Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations

Nonlinear Analysis 73 (2010) 1729–1738

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Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations Hui Yin a,∗ , Jiayi Hu b a

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

b

Department of Mathematics, South-Central University for Nationalities, Wuhan 430074, China

article

info

Article history: Received 17 August 2008 Accepted 27 April 2010 Keywords: Generalized Benjamin–Bona–Mahony–Burgers equation Traveling wave Exponential decay rate Space–time weighted energy method

abstract In this paper, we investigate the exponential time decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations ut − utxx − ν uxx + β ux + f (u)x = 0,

t > 0, x ∈ R

(E)

with prescribed initial data u(0, x) = u0 (x) → u± ,

as x → ±∞.

(I)

Here ν(> 0), β ∈ R are constants, u± are two given constants satisfying u+ 6= u− and the nonlinear function f (u) ∈ C 2 (R) is assumed to be either convex or concave. Based on the existence of traveling waves, the local stability and the algebraic decay rate to traveling waves of solutions to the Cauchy problem (E) and (I) established in Yin et al. (2007) [13], we show an exponential decay rate of the solutions to the Cauchy problem (E) and (I) toward the traveling waves mentioned above, by employing the space–time weighted energy method which was initiated by Kawashima and Matsumura in (1985) [14] and later elaborated by Matsumura and Nishihara (1994) [15] and Nishikawa (1998) [16]. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction When attempting to describe the propagation of small-amplitude long waves in a nonlinear dispersive media, it is frequently necessary to take account of dissipative mechanisms to accurately reflect real situations. Oftentimes the mechanisms leading to the degradation of the wave are quite complex and not well understood. In such cases one may be forced to rely upon ad hoc models of dissipation; cf. [1]. One of the equations that has gained some currency when the need to append dissipation to nonlinearity and dispersion arises in modeling unidirectional propagation of planar waves is ut − γ utxx − ν uxx + β ux + uux = 0,

(1.1)

where γ > 0, ν > 0 and β ∈ R are fixed constants and u = u(t , x) is a real-valued function of the two real variables t and x, which, in applications, are typically proportional to elapsed time and to distance in the direction of propagation, respectively. When ν = 0, γ = 1, (1.1) is the so-called regularized long-wave (RLW) equation proposed by Peregrine [2] and Benjamin et al. [3], and is called the Benjamin–Bona–Mahony (BBM) equation. This equation features a balance between the nonlinear

∗

Corresponding author. E-mail address: [email protected] (H. Yin).

0362-546X/$ – see front matter Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.04.078

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H. Yin, J. Hu / Nonlinear Analysis 73 (2010) 1729–1738

dispersive effect but takes no account of dissipation. If ν > 0, since the dispersive effect of (1.1) is the same as the BBM equation ut − γ utxx + β ux + uux = 0, while the dissipative effect is the same as the Burgers equation ut − ν uxx + β ux + uux = 0, (1.1) is called the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation. The BBM–Burgers equation (1.1) has been the subject of numerous investigations (a complete list of the literature in this direction is beyond the scope of this manuscript; the references cited in [4,3,1,5–7,2,8–11] or the text of Whitham [12] can help the interested reader into the literature). To go directly to the theme of this manuscript, we only review some former results closely related: For the pure initial-value problem for Eq. (1.1), i.e. to ask for a solution u(t , x) of (1.1) defined for (t , x) ∈ R+ × R, having a specified initial configuration u(0, x) = u0 (x) → u± ,

as x → ±∞

(1.2)

for x ∈ R, some excellent results concerning the existence of global smooth solution to the Cauchy problem (1.1), (1.2) and the large time behavior of the global smooth solution obtained above have been established for the case u− = u+ (in this case, without loss of generality, we can assume that u− = u+ = 0). Among them, by using energy estimates, a maximum principle, and a transformation of Cole–Hopf type, it is shown in [4] that the global smooth solution u(x, t ) of the Cauchy problem (1.1), (1.2) satisfies the following decay estimate 1

Z

+∞

u2 (x, t )dx =

lim t 2

t →∞

−∞

4ν 2 (v+ − 1)2

exp −2u2

+∞

Z

√ 2π ν

−∞



v + −1 1+ √ π

R +∞ u





exp −s2 ds

2 du,

(1.3)

where

  Z +∞ 1 v+ = exp − u0 (x)dx . 2ν −∞ R The estimate (1.3) implies that, if R u0 (x)dx 6= 0, the basic L2 decay estimate on u(t , x) is optimal, i.e. there exists a positive constant C > 0 such that 1

ku(t , x)kL2 ≥ C (1 + t )− 4 .

(1.4) q

On the other hand, Mei [6] considered the L -decay rates of solutions for the generalized Benjamin–Bona–Mahony–Burgers equations ut − utxx − ν uxx + β ux + f (u)x = 0, with initial data (1.2). For f (u) =

up+1 p+1

t > 0, x ∈ R,

, u− = u+ = 0, and

(1.5)

R R

u0 (x)dx = 0, it is shown in [6] that, among other things, the

basic L2 -decay estimate (1.3) can be improved to 3

ku(t , x)kL2 ≤ C (1 + t )− 4 .

