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Global Weak Solution for a generalized Camassa-Holm equation

In this paper we mainly investigate the Cauchy problem of a generalized Camassa-Holm equation. First by this relationship between the Degasperis-Procesi equation and the generalized Camassa-Holm equation, we then obtain two global existences result and two blow-up result. Then, we prove the existence and uniqueness of global weak solutions.


INTRODUCTION
Note that G(x) = 1 2 e −|x| and G(x) ⋆ f = (1 − ∂ 2 x ) −1 f for all f ∈ L 2 (R) and G ⋆ m = u. Then we can rewrite (1.1) as follows: (1.2) The equation (1.1) was proposed recently by Novikov in [42]. He showed that the equation (1.1) is integrable by using as definition of integrability the existence of an infinite hierarchy of quasi-local higher symmetries [42] and it belongs to the following class [42]: which has attracted much interest, particularly in the possible integrable members of (1.3).
The most celebrated integrable member of (1.3) is the well-known Camassa-Holm (CH) equation [4]: The CH equation can be regarded as a shallow water wave equation [4,15]. It is completely integrable. That means that the system can be transformed into a linear flow at constant speed in suitable actionangle variables (in the sense of infinite-dimensional Hamiltonian systems), for a large class of initial data [4,7,16]. It also has a bi-Hamiltonian structure [6,28], and admits exact peaked solitons of the form ce −|x−ct| with c > 0, which are orbitally stable [19]. It is worth mentioning that the peaked solitons present the characteristic for the traveling water waves of greatest height and largest amplitude and arise as solutions to the free-boundary problem for incompressible Euler equations over a flat bed, cf. [5,9,13,14,44].
The local well-posedness for the Cauchy problem of the CH equation in Sobolev spaces and Besov spaces was discussed in [10,11,20,21,43]. It was shown that there exist global strong solutions to the CH equation [8,10,11] and finite time blow-up strong solutions to the CH equation [8,10,11,12]. The existence and uniqueness of global weak solutions to the CH equation were proved in [17,51]. The global conservative and dissipative solutions of CH equation were investigated in [2,3].
The second celebrated integrable member of (1.3) is the famous Degasperis-Procesi (DP) equation [23]: The DP equation can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as for the CH shallow water equation [24]. The DP equation is integrable and has a bi-Hamiltonian structure [22]. An inverse scattering approach for the DP equation was presented in [18,38]. Its traveling wave solutions was investigated in [34,47].
Although the DP equation is similar to the CH equation in several aspects, these two equations are truly different. One of the novel features of the DP different from the CH equation is that it has not only peakon solutions [22] and periodic peakon solutions [56], but also shock peakons [37] and the periodic shock waves [27].
The third celebrated integrable member of (1.3) is the known Novikov equation [42]: The most difference between the Novikov equation and the CH and DP equations is that the former one has cubic nonlinearity and the latter ones have quadratic nonlinearity.
It was showed that the Novikov equation is integrable, possesses a bi-Hamiltonian structure, and admits exact peakon solutions u(t, x) = ± √ ce |x−ct| with c > 0 [32]. The local well-posedness for the Novikov equation in Sobolev spaces and Besov spaces was studied in [49,50,52,53]. The global existence of strong solutions under some sign conditions were established in [49] and the blow-up phenomena of the strong solutions were shown in [53]. The global weak solutions for the Novikov equation were studied in [33,48].
Recently, the Cauchy problem of (1.1) in the Besov spaces B s p,r , s > max{2 + 1 p , 5 2 }, 1 ≤ p, q ≤ ∞ and the critical Besov space B 5 2 2,1 has been studied in [45,46]. To our best knowledge, the global solution of (1.1) has not been studied yet. In this paper we first investigate the global solution of (1.2) with initial data in the Sobolev space H s , s > 5 2 and two new blow-up results. The main idea is based on the global solution of DP equation with initial data in the Sobolev space H s , s > 3 2 . Then, we prove the existence and uniqueness of global weak solution by using the method of approximation of smooth solutions and a regularization technique.
The paper is organized as follows. In Section 2 we introduce some preliminaries which will be used in the sequel. In Section 3 we show global existences and blow-up phenomena. Section 4 is devoted to the study of global weak solutions of (1.1).

Preliminaries
Theorem 2.1. [45] Let 1 ≤ p, r ≤ ∞, s > max{2 + 1 p , 5 2 }, and u 0 ∈ B s p,r . Then there exists some T > 0, such that (1.2) has a unique solution u in E s p,r (T ), where Moveover the solution depends continuously on the initial data u 0 .

