Journal of Applied Mathematics Volume 2012, Article ID 197672,20pages doi:10.1155/2012/197672

*Research Article*

**On the Blow-Up of Solutions of a Weakly**

**Dissipative Modified Two-Component Periodic** **Camassa-Holm System**

**Yongsheng Mi,**

^{1, 2}**Chunlai Mu,**

^{1}**and Weian Tao**

^{2}*1**College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China*

*2**College of Mathematics and Computer Sciences, Yangtze Normal University, Fuling,*
*Chongqing 408100, China*

Correspondence should be addressed to Yongsheng Mi,miyongshen@163.com Received 16 May 2012; Revised 24 July 2012; Accepted 30 July 2012

Academic Editor: Ferenc Hartung

Copyrightq2012 Yongsheng Mi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa- Holm equation. We first establish the local well-posedness result. Then we derive the precise blow- up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow- up results for strong solutions to the system.

**1. Introduction**

In this paper, we consider the Cauchy problem of the following weakly dissipative modified two-component Camassa-Holm system:

*m*_{t}*um** _{x}*2mu

_{x}*ρρ*

_{x}*λm*0,

*t >*0, x∈

*R,*

*ρ*

*t*

*ρu*

*x**λρ*0, *t >*0, x∈*R,*
*m0, x m*0x, *x*∈*R,*

*ρ0, x ρ*_{0}x, *x*∈*R,*
*mt, x*1 *mt, x,* *t*≥0, x∈*R,*

*ρt, x*1 *ρt, x,* *t*≥0, x∈*R,*

1.1

where*m* 1−*∂*^{2}* _{x}*u,

*ρ*1−

*∂*

^{2}

*ρ−*

_{x}*ρ*

_{0}, and

*λ*is a nonnegative dissipative parameter.

The Camassa-Holm equation 1 has been recently extended to a two-component integrable systemCH2

*m*_{t}*um** _{x}*2mu

_{x}*ρρ*

_{x}*,*

*t >*0, x∈

*R,*

*ρ*

*t*

*ρu*

*x*0, *t >*0, x∈*R,* 1.2

with*mu−u** _{xx}*, which is a model for wave motion on shallow water, where

*ut, x*describes the horizontal velocity of the fluid, and

*ρt, x*is in connection with the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. Moreover,

*u*and

*ρ*satisfy the boundary conditions:

*u*→ 0 and

*ρ*→ 1 as |x| → ∞. The system can be identified with the first negative flow of the AKNS hierarchy and possesses the interesting peakon and multikink solutions2. Moreover, it is connected with the time-dependent Schr ¨odinger spectral problem2. Popowicz3observes that the system is related to the bosonic sector of an

*N*2 supersymmetric extension of the classical Camassa-Holm equation. Equation1.2 with

*ρ*≡0 becomes the Camassa-Holm equation, which has global conservative solutions4 and dissipative solutions5.

Since the system was derived physically by Constantin and Ivanov6in the context of shallow water theoryalso by Chen et al. in2and Falqui et al. in7, many researchers have paid extensive attention to it. In 8, Escher et al. establish the local well-posedness and present the precise blow-up scenarios and several blow-up results of strong solutions to 1.2on the line. In6, Constantin and Ivanov investigate the global existence and blow-up phenomena of strong solutions of1.2 on the line. Later, Guan and Yin9obtain a new global existence result for strong solutions to1.2and get several blow-up results, which improve the recent results in6. Recently, they study the global existence of weak solutions to1.2 10. In11, Henry studies the infinite propagation speed for1.2. Gui and Liu12 establish the local well-posedness for1.2in a range of the Besov spaces, they also derive a wave breaking mechanism for strong solutions. Mustafa13gives a simple proof of existence for the smooth travelling waves for1.2. Hu and Yin14,15study the blow-up phenomena and the global existence of1.2on the circle.

Recently, the CH2 system was generalized into the following modified two-component Camassa-HolmMCH2system:

*m*_{t}*um** _{x}*2mu

*−gρρ*

_{x}

_{x}*,*

*t >*0, x∈

*R,*

*ρ*

_{t}*ρu*

*x*0, *t >*0, x∈*R,* 1.3

where*m* 1−*∂*^{2}* _{x}*u,

*ρ*1−

*∂*

^{2}

*ρ−*

_{x}*ρ*

_{0},

*u*denotes the velocity field,

*ρ*

_{0}is taken to be a constant, and

*g*is the downward constant acceleration of gravity in applications to shallow water waves. This MCH2 system admits peaked solutions in the velocity and average density, we refer this to 16 for details. There, the authors analytically identified the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. They found that wave breaking in the fluid velocity does not imply singularity in the pointwise density

*ρ*at the point of vertical slope. Some other recent work can be found in 17–25. We find that the MCH2 system is expressed in terms of an averaged or filtered density

*ρ*in analogy to the relation between momentum and velocity by setting

*ρ* 1−*∂*^{2}* _{x}*ρ−

*ρ*

_{0}, but it may not be integrable unlike the CH2 system. The important point here is that MCH2 has the following conservation law:

*R*

*u*^{2}*u*^{2}_{x}*ρ*^{2}*ρ*^{2}_{x}

*dx,* 1.4

which play a crucial role in the study of1.3. Noting that for the CH2 system, we cannot
obtain the conservation of*H*^{1}norm.

In general, it is quite diﬃcult to avoid energy dissipation mechanisms in a real world.

