### Abstract

This paper deals with the Cauchy problem for a generalized Camassa-Holm equation with high-order nonlinearities,

where

##### Keywords:

persistence properties; local well-posedness; weak solution### 1 Introduction

This work is concerned with the following one-dimensional nonlinear dispersive PDE:

where

Obviously, if

where the variable
*t* and in the spatial direction *x*, and *k* is a nonnegative parameter related to the critical shallow water speed [1]. The Camassa-Holm equation (1.2) is also a model for the propagation of axially symmetric
waves in hyperelastic rods (*cf.*[2]). It is well known that equation (1.2) has also a bi-Hamiltonian structure [3,4] and is completely integrable (see [5,6] and the in-depth discussion in [7,8]). In [9], Qiao has shown that the Camassa-Holm spectral problem yields two different integrable
hierarchies of nonlinear evolution equations, one is of negative order CH hierachy
while the other one is of positive order CH hierarchy. Its solitary waves are smooth
if
*cf.*[1]). The orbital stability of the peaked solitons is proved in [10], and the stability of the smooth solitons is considered in [11]. It is worth pointing out that solutions of this type are not mere abstractions:
the peakons replicate a feature that is characteristic for the waves of great height
- waves of largest amplitude that are exact solutions of the governing equations for
irrotational water waves (*cf.*[12-14]). The explicit interaction of the peaked solitons is given in [15] and all possible explicit single soliton solutions are shown in [16]. The Cauchy problem for the Camassa-Holm equation (1.2) has been studied extensively.
It has been shown that this problem is locally well-posed for initial data

For

Wazwaz [26,27] studied the solitary wave solutions for the generalized Camassa-Holm equation (1.3)
with

On the other hand, taking

The Novikov equation (1.4) possesses a matrix Lax pair, many conserved densities,
a bi-Hamiltonian structure as well as peakon solutions [34]. These apparently exotic waves replicate a feature that is characteristic of the
waves of great height-waves of largest amplitude that are exact solutions of the governing
equations for water waves, as far as the details are concerned [13,35,36]. The Novikov equation possesses the explicit formulas for multipeakon solutions [37]. It has been shown that the Cauchy problem for the Novikov equation is locally well-posed
in the Besov spaces and in Sobolev spaces and possesses the persistence properties
[38,39]. In [40,41], the authors showed that the data-to-solution map for equation (1.4) is not uniformly
continuous on bounded subsets of

Other integrable CH-type equations with cubic nonlinearity have been discovered:

where *γ* is a constant. equation (1.5) was independently proposed by Fokas [45], by Fuchssteiner [46], and Olver and Rosenau [47] as a new generalization of integrable system by using the general method of tri-Hamiltonian
duality to the bi-Hamiltonian representation of the modified Korteweg-de Vries equation.
Later, it was obtained by Qiao [48,49] from the two-dimensional Euler equations, where the variables

Applying the method of pseudoparabolic regularization, Lai and Wu [54] investigated the local well-posedness and existence of weak solutions for the following generalized Camassa-Holm equation with dissipative term:

where
*a*, *k*, *β* are constants. Hakkaev and Kirchev [55] studied the local well-posedness and orbital stability of solitary wave solution
for equation (1.6) with

Motivated by the results mentioned above, the goal of this paper is to establish the well-posedness of strong solutions and weak solutions for problem (1.1). First, we use Kato’s theorem to obtain the existence and uniqueness of strong solutions for equation (1.1).

**Theorem 1.1***Let*
*with*
*Then there exists a maximal*
*and a unique solution*
*to the problem* (1.1) *such that*

*Moreover*, *the solution depends continuously on the initial data*, *i*.*e*. *the mapping*

*is continuous*.

In [38,56,57], the spatial decay rates for the strong solution to the Camassa-Holm Novikov equation were established provided that the corresponding initial datum decays at infinity. This kind of property is so-called the persistence property. Similarly, for equation (1.1), we also have the following persistence properties for the strong solution.

**Theorem 1.2***Assume that*
*with*
*satisfies*

*for some*
*respectively*,
*then the corresponding strong solution*
*to equation* (1.1) *satisfies for some*

*uniformly in the time interval*

**Theorem 1.3***Assume that*
*and*
*with*
*satisfies*

*for some*
*respectively*,
*then the corresponding strong solution*
*to equation* (1.1) *satisfies for some*

*uniformly in the time interval*

**Remark 1.1** The notations mean that

Finally, we have the following theorem for the existence of a weak solution for equation (1.1).

**Theorem 1.4***Suppose that*
*with*
*and*
*Then there exists a life span*
*such that problem* (1.1) *has a weak solution*
*in the sense of a distribution and*

The plan of this paper is as follows. In the next section, the local well-posedness and persistence properties of strong solutions for the problem (1.1) are established, and Theorems 1.1-1.3 are proved. The existence of weak solutions for the problem (1.1) is proved in Section 3, and this proves Theorem 1.4.