(1.6)

Here p ≥ 1 is an integer and C is some positive constant. The above results are concentrated on the case of u− = u+ = 0. As to the case of u− 6= u+ , in [13] we get the existence and the nonlinear stability of traveling wave solutions to the Cauchy problem of the generalized BBM–Burgers equation (1.5), (1.2). Moreover, an algebraic time decay rate to the traveling wave solutions are obtained by employing the space–time k

weighted energy method. That is, the weighted H 2 -norm of the perturbation decays at the rate of (1 + t )− 2 provided that the initial perturbation decays at the rate |x|−k as |x| → +∞. The traveling waves of (1.2) and (1.5) are the solutions of (1.5) in the form u(t , x) = φ(x − st ) = φ(y), y = x − st , φ(y) → u± , as y → ±∞,



(1.7)

where s is the propagation speed of waves determined by the so-called Rankine–Hugoniot (R–H) condition s=β+

f (u+ ) − f (u− ) u+ − u−

.

(1.8)

Note that all the above results are concentrated on the stability and algebraic time decay rate of the solutions of the Cauchy problem (1.5), (1.2) of the generalized BBM–Burgers equation toward its traveling wave solutions, while fewer results have been obtained on the corresponding exponential time decay rate. Our main purpose in this manuscript is devoted to this problem for general flux function f (u) which is assumed to be either convex or concave for all u under our consideration.

H. Yin, J. Hu / Nonlinear Analysis 73 (2010) 1729–1738 α

∂iψ

1731

(y)

In fact, if the initial perturbation satisfies e 2 |y−y∗ | ∂ y0i ∈ L2 (R) (i = 0, 1, 2) for some α > 0, we can show in this manuscript that, under the assumption of the small initial perturbation and by employing the space–time weighted energy method which is essentially due to [14–16], there exist two positive constant C and b, which depends only on α , such that the following exponential convergence rate holds

ku(t , x) − φ(x − st + x0 )k2H 1 ≤ C e−bt . Rx Here ψ0 (x) = −∞ (u0 (y) − φ(y + x0 )) dy is the initial perturbation and x0 denotes the initial shift Z +∞ 1 x0 = (u0 (x) − φ(x)) dx. u+ − u− −∞

(1.9)

(1.10)

But for simplicity and without loss of generality, we will also let x0 = 0 in the rest of this paper. It is worth to pointing out that throughout this paper, we ask the nonlinear flux function f (u) to be either convex or concave, we are convinced that similar results hold for general smooth flux function f (u). The study for this case is left for the future. This paper is organized as follows. After stating some notations in the following, we give some preliminaries established in [13]. Then Section 3 is devoted to deducing the exponential time decay rates. 1.1. Notations Throughout this paper, without any ambiguity, we denote a generic positive constant by C or O(1) which may vary from line to line. If the dependence needs to be explicitly pointed out, then the notations Ci (i ∈ Z+ ) or C (·, ·) etc. are used. For function space, Lp = Lp (R) (1 ≤ p ≤ ∞) denotes the usual Lebesgue space on R = (−∞, +∞) with its norm

Z

|f (x)| dx p

kf k = Lp

 1p

(1 ≤ p < ∞),

kf kL∞ = sup |f (x)|, x∈R

R

and when p = 2, we write k · kL2 (R) = k · k. H l (R) denotes the usual l-th order Sobolev space with its norm

kf kH l (R) = kf kl =

l X

! 21 k∂xi f k2

.

i=0

For simplicity, kf (t , ·)kLp and kf (t , ·)kl are denoted by kf (t )kLp and kf (t )kl respectively. Let T be a positive constant and let B be a Banach space. Here, for any natural number k ≥ 0, C k ([0, T ]; B) denotes the space of B-valued k-times continuously differentiable functions on [0, T ] and L2 ([0, T ]; B) denotes the space of B-valued L2 -functions on [0, T ]. The corresponding spaces of B-valued functions on [0, ∞) are defined similarly. 2. Preliminaries In this section, we will give some preliminaries established in [13]. 2.1. Existence of traveling waves First, we state the existence of strictly monotonic traveling waves φ(x − st ) to the Cauchy problem of the generalized BBM–Burgers equation (1.5), (1.2) as follows: Lemma 2.1 (The Case s = 0, cf. [13]). Under the R–H condition (1.8), we can get that (i) If f is convex and u+ < u− , then there exists a stationary solution φ(x) of (1.5) with φ(±∞) = u± , unique up to a shift and φ(x) satisfying φx (x) < 0 and u+ < φ(x) < u− for all x ∈ R. (ii) If f is concave and u+ > u− , then there exists a stationary solution φ(x) of (1.5) with φ(±∞) = u± , unique up to a shift and φ(x) satisfying φx (x) > 0 and u− < φ(x) < u+ for all x ∈ R. Lemma 2.2 (The Case s 6= 0, cf. [13]). Under the R–H condition (1.8), we have (i) If f is convex and u± , s, β and ν satisfy

  u+ < u− , s > 0, 4s2 − 4s β − f 0 (u )
+ ν 2 ≥ 0, −

or

  u+ < u− , s < 0, 4s2 − 4s β − f 0 (u )
+ ν 2 ≥ 0, +

then there exists a stationary solution φ(x) of (1.5) with φ(±∞) = u± , unique up to a shift and φ(y) satisfying φy (y) < 0 and u+ < φ(y) < u− for all y ∈ R.