Ghidaglia26studies the long time behaviour of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system. Recently, Hu and Yin 27 study the blow-up and blow-up rate of solutions to a weakly dissipative periodic rod equation. In 28,29, Hu considered global existence and blow-up phenomena for a weakly dissipative two-component Camassa-Holm system on the circle and on the line. However,1.1on the circleperiodic casehas not been studied yet. The aim of this paper is to study the blow- up phenomena of the strong solutions to1.1. We find that the behavior of solutions to the weakly dissipative modified two-component periodic Camassa-Holm system1.1is similar to that of the modified two-component Camassa- Holm system1.3, such as the local well- posedness and the blow-up scenario. In addition, we also find that the blow-up rate of1.1is not aﬀected by the weakly dissipative term, but the occurrence of blow-up of1.1is aﬀected by the dissipative parameter.

This paper is organized as follows: In Section 2, we establish local well-posedness of the Cauchy problem associated with 1.1. In Section 3, we derive precise the blow- up scenario of strong solution and the blow-up rate. In Section 4, we discuss the blow-up phenomena of1.1.

**2. Local Well-Posedness**

In this section, by applying Kato’s semigroup theory 30, we can obtain the local well-
posedness for the Cauchy problem of1.1in*H** ^{s}*×

*H*

*,*

^{s}*s >*3/2, with with

*S*

*R/Z*the circle of unit length.

First, we introduce some notations. All spaces of functions are assumed to be over*S;*

for simplicity, we drop*S*in our notation for function spaces if there is no ambiguity. If*A*is an
unbounded operator, we denote by*DA*the domain of*A.*A;*B*denotes the commutator
of two linear operators*A*and*B.* · * _{X}* denotes the norm of Banach space

*X. We denote the*norm and the inner product of

*H*

*;*

^{s}*s*∈

*R*

^{}, by ·

*and·,·*

_{s}*, respectively.*

_{s}For convenience, we state here Kato’s theorem in the form suitable for our purpose.

Consider the following abstract quasilinear evolution equation:

*du*

*dt* *Au fu,* *t >*0, u0 *u*_{0}*.* 2.1

Let*X* and*Y* be Hilbert spaces such that*Y* is continuously and densely embedded in*X*and
let*Q*:*Y* → *X*be a topological isomorphism.*LY, X*denotes the space of all bounded linear
operator from*Y* to*X*and we write*LX, ifXY*.

**Theorem 2.1**see30. Assume that
i*Ay*∈*LY, Xfory*∈*Xwith*

*A*
*y*

−*Az*
*w*

*X* ≤*μ*_{1}*y*−*z*

*X*w_{Y}*,* *y, z, w*∈*Y,* 2.2
*andAy*∈*GX,*1, β*uniformly on bounded sets inY.*

ii*QAyQ*^{−1} *Ay By, whereBy*∈*LXis bounded, uniformly on bounded sets in*
*Y. Moreover,*

*B*
*y*

−*Bz*
*w*

*X*≤*μ*_{2}*y*−*z*

*Y*w_{X}*,* *y, z,*∈*Y, w*∈*X.* 2.3
iii*f* :*Y* → *Y* *and extends also to a map fromXintoX, f* *is bounded on bounded sets inY*

*and*

*f*
*y*

−*fz*

*Y* ≤*μ*_{3}*y*−*z*

*Y**,* *y, z*∈*Y,*
*f*

*y*

−*fz*

*X* ≤*μ*3*y*−*z*

*X**,* *y, z*∈*Y,* 2.4

*where,* *μ*_{1}*,* *μ*_{2}*, and* *μ*_{3} *depend only on max{y*_{X}*,*z* _{X}*}

*and*

*μ*

_{4}

*depends only on*max{y

_{Y}*,*z

*}. If the above conditions (i), (ii), and (iii) hold, given*

_{Y}*u*

_{0}∈

*Y, there is*

*a maximalT >0 depending only on*u0

_{Y}*and a unique solution u to*2.1

*such that*

*uu·, u*0∈*C0, T*;*Y*∩*C*^{1}0, T;*X.* 2.5

*Moreover, the mapu*_{0} → *u·, u*0*is continuous fromYtoC0, T*;*Y*∩*C*^{1}0, T;*X.*

We now provide the framework in which we will reformulate system1.1. With*m*
*u*−*u** _{xx}*,

*ργ*−

*γ*

*, and*

_{xx}*γρ*−

*ρ*

_{0}, we can rewrite1.1as follows:

*m**t**um**x*2mu*x**ργ**x**λm*0, *t >*0, x∈*R,*
*ρ*_{t}

*ρu*

*x**λρ*0, *t >*0, x∈*R,*
*m0, x u*_{0}x−*u*_{0,xx}x, *x*∈*R,*
*ρ0, x γ*0x−*γ*0,xxx, *x*∈*R,*
*mt, x*1 *mt, x,* *t*≥0, x∈*R,*

*ρt, x*1 *ρt, x,* *t*≥0, x∈*R.*

2.6

Note that if*px*:coshx−x−1/2/2 sinh1/2,*x*∈*R*is the kernel of1−*∂*^{2}_{x}^{−1}, where
xstands for the integer part of*x*∈*R, then*1−*∂*^{2}_{x}^{−1}*f* *p*∗*f*for all*f* ∈*L*^{2}S,*p*∗*mu,*

and*p*∗*ργ. Here we denote by*∗the convolution. Using this identity, we can rewrite2.6
as follows:

*u*_{t}*uu** _{x}*−∂

*x*

*p*∗

*u*^{2}1
2*u*^{2}* _{x}*1

2*γ*^{2}−1

2*γ*_{x}^{2} −*λu,* *t >*0, x∈*R,*
*γ*_{t}*uγ** _{x}*−p∗

*u*_{x}*γ*_{x}

*x**u*_{x}*γ*

−*λγ,* *t >*0, x∈*R,*
*u0, x u*_{0}x, *x*∈*R,*

*γ0, x γ*0x, *x*∈*R,*
*ut, x*1 *ut, x,* *t*≥0, x∈*R,*
*γt, x*1 *γt, x,* *t*≥0, x∈*R,*

