Global existence and decay rate of the Boussinesq–Burgers system with large initial data

J. Math. Anal. Appl. 439 (2016) 664–677

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Global existence and decay rate of the Boussinesq–Burgers system with large initial data Neng Zhu, Zhengrong Liu ∗ School of Mathematics, South China University of Technology, Guangzhou 510640, China

a r t i c l e

i n f o

Article history: Received 13 October 2015 Available online 9 March 2016 Submitted by H. Liu Keywords: Boussinesq–Burgers system Global existence Long-time behavior Algebraic decay rate Large initial data

a b s t r a c t In this paper, we study the global existence and the long-time asymptotic behavior of classical solutions to the Cauchy problem for the Boussinesq–Burgers system with large initial data. More precisely, we first show that the classical solutions exist globally in time with large initial data by using the Lp (p > 2) estimates other than the conventional L2 estimate. Then we prove that the global classical solutions converge to constant equilibrium states with an algebraic decay rate as time approaches infinity. © 2016 Published by Elsevier Inc.

1. Introduction and main results In this paper, we consider the following Boussinesq–Burgers system  ρt + wx + (wρ)x = ερxx , wt + ρx + wwx = δwxxt + μwxx ,

(1.1)

where ρ(x, t) and w(x, t) denote the height and the velocity of the free surface of the fluid above the bottom, respectively, δ > 0 is a parameter measuring the strength of the fluid dispersion, and ε, μ > 0 are the dissipative coefficients. This model was proposed in [17] to describe the propagation of bores. By contrast with the Korteweg–de Vries equation (KdV equation for short) and the Boussinesq system (see [3,4,17] and reference therein) which were both used to describe the propagation of weak bores, the KdV–Burgers equation and the Boussinesq–Burgers system not only contain the dispersion and nonlinearity but also incorporate the dissipation. The combination of the dispersion, nonlinearity and dissipation to predict the propagation of weak bores was suggested by the experimental results in [2,8,9]. The KdV–Burgers equation has been extensively studied for many years (see [7,20] and references therein). Also there have been some * Corresponding author. E-mail addresses: [email protected] (N. Zhu), [email protected] (Z. Liu). http://dx.doi.org/10.1016/j.jmaa.2016.03.018 0022-247X/© 2016 Published by Elsevier Inc.

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results about the Boussinesq–Burgers system (1.1). Under the variable transformation u(x, t) = 1 + ρ(x, t), the system (1.1) reduces to  ut + (uw)x = εuxx , wt + (u +

w2 2 )x

(1.2)

= δwxxt + μwxx .

For the bounded interval, Ding and Wang [6] established the global existence and the asymptotic behavior of classical solutions to the initial–boundary value problem for the Boussinesq–Burgers system (1.2) by utilizing a Lyapunov functional and the technique of Moser iteration. Jin and Liu [10] proved the global existence and the exponential decay rate of classical solutions to the Dirichlet boundary problem of the Boussinesq–Burgers system (1.2) by employing the Lp estimates. For the whole space R, when ε = μ > 0, the existence of traveling waves of the system (1.2) was verified in [17] with the bore-like initial data (u, w)(x, 0) = (u0 , w0 )(x) →

 (u+ , w+ ),

x → +∞,

(u− , w− ),

x → −∞.

(1.3)

However, the stability of traveling waves is left as an open problem. In this paper, we study the case of u ¯ = u+ = u− , w ¯ = w+ = w− as the first step. More precisely, our work is concerned with the analytical study of the Cauchy problem for the Boussinesq–Burgers system as follows ⎧ ⎪ ⎪ ⎨ut + (uw)x = εuxx ,

x ∈ R, t > 0,

w2 2 )x

= δwxxt + μwxx , wt + (u + ⎪ ⎪ ⎩(u, w)(x, 0) = (u , w )(x) → (¯ u, w), ¯ 0 0

x ∈ R, t > 0,

(1.4)

x → ∞.

When ε = μ = 0, the existence of classical solutions and weak solutions of system (1.2) has been established in [1] and [18], respectively. In general, due to high order nonlinearities, the long-time dynamics of large data classical solutions to the Cauchy problem is extremely difficult to deal with. The unboundedness of the domain makes the total mass infinite but still conserved, thus the special Sobolev-type inequality will no longer be valid for (1.4). This makes the approach developed in [6,10] for the initial–boundary value problem useless for the Cauchy problem. Inspired by ideas from [5,11,12,14,13,15,19], we will use the Lp (p > 2) estimates developed in [14] instead of the standard L2 -based energy estimate. Through these energy estimates, we will identify the global existence and the long-time asymptotic behavior of classical solutions of the Cauchy problem (1.4). In particular, we will show that the classical solutions converge to constant equilibrium states at an algebraic decay rate as time tends to infinity. The main results are stated as follows. ¯, w0 − w) ¯ ∈ H 1 (R) and u0 ≥ 0, Theorem 1.1 (Global existence and asymptotic behavior). Suppose (u0 − u u0 ≡ 0. Then for the Cauchy problem (1.4) there exists a unique global classical solution satisfying (u − u ¯, w − w) ¯ ∈ C([0, ∞); H 1 (R)) ∩ L2 ([0, ∞); H 2 (R)). Furthermore, the solution has the following asymptotic stability: sup |(u, w)(x, t) − (¯ u, w)| ¯ →0

as t → ∞.

x∈R

Remark 1.1. Since the bores are weak, it is physically meaningful to consider u(x, t) > 0.

