On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.

Obviously, for k 1 = 0, k 2 = −2, Eq. (1.1) is reduced to the Camassa-Holm (CH) equation [3,17] m t + um x + 2u x m = 0, m = u − u xx , which describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity [3,4]. u(t, x) stands for the fluid velocity at time t in the spatial x-direction, x ∈ R, and m(t, x) represents its potential density.
In the past few years, a large amount literature has devoted to the investigation of the CH equation, because it can describe both wave breaking phenomenon [5,8,9,14,24] (the solution remains bounded while the slope of u(t, x) becomes unbounded in finite time), and solitary waves interacting like solitons [3,11,12,13,25]. The well-posedness of the CH equation has been shown in [21,24,31] with the initial data u 0 ∈ H s (R), s > 3 2 . In particular, Danchin [14] has dealt with the initial-value problem of the CH equation for the initial data in the Besov space B s p,r , with 1 ≤ p, r ≤ +∞, s > max{1 + 1 p , 3 2 }. However, the Cauchy problem of the CH equation is not locally well-posed in H s (R), s < 3 2 . Indeed, the solution can not depend uniformly continuously with respect to the initial data [21]. On the other hand, the CH equation has the peaked solitions (peakons) of the form ϕ c (t, x) = ce −|x−ct| with the traveling speed c > 0. For the peakon solution, we know that it replicates a feature that is characteristic for the waves of great height-waves of largest amplitude that are exact solutions of the governing equations for water waves [6,7,10,33]. Constantin and Strauss [11] gave an impressive proof of stability of peakons by using the conservation laws.
For k 1 = −2, k 2 = 0, Eq. (1.1) becomes the following equation with cubic nonlinearity which was derived independently by Fokas [15], Fuchssteiner [19], Olver and Rosenau [26], and Qiao [27]. Eq. (1.2) regains attention due to its cuspon and peakon solution property and Lax pair [27], which may allow the initial value problem of (1.2) to be solved by the inverse scattering transform (IST) method. Unlike the CH equation, Eq. (1.2) admits not only new cusp solitons (cuspons), but also possesses weak kink solutions (u, u x , u t are continuous, but u xx has a jump at its peak point) [28,29]. It also has significant differences from the CH equation about the dynamics of the two-peakons and peakon-kink solutions [29]. Recently, the so called "white" solitons and "dark" ones of Eq. (1.2) have been presented in [32] and [22], respectively. In [2], the authors apply the geometric and analytic approaches to give a geometric interpretation to the variable m(t, x) and construct an infinite-dimensional Lie algebra of symmetries to Eq. (1.2).
In [20], the authors consider the formulation of the singularities of solutions and show that some solutions with certain initial date will blow up in finite time, then they discuss the existence of single peakon of the form ϕ c (t, x) = ± 3c 2 e −|x−ct| , c > 0, and multi-peakon solutions for Eq. (1.2). Very recently, the orbital stability of peakons for Eq. (1.2) has been proven in [30].
In the present paper, motivated by the study of the CH equation [14], our main work is to prove the local well-posedness to the Cauchy problem (1.1) in the nonhomogeneous Besov spaces. However, one of the differences with [14] is that we are required to deal with cubic nonlinearity in Besov spaces. Moreover, the nonlinear term "m x u 2 x " makes us have to solve a transport equation satisfied by m, rather than u. In contrast to the case of the CH equation with initial data u 0 in the Sobolev space H s (R), s > 3 2 , we can only prove the well-posedness result with the initial profile u 0 in H s (R), s > 5 2 . In our procedure, we have overcome the critical index case by the interpolation method when we applied the transport theory to Eq. (1.1). Another one of the differences with [14,18] is that Eq. (1.1) possesses the complicated mixed structure nonlinear structure (with both quadratic and cubic nonlinearity). To get the uniform boundedness of the approximate solutions {u (n) } n∈N , we have to handle the quadratic and cubic nonlinear terms together in Eq. (1.1). To overcome these difficulties, we need to consider two cases: the small initial data and the large one. Then with the local well-posedness result, we may naturally present a precise blow-up scenario to Eq. (1.1) by combining the blow-up criterion of the CH equation and the one of Eq. (1.2).