Global existence and blow-up phenomena
In this section, we begin by deriving the relationship between the Degasperis-Procesi equation and (1.1). Using this relationship, we then obtain the global existence results and the blow-up results for the system (1.1). We can rewrite the Degasperis-Procesi equation as It is easy to see that the Degasperis-Procesi equation transforms into the equation (1.1) under the transformation u → 2(2 − ∂ x )u.
and m 0 changes sign. Then, the corresponding solution to (1.1) blows up in finite time.
Then the corresponding solution to (1.1) blows up in finite time. Moreover, the maximal time of existence is estimated above by

Global weak solutions
In this section, we will investigate the existence and uniqueness of global weak solutions of (1.1) under some sign condition.
Proof. In order to prove Theorem 4.1, we proceed as the following five steps.
Step 1: We define a sequence of global solutions u n ∈ C([0, T ]; H 3 (R)) of (1.1) with a suitable approximation of the initial data u 0 ∈ H 1 (R) by smooth functions u n 0 .
Note that for n ≥ 1, Then, we derive Let u n be the global solution of (1.1) guaranteed by Theorem 3.1 with initial data u n 0 .
Step 2: We show that the sequence defined by this procedure converges pointwise a.e. to a function u ∈ H 1 loc (R + ; H 2 (R)) that satisfies (1.1) in the sense of distributions. For fix any T > 0, from (4.5)-(4.7), we verify that {u n } n≥1 and {u n x } n≥1 is uniformly bounded in the space H 1 ((0; T ) × R). Therefore we can get a subsequence such that and u n k → u a.e. on (0; T ) × R as n k → ∞, (4.10) For fixed t ∈ (0, T ), by (1.2), (4.2)-(4.7) and Young's inequality, we have , which is the space of functions with bounded variation and V(f ) is the total variation of f ∈ BV ((0, T ) × R).
Step 3: We prove that u ∈ C w (R + ; H 2 (R)). Due to (4.7), we have an uniform bound on u n k (t, ·) H 2 (R) for all t ∈ R + and all n k , which together with (1.2) leads to where ϕ(x) ∈ H −2 . Hence the family t → u n k (t, ·) ∈ H 2 (R) is weakly equicontinuous on [0, T ] for any T > 0. Making use of the Arzela-Ascoli theorem, implies that {u n k } contains a subsequence, denoted again by {u n k }, converging weakly in H 2 (R), uniformly in t. The previous discussion concludes the fact that the limit function u is weakly continuous from R + into H 2 (R).
Because of the weakly convergence in H 2 (R) for a.e. t ∈ R + , we have for a.e. t ∈ R + . Therefore u ∈ L ∞ (R + × R), which along with Theorem 3.1, we get which along with (4.14), leads to u x ∈ L ∞ (R + × R) and u xx ∈ L ∞ (R + × R).
Observe that In view of Lemma 2.5 with u ∈ W 1,∞ (R) and u(t, ) ∈ H 1 (R) ⊂ L 2 (R), we deduce which along with the fact that On the other hand, an integration by parts leads to The previous discussion enables us to get and G n (t) → 0, uniformly in t, as n → ∞, (4.31) for a.e. t ∈ R + .
Taking advantage of Lemma 2.2 and (4.30), we get Making full use of Hölder's inequality and Young's inequality, we obtain the existence of a constant K > 0 which only depends on u 0 H 1 (R) and 2m 0 − m ′ 0 M such that |G n (t)| ≤ K; t ∈ R + ; n ≤ 1, (4.33) for u, u x ∈ L ∞ (R + × R). Combining with (4.30)-(4.33), by Lebesgue's dominated convergence theorem, we term to Therefore, by (4.1), we have This implies that u ∈ C(R + ; H 1 (R)) and E is conserved along our solution.
Then t → ρ n * w tx = ρ n,x * w is uniformly bounded in L 1 (R), since Setting R → ∞, applying Lebesgue's dominated convergence theorem, yields that for all t ∈ R + , Differentiating the above relation with respect to time, we propose that Similarly, we infer that Therefore, convoluting ρ n with w, by (4.41) and (4.45), we get We estimate the terms on the right-hand side of (4.46), respectively. Due to (4.34)-(4.40), it follows that Setting w = z + y, we get By the same token with the last inequality, we obtain Thus, combining (4.47)-(4.51) with (4.46), we have Here Q n (t) satisfies the follow two conditions: (1) Q n (t) → 0 uniformly in t, as t → ∞, where H only depends on M .