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H. Yin, J. Hu / Nonlinear Analysis 73 (2010) 1729–1738

(ii) If f is concave and u± , s, β and ν satisfy

 u− < u+ , s > 0, 4s2 − 4s β − f 0 (u )
+ ν 2 ≥ 0, −

or

 u− < u+ , s < 0, 4s2 − 4s β − f 0 (u )
+ ν 2 ≥ 0, +

then there exists a traveling wave solution φ(y) = φ(x − st ) of (1.5) with φ(±∞) = u± , unique up to a shift and φ(y) satisfying φy (y) > 0 and u− < φ(y) < u+ for all y ∈ R. 2.2. Nonlinear stability of traveling waves In this subsection, we state the nonlinear stability of traveling waves to the Cauchy problem (1.5), (1.2). Without loss of generality, we suppose that f 00 (u) > 0 in the rest of this paper. In such a case, we have φy (y) < 0. To make the presentation simple, in the rest of this paper we let ∆

A=

max

00 f (u) ,

min

00 f (u) .

u∈[u+ −1,u− +1]

∆

B=

max

u∈[u+ −1,u− +1]

0 f (u) ,

and ∆

µ=

u∈[u+ −1,u− +1]

By using the following coordinate transformation t = t,

y = x − st ,

Eq. (1.5) can be rewritten as ut − utyy + suyyy − ν uyy − (s − β)uy + f (u)y = 0,

t > 0, y ∈ R.

(2.1)

Set the perturbation ψ(t , y) by

ψ(t , y) =

y

Z

(u(t , η) − φ(η)) dη,

or u(t , y) = φ(y) + ψy (t , y),

(2.2)

−∞

then from (2.1), (2.2), we can obtain the reformulated problem as follows

 ψt − ψtyy + Z sψyyy − (s − β)ψy − νψyy + [f (ψy + φ) − f (φ)] = 0, y

ψ(0, y) =

(u0 (η) − φ(η)) dη = ψ0 (y) → 0,

(2.3)

as y → ±∞.

−∞

Now we seek the solution of (2.3) in the set of function XM (t1 , t2 ) defined by



) ( ψ ∈ C [t1 , t2 ]; H 2 (R) ∩ C 1 [t1 , t2 ]; H 1 (R)
XM (t1 , t2 ) = ψ ψy (y, t ) ∈ C [t1 , t2 ]; H 1 (R) , sup kψ(t )k2 ≤ M [t ,t ] 1 2

for some positive constant M and the norm on XM (t1 , t2 ) is given by ∆

kψkX = sup kψ(t )k2 . t ∈[t1 ,t2 ]

Then by combining the monotonicity property of the traveling wave solutions with the analysis performed in [13], we obtain the nonlinear stability of the traveling waves to the reformulated Cauchy problem (2.3). Theorem 2.1 (Local Nonlinear Stability, cf. [13]). Let f 00 (u) > 0 and u+ < u− . If we assume further that ψ0 (y) ∈ H 2 (R) and there exists a positive constant ε such that kψ0 kH 2 < ε(1), then the Cauchy problem (2.3) admits a unique global solution ψ(t , y) ∈ XM (0, +∞) satisfying

kψ(t )k22 + kψt (t )k21 +

Z t 

ψy (s) 2 + kψt (s)k2 ds ≤ C0 kψ0 k2 , 1 2 1

(2.4)

0

for any t ≥ 0. Here C0 is some positive constant independent of t. Moreover we have

i

∂ ψ(t , y)

= 0, t →+∞ ∂ yi ∞ lim

i = 0, 1.

(2.5)

L

Besides, based on the monotonicity of the traveling wave solution φ(x − st ) obtained in Lemma 2.1 or Lemma 2.2 and k

the analysis in [13], we obtain the weighted H 2 -norm of the perturbation decays at the rate of (1 + t )− 2 provided that the initial perturbation decays at the rate |x|−k as |x| → +∞. After having the result mentioned above, we will turn to get an exponential time decay rate in the next section.

H. Yin, J. Hu / Nonlinear Analysis 73 (2010) 1729–1738

1733

3. Exponential time decay rate In this section, we mainly deduce an exponential time decay rate of solutions ψ(t , y) to the Cauchy problem (2.3) toward the traveling wave solution φ(x − st ). Now we state our main result as follows: Theorem 3.1 (Exponential Decay Rate). Let f 00 (u) > 0, u+ < u− , and ψ(t , y) be the global solution to the Cauchy problem (2.3). α

Suppose ψ0 (y) ∈ H 2 (R) and there exists a positive constant ε1 such that kψ0 kH 2 ≤ ε1 (1). If e 2 |y−y∗ | ∂∂yi ψ0 (y) ∈ L2 (R), (i = i

0, 1, 2) for some positive constant α > 0, we have for all t ≥ 0 that

kψ(t )k2H 2 ≤ C e−bt

Z

2 2 eα|y−y∗ | ψ02 + ψ0y + ψ0yy (y) dy.




(3.1)

R

Here C > 0 is a constant and b > 0 is a positive constant depending on α . More precisely, α and b are assumed to satisfy

  1  0 < α ≤ min 2

 0 < b ≤ α , 2C1

C1 ν C1 |s| + 4ν

, 1, 2

0<α+b≤

 ν , 3|s|

1 2C2

(3.2)

.