The entire paper is organized as follows. In Section 2, we present some facts on Besov spaces, some preliminary properties and the transport equation theory. In Section 3, we establish the local well-posedness result of Eq. (1.1) in Besov spaces. In Section 4, we derive a blow-up scenario for strong solutions to Eq. (1.1). In Section 5, we show that the existence of peakons which can be understand as weak solutions for Eq. (1.1).
N otation. In the following, we denote C > 0 a generic constant only depending on p, r, s. Since our discussion about Eq. (1.1) is mainly on the line R, for simplicity, we omit R in our notations of function spaces. And we denote the Fourier transform of a function u as Fu.

Preliminaries
In this section, we will recall some basic theory of the Littlewood-Paley decomposition and the transport equation theory on Besov spaces, which will play an important role in the sequel. One may get more details from [1,14]. and Then for all u ∈ S ′ (S ′ denotes the tempered distribution spaces), we can define the nonhomogeneous Littlewood-Paley decomposition of a distribution u.
where the localization operators are defined as follows: and Furthermore, we can define the low frequency cut-off operator S q as follows: If s = ∞, B ∞ p,r := s∈R B s p,r .
In order to state the local well-posedness result, we need to define the following spaces. Next, we list the following useful properties for Besov spaces.
(iii) Algebraic properties: if s > 0, B s p,r ∩ L ∞ is an algebra. Furthermore, B s p,r is an algebra, provided that s > 1 p or s ≥ 1 p and r = 1. (v) Complex interpolation: if u ∈ B s 1 p,r ∩ B s 2 p,r , then for all θ ∈ [0, 1], we have u ∈ B θs 1 +(1−θ)s 2 p,r . Moreover, where C is a constant independent of u and v.
(vii) Action of Fourier multipliers on Besov spaces: let m ∈ R and f be a S m -multiplier (i.e., f : R → R is a smooth function and satisfies that for each multi-index α, there exists a constant C α such that |∂ α f (ξ)| ≤ C α (1 + |ξ|) m−|α| , for ∀ξ ∈ R.) Then the operator f (D) is continuous from B s p,r to B s−m p,r . Now we state the following transport equation theory that is crucial to prove local wellposedness for Eq. (1.1).
Lemma 2.1. [1,14] (A priori estimate) Let 1 ≤ p, r ≤ +∞ and s > − min{ 1 Then there exists a constant C depending only on s, p, r such that the following statements hold where Lemma 2.2. [14] (Existence and uniqueness) Let p, r, s, f 0 and F be as in the statement of ) and the corresponding inequalities in Lemma 2.1 hold true. Moreover, if r < ∞, then f ∈ C([0, T ]; B s p,r ).

Local well-posedness
In this section, we shall study the local well-posedness of Eq. (1.1) in the nonhomogeneous Besov spaces. At first, we present a priori estimates about the solutions of Eq. (1.1), which can be applied to prove the uniqueness and continuity with the initial data in some sense.
0 ∈ B s p,r , and let u (12) := u (2) − u (1) and m (12) (2) If s = 4 + 1 p , then Proof. It is obvious that u (12) ∈ L ∞ ([0, T ]; B s p,r ) ∩ C([0, T ]; S ′ ) and u (12) , m (12) solves the following transport equation x m (12) . Applying Lemma 2.1 to the transport equation (3.1), we have 3) also holds true in view of the fact that B s−3 p,r is an algebra. Note that Therefore, inserting the above inequality and (3.3) into (3.2), we obtain Then, by Gronwall's inequality, we prove (1). Since we can not apply Lemma 2.1 to (3.1) for the critical case s = 4 + 1 p , we here use the interpolation method to deal with it.
In fact, we can choose θ = 1 Then, from the obtained result of (1) in this lemma, we get This completes the proof of Lemma 3.1.
