J. Math. Anal. Appl. 375 (2011) 502–509

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Journal of Mathematical Analysis andApplications

www.elsevier.com/locate/jmaa

Blow-up and global existence for a modified two-componentCamassa–Holm equation

Jingjing Liu ∗, Zhaoyang Yin

Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 September 2010Available online 1 October 2010Submitted by Goong Chen

Keywords:Modified two-component Camassa–HolmequationStrong solutionsBlow-upBlow-up rateGlobal existence

The present work is mainly concerned with blow-up phenomena and global existence for amodified two-component Camassa–Holm equation. Firstly, a new blow-up criterion and theprecise blow-up rate are presented. Then, a new global existence result for strong solutionsto the equation is given.

Crown Copyright © 2010 Published by Elsevier Inc. All rights reserved.

1. Introduction

In this paper we consider the following modified two-component Camassa–Holm equation:⎧⎪⎪⎨⎪⎪⎩

mt + umx + 2mux = −gρρ̄x, t > 0, x ∈ R,

ρt + (ρu)x = 0, t > 0, x ∈ R,

m(0, x) = m0(x), x ∈ R,

ρ(0, x) = ρ0(x), x ∈ R,

(1.1)

where (m0,ρ0) is a given initial profile, m = u − uxx and ρ = (1 − ∂2x )(ρ̄ − ρ̄0), g is a positive constant. System (1.1) was

recently introduced by Holm et al. in [22]. For convenience, we let g = 1 in this paper. The modified two-componentCamassa–Holm equation is written in terms of velocity u and locally averaged density ρ̄ (or depth, in the shallow-waterinterpretation) and ρ̄0 is taken to be constant.

For ρ ≡ 0, system (1.1) reduces to the Camassa–Holm equation, which was derived physically in [2] (found earlier in [19]as a bi-Hamiltonian generalization of the KdV equation) by approximating directly the Hamiltonian for Euler’s equation inthe shallow water region with u(t, x) representing the free surface above a flat bottom. The Camassa–Holm equation isalso a model for the propagation axially symmetric waves in hyperelastic rods [14]. It has a bi-Hamiltonian structure [5,19]and is completely integrable [2,6]. Its solitary waves are peaked [3], capturing thus the shape of solitary wave solutions tothe governing equations for water waves [11]. The orbital stability of the peaked solutions is proved in [12]. The explicitinteraction of the peaked solutions is given in [1]. The Cauchy problem and initial value problem for the periodic Camassa–Holm equation have been studied extensively. It has been established locally well-posedness, global existence and blow-up

* Corresponding author.E-mail addresses: jingjing830306@163.com (J. Liu), mcsyzy@mail.sysu.edu.cn (Z. Yin).

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J. Liu, Z. Yin / J. Math. Anal. Appl. 375 (2011) 502–509 503

phenomena in [8] and blow-up rate, blow-up set of breaking waves in [10]. The Cauchy problem and initial–boundary valueproblem for the Camassa–Holm equation also have been studied extensively [7,9,15,17,18,24,26,27].

Recently, two types of two-component Camassa–Holm equations have been studied in [4,13,16,20,23]. Moreover, themodified two-component Camassa–Holm equation has been studied in [21] on the line and in [25] on the circle respectively.The authors established the local well-posedness, derived precise blow-up scenarios and proved the existence of strongsolutions which blow-up in finite time. Furthermore, in [25] the authors discussed the blow-up rate and presented a newglobal existence result of strong solutions. However, the global existence of solutions to system (1.1) has not been discussedso far. Based on the steepening lemmas and a useful conservation law developed in [21], we will present a new blow-upresult, blow-up rate and a new global existence result of solutions to system (1.1).

The rest of this paper is organized as follows. In Section 2, we recall some preliminary results. In Section 3, we givea new blow-up criteria and a detailed blow-up rate proof. In Section 4, we will present a new global existence result forstrong solutions to system (1.1) with certain initial profiles.