Here positive constants C1 > 0, C2 > 0 are introduced by (3.3) and (3.26), respectively. Theorem 3.1 will be proved by the following a series of lemmas. Noticing u+ < φ(y) < u− , one can get that there is a unique number y∗ ∈ R such that φ(y∗ ) = ξ∗ and f 0 (ξ∗ ) = s − β =

f (u+ ) − f (u− ) u+ − u−

,

where ξ∗ is between u+ and u− . For such a y∗ , we first have the following result. Lemma 3.1. Let y∗ be chosen above, then for each α ≥ 0, there exists a positive constant C1 > 0, such that


− α sign (y − y∗ ) f 0 (φ(y)) − s + β + f 00 (φ(y)) φy (y) ≥ C1 α ≥ 0,

∀y ∈ R.

(3.3)

Proof. First, by virtue of f 00 (u) > 0 and φy (y) < 0, one can easily see that (3.3) is true for the case α = 0. Next, we mainly consider the case α > 0. To make the presentation simple, we let ∆

∆

b1 = min {|φ(y∗ − 1) − φ(y∗ )| , |φ(y∗ + 1) − φ(y∗ )|} ,

b2 =

min

y∈[y∗ −1,y∗ +1]

φy (y) .

Since

 

f 00 (u) > 0,

φ (y) < 0, y

s−β =

f (u+ ) − f (u− )

= f 0 (ξ∗ ) = f 0 (φ(y∗ )), u+ − u− φ(y) → u± , as y → ±∞,

we have ∆

I = −α sign (y − y∗ ) f 0 (φ(y)) − s + β + f 00 (φ(y)) φy (y)









= −α sign (y − y∗ ) f 0 (φ(y)) − f 0 (φ(y∗ )) + f 00 (φ(y)) φy (y) = −α sign (y − y∗ ) f 00 (θ ) (φ(y) − φ(y∗ )) + f 00 (φ(y)) φy (y) = α f 00 (θ ) |φ(y) − φ(y∗ )| + f 00 (φ(y)) φy (y) where θ = ηφ + (1 − η)ξ∗ , η ∈ (0, 1). Consequently, if |y − y∗ | ≥ 1, we have I ≥ αµ · min {|φ(y∗ − 1) − φ(y∗ )| , |φ(y∗ + 1) − φ(y∗ )|}

= αµb1 .

(3.4)

On the other hand, by virtue of f (u) > 0, φy (y) < 0, α > 0, when |y − y∗ | < 1, we can get that 00

µb2 I ≥ f 00 (φ(y)) φy (y) ≥ µ min φy = α · . α

|y−y∗ |<1

n

µb

o

Hence, if we choose C1 = min µb1 , α 2 , we can get (3.3) immediately for each α > 0 from (3.4) and (3.5). Then for all α ≥ 0 (3.3) is true. Thus Lemma 3.1 is proved. 

(3.5)

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H. Yin, J. Hu / Nonlinear Analysis 73 (2010) 1729–1738

Lemma 3.2. Let ψ(t , y) be the solution to the Cauchy problem (2.3). Under the assumptions in Theorem 3.1, if we choose kψ0 k2 sufficiently small such that kψ0 k2 ≤ √1C min νA , 1 , then we have 0

e

bt

Z e

α|y−y∗ |

ψ +ψ 2

2 y

(t , y) dy +




t

Z

e

R

Z

bτ

e

Z

2 (y) dy + b eα|y−y∗ | ψ02 + ψ0y




R

+α

t

Z

e

Z

bτ

e

α|y−y∗ |

ψ +ψ 2 ty

2 yy




ψ (τ , y) dydτ + α

e

bτ

Z

Z

eb τ

eα|y−y∗ | ψ 2 (τ , y) dydτ

R

0

eα|y−y∗ | ψ 2 + ψy2 (τ , y) dydτ




R

0 t

Z

t

Z

2 y

R

0

≤C

α|y−y∗ |



(τ , y) dydτ .

(3.6)

R

0

Here C > 0 is a constant independent of kψ0 k2 . Proof. First, multiplying the first equation of (2.3) by 2ebt eα|y−y∗ | ψ , we arrive at



+ 2ν ebt eα|y−y∗ | ψy2 + ebt eα|y−y∗ | −α sign (y − y∗ ) f 0 (φ) − s + β + f 00 (φ) φy ψ 2    + ebt eα|y−y∗ | −2ψψty + 2sψψyy − sψy2 − (s − β)ψ 2 − 2νψψy + f 0 (φ)ψ 2 y
= bebt eα|y−y∗ | ψ 2 + ψy2 − sα sign (y − y∗ ) ebt eα|y−y∗ | ψy2

ebt eα|y−y∗ | ψ 2 + ψy2






t

+ 2sα sign (y − y∗ ) ebt eα|y−y∗ | ψψyy − 2αν sign (y − y∗ ) ebt eα|y−y∗ | ψψy − 2α sign (y − y∗ ) ebt eα|y−y∗ | ψψty − ebt eα|y−y∗ | f 00 (ξ1 )ψψy2 ,

(3.7)

where ξ1 is between φ and φ + ψy . And we used the following identity

   f 00 (ξ1 ) 2 0 2ψ f φ + ψy − f (φ) = 2ψ f (φ)ψy + ψy 2  0 = f (φ)ψ 2 y − f 00 (φ)φy ψ 2 + f 00 (ξ1 )ψψy2 . 




