Journal of Applied Mathematics

Volume 2013 (2013), Article ID 547261, 11 pages

http://dx.doi.org/10.1155/2013/547261

Research Article

Well-Posedness, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation

Yongsheng Mi1,2 and Chunlai Mu²

¹College of Mathematics and Computer Sciences, Yangtze Normal University, Chongqing, Fuling 408100, China

²College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

Received 20 March 2013; Revised 27 June 2013; Accepted 3 July 2013

Academic Editor: Michael Meylan

Copyright © 2013 Yongsheng Mi and Chunlai Mu. 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.

Abstract.

We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions.

1. Introduction

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

where ,, and is a nonnegative dissipative parameter.

The Camassa-Holm equation [1] has been recently extended to a two-component integrable system (CH2)

with , which is a model for wave motion on shallow water, where describes the horizontal velocity of the fluid and is in connection with the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. Moreover, and satisfy the boundary conditions: and as . The system can be identified with the first negative flow of the AKNS hierarchy and possesses the interesting peakon and multikink solutions [2]. Moreover, it is connected with the time-dependent Schrödinger spectral problem [2]. Popowicz [3] observes that the system is related to the bosonic sector of an supersymmetric extension of the classical Camassa-Holm equation. Equation (2) with becomes the Camassa-Holm equation, which has global conservative solutions [4] and dissipative solutions [5]. For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water equations, the reader is referred to [6–8] and the references therein.

Since the system was derived physically by Constantin and Ivanov [9] in the context of shallow water theory (also by Chen et al. in [2] and Falqui in [10]), many researchers have paid extensive attention to it. In [11], Escher et al. establish the local well-posedness and present the precise blow-up scenarios and several blow-up results of strong solutions to (2) on the line. In [9], Constantin and Ivanov investigate the global existence and blow-up phenomena of strong solutions of (2) on the line. Later, Guan and Yin [12] obtain a new global existence result for strong solutions to (2) and get several blow-up results, which improve the recent results in [9]. Recently, they study the global existence of weak solutions to (2) [13]. In [14], Henry studies the infinite propagation speed for (2). Gui and Liu [15] establish the local well-posedness for (2) in a range of the Besov spaces; they also derive a wave-breaking mechanism for strong solutions. Mustafa [16] gives a simple proof of existence for the smooth travelling waves for (2). Hu and Yin [17, 18] study the blow-up phenomena and the global existence of (2) on the circle.

Recently, the CH2 system was generalized into the following modified two-component Camassa-Holm (MCH2) system:

where ,, denotes the velocity field, is taken to be a constant, and is the downward constant acceleration of gravity in applications to shallow water waves. This MCH2 system does admit peaked solutions in the velocity and average density; we refer this to [19] 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 at the point of vertical slope. Some other recent works can be found in [20–31]. We find that the MCH2 system is expressed in terms of an averaged or filtered density in analogy to the relation between momentum and velocity by setting , but it may not be integrable unlike the CH2 system. The important point here is that MCH2 has the following conservation law , which plays a crucial role in the study of (3), noting that, for the CH2 system, we cannot obtain the conservation of norm.

In general, it is quite difficult to avoid energy dissipation mechanisms in the real world. Ghidaglia [32] studies the long-time behavior of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system. Recently, Hu and Yin [33] study the blow-up and blow-up rate of solutions to a weakly dissipative periodic rod equation. In [34, 35], 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) on the line (nonperiodic case) has not been studied yet. The aim of this paper is to study the blow-up phenomena and asymptotic profile of the strong solutions to (1). We find that asymptotic profile of solutions to the weakly dissipative modified two-component periodic Camassa-Holm system (1) is similar to that of the modified two-component Camassa-Holm system (3), such as the local well-posedness and the blow-up scenario. In addition, we also find that the blow-up rate of (1) is not affected by the weakly dissipative term, but the occurrence of blow-up of (1) is affected by the dissipative parameter.

This paper is organized as follows. In Section 2, we present some notations and establish the local well-posedness for system (1) by applying Kato's semigroup approach to nonlinear hyperbolic evolution equations. In Section 3, we prove a precise blow-up scenario result. In Section 4, we present the blow-up results for strong solutions to (1) provided that the initial data satisfy appropriate conditions and we derive a blow-up rate estimate result. Finally, we consider the asymptotic behavior of solutions.