Notation. In this paper, we identify all spaces of functions with function spaces over R, for simplicity, we drop R from ournotation. For 1 � p � ∞, the norm in the Lebesgue space L p(R) will be written by ‖ · ‖Lp , while ‖ · ‖Hs , s � 0 will stand forthe norm in the classical Sobolev spaces Hs(R) and ‖ · ‖Hs×Hs , s � 0 will represent the norm in the classical Sobolev spacesHs(R) × Hs(R).

2. Preliminaries

In this section, we will recall the local well-posedness for the equivalent system of system (1.1) in Hs × Hs , s > 32 , the

precise blow-up scenarios for strong solutions to the system and a useful conversation law.We first introduce the equivalent system of system (1.1). With m = u − uxx , ρ = γ − γxx and γ = ρ̄ − ρ̄0, we can rewrite

system (1.1) as follows:⎧⎪⎪⎨⎪⎪⎩

mt + mxu + 2mux = −ργx, t > 0, x ∈ R,

ρt + (uρ)x = 0, t > 0, x ∈ R,

m(0, x) = u0(x) − u0,xx(x), x ∈ R,

ρ(0, x) = γ0 − γ0,xx, x ∈ R.

(2.1)

Note that if p(x) := 12 e−|x| , x ∈ R, then (1 − ∂2

x )−1 f = p ∗ f for all f ∈ L2, p ∗m = u and p ∗ρ = γ . Here we denote by ∗ theconvolution. Using this identity, we can rewrite system (2.1) as follows:⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

ut + uux = −∂x p ∗(

u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

), t > 0, x ∈ R,

γt + uγx = −p ∗ ((uxγx)x + uxγ

), t > 0, x ∈ R,

u(0, x) = u0(x), x ∈ R,

γ (0, x) = γ0(x), x ∈ R,

(2.2)

or the equivalent form:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

ut + uux = −∂x(1 − ∂2

x

)−1(

u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

), t > 0, x ∈ R,

γt + uγx = −(1 − ∂2

x

)−1((uxγx)x + uxγ

), t > 0, x ∈ R,

u(0, x) = u0(x), x ∈ R,

γ (0, x) = γ0(x), x ∈ R.

(2.3)

We then have the following useful lemmas which will be used in the sequel.

Lemma 2.1. (See [21].) Given z0 = (u0, γ0) ∈ Hs × Hs, s > 32 , there exist a maximal T = T (‖z0‖Hs×Hs ) > 0, and a unique solution

z = (u, γ ) to system (2.2) such that

z = z(·, z0) ∈ C([0, T ); Hs × Hs) ∩ C1([0, T ); Hs−1 × Hs−1).

Moreover, the solution depends continuously on the initial data, i.e. the mapping

z0 → z(·, z0) : Hs × Hs → C([0, T ); Hs × Hs) ∩ C1([0, T ); Hs−1 × Hs−1)

is continuous. Furthermore, the maximal T may be chosen independent of s.

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Lemma 2.2. (See [21].) Let z0 ∈ Hs × Hs, s > 32 and let T > 0 be the maximal existence time of the corresponding solution z = (u, γ )

to system (1.1). Then we have

E(t) =∫R

(u2 + u2

x + γ 2 + γ 2x

)dx =

∫R

(u2

0 + u20,x + γ 2

0 + γ 20,x

)dx. (2.4)

We know that if z ∈ Hs × Hs , s > 32 , then by the above lemma we have

∥∥u(t, ·)∥∥2L∞ + ∥∥γ (t, ·)∥∥2

L∞ � 1

2‖u‖2

H1 + 1

2‖γ ‖2

H1

= 1

2

(‖u0‖2H1 + ‖γ0‖2

H1

) = 1

2‖z0‖2

H1×H1 , (2.5)

for all t ∈ [0, T ).

Lemma 2.3. (See [21].) Let z0 ∈ Hs × Hs, s � 2 and let T > 0 be the maximal existence time of the corresponding solution z = (u, γ )

to system (2.2) guaranteed by Lemma 2.1. Then we have

∥∥γx(t, ·)∥∥

L∞ � 1

2‖z0‖2

H1×H1t + ‖γ0,x‖L∞ . (2.6)

Lemma 2.4. (See [21].) Let z0 = ( u0γ0

) ∈ Hs × Hs, s > 32 be given and assume that T is the maximal existence time of the corresponding

solution z = ( uγ

)of system (2.3) with the initial data z0 . Then the corresponding solution blows up in finite time if and only if

lim supt→T

∥∥ux(t, ·)∥∥

L∞ = +∞.