2.7

or we can write it in the following equivalent form:

*u**t**uu**x*−∂*x*

1−*∂*^{2}_{x}_{−1}
*u*^{2}1

2*u*^{2}* _{x}*1
2

*γ*

^{2}−1

2*γ*_{x}^{2} −*λu,* *t >*0, x∈*R,*
*γ*_{t}*uγ** _{x}*−∂

*x*

1−*∂*^{2}_{x}_{−1}
*u*_{x}*γ*_{x}

−

1−*∂*^{2}_{x}_{−1}

*u*_{x}*γ*−*λγ,* *t >*0, x∈*R,*
*u0, x u*_{0}x, *x*∈*R,*

*γ0, x γ*0x, *x*∈*R,*
*ut, x*1 *ut, x,* *t*≥0, x∈*R,*
*γt, x*1 *γt, x,* *t*≥0, x∈*R.*

2.8

**Theorem 2.2. Given***z*0 *zx,*0 u0*, γ*0 ∈ *H** ^{s}*×

*H*

^{s}*s >*3/2, then there exist a maximal

*T*

*T*z0

*>0 and a unique solutionz*u, γ

*to*1.1

*or*2.7

*such that*

*zz·, z*0∈*C0, T*;*H** ^{s}*×

*H*

*∩*

^{s}*C*

^{1}

0, T;*H** ^{s−1}*×

*H*

^{s−1}*.* 2.9

*Moreover, the solution depends continuously on the initial data, that is, the mappingz*0 → *z·, z*0:
*H** ^{s}*×

*H*

*→*

^{s}*C0, T*;

*H*

*×*

^{s}*H*

*∩*

^{s}*C*

^{1}0, T;

*H*

*×*

^{s−1}*H*

^{s−1}*is continuous and the maximal time of*

*existenceT >0 can be chosen to be independent ofs.*

The remainder of this section is devoted to the proof of Theorem2.2.

Let*z*_{u}

*γ*

,*Az *

*u∂**x**,* 0
0, u∂*x*

and

*fz *

⎛

⎝−∂*x*

1−*∂*^{2}_{x}_{−1}
*u*^{2} 1

2*u*^{2}* _{x}*1
2

*γ*

^{2}−1

2*γ*_{x}^{2} −*λu*

−∂*x*

1−*∂*^{2}_{x}_{−1}
*u**x**γ**x*

−

1−*∂*^{2}_{x}_{−1}

*u**x**γ*−*λγ.*

⎞

⎠*.* 2.10

Set*Y* *H** ^{s}* ×

*H*

*,*

^{s}*X*

*H*

*×*

^{s−1}*H*

*,Λ 1−*

^{s−1}*∂*

^{2}

_{x}^{1/2}and

*Q*

_{Λ}

_{0}

0 Λ

. Obviously, *Q* is an
isomorphism of *H** ^{s}* ×

*H*

*onto*

^{s}*H*

*×*

^{s−1}*H*

*. In order to prove Theorem 2.2 by applying Theorem2.1, we only need to verify*

^{s−1}*Az*and

*fz*which satisfy the conditionsi–iii.

We break the argument into several lemmas.

**Lemma 2.3. The operator**Az

*u∂**x**,* 0
0, u∂*x*

*, withz*∈*H** ^{s}*×H

^{s}*,s >*3/2, belongs to

*GL*

^{2}×L

^{2}

*,*1, β.

**Lemma 2.4. The operator**Az

*u∂**x**,* 0
0, u∂*x*

*, withz* ∈ *H** ^{s}*×

*H*

^{s}*,s >*3/2, belongs to

*GH*

*×*

^{s−1}*H*

^{s−1}*,*1, β.

**Lemma 2.5.** *Az *

*u∂**x**,* 0
0, u∂*x*

*, withz*∈*H** ^{s}*×H

^{s}*,s >*3/2. The operator

*Az*∈

*LH*

*×H*

^{s}

^{s}*, H*

*×*

^{s−1}*H*

*. Moreover,*

^{s−1}*A*
*y*

−*Az*
*w*

*H** ^{s−1}*×H

*≤*

^{s−1}*μ*

_{1}

*y*−

*z*

*H** ^{s}*×H

*w*

^{s}

_{H}*s*×H

^{s}*,*

*y, z, w*∈ ×H

^{s}*.*2.11

**Lemma 2.6. The operator**Bz Q, AzQ^{−1}*withz*∈*H** ^{s}*×

*H*

^{s}*,s >*3/2. Then

*Bz*∈

*LH*

*×*

^{s−1}*H*

^{s−1}*and*

*B*
*y*

−*Bz*
*w*

*H** ^{s−1}*×H

*≤*

^{s−1}*μ*

_{2}

*y*−

*z*

*H** ^{s}*×H

*w*

^{s}

_{H}*s−1*×H

^{s−1}*,*2.12

*fory, z*∈*H** ^{s}*×

*H*

^{s}*andw*∈

*H*

*×*

^{s−1}*H*

^{s−1}*.*

The proof of the above five lemmas can be done similarly as in8, therefore we omit it here.

Hence, according to Kato’s theoremTheorem2.1, in order to prove Theorem2.2, we only need to verify conditioniii, that is, we need to prove the following lemma.