Integrating the identity (3.7) with respect to t, y over [0, t ] × R and then, using Lemma 3.1 and noticing |sign (y − y∗ )| ≤ 1, we obtain ebt

Z

eα|y−y∗ | ψ 2 + ψy2 (t , y) dy + 2ν




Z

R

Z ≤

e

α|y−y∗ |

ψ +ψ Z 2 0

R t

Z + |s|α

e

bτ

e

t

Z + 2|s|α | Z

t




eb τ

Z

(y) dy + b

ebτ

Z

Z

e

bτ

Z

Z

eα|y−y∗ | ψ 2 (τ , y) dydτ R




R

ψ (τ , y) dydτ + 2α |

t

Z

Z

ebτ

eα|y−y∗ | ψψty (τ , y) dydτ



{z

}

I1



t

Z

eb τ

Z

eα|y−y∗ | ψψy (τ , y) dydτ





R

0

}

I2



R

0

eα|y−y∗ | ψψyy (τ , y) dydτ + 2αν

{z Z

eb τ

eα|y−y∗ | ψ 2 + ψy2 (τ , y) dydτ

2 y



t

Z 0

R

0

eb τ

eα|y−y∗ | ψy2 (τ , y) dydτ + C1 α

R

0 t 0

α|y−y∗ |

R

0

+

2 0y

t

|

{z I3

eα|y−y∗ | f 00 (ξ1 )ψψy2 (τ , y) dydτ .





{z

}

} (3.8)

R

|0

I4

Now we estimate the last four terms I1 − I4 in the right-hand side of (3.8). To this end, by using the Cauchy–Schwarz inequality, we have I1 ≤ I2 ≤ I3 ≤

C1 4 C1 4 C1 4

α

t

Z

eb τ

Z R

0

α

t

Z

eb τ

Z

t

Z 0

eα|y−y∗ | ψ 2 (τ , y) dydτ + R

0

α

eα|y−y∗ | ψ 2 (τ , y) dydτ +

eb τ

Z

eα|y−y∗ | ψ 2 (τ , y) dydτ + R

4 C1 4 C1 4 C1

α

t

Z

ebτ

Z

s2 α

2 eα|y−y∗ | ψty (τ , y) dydτ , R

0 t

Z

eb τ

t

Z 0

2 eα|y−y∗ | ψyy (τ , y) dydτ , R

0

ν2α

Z

ebτ

Z

eα|y−y∗ | ψy2 (τ , y) dydτ . R

(3.9)

H. Yin, J. Hu / Nonlinear Analysis 73 (2010) 1729–1738

On the other hand, by choosing kψ0 k2 ≤ √1C min

ν

p

sup kψ(τ )kL∞ ≤ sup kψ(τ )k1 ≤

τ ∈[0,t ]

τ ∈[0,t ]

, 1 , from (2.4) and the Sobolev inequality, we have

A

0

1735



ν

C0 kψ0 k2 ≤

A

.

(3.10)

From (3.10) we can estimate I4 as follows τ ∈[0,t ] t

Z

≤ν

t

Z

I4 ≤ A sup kψ(τ )kL∞

Z

eb τ

Z

eb τ

eα|y−y∗ | ψy2 (τ , y) dydτ R

0

eα|y−y∗ | ψy2 (τ , y) dydτ .

(3.11)

R

0

Substituting (3.9), (3.11) into (3.8), we obtain ebt

Z

eα|y−y∗ | ψ 2 + ψy2 (t , y) dy +




R

 + Z

4

ν − |s|α −

C1

ν α 2

t

Z

e

bτ

C1 4

Z

ebτ




C1

α

t

Z

e

bτ

Z e

α|y−y∗ |

t

Z

ebτ

Z

eα|y−y∗ | ψ 2 + ψy2 (τ , y) dydτ




R

ψ (τ , y) dydτ + 2 ty

R

0

eα|y−y∗ | ψ 2 (τ , y) dydτ R

0

0

4

Z

eα|y−y∗ | ψy2 (τ , y) dydτ

R

+

t

Z

R

0

2 eα|y−y∗ | ψ02 + ψ0y (y) dy + b

≤

α

4

s α 2

C1

t

Z

e

bτ

Z

2 eα|y−y∗ | ψyy (τ , y) dydτ .

(3.12)

R

0

If we choose α > 0 suitably small satisfying

 |s| +

4 C1

ν

2



ν

α≤

2

,

0<α≤

or

C1 ν 2C1 |s| + 8ν 2

,

(3.13)

then we can immediately get (3.6) from (3.12) and (3.13). This completes the proof of Lemma 3.2. The following lemmas are devoted to control the term of (3.6).

Rt

e

0

bτ

R R

α|y−y∗ |

ψ +ψ 2 ty

e

2 yy






(τ , y) dydτ in the right-hand side

Lemma 3.3. Let ψ(t , y) be the solution to the Cauchy problem (2.3). Under the assumptions in Theorem 3.1, if we choose kψ0 k2 sufficiently small such that kψ0 k2 ≤ √1C min νA , 1 , then we have 0

t

Z

ebτ

Z

2 eα|y−y∗ | ψt2 + ψty (τ , y) dydτ ≤ C




R

0

t

Z

eb τ

Z

2 eα|y−y∗ | ψy2 + ψyy (τ , y) dydτ .