(1.5)

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Remark 1.2. Unlike the bounded region case, when u ¯ = 0, the Poincaré inequality f L2 ≤ fx L2 cannot be useful to the energy estimate derived from the nonlinearity term for the first equation of the system in the whole space, then we cannot get uniformly bounded basic estimates independent of time. In order to obtain uniform basic estimates independent of time, we should consider the case of u ¯ > 0. Our next goal is to establish the explicit decay rate of classical solutions to (1.4). Unlike the bounded interval case, in which low frequencies of the system are often controlled by high frequencies leading to the exponential decay rate of the solution, the decay rate in the whole space is usually algebraic. To overcome this difficulty, we use the method proposed in [14] to study the explicit decay rate of classical solutions to (1.4). We define the following anti-derivatives: x

x (u(y, t) − u ¯)dy,

φ(x, t) = −∞

(w(y, t) − w)dy. ¯

ψ(x, t) =

(1.6)

−∞

By integrating the perturbed system with respect to x over (−∞, x], we obtain the following Cauchy problem: ⎧ ⎪ ¯ x+u ¯ψx = εφxx , φ + φx ψx + wφ ⎪ ⎨ t ⎪ ⎪ ⎩

ψt + φx +

2 ψx 2

+ wψ ¯ x = δψxxt + μψxx , x x ¯)dy, −∞ (w0 (y) − w)dy). ¯ (φ, ψ)(x, 0) = (φ0 , ψ0 )(x) = ( −∞ (u0 (y) − u

(1.7)

By studying the perturbed system (1.7), we have the following results. Theorem 1.2 (Algebraic decay rate). Assume that (φ0 , ψ0 )(x) ∈ H 2 (R) and there exists a constant ξ0 > 0 small enough such that φ0 2 + ψ0 2 + ψx0 2 ≤ ξ0 . Then there exists a constant C > 0 which is independent of time such that the solution obtained in Theorem 1.1 satisfies 2

 (1 + t)k ∂xk−1 (u(t) − u ¯) 2L2 + ∂xk−1 (w(t) − w) ¯ 2L2

k=1

+

2 

t

 (1 + τ )k ∂xk u(τ ) 2L2 + ∂xk w(τ ) 2L2 dτ ≤ C.

(1.8)

k=1 0

Especially, it holds that (u(x, t) − u ¯, w(x, t) − w) ¯ L∞ ≤ C(1 + t)− 8 . 3

2. Proof of Theorem 1.1 In this section, we prove Theorem 1.1. We seek the solution of (1.4) in the space: X(0, T ) = {(u, w) : (u − u ¯, w − w) ¯ ∈ C([0, T ]; H 1 (R)), (ux , wx ) ∈ L2 ((0, T ); H 1 (R))}. Thus Theorem 1.1 is a consequence of the following proposition.

(1.9)

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Proposition 2.1. There exists a unique global solution (u, w) ∈ X(0, ∞) to (1.4) such that ∞ (u − u ¯, w −

w) ¯ 2H 1

(ux , wx )(τ ) 2H 1 dτ ≤ C,

+ 0

where C > 0 is a constant independent of t. Next, we prove Proposition 2.1 by combining a unique local solution and the a priori estimates. The construction on the local existence of solutions is standard, and is based on an iteration argument and the fixed point theorem (cf. [16]). We omit the details for brevity. We give the following local existence of classical solutions to (1.4). ¯, w0 − w) ¯ ∈ H 1 (R) and u0 ≥ 0, u0 ≡ 0. Then there exists a Lemma 2.2 (Local existence). If (u0 − u positive constant T0 such that the Cauchy problem (1.4) admits a unique classical solution (u, w) ∈ X(0, T0 ) satisfying (u(·, t) − u ¯, w(·, t) − w) ¯ H 1 ≤ 2 (u0 − u ¯, w0 − w) ¯ H1 ,

for all 0 ≤ t ≤ T0 .

(2.1)

To prove the Proposition 2.1, we should establish the a priori estimates of the local solution. Lemma 2.3 ( A priori estimates). Suppose the Cauchy problem (1.4) has a solution (u, w) ∈ X(0, T ) for some T > 0. Then there exists a constant C > 0 independent of T such that t (u − u ¯, w −

w) ¯ 2H 1

(ux , wx )(τ ) 2H 1 dτ ≤ C.