Lemma 2.5. (See [21].) Let z0 = ( u0γ0

) ∈ Hs × Hs, s > 32 , and let T be the maximal existence time of the solution z = ( u

γ

)to system (2.2)

with the initial data z0 . Then the corresponding solution blows up in finite time if and only if

lim inft→T

infx∈R

{ux(t, x)

} = −∞.

Consider now the following initial value problem{qt = u(t,q), t ∈ [0, T ),

q(0, x) = x, x ∈ R,(2.7)

where u denotes the first component of the solution z to system (1.1). Applying classical results in the theory of ordinarydifferential equations, one can obtain the following results on q.

Lemma 2.6. (See [21].) Let u ∈ C([0, T ); Hs) ∩ C1([0, T ); Hs−1), s � 2. Then problem (2.7) has a unique solution q ∈ C1([0, T ) ×R;R). Moreover, the map q(t, ·) is an increasing diffeomorphism of R with

qx(t, x) = exp

( t∫0

ux(s,q(s, x)

)ds

)> 0, ∀(t, x) ∈ [0, T ) × R.

Lemma 2.7. (See [21].) Let z0 = ( u0γ0

) ∈ Hs × Hs, s > 32 be given and assume that T is the maximal existence time of the corresponding

solution z = ( uγ

)of system (2.3) with the initial data z0 . Then we have

ρ(t,q(t, x)

)qx(t, x) = ρ0(x), ∀(t, x) ∈ [0, T ) × R.

3. Blow-up and blow-up rate

In this section, we prove that there exist strong solutions to system (2.2) which do not exist globally in time and givemore insight into the blow-up mechanism for the wave-breaking solution to system (2.2).

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Theorem 3.1. Let z0 = ( u0γ0

) ∈ Hs × Hs, s > 32 , and T be the maximal time of the solution z = ( u

γ

)to system (2.2) with the initial

data z0 . If z0 satisfies∫R

u30x dx < −√

6‖z0‖3H1×H1 ,

then the corresponding solution to system (2.2) blows up in finite time.

Proof. By Lemma 2.1 and a simple density argument, we only need to show that the above theorem holds for s = 3.Differentiating the first equation of system (2.2) with respect to x, in view of ∂2

x p ∗ f = p ∗ f − f , we have

utx + uuxx = −1

2u2

x + u2 + 1

2γ 2 − 1

2γ 2

x − p ∗(

u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

). (3.1)

Then, it follows that

d

dt

∫R

u3x dx =

∫R

3u2x uxt dx

= 3∫R

u2x

(−uuxx + u2 − 1

2u2

x + 1

2γ 2 − 1

2γ 2

x − p ∗(

u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

))dx

= 3∫R

(−uu2

x uxx + u2x u2 − 1

2u4

x + 1

2u2

xγ2 − 1

2u2

xγ2x − u2

x p ∗(

u2 + 1

2u2

x + 1

2γ 2

)+ 1

2u2

x p ∗ γ 2x

)dx

� 3∫R

(−uu2

x uxx + u2x u2 − 1

2u4

x + 1

2u2

xγ2 + 1

2u2

x p ∗ γ 2x

)dx

= −1

2

∫R

u4x dx + 3

∫R

u2x u2 dx + 3

2

∫R

u2xγ

2 dx + 3

2

∫R

u2x p ∗ γ 2

x dx.