2. Local Well-Posedness

We now provide the framework in which we will reformulate system (1). With ,, and , we can rewrite (1) as follows:

Note that if ,, then for all ,, and . Here, we denote by the convolution. Using this identity, we can rewrite (5) as follows:

or we can write it in the equivalent form

The local well-posedness of the Cauchy problem (5) in Sobolev spaces with can be obtained by applying Kato's theorem [23, 36]. As a result, we have the following well-posedness result.

Theorem 1. Given , , then there exist a maximal and a unique solution to (3) (or (6)) such that

Moreover, the solution depends continuously on the initial data; that is, the mapping is continuous.

3. The Precise Blow-Up Scenario

In this section, we present the precise blow-up scenarios for solutions to (6).

Theorem 2. Let ,, be given and assume that is the maximal existence time of the corresponding solution to (6) with initial data ; if there exists such that

then the norm of does not blow up on .

The proof of the theorem is similar to the proof of Theorem 3 in [22]; we omit it here.

Consider the following differential equation:

By applying the classical results on the theory of ordinary differential equations, we may derive the following properties of the solution of (9), which are crucial in the proof of global existence and blow-up of solutions.

Lemma 3 (see [23]). Let ,, and let be the maximal existence time of the corresponding solution to (9). Then (9) has a unique solution . Moreover, the map is an increasing diffeomorphism of with

Lemma 4. Let with , and let be the maximal existence time of the corresponding solution to (5). Then, one has

Moreover, if there exists , such that for all , then

Proof. Differentiating the left-hand side of (6) with respect to , in view of (9) and the second equation of (5), we have

Solving the equation, we get (11).

By Lemma 3, in view of (11) and the assumption of the lemma, we obtain

The following result is proved only with regard to , since we can obtain the same conclusion for the general case by using Theorem 1 and a simple density argument.

We now present a precise blow-up scenario for strong solutions to (5).

Theorem 5. Let , , and let be the maximal existence of the corresponding solution to (6). Then, the solution blows up in finite time if and only if

or

Proof. Multiplying the first equation in (5) by and integrating by parts, we obtain

Repeating the same procedure to the second equation in (5), we get

A combination of (17) and (18) yields

Differentiating the first equation in (5) with respect to , multiplying by , and then integrating over , we obtain

Similarly,

A combination of (17)–(21) yields

Assume that there exist and such that and for all ; then it follows from Lemma 4 that

Therefore,

The previous discussion shows that if there exist and such that and for all , then there exist two positive constants and such that the following estimate holds:

This inequality, Sobolev's embedding theorem, and Theorem 2 guarantee that the solution does not blow up in finite time.

On the other hand, we see that, if

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

4. Blow-Up Results and Blow-Up Rate Estimate

In this section, we investigate the blow-up phenomena of strong solutions to (6). We now present the first blow-up result.

Lemma 6. Let ,, and let be the maximal existence time of the solution to (6) with the initial data . Then, for all , one has

Moreover,

Proof. Denote

In view of the identity , we can obtain, from (6),

Therefore, an integration by parts yields

Thus, the statement of the conservation law follows. The remaining part of this lemma can be easily deduced from the conservation law. The proof of the lemma is complete.

Lemma 7 (see [37]). Let and . Then, for every , there exists at least one point with

The function is absolutely continuous on with

Theorem 8. Let , and let be the maximal existence time of the solution to the (6) with the initial data . If there exists some such that

then the existence time is finite and the slope of tends to negative infinity as goes to while remains uniformly bounded on .

Proof. As mentioned earlier, here we only need to show that the previous theorem holds for . Differentiating the first equation of (6) with respect to , in view of , we have

Note that

We know that and

By (35) and (36) and the previous estimates, we deduce that

in view of Lemma 6. Take

and define . It then follows from (38) that on ,

Note that if , then , . Therefore, we can solve the previous inequality to obtain

Due to , then there exists , and , such that . Applying Theorem 5, the solution does not exist globally in time.

Next, we give a blow-up result if and are odd.

Theorem 9. Let ,, and let be the maximal existence time of the solution to (6) with the initial data . If and are odd, and furthermore

then is finite and as goes to .