+

(2.2)

0

2.1. Proof of the a priori estimates In this subsection, we prove Lemma 2.3 to complete the proof of Theorem 1.1. Firstly, we show the basic energy estimates derived from the Lyapunov functional, and denote ci as a positive constant independent of time. Lemma 2.4 (Basic estimates). Assume that (u, w) is the solution to (1.4) with u ¯ > 0. Then for all t ∈ (0, Tmax ), the classical solution (u, w) to (1.4) satisfies t w −

w ¯ 2L2

+

δ wx 2L2

t  wx (τ ) 2L2 dτ

+μ 0

≤ C( u0 −

u ¯ 2L2

+ w0 −

+ε 0

w ¯ 2L2

+

R

u2x dxdτ u

δ wx0 2L2 ),

(2.3)

where C is a positive constant independent of t. Proof. Multiplying the first equation of (1.4) by ln u −ln u ¯ and integrating by parts with respect to x over R, we have ⎞ ⎛   2  d ⎝ ux  ⎠ dx = 0, (2.4) η(u) − η(¯ u) − η (¯ u)(u − u ¯)dx − ux wdx + ε dt u R

where η(z) =

z 0

ln θ dθ.

R

R

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Letting u ˜=u−u ¯ and w ˜ = w − w, ¯ and substituting u ˜ and w ˜ into system (1.4), we obtain 

uw) ˜ x + w˜ ¯ ux + u ¯w ˜x = ε˜ uxx , u ˜t + (˜

(2.5)

2

˜x + ( w˜2 )x + w ¯w ˜x = δ w ˜xxt + μw ˜xx . w ˜t + u Multiplying the second equation of (2.5) by w ˜ and integrating by parts, we get 1 d ( w ˜ 2L2 + δ w ˜x 2L2 ) − 2 dt



 w ˜x2 dx = 0.

w ˜x u ˜dx + μ R

(2.6)

R

The combination of (2.4) and (2.6) gives ⎞

⎛



d ⎝ dt

R

1 δ ˜ 2L2 + w ˜x 2L2 ⎠ + μ w η(u) − η(¯ u) − η (¯ u)(u − u ¯)dx + w ˜x 2L2 + ε 2 2 

 R

u2x dx = 0, u

(2.7)

where 

 ux wdx + R

 w ˜x u ˜dx = w ¯

R

u ˜x dx = 0 R

is used. For any t > 0, integrating (2.7) over [0, t], we derive  R

1 δ ˜ 2L2 + w ˜x 2L2 + μ η(u) − η(¯ u) − η (¯ u)(u − u ¯)dx + w 2 2 

 = R

t

t  w ˜x (τ ) 2L2 dτ

+ε

0

0

R

u2x dxdτ u

1 δ ˜x0 2L2 . η(u0 ) − η(¯ u) − η  (¯ u)(u0 − u ¯)dx + w˜0 2L2 + w 2 2

(2.8)

Since u > 0 and u ¯ > 0, there exists a u∗ satisfying 0 < u∗ ∈ (min{u, u ¯}, max{u, u ¯}) such that η(u) − η(¯ u) − η  (¯ u)(u − u ¯) =

η  (u∗ ) (u − u ¯)2 (u − u ¯ )2 = ≥ 0. ∗ 2 2u

(2.9)

On the other hand, due to the conditions 0 < u ¯ < ∞ and u0 ≥ 0, it holds that η(u0 ) − η(¯ u) − η  (¯ u)(u0 − u ¯) =

(u0 − u ¯ )2 ≤ c1 (u0 − u ¯ )2 . ∗ 2u

(2.10)

Substituting (2.9) and (2.10) into (2.8), one has (2.3). Thus the proof of Lemma 2.4 is completed. 2 ¯ > 0. Then it follows that Lemma 2.5 (L2 -estimate). Assume that (u, w) is the solution to (1.4) with u t ˜ u 2L2

+

w ˜ 2L2

+

w ˜x 2L2

( ˜ ux (τ ) 2L2 + w ˜x (τ ) 2L2 )dτ ≤ C,

+ 0

where C is a positive constant independent of t.

(2.11)

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Proof. Firstly, multiplying the first equation of (2.5) by u ˜ and the second equation by u ¯w, ˜ adding the results and integrating by parts with respect to x, we end up with 1 d 2 2 2 2 2 ( ˜ u L2 + u ¯ w ˜ L2 + δ u ¯ w ˜x L2 ) + ε ˜ ux L2 + μ¯ u w ˜ x L2 = 2 dt

 u ˜w˜ ˜ ux dx.

(2.12)

R

Secondly, we multiply the first equation of (2.5) by u ˜2 and integrate by parts to derive ⎛ ⎞     1 d ⎝ u ˜3 dx⎠ + 2ε u ˜u ˜2x dx = 2 u ˜2 w˜ ˜ ux dx + 2¯ u u ˜w˜ ˜ ux dx. 3 dt R

R

R

(2.13)

R

Thirdly, multiplying the first equation of (2.5) by with u ˜3 and integrating by parts, one has 1 d ˜ u 4L4 + 3ε ˜ uu ˜x 2L2 = 3 4 dt



 u ˜3 w˜ ˜ ux dx + 3¯ u

R

u ˜2 w˜ ˜ ux dx.

