Blow-Up for a Periodic Two-Component Camassa-Holm Equation with Generalized Weakly Dissipation

In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria and the blow-up rate are established by applying monotonicity. Finally, the global existence results for solutions to the Cauchy problem of equation are proved by structuring functions.


Introduction
In this paper, we consider the Cauchy problem of periodic two-component Camassa-Holm equation with a generalized weakly dissipation: , , 1 , 0, , where 0 λ ≥ and k is a fixed constant; σ is a free parameter.
It is well known that the two-component integrable Camassa , u t x standing for the fluid velocity at time 0 t ≥ in the spatial x direction [1], ( ) , t x ρ is in connection with the horizontal deviation of the surface from equilibrium (i.e. amplitude). Equation (1.2) possesses a bi-Hamiltonian structure [2] and the solution interaction of peaked travelling waves and wave breaking [1] [2] [3]. It is completely integrable [3] and becomes the Camassa-Holm equation when 0 ρ = .
Equation (1.2) was derived physically by Constantin and Ivanov [4] in the context of shallow water theory. As soon as this equation was put forward, it attracted attention of a large number of researchers. Escher et al. [5] established the local well-posedness and present the blow-up scenarios and several blow-up results of strong solutions to Equation (1.2). Constantin and Ivanov [6] investigated the global existence and blow-up phenomena of strong solutions of Equation (1.2). Guan and Yin [7] obtained a new global existence result for strong solutions to Equation (1.2) and several blow-up results, which improved the results in [6]. Gui and Liu [8] established the local well-posedness for Equation (1.2) in a range of the Besov spaces, they also characterized a wave breaking mechanism for strong solutions. Hu and Yin [9] [10] studied the blow-up phenomena and the global existence of Equation (1.2).
Dissipation is an inevitable phenomenon in real physical word. It is necessary to study periodic two-Camassa-Holm equation with a generalized weakly dissipation. Hu and Yin [11] study the blow-up of solutions to a weakly dissipative periodic rod equation. Hu considered global existence and blow-up phenomena for a weakly dissipative two-component Camassa-Holm system [12] [13]. The purpose of this paper is to study the blow-up phenomenon of the solutions of Furthermore, we also learn the blow-up rate of solutions. In Section 4, we address the global existence of Equation (1.1).

Local Well-Posedness
Let us introduce some notations, the S R Z = is the circle of unit length, the [ ] x stands for the integer part of x R ∈ , the * stands for the convolution, the X ⋅ is used to represent the norm of Banach space X.
In this section, we investigate the local well-posedness for the Cauchy problem of Equation (1.1) by applying Kato's theory [14] in For convenience we recall the Kato's theorem in the suitable form for our purpose. Consider the following abstract quasilinear evolution equation: There are two Hilbert's spaces X and Y, Y is continuously and densely embedded in X and : Q Y X → is a topological isomorphism, the ( ) , L Y X stands for the space of all bounded linear operator from Y to X. 3 If the 1), 2), 3) hold, given 0 u Y ∈ , there is a maximal 0 T > depending only on 0 Y u and a unique solution u of Equation (2.1) such that Moreover, the map Then Equation (1.1) can be rewritten as The [15] shows that Q is an isomorphism from It is sufficiently to verify ( )

Blow-Up
This section will establish a blow-up criterion for solution of Equation Consider the following equation of trajectory: Hence Journal of Applied Mathematics and Physics , then for every The second equation of Equation (1.1) can be rewritten as According to (3.6) and (3.7) where the constant ( ) where the best constant c is ( )

3) For every
( ) s ≥ , and T be the maximal time of existence, then ( ) Proof: The theorem 2.2 and a density argument imply that it is sufficient to prove the desired estimates for 3 s = .
Differentiate the first equation of Equation (2.7) with respect to x From the Fermat's lemma, we know From (3.11) and the second equation of Equation (1.1), we obtain Similarly, we turn to the lower bound of f From the second Equation of (3.15), we have By contradictory arguement, there exists From (3.21), we know ( ) ( ) On the other hand, from the first Equation of (3.15), we have ( ) It yields a contradiction, then the proof of the Lemma 3.4 is complete.
The similar process to (3.16) leads to Hence we know From (3.32), we have Integrating (3.34) over [ ) The proof of the Theorem 3.3 is complete.
The proof of the theorem 3.4 is complete.
Next we will show the blow-up rate of solutions and the result shows: the blow-up rate is not affected by the weakly dissipation.

of Applied Mathematics and Physics
Since ε is arbitrary, so is not effected by the weakly dissipation.

Global Existence
In this section, we provide a sufficient condition for the global solution of Equation (1.1) in the case 0 2 σ < < .    Here we apply Young's inequality