* Lemma 2.7. Letz*∈

*H*

*×*

^{s}*H*

^{s}*,s >*3/2 and

*f*z

⎛

⎜⎜

⎜⎜

⎝

−∂*x*

1−*∂*^{2}_{x}_{−1}
*u*^{2} 1

2*u*^{2}* _{x}*1
2

*γ*

^{2}−1

2*γ*_{x}^{2} −*λu*

−∂*x*

1−*∂*^{2}_{x}_{−1}
*u**x**γ**x*

−

1−*∂*^{2}_{x}_{−1}

*u**x**γ*−*λγ*

⎞

⎟⎟

⎟⎟

⎠*.* 2.13

*Thenfis bounded on bounded sets inH** ^{s}*×

*H*

^{s}*and satisfies*

afy−*fz*_{H}*s*×H* ^{s}*≤

*μ*

_{3}y−

*z*

_{H}*s*×H

^{s}*,*

*y, z*∈

*H*

*×*

^{s}*H*

^{s}*,*bfy−

*fz*

_{H}*s−1*×H

*≤*

^{s−1}*μ*4y−

*z*

_{H}*s−1*×H

^{s−1}*,*

*y, z*∈

*H*

*×*

^{s}*H*

^{s}*.*

*Proof. Lety, z*∈*H** ^{s}*×

*H*

*,*

^{s}*s >*3/2. Since

*H*

*is a Banach algebra, it follows that*

^{s−1}*f*

*y*

−*fz*

*H** ^{s}*×H

^{s}≤
−∂*x*

1−*∂*^{2}_{x}_{−1}

*y*^{2}_{1}−*u*^{2}
1

2

*y*^{2}_{1x}−*u*^{2}* _{x}*
1

2

*y*^{2}_{2}−*γ*^{2}

−1 2

*y*_{2x}^{2} −*γ*_{x}^{2} −*λ*

*y*_{1}−*u*

*H*^{s}

−∂*x*

1−*∂*^{2}_{x}_{−1}

*y*_{1x}*y*_{2x}−*u*_{x}*γ*_{x}

−

1−*∂*^{2}_{x}_{−1}

*y*_{1x}*y*_{2}−*u*_{x}*γ*

−*λ*

*y*_{2}−*γ*

*H*^{s}

≤*y*_{1}−*u*

*y*_{1}*u*

*H** ^{s−1}*1

2*y*_{1x}−*u*_{x}

*y*_{1x}*u*_{x}

*H** ^{s−1}* 1

2*y*_{2}−*γ*

*y*_{2}*γ*

*H*^{s−1}

1

2*y*2x−*γ**x*

*y*2x*γ**x*

*H*^{s−1}*λy*1−*u*

*H*^{s}*u**x*

*y*2x−*γ**x*

*H*^{s−1}

*y*_{1x}−*u*_{x}*y*_{2x}

*H*^{s−1}*u*_{x}

*y*_{2}−*γ*

*H*^{s−2}*y*_{1x}−*u*_{x}*y*_{2}

*H*^{s−2}*λy*_{2}−*γ*

≤*cy*_{1}−*u*

*H*^{s−1}*y*_{1}*u*

*H** ^{s−1}*1

2*y*_{1}−*u*

*H*^{s}*y*_{1}*u*

*H** ^{s}*1

2*y*_{2}−*γ*

*H*^{s−1}*y*_{2}*γ*

*H*^{s−1}

1

2*y*2−*γ*

*H*^{s}*y*2*γ*

*H*^{s−1}*λy*1−*u*

*H** ^{s}*u

_{H}

^{s}*y*2−

*γ*

*H*^{s}*y*1−*u*

*H*^{s}*y*2

*H*^{s}

*λy*_{2}−*γ*

*H** ^{s}*u

_{H}*s−1*

*y*

_{2}−

*γ*

*H*^{s−2}*y*_{1}−*u*

*H*^{s−1}*y*_{2}

*H*^{s−2}

≤*cy*

*H** ^{s}*×H

*z*

^{s}

_{H}

^{s}_{×H}

^{s}*λy*−

*z*

*H** ^{s}*×H

^{s}*.*

2.14

This provesa. Taking*y*0 in the above inequality, we obtain that*f*is bounded on bounded
set in*H** ^{s}*×