(2.14)

R

Then the operation (2.12) × 2¯ u − (2.13) gives ⎛ 1 d ⎝ ¯2 w ˜ 2L2 + δ u ¯ 2 w ˜x 2L2 − u ¯ ˜ u 2L2 + u dt 3  = −2



⎞ u ˜3 dx⎠ + 2ε¯ u ˜ ux 2L2 + 2μ¯ u2 w ˜x 2L2 − 2ε

R

 u ˜u ˜2x dx R

u ˜2 w˜ ˜ ux dx.

(2.15)

R

Adding (2.14) to (2.15) × 32 u ¯, we obtain d G(t) + K(t) = 3 dt

 u ˜3 w˜ ˜ ux dx,

(2.16)

R

where 3¯ u2 3¯ u3 3δ u ¯3 u ¯ ˜ u 2L2 + w ˜ 2L2 + w ˜x 2L2 − G(t) = 2 2 2 2 =u ¯2 ˜ u 2L2

 R

1 u 4L4 u ˜3 dx + ˜ 4

1 1 3¯ u3 3δ u ¯3 uu ˜−u ˜2 2L2 + ˜ u 4L4 + w ˜ 2L2 + w ˜x 2L2 , + 2¯ 8 8 2 2

(2.17)

and  2

K(t) = 3ε¯ u

˜ ux 2L2

+ 3μ¯ u

3

w ˜x 2L2

− 3ε¯ u

u ˜u ˜2x dx + 3ε ˜ uu ˜x 2L2 R

2

=

3ε¯ u 3ε 3ε ˜ ux 2L2 + ¯ uu ˜x − u uu ˜x 2L2 + 3μ¯ ˜u ˜x 2L2 + ˜ u3 w ˜x 2L2 . 2 2 2

Utilizing the Cauchy–Schwarz inequality and w ˜ 2L2 ≤ C in (2.3), we have        ε 9 2 2 3 3 u  ≤ ˜ ˜x 2L2 + ˜ u w ˜ w˜ ˜ u dx ˜ L2 x   2 uu 2ε   R

(2.18)

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ε 9 ˜ uu ˜x 2L2 + ˜ u 4L∞ w ˜ 2L2 2 2ε ε uu ˜x 2L2 + c2 ˜ ≤ ˜ u 4L∞ . 2 ≤

(2.19)

Substituting (2.19) into (2.16) and using (2.17) and (2.18), we get 3ε¯ u2 3ε d G(t) + ˜ ux 2L2 + ¯ uu ˜x − u ˜u ˜x 2L2 + ε ˜ uu ˜x 2L2 + 3μ¯ u3 w ˜x 2L2 ≤ c3 ˜ u 4L∞ . dt 2 2

(2.20)

For the estimate of ˜ u 4L∞ on the right hand side of (2.20), using the Hölder’s inequality and the Cauchy– Schwarz inequality, we have x u ˜4 (x, t) = 4

⎞ 12 u ˜2x dx⎠ u ˜+u ¯

⎛ ⎞ 12 ⎛   3 6 ⎝ ⎠ ⎝ u ˜ u ˜x dx ≤ 4 u ˜ (˜ u+u ¯)dx

−∞

R

R

⎞ 12 ⎛

⎛



≤ 4 ˜ u 2L∞ ⎝

R



u ˜2 (˜ u+u ¯)dx⎠ ⎝

R

u ˜2x u ˜+u ¯

⎛

⎞⎛

1 u 4L∞ + 8 ⎝ ≤ ˜ 2

u ˜2 (˜ u+u ¯)dx⎠ ⎝

 R

 R

⎞ 12 dx⎠

⎞ u ˜2x dx⎠ u ˜+u ¯

⎞ ⎛  2 1 u ˜ x ≤ ˜ u 4L∞ + (4 ˜ dx⎠ , u 4L4 + (4 + 8¯ u) ˜ u 2L2 ) ⎝ 2 u ˜+u ¯ R

which implies ⎛



˜ u 4L∞ ≤ (8 ˜ u 4L4 + (8 + 16¯ u) ˜ u 2L2 ) ⎝

R

⎞ u ˜2x dx⎠ . u ˜+u ¯

(2.21)

Substituting (2.21) into (2.20) and using (2.17), it follows that 3ε¯ u2 3ε d G(t) + ˜ ux 2L2 + ¯ uu ˜x − u ˜u ˜x 2L2 + ε ˜ uu ˜x 2L2 + 3μ¯ u3 w ˜x 2L2 dt 2 2 ⎞ ⎛  2 u ˜ x dx⎠ . ≤ c3 (8 ˜ u 4L4 + (8 + 16¯ u) ˜ u 2L2 ) ⎝ u ˜+u ¯