Next, we use the classical Friedrichs regularization method to construct the approximation solutions to Eq. (1.1).

(3.4)
Moreover, there exists a T > 0 such that the solutions satisfying the following properties.
(1) {u (n) } n∈N is uniformly bounded in E s p,r (T ). Proof. By the definition of S q , we know that all the data S n+1 u 0 ∈ B ∞ p,r . Thus, from Lemma 2.2, we deduce by induction that for all n ∈ N, Eq. (3.4) has a global solution belonging C(R + ; B ∞ p,r ). For s > max{2 + 1 p , 5 2 , 3 − 1 p } and s = 4 + 1 p , by Lemma 2.1, we obtain Since s > 2 + 1 p , we know that B s−2 p,r is an algebra. Thus, we have Inserting the above inequalities into (3.5), we obtain In order to prove the uniform boundedness of {u (n) } n∈N , we shall divide our discussion into two parts. When 2 u 0 B s p,r < 1, we choose a T 1 > 0 such that and suppose by induction that for all t ∈ [0, Noting that t ≤ T 1 < Inserting the above inequality and (3.8) into (3.6) yields where we used the following fact that in the last inequality. Thus, we prove (3.7) for the case 2 u 0 B s p,r < 1. On the other hand, when 2 u 0 B s p,r ≥ 1, we choose a T 2 > 0 such that By (3.11), we find that Inserting the above inequality and (3.11) into (3.6), we obtain where we used the following fact that in the last inequality. Thus, we prove (3.10) for the case 2 u 0 B s p,r ≥ 1. Therefore, from the above discussion of the two cases, choosing T = min{T 1 , T 2 } > 0, combining (3.9) and (3.12), we have proved that {u (n) } n∈N is uniformly bounded in C([0, T ]; B s p,r ). Using Proposition 2.2 (vi) and the fact B s−2 p,r is an algebra as s > 2 + 1 p , we have is uniformly bounded, which yields that the sequence {u (n) } n∈N is uniformly bounded in E s p,r (T ). Now it suffices to prove that Indeed, from Eq. (3.4), for all n, l ∈ N, we have x )m (n) . Applying Lemma 2.1 again, for all t ∈ [0, T ], we have Similar to the proof of the estimate of F (τ ) B s−3 p,r in Lemma 3.1, for s > max{2 + 1 p , 5 2 , 3 − 1 p } and s = 4 + 1 p , we also obtain Inserting the above inequality into (3.13), we have Note that {u (n) } n∈N is uniformly bounded in E s p,r (T ) and Hence, there exists a constant C T independent of n, l such that for all t ∈ [0, T ] Arguing by induction with respect to the index n, we deduce which yields the desired result. Finally, we can apply the interpolation method, which is similar to the proof in Lemma 3.1, to the critical case s = 4 + 1 p . We here omit the details. Therefore, we complete the proof of Lemma 3.2.
Based on the above preparations, we are in position to state the local existence result of the Cauchy problem (1.1). Theorem 3.1. Suppose that 1 ≤ p, r ≤ ∞, s > max{2 + 1 p , 5 2 , 3 − 1 p } and u 0 ∈ B s p,r . Then there exists a time T > 0 such that the Cauchy problem (1.1) has a unique solution u ∈ E s p,r (T ), and the mapping u 0 → u is continuous from B s p,r into for all s ′ < s if r = ∞ and s ′ = s otherwise.
Proof. According to Lemma 3.2, {u (n) } n∈N is a Cauchy sequence in C([0, T ]; B s−1 p,r ), so it converges to some function u ∈ C([0, T ]; B s−1 p,r ). Thanks to Lemma 3.2 and Proposition 2.2 (iv) Fatou lemma, we have that u ∈ L ∞ ([0, T ]; B s p,r ). Thus, by the interpolation method, for all s ′ < s, we find that u ∈ C([0, T ]; B s ′ p,r ). Taking limit in Eq. (3.4), we conclude that u solves Eq. (1.1) in the sense of u ∈ C([0, T ]; B s ′ −1 p,r ), for all s ′ < s. Since u ∈ L ∞ ([0, T ]; B s p,r ) and the fact B s p,r is an algebra as s > 2 + 1 p , the right-hand side of the following equation Remark 3.1. We know that nonhomogeneous Besov spaces contain Sobolev spaces. In fact, by Fourier-Plancherel formula, we find that the Besov space B s 2,2 (R) coincides with the Sobolev space H s (R). Therefore, Theorem 3.1 implies that under the assumption u 0 ∈ H s (R), s > 5 2 , we can obtain the local well-posedness result to Eq. (1.1).