Proof. As mentioned earlier, here we only need to show that the previous theorem holds for . Note that (6) is the invariant under the transformation and . Thus, we deduce that if and are odd, then and are odd for any . By continuity with respect to of and , we have

Hence, in view of (35) and Lemma 6, we obtain

Take

and define . It then follows from (38) that on ,

Note that if , then , for all . Therefore, we can solve the previous inequality to obtain

Due to , then there exists , and , such that . Applying Theorem 5, the solution does not exist globally in time.

Next, we give more insight into the blow-up rate for the wave-breaking solutions to (6).

Theorem 10. Let ,, be the corresponding solution to (6) with initial data and satisfy for all , and be the maximal existence time of the solution. Then let one has

Proof. By Lemma 6, we get the uniform bound of . Set . By the proof of Theorem 8 (or Theorem 9), we find a constant such that

where depends only on and . It follows that

Choose . Since by Theorem 5, there is some with and . Let us first prove that

Since is locally Lipschitz, there is some such that

Note that is locally Lipschitz and therefore absolutely continuous. Integrating the previous relation on yields that

It follows from the previous inequality that

By (50)-(51), we infer that

For , integrating (55) on   to get

Since on , it follows that

By the arbitrariness of , the statement of the theorem follows.

5. Asymptotic Profile

In this section, we focus on the persistence property of the solution to (6) in -space. Precisely, we give an asymptotic description on how the solutions behave under the initial values possess algebraic decay at infinity. Recently, the asymptotic behavior for the celebrated Camassa-Holm equation was investigated in [38]. We notice that in [39], the authors showed that the solution of the Camassa-Holm equation and its first-order spatial derivative retain exponential decay at infinity as their initial values behave. After all, the exponential decay of initial value is a faster way; this motivates us to establish the decay rate of solution if its initial value decays algebraically. We show that the strong solution of (6) corresponding to initial data with a slower algebraically decaying way will keep this behavior in the -variable at infinity in its lifespan. In order to achieve our result, we first recall the following lemma.

Lemma 11 (see [40]). For a function defined next, there exists a constant which only depends on , such that for any positive integer

where

Theorem 12. Assume that with satisfies that for some

Then, the corresponding strong solution to (6) satisfies that

uniformly in the time interval .

Notation. One has

where is a nonnegative constant. In order to shorten the presentation in the sequel, we introduce

Proof. The first step is devoted to giving estimates on and , where is the standard norm.

Multiplying the first equation of (6) by with and integrating both sides with respect to variable, we obtain

The first term in (64) is

for the second term of (64), we have

It follows from the Höder inequality that

Therefore,

Similarly, for the estimate of , we have by multiplying and integration

By the Sobolev embedding theorem, there exists a constant such that

In view of Gronwall's inequality, we have the estimate

We can take limits as goes to infinity to obtain

The second step is to establish estimates for and by using the same method as previously mentioned. Differentiating the first equation of (6) with respect to produces the following equation:

Multiplying (73) by , and then integrating by parts, one obtains

Similarly, one can obtain

For the second equation of (6), we may get

and by Gronwall's inequality

In order to arrive at our result, we introduce a weighted continuous function which is independent on as follows:

where ,,. It is trivial that

where the derivative is with respect to the variable . From the first equation of (6) and (73), we have

Next, in order to obtain the estimates on and , we apply a similar technique that was used before for and step by step to (80) and (81). For (81), we need to eliminate the term with the second-order derivative in order to attain the estimate for . Using integration by parts, we obtain

where (79) is used directly. Therefore, with these preparations, it holds that

For the second equation of (6), we have

Therefore,

Then by (85), we obtain

On the other hand, for a suitable function , one obtains, due to Lemma 11,

Similarly,

Note that there are two additional quantities in (86) to be dealt with. Let us estimate first. One has

It is similar to the remaining . Then, combining (87)–(89) with (83) and (86), it follows that there exists a constant such that

where

are introduced just for simplicity. Next for any ,, and , we have, by Gronwall's inequality,

Finally, passing limit as goes to infinity in the previous inequality, we obtain

We complete the proof.

Acknowledgments

The authors would like to say thanks to Professor Zhaoyang Yin and Professor Yong Zhou for sending several of their papers to the authors. This work was partially supported by NSF of China (11371384), NSF of Chongqing (cstc2013jcyjA0940), and NSF of Fuling (FLKJ, 2013ABA2036).

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