By (2.5), we have

∥∥p ∗ γ 2x

∥∥L∞ � ‖p‖L∞

∥∥γ 2x

∥∥L1 = 1

2

∥∥γ 2x

∥∥L1 ,∫

R

u2x u2 dx �

∥∥u2∥∥

L∞

∫R

u2x dx � 1

2‖z0‖4

H1×H1 ,

∫R

u2xγ

2 dx �∥∥γ 2

∥∥L∞

∫R

u2x dx � 1

2‖z0‖4

H1×H1 ,

and ∫R

u2x p ∗ γ 2

x dx � 1

2

∥∥γ 2x

∥∥L1

∫R

u2x dx � 1

2‖z0‖4

H1×H1 .

Thus

d

dt

∫R

u3x dx � −1

2

∫R

u4x dx + 3‖z0‖4

H1×H1 .

Using the following inequality∣∣∣∣∫R

u3x dx

∣∣∣∣ �( ∫

R

u4x dx

) 12( ∫

R

u2x dx

) 12

�( ∫

R

u4x dx

) 12

‖z0‖H1×H1 ,

and letting

m(t) =∫

u3x dx,

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506 J. Liu, Z. Yin / J. Math. Anal. Appl. 375 (2011) 502–509

we have

d

dtm(t) � − 1

2‖z0‖2H1×H1

m2(t) + 3‖z0‖4H1×H1

= − 1

2‖z0‖2H1×H1

(m(t) + √

6‖z0‖3H1×H1

)(m(t) − √

6‖z0‖3H1×H1

).

Note that if m(0) < −√6‖z0‖3

H1×H1 then m(t) < −√6‖z0‖3

H1×H1 for all t ∈ [0, T ). From the above inequality we obtain

m(0) + √6‖z0‖3

H1×H1

m(0) − √6‖z0‖3

H1×H1

e√

6‖z0‖H1×H1 t − 1 �2√

6‖z0‖3H1×H1

m(t) − √6‖z0‖3

H1×H1

� 0.

Since 0 <m(0)+√

6‖z0‖3H1×H1

m(0)−√6‖z0‖3

H1×H1< 1, then there exists

0 < T � 1√6‖z0‖H1×H1

lnm(0) − √

6‖z0‖3H1×H1

m(0) + √6‖z0‖3

H1×H1

,

such that limt→T m(t) = −∞. On the other hand,∣∣∣∣∫R

u3x dx

∣∣∣∣ � ‖ux‖L∞∫R

u2x dx � ‖ux‖L∞‖z0‖2

H1×H1 .

Applying Lemma 2.4, the solution z blows up in finite time. �Lemma 3.1. (See [9].) Let t0 > 0 and v ∈ C1([0, t0); H2). Then for every t ∈ [0, t0) there exists at least one point ξ(t) ∈ R with

m(t) := minx∈R

[vx(t, x)

] = vx(t, ξ(t)

),

and the function m is almost everywhere differentiable on (0, t0) with

d

dtm(t) = vtx

(t, ξ(t)

)a.e. on (0, t0).

Next, we use the previous lemma to give more insight into the blow-up mechanism for the wave-breaking solution tosystem (2.2).

Theorem 3.2. Let z0 = ( u0γ0

) ∈ Hs × Hs, s > 32 , and let T > 0 be the maximal existence time of the solution z = ( u

γ

)to system (2.2)

with the initial data z0 . If T is finite, we have

limt→T

(T − t)minx∈R

ux(t, x) = −2.

Proof. Applying Lemma 2.1 and a simple density argument, we only need to show that the above theorem holds for somes > 3

2 . Here we assume s = 3 to prove the above theorem.We already know by Lemma 2.5 that

lim inft→T

infx∈R

{ux(t, x)

} = −∞.

Define now

m(t) := minx∈R

[ux(t, x)

], t ∈ [0, T ),

and let ξ(t) ∈ R be a point where this minimum is attained. Clearly uxx(t, ξ(t)) = 0 since u(t, ·) ∈ H3 ⊂ C2. Evaluating (3.1)at (t, ξ(t)) and using Lemma 3.1, we obtain that the relation

utx + 1

2u2

x = u2 + 1

2γ 2 − 1

2γ 2

x − p ∗(

u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

)holds at the point x = ξ(t), i.e.

d

dtm + 1

2m2 = u2(t, ξ(t)

) + 1

2γ 2(t, ξ(t)

) − 1

2γ 2

x

(t, ξ(t)

) − (p ∗ f )(t, ξ(t)

), (3.2)

where f = u2 + 1 u2x + 1 γ 2 − 1 γ 2

x .

