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

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.

The Camassa-Holm equation [1] has been recently extended to a two-component integrable system (CH2)   +   + 2  =   ,  > 0,  ∈ R, 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:  → 0 and  → 1 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 timedependent Schrödinger spectral problem [2].Popowicz [3] observes that the system is related to the bosonic sector of an  = 2 supersymmetric extension of the classical Camassa-Holm equation.Equation (2) with  ≡ 0 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][7][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 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.

Local Well-Posedness
We now provide the framework in which we will reformulate system (1).With  =  −   ,  =  −   , and  =  −  0 , we can rewrite (1) as follows: ),  *  = , 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  > 5/2 can be obtained by applying Kato's theorem [23,36].As a result, we have the following well-posedness result.

The Precise Blow-Up Scenario
In this section, we present the precise blow-up scenarios for solutions to (6).
The proof of the theorem is similar to the proof of Theorem 3 in [22]; we omit it here.
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  0 ∈   ,  ≥ 5/2, and let  be the maximal existence time of the corresponding solution (, ) to (9).Then (9) has a unique solution  ∈  1 ([0, ) × R, R).Moreover, the map (, ⋅) is an increasing diffeomorphism of R with 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  = 3, since we can obtain the same conclusion for the general case  > 5/2 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  0 = ( 0 ,  0 ) ∈   ×  ,  > 5/2, 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 lim sup 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 R, we obtain Similarly, A combination of ( 17)-( 21) yields Assume that there exist  1 > 0 and  2 > 0 such that (, ) ≥  1 and ‖  (, ⋅)‖  ∞ ≤  2 for all (, ) ∈ [0, ) × ; then it follows from Lemma 4 that Therefore, The previous discussion shows that if there exist  1 > 0 and  2 > 0 such that   (, ) ≥  1 and ‖  (, ⋅)‖ ≤  2 for all (, ) ∈ [0, ) × R, 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 lim inf then, by Sobolev's embedding theorem, the solution will blow up in finite time.This completes the proof of the theorem.

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  0 = ( 0 ,  0 ) ∈   ×   ,  > 5/2, and let  be the maximal existence time of the solution  = (, ) to (6) with the initial data  0 .Then, for all  ∈ [0, ), one has Moreover, Proof.Denote In view of the identity − 2   *  =  −  * , 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.
Theorem 8. Let  0 = ( 0 ,  0 ) ∈   ×   ,  > 5/2, and let  be the maximal existence time of the solution  = (, ) to the (6) with the initial data  0 .If there exists some  0 ∈ R such that then the existence time  is finite and the slope of  tends to negative infinity as  goes to  while  remains uniformly bounded on [0, ].
Next, we give more insight into the blow-up rate for the wave-breaking solutions to (6).

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.
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   (R) norm.
Multiplying the first equation of ( 6) by  2−1 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 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  2−1

𝑥
, and then integrating by parts, one obtains For the second equation of ( 6), we may get In order to arrive at our result, we introduce a weighted continuous function which is independent on  as follows: where  ∈ (0, 1],  ∈  + ,  > 2. It is trivial that where the derivative is with respect to the variable .From the first equation of ( 6) and (73), we have where We complete the proof.