(2.22)

R

By the definition of (2.17), one can easily see c3 (8 ˜ u 4L4 + (8 + 16¯ u) ˜ u 2L2 ) ≤ c4 G(t). Hence we rewrite (2.22) as 3ε¯ u2 3ε d G(t) + ˜ ux 2L2 + ¯ uu ˜x − u ˜u ˜x 2L2 + ε ˜ uu ˜x 2L2 + 3μ¯ u3 w ˜x 2L2 dt 2 2 ⎞ ⎛  2 u ˜x dx⎠ . ≤ c4 G(t) ⎝ u ˜+u ¯ R

(2.23)

N. Zhu, Z. Liu / J. Math. Anal. Appl. 439 (2016) 664–677

Applying the Gronwall’s inequality to (2.23) and using

t 0

u2x dxdτ R u

671

≤ C in (2.3), we get

1 δ ˜0 2L2 + w ˜x0 2L2 )}. G(t) ≤ G(0) exp{c4 (C u0 − u ¯ 2L2 + w 2 2

(2.24)

Substituting (2.24) into (2.23) and integrating over [0, t] for any t > 0, we obtain t  G(t) +

 3ε 3ε¯ u2 ˜ ux (τ ) 2L2 + (¯ uu ˜x − u ˜u ˜x )(τ ) 2L2 + ε ˜ uu ˜x (τ ) 2L2 + 3μ¯ u3 w ˜x (τ ) 2L2 dτ ≤ c5 . (2.25) 2 2

0

Substituting (2.17) into (2.25) gives (2.11). Therefore, we complete the proof of Lemma 2.5. 2 Lemma 2.6 (H 1 -estimate). Assume that (u, w) is the solution to (1.4). Then it holds that t ˜ ux 2L2

+

w ˜x 2L2

+

w ˜xx 2L2

( ˜ uxx (τ ) 2L2 + w ˜xx (τ ) 2L2 )dτ ≤ C,

+

(2.26)

0

where C is a positive constant independent of t. Proof. Taking spatial derivatives of (2.5), we have 

uw) ˜ xx + w˜ ¯ uxx + u ¯w ˜xx = ε˜ uxxx , u ˜xt + (˜ 2

˜xx + ( w˜2 )xx + w ¯w ˜xx = δ w ˜xxxt + μw ˜xxx . w ˜xt + u

(2.27)

Multiplying the first equation of (2.27) by u ˜x and the second equation by u ¯w ˜x , adding the results and integrating by parts with respect to x, we obtain 1 d ( ˜ ux 2L2 + u ¯ w ˜x 2L2 + δ u ¯ w ˜xx 2L2 ) + ε ˜ uxx 2L2 + μ¯ u w ˜xx 2L2 2 dt   = (˜ uw) ˜ xu ˜xx dx + u ¯ w ˜w ˜x w ˜xx dx. R

(2.28)

R

Next, we estimate the right hand side of (2.28). Using the Cauchy–Schwarz inequality and the Sobolev inequality f 2L∞ ≤ 2 f L2 fx L2 and the estimate ˜ u 2L2 + w ˜ 2L2 + w ˜x 2L2 ≤ C (see (2.11)), we have       ε 1  (˜ uxx 2L2 + (˜ uw) ˜ x 2L2 ˜ xu ˜xx dx ≤ ˜  uw) 2ε   2 R

≤ ≤ ≤ ≤ ≤ and

ε ˜ uxx 2L2 2 ε ˜ uxx 2L2 2 ε ˜ uxx 2L2 2 ε ˜ uxx 2L2 2 ε ˜ uxx 2L2 2

1 u 2L∞ w + ( ˜ ˜x 2L2 + w ˜ 2L∞ ˜ ux 2L2 ) ε 2 u L2 ˜ + ( ˜ u x L2 w ˜x 2L2 + w ˜ L2 w ˜x L2 ˜ ux 2L2 ) ε + c6 ˜ u x L2 w ˜x L2 ( ˜ u x L2 + w ˜ x L2 ) + c7 ˜ ux 2L2 w ˜x 2L2 + c8 ( ˜ ux 2L2 + w ˜x 2L2 ) + c9 ( ˜ ux 2L2 + w ˜x 2L2 ),

(2.29)

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       μ¯ u ¯ u  ≤ u w ¯ ˜xx 2L2 + w ˜ 2L∞ w w ˜ w ˜ w ˜ dx ˜x 2L2 x xx   2 2μ   R

≤

μ¯ u w ˜xx 2L2 + c10 w ˜x 2L2 . 2

(2.30)