*H*

*.*

^{s}Next, we proveb. Note that*H** ^{s−1}*is a Banach algebra. Then, we have

*f*

*y*

−*fz*

*H** ^{s−1}*×H

^{s−1}≤
−∂*x*

1−*∂*^{2}_{x}_{−1}

*y*^{2}_{1}−*u*^{2}
1

2

*y*^{2}_{1x}−*u*^{2}* _{x}*
1

2

*y*^{2}_{2}−*γ*^{2}

−1 2

*y*_{2x}^{2} −*γ*_{x}^{2} −λ

*y*_{1}−*u*

*H*^{s−1}

−∂*x*

1−*∂*^{2}_{x}_{−1}

*y*_{1x}*y*_{2x}−*u*_{x}*γ*_{x}

−

1−*∂*^{2}_{x}_{−1}

*y*_{1x}*y*_{2}−*u*_{x}*γ*

−*λ*

*y*_{2}−*γ*

*H*^{s−1}

≤*y*_{1}−*u*

*y*_{1}*u*

*H** ^{s−2}*1

2*y*_{1x}−*u*_{x}

*y*_{1x}*u*_{x}

*H** ^{s−2}* 1

2*y*_{2}−*γ*

*y*_{2}*γ*

*H*^{s−2}

1

2*y*_{2x}−*γ*_{x}

*y*_{2x}*γ*_{x}

*H*^{s−2}*λy*_{1}−*u*

*H*^{s−1}*u*_{x}

*y*_{2x}−*γ*_{x}

*H*^{s−2}

*y*_{1x}−*u*_{x}*y*_{2x}

*H*^{s−2}*u*_{x}

*y*_{2}−*γ*

*H*^{s−3}*y*_{1x}−*u*_{x}*y*_{2}

*H*^{s−3}*λy*_{2}−*γ*

≤*cy*_{1}−*u*

*H*^{s−2}*y*_{1}*u*

*H** ^{s−1}*1

2*y*_{1}−*u*

*H*^{s−1}*y*_{1}*u*

*H** ^{s−1}*1

2*y*_{2}−*γ*

*H*^{s−2}*y*_{2}*γ*

*H*^{s−2}

1

2*y*_{2}−*γ*

*H*^{s−1}*y*_{2}*γ*

*H*^{s−2}*λy*_{1}−*u*

*H** ^{s−1}*u

_{H}*s*

*y*

_{2}−

*γ*

*H*^{s−1}

*y*_{1}−*u*

*H*^{s−1}*y*_{2}

*H*^{s−1}*λy*_{2}−*γ*

*H** ^{s−1}*u

_{H}*s−2*

*y*

_{2}−

*γ*

*H*^{s−3}*y*_{1}−*u*

*H*^{s−2}*y*_{2}

*H*^{s−3}

≤*cy*

*H** ^{s}*×H

*z*

^{s−1}

_{H}*s−1*×H

^{s−1}*λy*−

*z*

*H** ^{s−1}*×H

^{s−1}*.*

2.15

This provesband completes the proof of the Lemma2.7.

*Proof of Theorem2.2. Combining Theorem*2.1and Lemmas2.3–2.7, we can get the statement
of Theorem2.2.

**3. The Precise Blow-Up Scenario and Blow-Up Rate**

In this section, we present the precise blow-up scenario and the blow-up rate for strong solutions to2.7.

**Lemma 3.1. Let**z_{0} u0*, γ*_{0}∈*H** ^{s}*×

*H*

^{s}*,s >*3/2, and let

*T*

*be the maximal existence time of the*

*solutionz*u, γ

*to*2.7

*with the initial dataz*

_{0}

*. Then for allt*∈0, T, we have

ut,·^{2}* _{H}*1

*γt,*·

^{2}

*H*^{1}*e*^{−2λt}

u0^{2}* _{H}*1

*γ*0

^{2}

*H*^{1}

*.* 3.1

*Proof. Denote*

*f*
*u, γ*

*u*^{2}1
2*u*^{2}* _{x}*1

2*γ*^{2}−1

2*γ*_{x}^{2}*,* *g* *g*
*u, γ*

*u*_{x}*γ*_{x}

*x**u*_{x}*γ.* 3.2

In view of the identity−∂^{2}_{x}*p*∗*ff*−*p*∗*f*, we can obtain from2.7,

*u**tx*−u^{2}* _{x}*−

*uu*

*xx*

*f*−

*p*∗

*f,*

*γ*

*tx*−u

*x*

*γ*

*x*−

*uγ*

*xx*−

*∂*

*x*

*p*∗

*g.*3.3

Therefore, an integration by parts yields

1 2

*d*
*dt*

u^{2}* _{H}*1

*γ*

^{2}

*H*^{1}

*R*

*uu**t**u**x**u**tx**γγ**t**γ**x**γ**tx*

*dx*

*R*

*u*

−uu*x*−*∂*^{2}_{x}*p*∗*f*−*λu*
*u*_{x}

−u^{2}* _{x}*−

*uu*

_{xx}*f*−

*p*∗

*f*−

*λu*

_{x}*γ*

−uγ*x*−*p*∗*g*−*λγ*
*γ*_{x}

−uγ*x*−*uγ** _{xx}*−

*∂*

_{x}*p*∗

*g*−

*λγ*

_{x}

*R*

−1
2*u*^{3}_{x}*u**x*

*u*^{2} 1

2*u*^{2}* _{x}*1
2

*γ*

^{2}−1

2*γ*_{x}^{2} −*uγγ**x*−*γ*

*u**xx**γ**x**u**x**γ*

−uγ*x**γ*_{x}^{2}−*uγ**x**γ**xx*−*λ*

*u*^{2}*u*^{2}_{x}*γ*^{2}*γ*_{x}^{2}
*dx*
−λ

*R*

*u*^{2}*u*^{2}_{x}*γ*^{2}*γ*_{x}^{2}
*dx.*

3.4

Thus, the statement of the conservation law follows.

**Lemma 3.2**see31. i*For everyf* ∈*H*^{1}S, we have

*x∈0,1*max*f*^{2}x≤ *e*1
2e−1*f*^{2}

*H*^{1}*,* 3.5

*where the constant*e1/2e−1*is sharp.*

ii*For everyf*∈*H*^{3}S, we have

*x∈0,1*max *f*^{2}x≤*cf*^{2}

*H*^{1}*,* 3.6

*with the best possible constantclying within the range*1,13/12. Moreover, the best constant*cis*
e1/2e−1.

So, if*z*∈*H*^{3}×*H*^{3}, then by Lemmas3.1and3.2, we have
ut,·^{2}* _{L}*∞

*γt,*·

^{2}

*L*^{∞} ≤ *e*1

2e−1u^{2}* _{H}*1

*e*1 2e−1

*γ*

^{2}

*H*^{1}

*e*1
2e−1

u0^{2}* _{H}*1

*γ*

_{0}

^{2}

*H*^{1}

*e*1

2e−1z0^{2}* _{H}*1×H

^{1}

*,*

3.7

for all*t*∈0, T.

* Theorem 3.3. Letz*0 u, γ ∈

*H*

*×*

^{s}*H*

^{s}*,s >*3/2 be given and assume that

*T*

*is the maximal*

*existence time of the corresponding solutionz*u, γ

*to*2.7

*with initial dataz*

_{0}

*, if there exists*

*M >0 such that*

u*x*t,·* _{L}*∞

*γ*

*x*t,·

*L*^{∞} ≤*M,* *t*∈0, T, 3.8

*then theH** ^{s}*×

*H*

^{s}*norm ofzt,*·

*does not blow-up on*0, T.