### 2 Well-posedness and persistence properties of strong solutions

**Notation** The space of all infinitely differentiable functions
*p* be any constant with
*f* such that
*s*, let

where

*Proof of Theorem 1.1* To prove well-posedness we apply Kato’s semigroup approach [58]. For this, we rewrite the Cauchy problem of equation (1.1) as follows for the transport
equation:

where

*Proof of Theorem 1.2* We introduce the notation
*x* variable by multiplying the first equation of (2.1) by

Note that the estimates

and

are true. Moreover, using Hölder’s inequality

From equation (2.2) we can obtain

Since

where we use

Because of Gronwall’s inequality, we get

Next, we will give estimates on
*x*-variable produces the equation

Multiplying this equation by
*x*-variable, and using integration by parts:

From the above inequalities, we also can get the following inequality:

where we use

We shall now repeat the arguments using the weight

where
*N* we have

Using the notation

and from (2.3), we also obtain

We need to eliminate the second derivatives in the second term in the above equality. Thus, combining integration by parts and equation (2.4) we find

Hence, as in the weightless case, we have

A simple calculation shows that there exists

Thus, we have

and

Using the same method, we can estimate the other terms:

and

Thus, it follows that there exists a constant
*M*, *m*, *n*, *k*, *a*, and *T*, such that

Hence, for any

Finally, taking the limit as *N* goes to infinity we find that for any

which completes the proof of Theorem 1.2. □

Next, we give a simple proof for Theorem 1.3.

*Proof of Theorem 1.3* We should use Theorem 1.3 to prove this theorem.

For any

From Theorem 1.2, it follows that

and so

We shall show that the last term in equation (2.5) is

From the given condition and Theorem 1.2. we know

we have

Thus

From equation (2.5) and

By the arbitrariness of

uniformly in the time interval

### 3 Existence of solution of the regularized equation

In order to prove Theorem 1.4, we consider the regularized problem for equation (1.1) in the following form:

where
*a*, *k* are constants. One can easily check that when

Before giving the proof of Theorem 1.4, we give several lemmas.

**Lemma 3.1** (See [54])

*Let**p**and**q**be real numbers such that*
*Then*

**Lemma 3.2***Let*
*with*
*Then the Cauchy problem* (3.1) *has a unique solution*
*where*
*depends on*
*If*
*the solution*
*exists for all time*. *In particular*, *when*
*the corresponding solution is a classical globally defined solution of problem* (3.1).

*Proof* First, we note that, for any
*s*, the integral operator

defines a bounded linear operator on the indicated Sobolev spaces.

To prove the existence of a solution to the problem (3.1), we apply the operator to both sides of equation (3.1) and then integrate the resulting equations with regard
to *t*. This leads to the following equations:

Suppose that is the operator in the right-hand side of equation (3.2). For fixed

Since

Since

and

where
*n*. Suppose that both *u* and *v* are in the closed ball
*R* about the zero function in

where
*C* only depend on *a*, *k*, *m*, *n*. Choosing *T* sufficiently small such that

Taking *T* sufficiently small so that
*u* in

For
*u*, integrating with respect to *x*, one derives

from which we have the conservation law

The global existence result follows from the integral from equation (3.2) and equation (3.3). □

Now we study the norms of solutions of equation (3.1) using energy estimates. First, recall the following two lemmas.

**Lemma 3.3** (See [58])

*If*
*then*
*is an algebra*, *and*

*here**c**is a constant depending only on**r*.

**Lemma 3.4** (See [58])

*If*
*then*

*where*
*denotes the commutator of the linear operators**A**and**B*, *and**c**is a constant depending only on**r*.

**Theorem 3.1***Suppose that*, *for some*
*the functions*
*are a solution of equation* (3.1) *corresponding to the initial data*
*Then the following inequality holds*:

*For any real number*
*there exists a constant**c**depending only on**q**such that*

*For*
*there is a constant**c**independent of**ϵ**such that*

*Proof* Using

Since

For any
*x* again,using integration by parts, one obtains

We will estimate the terms on the right-hand side of (3.7) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.3 and 3.4, we have

Using the above estimate to the second term on the right-hand side of equation (3.7) yields

For the fourth term on the right-hand side of equation (3.7), using the Cauchy-Schwartz inequality and Lemma 3.3, we obtain

For the last term on the right-hand side of equation (3.7), using Lemma 3.3 repeatedly results in

It follows from equations (3.7)-(3.11) that there exists a constant *c* depending only on *a*, *m*, *n*, *s* such that

Integrating both sides of the above inequality with respect to *t* results in inequality (3.5).

To estimate the norm of

Applying

For the right-hand of equation (3.13), we have

and

Since

Using Lemma 3.3,

and

By the Cauchy-Schwartz inequality and Lemma 3.3, we get

Substituting equations (3.14)-(3.19) into equation (3.13) yields the inequality

with a constant

For a real number *s* with
*ϕ* satisfies
*ϵ* satisfying

has a unique solution
*ϵ*.

For an arbitrary positive Sobolev exponent

**Lemma 3.5***For*
*with*
*and*
*the following estimates hold for any**ϵ**with*

*where**c**is a constant independent of**ϵ*.

*Proof* This proof is similar to that of Lemma 5 in [60] and Lemma 4.5 in [61], we omit it here. □

**Remark 3.1** For

where
*ϵ*.

**Theorem 3.2***If*
*with*
*such that*
*Let*
*be defined as in the system* (3.21). *Then there exist two constants*, *c**and*
*which are independent of**ϵ*, *such that*
*of problem* (3.21) *satisfies*
*for any*

*Proof* Using the notation
*x* give rise to

Letting
*x*, and using

we find the equality

Applying Hölder’s inequality, we get

where

Since
*t* and taking the limit as

Using the algebraic property of