(3.14)

R

0

Here C > 0 is a constant independent of kψ0 k2 . Proof. Multiplying the first equation of (2.3) by ebt eα|y−y∗ | ψt , we have 2 ebt eα|y−y∗ | ψt2 + ψty + ebt eα|y−y∗ | −ψt ψty + sψt ψyy







y

+ α sign (y − y∗ ) ebt eα|y−y∗ | ψt ψty − sebt eα|y−y∗ | ψty ψyy − ν ebt eα|y−y∗ | ψt ψyy
− sα sign (y − y∗ ) ebt eα|y−y∗ | ψt ψyy + f 0 (ξ2 ) − s + β ebt eα|y−y∗ | ψt ψy = 0,

(3.15)

where ξ2 is between φ and φ + ψy . And we used the following identity

  ψt f (φ + ψy ) − f (φ) = f 0 (ξ2 )ψt ψy . Integrating the identity (3.15) with respect to t, y over [0, t ] × R and noticing |sign (y − y∗ )| ≤ 1, we arrive at t

Z

e 0

bτ

Z e

α|y−y∗ |

ψ +ψ 2 t

2 ty




R

t

Z + | s| |

0

ebτ

Z

(τ , y) dydτ ≤ α |

t

Z

e

bτ

Z

eα|y−y∗ | ψt ψty (τ , y) dydτ



R

0

{z



R

+ (B + |s| + |β|) |

t

Z 0

{z

}

I6

t

Z 0

eb τ

Z

|

I8

Z

eα|y−y∗ | ψt ψyy (τ , y) dydτ





}



R I7

R

{z

eb τ

{z

eα|y−y∗ | ψt ψy (τ , y) dydτ .



}

I5

eα|y−y∗ | ψty ψyy (τ , y) dydτ + (|s|α + ν)





} (3.16)

1736

H. Yin, J. Hu / Nonlinear Analysis 73 (2010) 1729–1738

Now we estimate the terms I5 –I8 in the right-hand side of (3.16). To this end, we have by employing the Cauchy–Schwarz inequality that

α

I5 ≤

Z

t

Z

0 t

2 1

I6 ≤

2

ebτ

Z

t

2 eα|y−y∗ | ψty (τ , y) dydτ +

Z

ebτ

s2

t

Z

2

Z

ebτ

2 eα|y−y∗ | ψyy (τ , y) dydτ , R

0

eα|y−y∗ | ψt2 (τ , y) dydτ + (|s|α + ν)2

t

Z

ebτ

t

Z

R

Z

4




R

0

1

2 eα|y−y∗ | ψt2 + ψty (τ , y) dydτ ,

R

Z

4

I8 ≤

Z

0

1

I7 ≤

eb τ

ebτ

2 eα|y−y∗ | ψyy (τ , y) dydτ ,

(3.17)

R

0 t

Z

eα|y−y∗ | ψt2 (τ , y) dydτ + (B + |s| + |β|)2

R

0

Z

Z

ebτ

eα|y−y∗ | ψy2 (τ , y) dydτ . R

0

Substituting (3.17) into (3.16), we get 1 2

t

Z

(1 − α)

Z

ebτ




R

0

t

Z

2 eα|y−y∗ | ψt2 + ψty (τ , y) dydτ ≤ C

0

ebτ

Z

2 eα|y−y∗ | ψy2 + ψyy (τ , y) dydτ .




(3.18)

R

If we choose α > 0 suitably small such that 1

0<α≤

,

2

(3.19)

then we can immediately get (3.14) from (3.18) and (3.19). Thus Lemma 3.3 is proved.



Combining the estimates (3.6) and (3.14), we can get the following result Lemma 3.4. Let ψ(t , y) be the solution to the Cauchy problem (2.3). Under the assumptions in Theorem 3.1, if we choose kψ0 k2 sufficiently small such that kψ0 k2 ≤ √1C min νA , 1 , then we have 0

e

bt

Z e

α|y−y∗ |

ψ + ψy (t , y) dy +
2

2

R

t

Z

e

Z

bτ

e

Z

2 eα|y−y∗ | ψ02 + ψ0y (y) dy + b




+α

t

Z

R

e

bτ

Z e

α|y−y∗ |

t

Z

e

bτ

eb τ

Z

Z

eα|y−y∗ | ψ 2 (τ , y) dydτ

R

0

eα|y−y∗ | ψ 2 + ψy2 (τ , y) dydτ




R

0 t

Z

ψ (τ , y) dydτ + α 2 y

R

0

≤C

α|y−y∗ |



ψ + ψyy (τ , y) dydτ . 2 y


2

(3.20)

R

0

Here C > 0 is a constant independent of kψ0 k2 . From the above inequality, to close the energy estimate, we only need to get an estimate on the term 2 ψyy (τ , y) dydτ and this is the main content of our next lemma.