The proof of the theorem is similar to the proof of Theorem 3.1 in20, we omit it here.

Consider the following diﬀerential equation equation:

*dqx, t*
*dt* *u*

*qx, t, t*

*,* *t*∈0, T,
*q0, t x,* *x*∈*R,*

3.9

where*u*denotes the first component of the solution*z*to2.7. Applying classical results in
the theory of ordinary diﬀerential equations, one can obtain the following result on*q*which
is crucial in the proof of blow-up scenario.

**Lemma 3.4**see8. Let*u*_{0} ∈ ∩C^{1}0, T;*H** ^{s−1}*,

*s >*3/2, and

*Tbe the maximal existence time of*

*the corresponding solutionut, xto*3.7. Then3.7

*has a unique solutionq*∈

*C*

^{1}0, T×

*R, R.*

*Moreover, the mapqt,*·*is an increasing diﬀeomorphism ofRwith*

*q**x*x, t exp
_{t}

0

*u**x*

*qx, s, s*
*ds*

*>*0, *q**x*x,0 1, x∈*R,* 0≤*t < T.* 3.10

The following result is proved only with regard to *r* 3, since we can obtain the same
conclusion for the general case*r >*3/2 by using Theorem2.1and a simple density argument.

We now present a precise blow-up scenario for strong solutions to2.6.

**Theorem 3.5. Let**y_{0} u0*, γ*_{0} ∈ *H** ^{s}*×

*H*

^{s}*,s >*3/2, and let

*T*

*be the maximal existence of the*

*corresponding solutionz*u, γ

*to*2.7. Then the solution blows up in finite time if and only if

lim inf

*t→**T,x∈R**u**x*t, x −∞ *or lim sup*

*t→**T*

*γ**x*t,·

*L*^{∞}

∞. 3.11

*Proof. Multiplying the first equation in* 2.6 by*m* *u*−*u** _{xx}* and integrating by parts, we
obtain

*d*
*dt*

*S*

*m*^{2}*dx*2

*S*

*mm**t**dx*2

*S*

*m*

−um*x*−2mu*x*−*ργ**x*

*dx*−2λ

*S*

*m*^{2}*dx*
−3

*S*

*m*^{2}*u*_{x}*dx*−2

*S*

*mργ*_{x}*dx*−2λ

*S*

*m*^{2}*dx.*

3.12

Repeating the same procedure to the second equation in2.6we get
*d*

*dt*

*S*

*ρ*^{2}*dx*−

*S*

*ρ*^{2}*u**x*−2λ

*S*

*ρ*^{2}*dx.* 3.13

A combination of3.7and3.9yields
*d*

*dt*

*S*

*m*^{2}*ρ*^{2}

*dx*−3

*S*

*m*^{2}*u*_{x}*dx*−2

*S*

*mργ*_{x}*dx*−

*S*

*ρ*^{2}*u** _{x}*−2λ

*S*

*m*^{2}*ρ*^{2}

*dx.* 3.14

Diﬀerentiating the first equation in2.6with respect to*x, multiplying bym**x* *u**x*−*u**xxx*,
then integrating over*S, we obtain*

