T, Wang Z A. Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis. SIAM J Appl Math, 2009, 70: 1522–1541 [21] Wang Z A, Zhu C J. Stability of the rarefaction wave for the generalized kdv-Burgers equation. Acta Math Sci, 2002, 22B(3): 319–328