Rt 0

eb τ

R R

eα|y−y∗ |

Lemma 3.5. Let ψ(t , y) be the solution to the Cauchy problem (2.3). Under the assumptions in Theorem 3.1, if we choose kψ0 k2 sufficiently small such that kψ0 k2 ≤ √1C min νA , 1 , then we have 0

e

bt

Z e

α|y−y∗ |

ψ + ψyy (t , y) dy +
2

2 y

t

Z

e

R

bτ

2 eα|y−y∗ | ψyy (τ , y) dydτ

R

0

Z

Z

2 2 eα|y−y∗ | ψ02 + ψ0y + ψ0yy (y) dy + α




≤C R

+b 0

eb τ

Z

eb τ

Z

2 eα|y−y∗ | ψ 2 + ψy2 + ψyy (τ , y) dydτ




2 eα|y−y∗ | ψy2 + ψyy (τ , y) dydτ




R

0 t

Z

t

Z



.

(3.21)

R

Here C > 0 is a constant independent of kψ0 k2 . Proof. Differentiating the first equation of (2.3) with respect to y once, and multiplying the result by 2ebt eα|y−y∗ | ψy , we get 2 ebt eα|y−y∗ | ψy2 + ψyy




t


2 2 − bebt eα|y−y∗ | ψy2 + ψyy + 2ν ebt eα|y−y∗ | ψyy

  + ebt eα|y−y∗ | −2ψy ψtyy + 2α sign (y − y∗ ) ψy ψty + 2sψy ψyyy  2 − sψyy − 2sα sign (y − y∗ ) ψy ψyy − 2νψy ψyy + 2f 0 (ξ3 )ψy2 y − 2α sign (y − y∗ ) ebt eα|y−y∗ | ψyy ψty − 2α 2 (sign (y − y∗ ))2 ebt eα|y−y∗ | ψt ψty

H. Yin, J. Hu / Nonlinear Analysis 73 (2010) 1729–1738

1737

2 + 3sα sign (y − y∗ ) ebt eα|y−y∗ | ψyy + 2sα 2 (sign (y − y∗ ))2 ebt eα|y−y∗ | ψy ψyy

+ 2να sign (y − y∗ ) ebt eα|y−y∗ | ψy ψyy − 2ebt eα|y−y∗ | f 0 (ξ3 )ψy ψyy − 2α sign (y − y∗ ) ebt eα|y−y∗ | f 0 (ξ3 )ψy2 = 0, where ξ3 is between φ and φ + ψy . Integrating the above identity with respect to t, y over [0, t ] × R, and noticing |sign (y − y∗ )| ≤ 1, we can get ebt

Z

2 eα|y−y∗ | ψy2 + ψyy (t , y) dy + 2ν




R

Z ≤

e

α|y−y∗ |

ψ +ψ Z 2 0y

R t

Z

2 0yy

ebτ

+ 3|s|α




(y) dy + b

Z

bτ

2 eα|y−y∗ | ψy2 + ψyy (τ , y) dydτ




e

R

0

e

Z

bτ

e




t

Z

e

bτ

Z

eα|y−y∗ | ψy2 (τ , y) dydτ R

eα|y−y∗ | ψy ψyy (τ , y) dydτ





R

0

{z

}

I9

α|y−y∗ |

ψyy ψty (τ , y) dydτ + 2α 2 {z } |

R

0

Z

ebτ

0

+ 2 |s|α + |s| + |β| + να | t

t

Z

R

2

+ 2α |

2 eα|y−y∗ | ψyy (τ , y) dydτ R

0 t

Z

Z

eb τ

2 eα|y−y∗ | ψyy (τ , y) dydτ + 2Bα

0

Z

t

Z

t

Z

e

bτ

Z

eα|y−y∗ | ψy ψty (τ , y) dydτ .





(3.22)

R

0

{z

I10

}

I11

Now we estimate the terms I9 − I11 respectively. By employing the Cauchy–Schwarz inequality, we have

ν

I9 ≤

2

ν

ebτ

Z

1

2 (τ , y) dydτ + eα|y−y∗ | ψyy

ν

R

0 t

|s|α 2 + |s| + |β| + να


2

Z

t

ebτ

Z

eα|y−y∗ | ψy2 (τ , y) dydτ ,

R

0

Z t Z 2 (τ , y) dydτ , α2 ebτ eα|y−y∗ | ψty 2 0 ν R 0 R Z t Z Z t Z 2 (τ , y) dydτ . ebτ eα|y−y∗ | ψty ≤ α2 ebτ eα|y−y∗ | ψy2 (τ , y) dydτ + α 2

I10 ≤ I11

t

Z Z

ebτ

Z

2 (τ , y) dydτ + eα|y−y∗ | ψyy

2

R

0

(3.23)

R

0

Substituting (3.23) into (3.22), we obtain ebt

Z

2 eα|y−y∗ | ψy2 + ψyy (t , y) dy + (ν − 3|s|α)




t

Z

R

eb τ

Z

Z ≤

e

α|y−y∗ |

ψ +ψ 2 0y

2 0yy




(y) dy + b

t

Z

R

bτ

+ Cα

eb τ

Z




R

2 eα|y−y∗ | ψty (τ , y) dydτ + C R

0

2 eα|y−y∗ | ψy2 + ψyy (τ , y) dydτ

e 0

t

Z

Z

2 eα|y−y∗ | ψyy (τ , y) dydτ R

0

t

Z

eb τ

Z

eα|y−y∗ | ψy2 (τ , y) dydτ .