*d*
*dt*

*S*

*m*^{2}_{x}*dx*−5

*S*

*m*^{2}_{x}*u*_{x}*dx*2

*S*

*m*^{2}*u*_{x}*dx*−2

*S*

*m*_{x}*ρ*_{x}*γ*_{x}*dx*

−2

*S*

*m**x**ργ**xx**dx*−2λ

*S*

*m*^{2}_{x}*dx.*

3.15

Similarly,

*d*
*dt*

*S*

*ρ*^{2}_{x}*dx*−3

*S*

*ρ*^{2}_{x}*u**x**dx*

*S*

*ρ*^{2}*u**xxx**dx*−2λ

*S*

*ρ*^{2}_{x}*dx.* 3.16

A combination of3.12–3.16yields
*d*

*dt*

*S*

*m*^{2}*ρ*^{2}*m*^{2}_{x}*ρ*^{2}_{x}*dx*
−

*S*

*m*^{2}*u**x**dx*−5

*S*

*m*^{2}_{x}*u**x**dx*−2

*S*

*mργ**x**dx*−2

*S*

*m**x**ρ**x**γ**x**dx*−2λ

*S*

*m*^{2}*ρ*^{2}
*dx*

−2

*S*

*m*_{x}*ργ*_{xx}*dx*−

*S*

*ρ*^{2}*u*_{x}*dx*−3

*S*

*ρ*^{2}_{x}*u*_{x}*dx*

*S*

*ρ*^{2}*u*_{xxx}*dx*−2λ

*S*

*m*^{2}_{x}*ρ*_{x}^{2}
*dx*
−

*S*

*m*^{2}*u*_{x}*dx*−5

*S*

*m*^{2}_{x}*u*_{x}*dx*−

*S*

*ρ*^{2}*u*_{x}*dx*−3

*S*

*ρ*^{2}_{x}*u*_{x}*dx*−2λ

*S*

*m*^{2}*ρ*^{2}
*dx*

*S*

*ρ*^{2}*u*_{xxx}*dx*−2

*S*

*mργ*_{x}*dx*−2

*S*

*m*_{x}*ρ*_{x}*γ*_{x}*dx*−2

*S*

*m*_{x}*ργ*_{xx}*dx*−2λ

*S*

*m*^{2}_{x}*ρ*_{x}^{2}
*dx.*

3.17
Assume that there exists*M*1 *>*0 and*M*2 *>*0 such that*u**x*t, x≥ −M1andγ*x*t,·* _{L}*∞ ≤

*M*2

for allt, x∈0, T×*R, then it follows from Lemma*2.4that
*ρt,*·

*L*^{∞} ≤*e*^{M}^{1}^{T}*ρ*_{0}·

*L*^{∞}*.* 3.18

Therefore,
*d*
*dt*

*S*

*m*^{2}*ρ*^{2}*m*^{2}_{x}*ρ*^{2}_{x}*dx*

≤5M1

*S*

*m*^{2}*ρ*^{2}*m*^{2}_{x}*ρ*^{2}_{x}*dx*

*M*2*e*^{M}^{1}^{T}*ρ*0·

*L*^{∞}
*S*

*m*^{2}*ρ*^{2}*m*^{2}_{x}*ρ*^{2}_{x}*u*^{2}_{xxx}*γ*_{xx}^{2}
*dx*

≤5M1

*S*

*m*^{2}*ρ*^{2}*m*^{2}_{x}*ρ*^{2}_{x}*dx*2

*M*2*e*^{M}^{1}^{T}*ρ*0·

*L*^{∞}
*S*

*m*^{2}*ρ*^{2}*m*^{2}_{x}*ρ*^{2}_{x}*dx*

≤

5M_{1}2

*M*_{2}*e*^{M}^{1}^{T}*ρ*_{0}·

*L*^{∞}
*S*

*m*^{2}*ρ*^{2}*m*^{2}_{x}*ρ*^{2}_{x}*dx.*

3.19

The above discussion shows that if there exist*M*1 *>*0 and*M*2 *>*0 such that*u**x*t, x≥ −M1

andγ*x*t,· ≤ *M*_{2}for allt, x∈0, T×*S, then there exist two positive constantsK*and*k*
such that the following estimate holds

ut,·^{2}_{H}*s**γt,*·^{2}

*H** ^{s}* ≤

*Ke*

^{kt}*,*

*t*∈0, T. 3.20 This inequality, Sobolev’s embedding theorem and Theorem 3.3guarantee that the solution does not blow-up in finite time.

On the other hand, we see that if lim inf

*t→**T,x∈R**u** _{x}*t, x −∞ or lim sup

*t→**T*

*γ** _{x}*t,·

*L*^{∞}

∞, 3.21

then by Sobolev’s embedding theorem, the solution will blow-up in finite time. This completes the proof of the theorem.

**Lemma 3.6**see32. Let*T >0 andv*∈*C*^{1}0, T;*H*^{2}. Then for every*t*∈0, T, there exists at
*least one pointξ*∈*Rwith*

*ζt*:inf

*x∈R*v*x*t, x *v** _{x}*t, ξt. 3.22

*The functionζtis absolutely continuous on*0, T

*with*

*dζ*

*dt* *v** _{tx}*t, ξt, a.e., on0, T. 3.23

**Theorem 3.7. Let**z_{0}u0

*, γ*

_{0}∈

*H*

*×*

^{s}*H*

^{s}*,s >*3/2, z u, γ

*be the corresponding solution to*2.7

*with initial dataz*0

*and satisfies*γ

*x*t, x

*∞ ≤*

_{L}*M, for all*t, x∈0, T×

*S,T*

*be the maximal*

*existence time of the solution. Then we have*

*t→*lim*T*

*x∈R*inf*u** _{x}*t, xT−

*t*−2. 3.24

*Proof. Applying Theorems*2.1and a simple density argument, we only need to show that the
above theorem holds for some*s >*3/2. Here, we assume*s*3 to prove the above theorem.

Define now

*gt *inf

*x∈s**u**x*t, x, *t*∈0, T, 3.25
and let*ξ*∈*S*be a point where this minimum is attained. Clearly,*u**xx*t, ξt 0 since*ut,*·∈
*H*^{3} ⊂*C*^{2}S. Diﬀerentiating the first equation of2.7with respect to*x, in view of∂*^{2}_{x}*p*∗*f*
*p*∗*f*−*f, we have*

*u*_{tx}*uu** _{xx}*−1

2*u*^{2}_{x}*u*^{2}1
2*γ*^{2}−1

2*γ*_{x}^{2}−*p*∗

*u*^{2} 1
2*u*^{2}* _{x}*1

2*γ*^{2}−1

2*γ*_{x}^{2} −*λu*_{x}*.* 3.26

Evaluating3.26at*ξt*and using Lemma3.6, we obtain
*d*

*dtgt * 1

2*g*^{2}t *λgt u*^{2}t, ξt 1

2*γ*^{2}t, ξt−1

2*γ*_{x}^{2}t, ξt−
*p*∗*f*

t, ξt, 3.27
where*f* *u*^{2} 1/2u^{2}* _{x}* 1/2γ

^{2}−1/2γ

_{x}^{2}. By Lemma3.1and Young’s inequality, we have for all

*t*∈0, Tthat

*p*∗*f*

*L*^{∞}≤ *G** _{L}*∞

*u*^{2} 1
2*u*^{2}* _{x}*1

2*γ*^{2}−1
2*γ*_{x}^{2}

*L*^{1}

≤ cosh1/2 2 sinh1/2

u^{2}* _{H}*1

*γ*

^{2}

*H*^{1}

cosh1/2

2 sinh1/2z^{2}* _{H}*1×H

^{1}≤ cosh1/2

2 sinh1/2z0^{2}* _{H}*1×H

^{1}

*.*

3.28

This relation together with3.7andγ*x*t, x* _{L}*∞ ≤

*M*implies that there is a constant

*K >*0 such that

*g*^{}t 1

2*g*t *λgt*

≤*K,* 3.29

where*K*depends only onu0* _{H}*1andγ0

*1. It follows that*

_{H}−K−1

2*λ*^{2} ≤*g*^{}t 1
2

*gt λ*_{2}

≤*K*1

2*λ*^{2} a.e., on0, T. 3.30
Choose ∈0,1/2. Since lim inf*t*→Tyt *λ *−∞by Theorem3.5, there is some*t*0∈0, T
with*gt*0 *λ <*0 andgt0 *λ*^{2}*> K* 1/2λ^{2}*/ . Let us first prove that*