Differentiating Eq. (1.1) with respect to x, we deduce Multiplying the above equation by m x , and integrating with respect x over R, we have Assume that T < ∞ and there exists N 1 , N 2 > 0 such that mu x ≥ −N 1 , u x ≥ −N 2 for all (t, x) ∈ [0, T )×R. Let us choose N, k > 0 such that N := max{N 1 , N 2 } and k := max{−k 1 , −k 2 }. It then follows that Applying Gronwall's lemma to the above inequality implies for t ∈ [0, T ), Differentiating Eq. (1.1) with respect to x twice, we deduce Multiplying the above equation by m xx , integrating with respect to x over R, we have Combining (4.1)-(4.2) and (4.4), we obtain If mu x and u x are bounded from below on [0, T ) × R, i.e., there exists N 1 , N 2 > 0 such that Similarly, we can choose N, k > 0 such that N := max{N 1 , N 2 } and k := max{−k 1 , −k 2 }. Then, by (4.3) and the above equality, we get Hence, applying Gronwall's inequality implies that for all t ∈ [0, T ) The above inequality and Sobolev's embedding theorem ensure that u(t, x) does not blow up in finite time.
On the other hand, by Sobolev's imbedding theorem, we find that if lim inf

The existence of peaked solutions
In order to understand the meaning of a peaked solution to Eq. (1.1), we first rewrite Eq. (1.1) as From the above two facts, we can then define the notion of weak solutions as follows. where C 1 satisfies 1 3 k 1 C 2 1 + 1 2 k 2 C 1 + c = 0, is a global weak solution to Eq. (1.1) in the sense of Definition 5.1. Moreover, for every time t ≥ 0, the peaked solutions ϕ c (t, x) belongs to Remark 5.1. (i) For k 1 = 0, k 2 = 0, we have C 1 = − 2c k 2 . In particular, if k 1 = 0, k 2 = −2, then we obtain the single peakon ϕ c (t, x) = ce −|x−ct| for the CH equation.
(ii) For k 1 = 0, we easily get If 3k 2 2 − 16k 1 c ≥ 0, then the coefficient C 1 of the peakons ϕ c is a real number. For example, if we choose k 1 = −2, k 2 = 0, and c > 0, then we obtain the single peakon ϕ c (t, x) = ± 3 2 ce −|x−ct| of the modified CH equation (1.2). If 3k 2 2 − 16k 1 c < 0, then the coefficient C 1 of the peakons ϕ c is a complex number. In [28], the authors call it as a complex peakon, i.e., the peakon has the complex coefficient. Thus, we can propose here the complex peakon for Eq. (1.1), which is not presented in both the CH equation and the modified CH equation (1.2).
Proof. For any test function φ(·) ∈ C ∞ c (R), using integration by parts, we infer Thus, for all t ≥ 0, we have ∂ x ϕ c (t, x) = −sign(x − ct)ϕ c (t, x), (5.2) in the sense of distribution S ′ (R). Hence, the peaked solutions ϕ c (t, x) belongs to H 1 ∩ W 1,∞ . The same computation as in (5.2), for all t ≥ 0, yields, Form the definition of ϕ c and C 1 satisfying 1 3 k 1 C 2 1 + 1 2 k 2 C 1 + c = 0, for x > ct, we have sign(x − ct)ϕ c (c + Similarly, for x ≤ ct, we find On the other hand, similar to Definition 2.1, we derive