Substituting (2.29) and (2.30) into (2.28), we end up with d ( ˜ ux 2L2 + u ¯ w ˜x 2L2 + δ u ¯ w ˜xx 2L2 ) + ε ˜ uxx 2L2 + μ¯ u w ˜xx 2L2 ≤ c11 ( ˜ ux 2L2 + w ˜x 2L2 ). (2.31) dt t After integrating (2.31) over [0, t] and using 0 ( ˜ ux (τ ) 2L2 + w ˜x (τ ) 2L2 )dτ ≤ C in (2.11), we obtain (2.26). Hence we complete the proof of Lemma 2.6. 2 Lemma 2.3 is a direct consequence of the combination of Lemmas 2.4–2.6. Then the proof of the a priori estimates in Lemma 2.3 is completed. Thus Proposition 2.1 is proved by using a unique local solution in Lemma 2.2 and the a priori estimates obtained in Lemma 2.3. With the above results, we are now in a position to prove Theorem 1.1. 2.2. Proof of Theorem 1.1 Proposition 2.1 can be obtained by combining the existence of local solutions and the a priori estimates obtained in Lemma 2.3. Theorem 1.1 is a consequence of the Proposition 2.1. This completes the proof of the global existence in Theorem 1.1. Next, we prove (1.5). In fact, from (2.2), one has (u(·, t) − u ¯, w(·, t) − w) ¯ L2 ≤ C and (ux , wx )(·, t) L2 → 0 as t → ∞. Hence, for all x ∈ R, it holds that x (u(x, t) − u ¯) = 2

(u(y, t) − u ¯)(u(y, t) − u ¯)y dy

2

−∞

⎞ 12 ⎛ ⎞ 12 ⎛   ¯)2 dy ⎠ ⎝ u2y dy ⎠ ≤ 2 ⎝ (u(y, t) − u R

R

≤ 2C ux (·, t) L2 → 0 as t → ∞, which implies sup |u(x, t) − u ¯| → 0 as t → ∞. Similarly, we can prove sup |w(x, t) − w| ¯ → 0 as t → ∞. Thus x∈R

x∈R

(2.2) is proved and the proof of Theorem 1.1 is completed. 3. Proof of Theorem 1.2 This section is devoted to studying the decay rate of the solution obtained in Theorem 1.1. In fact, we only need to study the transformed system (1.7). Following the standard argument, we carry out energy estimates under the a priori assumption: sup 0≤t≤T



 φ(t) 2L2 + ψ(t) 2L2 ≤ ξ.

In the following derivations, we denote Ci as a positive constant which is independent of time.

(3.1)

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Lemma 3.1 (L2 -estimate). Assume that (φ, ψ) is the solution to the Cauchy problem of (1.7). Then it holds that t φ 2L2 + u ¯ ψ 2L2 + u ¯δ ψx 2L2 +

( φx (τ ) 2L2 + ψx (τ ) 2L2 )dτ 0

≤

C( φ0 2L2

+

u ¯ ψ0 2L2

+u ¯δ ψx0 2L2 ),

(3.2)

where C is a positive constant independent of t. ¯ψ, adding the Proof. Multiplying the first equation of (1.7) by φ and the second equation of (1.7) by u results and integrating by parts with respect to x, we have 1 d ( φ 2L2 + u ¯ ψ 2L2 + u ¯δ ψx 2L2 ) + ε φx 2L2 + μ¯ u ψx 2L2 2 dt   u ¯ = − φx ψx φdx − ψψx2 dx 2 R

R

u ¯ ψ L∞ ψx 2L2 2 √ √ 1 1 1 1 2¯ u 2 2 ψ L2 2 ψx L2 2 ψx 2L2 ≤ 2 φ L2 φx L2 φx L2 ψx L2 + 2 ≤ φ L∞ φx L2 ψx L2 +

1

1

≤ C1 ξ 4 φx L2 ψx L2 + C2 ξ 4 ψx 2L2 1

≤ C3 ξ 4 ( φx 2L2 + ψx 2L2 ),

(3.3)

where we have used the definition of φ and ψ, and φx L2 = u − u ¯ L2 ≤ C and ψx L2 = w − w ¯ L2 ≤ C from Theorem 1.1, and the Hölder’s inequality and the a priori assumption (3.1). By choosing ξ sufficiently small, we update (3.3) as 1 d ( φ 2L2 + u ¯ ψ 2L2 + u ¯δ ψx 2L2 ) + C4 ( φx 2L2 + ψx 2L2 ) ≤ 0, 2 dt

(3.4)

which implies that (3.2) holds true for any time if φ0 2L2 + ψ0 2L2 + ψx0 2L2 is sufficiently small. Then the proof of Lemma 3.1 is completed. 2 Remark 3.1. If the initial data φ0 2L2 + ψ0 2L2 + ψx0 2L2 is sufficiently small, then from Lemma 3.1, we obtain φ(t) 2L2 + ψ(t) 2L2 ≤

ξ . 2

Hence the a priori estimate (3.1) is closed. Next, we carry out the time-weighted energy estimates and get the algebraic decay rate by the progressive argument. Lemma 3.2 (H 1 -estimate). Assume that (φ, ψ) is the solution to the Cauchy problem of (1.7). Then it follows that

N. Zhu, Z. Liu / J. Math. Anal. Appl. 439 (2016) 664–677

674



(1 + t)

φx 2L2

+

u ¯ ψx 2L2

+

u ¯δ ψxx 2L2



t +

 (1 + τ ) φxx (τ ) 2L2 + ψxx (τ ) 2L2 dτ ≤ C,

(3.5)

0

where C is a positive constant independent of t. Proof. Taking spatial derivatives of (1.7), we have 

¯ xx + u ¯ψxx = εφxxx , φxt + φxx ψx + φx ψxx + wφ ψxt + φxx + (

2 ψx 2 )x

+ wψ ¯ xx = δψxxxt + μψxxx .