*gt λ*2

*>* 1
*K*1

2*λ*^{2} *,* *t*∈t0*, T*. 3.31

Since*g*is locally Lipschitz, there is some*δ >*0 such that
*gt λ*2

*>* 1
*K*1

2*λ*^{2} *,* *t*∈t0*, t*_{0}*δ.* 3.32
Note that*g* is locally Lipschitzit belongs to*W*_{loc}^{1,∞}sby Lemma3.6and therefore
absolutely continuous. Integrating the previous relation ont0*, t*_{0}*δ*yields that

*gt*0*δ λ*≤*gt*0 *λ <*0. 3.33

It follows from the above inequality that
*gt*0*δ λ*2≥

*gt*0 *λ*2

*>* 1
*K*1

2*λ*^{2} *.* 3.34

The obtained contradiction completes the proof of the relation3.31. By 3.30-3.31, we infer

1

2 − ≤ − *g*^{}t
m*λ*^{2} ≤ 1

2 * ,* a.e.on0, T. 3.35

For*T* ∈t0*, T*, integrating3.35ont, Tto get
1

2 − T−*t*≤ − 1
*gt λ* ≤

1

2 T−*t,* *t*∈t0*, T*. 3.36
Since*g*t *λ <*0 ont0*, T*, it follows that

1

1/2 ≤ −

*g*t *λ*

T−*t*≤ 1

1/2 *,* *t*∈t0*, T*. 3.37
By the arbitrariness of ∈0,1/2, the statement of the theorem follows.

**4. Blow-Up**

In this section, we discuss the blow-up phenomena of2.7and prove that there exist strong solutions to2.7which do not exist globally in time.

* Theorem 4.1. Letz*0 u0

*, γ*0 ∈

*H*

*×*

^{s}*H*

^{s}*,s >*3/2 and

*T*

*be the maximal existence time of the*

*solutionz*u, γ

*to*2.7

*with the initial dataz*

_{0}

*. If there exists somex*

_{0}∈

*Ssuch that*

*u*^{}_{0}x0*<*−λ−

*λ*^{2}
*e*1

*e*−1 cosh1/2

2 sinh1/2 z0^{2}* _{H}*1×H

^{1}

*,*4.1

*then the existence timeT*

*is finite and the slope ofutends to negative infinity astgoes toT*

*whileu*

*remains uniformly bounded on*0, T.

*Proof. As mentioned earlier, here we only need to show that the above theorem holds for*
*s*3. Diﬀerentiating the first equation of2.7with respect to*x, in view of∂*^{2}_{x}*p*∗*fp*∗*f*−*f,*
we have

*u*_{tx}*uu** _{xx}*−1

2*u*^{2}_{x}*u*^{2}1
2*γ*^{2}−1

2*γ*_{x}^{2}−*p*∗

*u*^{2} 1
2*u*^{2}* _{x}*1

2*γ*^{2}−1

2*γ*_{x}^{2} −*λu*_{x}*.* 4.2
Define now

*gt*:min

*x∈S*u*x*t, x, *t*∈0, T, 4.3
and let*ξt*∈*S*be a point where this minimum is attained. It follows that

*gt u**x*t, ξt. 4.4

Clearly*u**xx*t, ξt 0 since*ut,*·∈*H*^{3}S⊂*C*^{2}S. Evaluating4.2at*ξt, we obtain*

*u** _{tx}*t, ξt 1

2*u*^{2}* _{x}*t, ξt

*λu*

*t, ξt*

_{x}*u*

^{2}t, ξt 1

2*γ*^{2}t, ξt− 1

2*γ*_{x}^{2}t, ξt

−*p*∗

*u*^{2}1
2*u*^{2}* _{x}*1

2*γ*^{2}−1

2*γ*_{x}^{2} t, ξt

≤*u*^{2}t, ξt 1

2*γ*^{2}t, ξt 1

2*p*∗*γ*_{x}^{2}t, ξt

≤ *e*1

2e−1z0^{2}* _{H}*1×H

^{1}cosh1/2 4 sinh1/2γ

_{x}^{2}

*L*^{1}

≤

*e*1

2e−1 cosh1/2

4 sinh1/2 z0^{2}* _{H}*1×H

^{1}

*,*

4.5

here, we used Lemma3.2and
p∗*γ*_{x}^{2}

*L*^{∞}≤*p*

*L*^{∞}

γ_{x}^{2}

*L*^{1} cosh1/2
2 sinh1/2γ_{x}^{2}

*L*^{1}*.* 4.6

Inequality4.5and Lemma 3.4 imply

*d*

*dtgt *1

2*g*^{2}t *λgt*≤

*e*1

2e−1 cosh1/2

4 sinh1/2 z0^{2}* _{H}*1×H

^{1}

*,*4.7

that is,

*d*

*dtgt*≤ −1

2*g*^{2}t−*λgt *

*e*1

2e−1 cosh1/2

4 sinh1/2 z0^{2}* _{H}*1×H

^{1}

*,*4.8

Take

*K*:

*e*1

2e−1 cosh1/2

4 sinh1/2z0* _{H}*1×H

^{1}

*.*4.9

It then follows that

*g*^{}t≤ −1

2*g*^{2}t−*λgK*^{2}
−1

2

*gt λ*

*λ*^{2}2K^{2}

*gt λ*−

*λ*^{2}2K^{2}
*.*

4.10