(3.24)

R

0

If we choose α > 0 such that 0<α≤

ν

6|s|

,

(3.25)

2 dydτ in Lemmas 3.2 and 3.3, we can immedithen from the estimates on 0 ebτ R eα|y−y∗ | ψy2 dydτ and 0 ebτ R eα|y−y∗ | ψty ately get (3.21) from (3.24) and (3.25). This completes the proof of Lemma 3.5. 

Rt

Rt

R

R

Having obtained Lemmas 3.4 and 3.5, to complete the proof of Theorem 3.1, we can get by performing (3.20) + (3.21) that Z Z Z t

2 eα|y−y∗ | ψ 2 + ψy2 + ψyy (t , y) dy +

ebt




R

eb τ




R

0

Z

t

+α e 0 Z

bτ

Z e

α|y−y∗ |

ψ (τ , y) dydτ 2

R

2 2 eα|y−y∗ | ψ02 + ψ0y + ψ0yy (y) dy + b

≤ C2




R

+α

2 eα|y−y∗ | ψy2 + ψyy (τ , y) dydτ

0

eb τ

Z

eα|y−y∗ | ψy2 + ψyy (τ , y) dydτ


2

R

Here C > 0 is a constant independent of kψ0 k2 .



ebτ

Z

2 eα|y−y∗ | ψ 2 + ψy2 + ψyy (τ , y) dydτ




R

0 t

Z

t

Z

.

(3.26)

1738

H. Yin, J. Hu / Nonlinear Analysis 73 (2010) 1729–1738

Under the conditions (3.13), (3.19) and (3.25), we easily see that if we choose the positive constants α > 0 and b > 0 suitably such that 0
α 2C2

,

0<α+b≤

1 2C2

,

(3.27)

then from (3.26), we can immediately obtain the following estimate ebt

Z

2 eα|y−y∗ | ψ 2 + ψy2 + ψyy (t , y) dy ≤ C




R

Hence, if

2 2 eα|y−y∗ | ψ02 + ψ0y + ψ0yy (y) dy.




R

α i e 2 |y−y∗ | ∂∂yi

kψ(t )k2H 2

Z

ψ0 (y) ∈ L2 (R), (i = 0, 1, 2), and kψ0 k2 ≤ Z
2 2 −bt (y) dy + ψ0yy eα|y−y∗ | ψ02 + ψ0y ≤ Ce

√1

C0

 min νA , 1 , we have the decay rate

R

holds for all t ≥ 0, where constants α , b satisfy

  1  0 < α ≤ min 2

 0 < b ≤ α , 2C1

C1 ν

 ν , 1, , C1 |s| + 4ν 2 3|s| 0<α+b≤

This completes the proof of Theorem 3.1.

1

2C2

.



Acknowledgements The authors are grateful to the anonymous referee for her/his helpful comments which improved both the mathematical results and the way to present them. This work was supported by two grants from the National Natural Science Foundation of China under contracts 10431060 and 10329101 respectively. References [1] J.L. Bona, W.G. Pritchard, L.R. Scott, Solitary-wave interaction, Phys. Fluids 13 (3) (1980) 438. [2] D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25 (1966) 321–330. [3] T.B. Benjamin, J.L. Boan, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A 272 (1972) 47–78. [4] C.J. Amick, J.L. Bona, M.E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations 81 (1) (1989) 1–49. [5] B.L. Guo, C.X. Miao, On inhomogeneous GBBM equations, J. Partial Differential Equations 8 (1995) 193–204. [6] M. Mei, Lq -decay rates of solutions for Benjamin–Bona–Mahony–Burgers equations, J. Differential Equations 158 (1999) 314–340. [7] P.I. Naumkin, Large-time asymptotic of a step for the Benjamin–Bona–Mahony–Burgers equation, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 1–18. [8] L.H. Zhang, Decay of solutions of generalized Benjamin–Bona–Mahony–Burgers equations in -space dimensions, Nonlinear Anal. TMA 25 (1995) 1343–1369. [9] H.J. Zhao, Optimal temporal decay estimates for the solution to the generalized BBM–Burgers equations with dissipative term, Appl. Anal. 75 (1–2) (2000) 85–105. [10] H.J. Zhao, Existence and convergence of solutions for the generalized BBM–Burgers equations with dissipative term II: The multidimensional case, Appl. Anal. 75 (1–2) (2000) 107–135. [11] H.J. Zhao, B.J. Xuan, Existence and convergence of solutions for the generalized BBM–Burgers equations with dissipative term, Nonlinear Anal. TMA 28 (11) (1997) 1835–1849. [12] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [13] H. Yin, Shuyue Chen, Jing Jin, Convergence rate to traveling waves for generalized Benjamin–Bona–Mahony–Burgers equations, Z. Angew. Math. Phys. 58 (2007) 1–33. [14] S. Kawashima, A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985) 97–127. [15] A. Matsumura, K. Nishihara, Asymptotic stability of travelling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys. 165 (1994) 83–96. [16] M. Nishikawa, Convergence rates to the travelling wave for viscous conservation laws, Funkcial Ekvac. 41 (1998) 107–132.