(3.6)

Multiplying the first equation of (3.6) by (1 + t)φx and the second equation of (3.6) by u ¯(1 + t)ψx , adding the results and integrating by parts with respect to x, we see that 

1 d ¯ ψx 2L2 + δ u ¯ ψxx 2L2 + ε(1 + t) φxx 2L2 + μ¯ u(1 + t) ψxx 2L2 (1 + t) φx 2L2 + u 2 dt   1

= φx 2L2 + u ¯ ψx 2L2 + δ u ¯ ψxx 2L2 − (1 + t) (φx ψx )x φx dx. 2

(3.7)

R

Using the Cauchy–Schwarz inequality and the Gagliardo–Nirenberg type interpolation inequality fx L4 ≤ 1

1

C f L2 ∞ fxx L2 2 , and the estimate φx L2 = u − u ¯ L2 ≤ C from Theorem 1.1, and the a priori assumption (3.1), we can estimate the last term on the right hand side of (3.7). Thus it holds that         −(1 + t) (φx ψx )x φx dx ≤ (1 + t) |φ2x ψxx |dx   2   R

R

μ¯ u(1 + t) (1 + t) ≤ ψxx 2L2 + φx 4L4 2 8μ¯ u μ¯ u(1 + t) ψxx 2L2 + C5 (1 + t) φ 2L∞ φxx 2L2 2 μ¯ u(1 + t) ψxx 2L2 + C6 (1 + t) φ L2 φx L2 φxx 2L2 ≤ 2 1 μ¯ u(1 + t) ψxx 2L2 + C7 (1 + t)ξ 2 φxx 2L2 . ≤ 2 ≤

(3.8)

Choosing ξ sufficiently small and combining (3.7) and (3.8), we end up with 

d ¯ ψx 2L2 + δ u ¯ ψxx 2L2 + C8 (1 + t)( φxx 2L2 + ψxx 2L2 ) (1 + t) φx 2L2 + u dt ≤ C9 ( φx 2L2 + u ¯ ψx 2L2 + δ u ¯ ψxx 2L2 ).

(3.9)

Integrating (3.9) over time and using (3.2) and Theorem 1.1, we obtain (3.5). Thus the proof of Lemma 3.2 is completed. 2 Lemma 3.3 (H 2 -estimate). Assume that (φ, ψ) is the solution to the Cauchy problem of (1.7). Then it holds that

 (1 + t) φxx 2L2 + u ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 +

t

2

0

where C is a positive constant independent of t.

 (1 + τ )2 φxxx (τ ) 2L2 + ψxxx (τ ) 2L2 dτ ≤ C, (3.10)

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Proof. Taking spatial derivatives of (3.6), we have 

¯ xxx + u ¯ψxxx = εφxxxx , φxxt + (φx ψx )xx + wφ ψxxt + φxxx + (

2 ψx 2 )xx

(3.11)

+ wψ ¯ xxx = δψxxxxt + μψxxxx .

Multiplying the first equation of (3.11) by (1 +t)φxx and the second equation of (3.11) by u ¯(1 +t)ψxx , adding the results and integrating by parts with respect to x, and employing φx 2L∞ ≤ 2 φx L2 φxx L2 ≤ C and ψx 2L∞ ≤ 2 ψx L2 ψxx L2 ≤ C from Theorem 1.1, we have 



1 d 2 ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 + (1 + t) ε φxxx 2L2 + μ¯ u ψxxx 2L2 (1 + t) φxx L2 + u 2 dt   ψ2 1 ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 ) − (1 + t) (φx ψx )xx φxx dx − u ¯(1 + t) ( x )xx ψxx dx = ( φxx 2L2 + u 2 2 R

1 = ( φxx 2L2 + u ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 ) + (1 + t) 2  +u ¯(1 + t)

R

 (φxx ψx + φx ψxx )φxxx dx R

ψx ψxx ψxxx dx R

1 ε(1 + t) μ¯ u(1 + t) ≤ ( φxx 2L2 + u φxxx 2L2 + ψxxx 2L2 ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 ) + 2 2 2 + C10 (1 + t)( φx 2L∞ ψxx 2L2 + ψx 2L∞ φxx 2L2 ) + C11 (1 + t) ψx 2L∞ ψxx 2L2 ≤

1 ε(1 + t) μ¯ u(1 + t) ( φxx 2L2 + u φxxx 2L2 + ψxxx 2L2 ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 ) + 2 2 2 + C12 (1 + t)( φxx 2L2 + ψxx 2L2 ),

(3.12)

which implies 



d ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 + (1 + t) ε φxxx 2L2 + μ¯ u ψxxx 2L2 (1 + t) φxx 2L2 + u dt ≤ ( φxx 2L2 + u ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 ) + C13 (1 + t)( φxx 2L2 + ψxx 2L2 ).