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J. Liu, Z. Yin / J. Math. Anal. Appl. 375 (2011) 502–509 507

By Lemma 2.2 and Young’s inequality, we have for all t ∈ [0, T ) that

‖p ∗ f ‖L∞ � ‖p‖L∞∥∥∥∥u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

∥∥∥∥L1

� 1

2

(‖u‖2H1 + ‖γ ‖2

H1

)= 1

2‖z‖2

H1×H1 = 1

2‖z0‖2

H1×H1 .

This relation together with (2.5) and Lemma 2.3 imply that there is a constant K > 0 such that∥∥∥∥u2(t, ξ(t)) + 1

2γ 2(t, ξ(t)

) − 1

2γ 2

x

(t, ξ(t)

) − [p ∗ f ](t, ξ(t))∥∥∥∥

L∞� K , t ∈ [0, T ).

From (3.2) and the previous relation we obtain

−K � d

dtm + 1

2m2 � K a.e. on (0, T ). (3.3)

Let ε ∈ (0, 12 ). Since lim inft→T m(t) = −∞ by Lemma 2.5, there is some t0 ∈ (0, T ) with m(t0) < 0 and m2(t0) > K

ε . Letus first prove that

m2(t) >K

ε, t ∈ [t0, T ). (3.4)

Since m is locally Lipschitz (it belongs to W 1,∞loc (R) by Lemma 3.1) there is some δ > 0 such that

m2(t) >K

ε, t ∈ [t0, t0 + δ).

Pick δ > 0 maximal with this property. If δ < T − t0 we would have m2(t0 + δ) = Kε while

dm

dt� −1

2m2 + K < −1

2m2 + εm2 < 0 a.e. on (t0, t0 + δ).

Being locally Lipschitz, the function m is absolutely continuous and therefore we would obtain by integrating the previousrelation on [t0, t0 + δ] that

m(t0 + δ) � m(t0) < 0,

which on its turn would yield

m2(t0 + δ) � m2(t0) >K

ε.

The obtained contradiction completes the proof of relation (3.4).A combination of (3.3) and (3.4) enables us to infer

1

2+ ε � −

dmdt

m2� 1

2− ε a.e. on (0, T ). (3.5)

Since m is locally Lipschitz on [0, T ) and (3.4) holds, it is easy to check that 1m is locally Lipschitz on (t0, T ). Differentiating

the relation m(t) · 1m(t) = 1, t ∈ (t0, T ), we get

d

dt

(1

m

)= −

dmdt

m2a.e. on (t0, T ),

with 1m absolutely continuous on (t0, T ). For t ∈ (t0, T ), integrating (3.5) on (t, T ) to obtain(

1

2+ ε

)(T − t) � − 1

m(t)�

(1

2− ε

)(T − t), t ∈ (t0, T ),

that is,

112 + ε

� −m(t)(T − t) � 112 − ε

, t ∈ (t0, T ).

By the arbitrariness of ε ∈ (0, 12 ) the statement of Theorem 3.2 follows. �

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4. Global existence

Not all strong solutions develop singularities in finite time. The following theorem will show that system (1.1) admitsglobal solutions.

Theorem 4.1. Let z0 = ( u0γ0

) ∈ Hs × Hs, s > 32 , and T be the maximal time of the solution z = ( u

γ

)to system (2.2) with the initial

data z0 . If ρ0(x) �= 0 for all x ∈ R, then the corresponding solution z exists globally in time.

Proof. Applying Lemma 2.1 and a simple density argument, we only need to show that the above theorem holds for somes > 3

2 . Here we assume s = 3 to prove the above theorem.By Lemma 2.6, we know that q(t, ·) is an increasing diffeomorphism of R with

qx(t, x) = exp

( t∫0

ux(s,q(s, x)

)ds

)> 0, ∀(t, x) ∈ [0, T ) × R.