(3.13)

Integrating (3.13) over time and using (3.5) and Theorem 1.1, we update (3.13) as

(1 + t)

φxx 2L2

+

u ¯ ψxx 2L2

+

δu ¯ ψxxx 2L2



t (1 + τ )( φxxx (τ ) 2L2 + ψxxx (τ ) 2L2 )dτ ≤ C14 .

+

(3.14)

0

Combining (3.5) and (3.14), and using the Sobolev inequality f 2L∞ ≤ 2 f L2 fx L2 , we obtain φx 2L∞ ≤ C(1 + t)−1 and ψx 2L∞ ≤ C(1 + t)−1 .

(3.15)

Similarly, we get



 1 d ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 + (1 + t)2 ε φxxx 2L2 + μ¯ u ψxxx 2L2 (1 + t)2 φxx 2L2 + u 2 dt ε(1 + t)2 μ¯ u(1 + t)2 (1 + t) ( φxx 2L2 + u φxxx 2L2 + ψxxx 2L2 ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 ) + ≤ 2 2 2 + C15 (1 + t)2 ( φx 2L∞ ψxx 2L2 + ψx 2L∞ φxx 2L2 ) + C16 (1 + t)2 ψx 2L∞ ψxx 2L2 . (3.16)

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Substituting (3.15) into (3.16), one has



 d (1 + t)2 φxx 2L2 + u ¯ ψxx 2L2 + δ u ¯ ψxxx 2L2 + (1 + t)2 ε φxxx 2L2 + μ¯ u ψxxx 2L2 dt ≤ C17 (1 + t)( φxx 2L2 + ψxx 2L2 + ψxxx 2L2 ).

(3.17)

Thus the combination of (3.5), (3.14) and (3.17) gives (3.10). The proof of Lemma 3.3 is completed. 2 3.1. Proof of Theorem 1.2 The combination of Lemma 3.2 and Lemma 3.3 gives the algebraic decay rate of the solution to the Cauchy problem (1.7). Using the definition of anti-derivatives in (1.6), we can derive the algebraic decay rate (1.8) of classical solutions to the Cauchy problem (1.4). From (1.6), (3.5) and (3.10), we obtain u(x, t) − u ¯ L2 ≤ C18 (1 + t)− 2 , 1

and (u(x, t) − u ¯)x L2 ≤ C19 (1 + t)− 4 , 1

which implies u(x, t) − u ¯ L∞ ≤ ≤

√ √

1

1

2 u(x, t) − u ¯ L2 2 (u(x, t) − u ¯)x L2 2 2 C18 (1 + t)− 4 C19 (1 + t)− 8 1

1

≤ C20 (1 + t)− 8 , 3

where we have used the Hölder’s inequality. Similarly, we can prove w(x, t) − w ¯ L∞ ≤ C21 (1 + t)− 8 . Therefore (1.9) is proved and the proof of Theorem 1.2 is completed. 3

Acknowledgments We would like to express our sincere gratitude to the anonymous referee for the detailed and helpful comments and suggestions. We would also like to express our sincere thanks to Dr. Hai-Yang Jin for his valuable suggestions which have improved the paper. This work is supported by the National Natural Science Foundation of China (No. 11571116). References [1] C.J. Amick, Regularity and uniqueness of solutions to the Boussinesq system of equations, J. Differential Equations 54 (1984) 231–247. [2] J.L. Bona, W.G. Pritchard, L.R. Scott, An evaluation of a model equation for water waves, Philos. Trans. R. Soc. Lond. Ser. A 302 (1981) 457–510. [3] M. Chen, C.W. Curtis, B. Deconinck, C.W. Lee, N. Nguyen, Spectral stability of stationary solutions of a Boussinesq system describing long waves in dispersive media, SIAM J. Appl. Dyn. Syst. 9 (2010) 999–1018. [4] M. Chen, O. Goubet, Long-time asymptotic behavior of dissipative Boussinesq systems, Discrete Contin. Dyn. Syst. 17 (2007) 509–528. [5] C.M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Ration. Mech. Anal. 29 (1968) 241–271. [6] W. Ding, Z.A. Wang, Global existence and asymptotic behavior of the Boussinesq–Burgers system, J. Math. Anal. Appl. 424 (2015) 584–597. [7] R. Duan, H.J. Zhao, Global stability of strong rarefaction waves for the generalized KdV–Burgers equation, Nonlinear Anal. 66 (2007) 1100–1117.

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