Moreover,

infx∈R

ux(t, x) = infx∈R

ux(t,q(t, x)

), ∀t ∈ [0, T ). (4.1)

Set M(t, x) = ux(t,q(t, x)) and α(t, x) = ρ(t,q(t, x)) for t ∈ [0, T ) and x ∈ R. By system (2.2) and problem (2.7), we have

∂M

∂t= (utx + uuxx)

(t,q(t, x)

)and

∂α

∂t= −αM. (4.2)

Then by (3.1) we get

Mt =(

−1

2M2 + u2 + 1

2γ 2 − 1

2γ 2

x − p ∗(

u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

))(t,q). (4.3)

Note that from (2.5) and Lemma 2.3 we can obtain∣∣∣∣u2(t,q) − 1

2γ 2

x (t,q) − p ∗(

u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

)(t,q)

∣∣∣∣�

∥∥u(t, ·)∥∥2L∞ + 1

2

∥∥γx(t, ·)∥∥2

L∞ +∥∥∥∥p ∗

(u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

)(t, y)

∥∥∥∥L∞

� ‖u‖2L∞ + 1

2‖γx‖2

L∞ + ‖p‖L∞ ·∥∥∥∥u2 + 1

2u2

x + 1

2γ 2 − 1

2γ 2

x

∥∥∥∥L1

� ‖z0‖2H1×H1 + 1

2

(1

2‖z0‖2

H1×H1 T + ‖γ0,x‖L∞)2

:= K .

If we write f (t, x) = u2(t,q) − 12 γ 2

x (t,q) − p ∗ (u2 + 12 u2

x + 12 γ 2 − 1

2 γ 2x )(t,q), then

∣∣ f (t, x)∣∣ � K (4.4)

and

Mt = −1

2M2 + 1

2γ 2 + f (t, x), ∀(t, x) ∈ [0, T ) × R. (4.5)

By Lemmas 2.6–2.7, we know that α(t, x) has the same sign with α(0, x) = ρ0(x) for every x ∈ R. Assume that thecorresponding solution exists in finite time. Then by Lemma 2.5 we have lim inft→T infx∈R M(t, x) = −∞, where T < ∞ isthe maximal existence time. We will show this does not occur in the following.

Next, we consider the following Lyapunov function first introduced in [13]:

w(t, x) = α(t, x)α(0, x) + α(0, x)

α(t, x)

(1 + M2), (t, x) ∈ [0, T ) × R. (4.6)

By Sobolev’s imbedding theorem, we have

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J. Liu, Z. Yin / J. Math. Anal. Appl. 375 (2011) 502–509 509

0 < w(0, x) = α2(0, x) + 1 + M2(0, x)

= (γ0 − γ0,xx)2 + 1 + u2

0,x

� 2(γ 2

0 + γ 20,xx

) + 1 + u20,x

� 2C1‖γ0‖2H3 + 2C2‖γ0,xx‖2

H1 + C3‖u0,x‖2H2 + 1

� C‖z0‖2H3×H3 + 1. (4.7)

Differentiating (4.6) with respect to t and using (4.4)–(4.5), we obtain

∂ w

∂t(t, x) = 2

α(0, x)

α(t, x)M(t, x)

(f (t, x) + 1

2

)

�(

K + 1

2

)α(0, x)

α(t, x)

(1 + M2)

�(

K + 1

2

)w(t, x).

By Gronwall’s inequality, the above inequality and (4.7), we have

w(t, x) � w(0, x)exp

{(K + 1

2

)t

}

�(C‖z0‖2

H3×H3 + 1)

exp

{(K + 1

2

)t

}for all (t, x) ∈ [0, T )×R. This implies that w(t, x) does not blow up in finite time. So neither |M(t, x)| → +∞ nor |α(t, x)| →+∞ will occur. This contradicts our assumption. Thus the corresponding solution exists globally in time. �Acknowledgments

This work was partially supported by NNSFC (No. 10971235), RFDP (No. 200805580014), NCET-08-0579 and the key project of Sun Yat-sen University.The authors are grateful to the reviewers for the valuable comments and suggestions.

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