Several Dynamic Properties of Solutions to a Generalized Camassa-Holm Equation

For a nonlinear generalization of the Camassa-Holm equation, we investigate the dynamic properties of solutions for the equation under the assumption that the initial value 𝑢 0 (𝑥) lies in the space 𝐻 1 (𝑅) . A one-sided upper bound estimate on the first-order spatial derivative, 𝐿 𝑝 bound estimate, and a space-time higher-norm estimate for the solutions are obtained.


Introduction
Hakkaev and Kirchev [1] investigated the following generalized Camassa-Holm equation: where is an integer. When = 1, (1) becomes the Camassa-Holm model (see [2]). The local well-posedness in the Sobolev space with > 3/2 is established, and sufficient conditions for the stability and instability of the solitary wave solutions are given in [1]. However, the estimate of strong solutions and one-sided upper bound estimate on the first-order spatial derivative for the solutions are not discussed in [1]. This constitutes the objective of this work.
Like the Camassa-Holm equation (see [1,2]), (1) has the conservation law ∫ ( 2 ( , ) + 2 ( , )) = ∫ ( 2 (0, ) + 2 (0, )) , which plays an important role in our further investigations. In fact, many scholars have paid their attentions to the study of the Camassa-Holm equation. The existence of global weak solutions is established in Constantin and Escher [3], Constantin and Molinet [4], Xin and Zhang [5], and Coclite et al. [6]. It was shown in Constantin and Escher [7] that the blowup occurs in the form of breaking waves. Namely, the solution remains bounded, but its slope becomes unbounded in finite time. After wave breaking, the solution is continued uniquely either as a global conservative weak solution [8,9] or as a global dissipative solution [10,11]. Exact traveling wave solutions for the Camassa-Holm equation are presented in [12]. For other methods to investigate the problems involving various dynamic properties of the Camassa-Holm equation, the reader is referred to [13][14][15][16] and the references therein.
In this paper, we investigate several dynamic properties of strong solutions for the generalized Camassa-Holm equation (1) in the case where is an odd natural number and the assumption 0 ( ) ∈ 1 ( ). The results obtained in this work include a one-sided upper bound estimate on the firstorder derivatives of the solution, a space-time higher-norm estimate, and the (2 ≤ < ∞) bound estimate.
The rest of this paper is organized as follows. Section 2 states the main result. Several lemmas are given in Section 3 where the proof of main result is completed.

Main Result
Let be a nonnegative integer and = 2 + 1. In this case, the Cauchy problem for (1) is written in the form which is equivalent to where the operator Λ 2 = 1 − 2 / 2 and We introduce a result presented in [1] for problem (3).
Now we state the main result of this paper.
Then the solution of problem (3) has the following properties.
(a) There exists a positive constant 0 depending on and ‖ 0 ‖ 1 ( ) such that the one-sided ∞ norm estimate on the first-order spatial derivative holds > 0, and , ∈ , < . Then there exists a positive constant 1 depending only on ‖ 0 ‖ 1 ( ) , , , , , and such that the following estimate holds: (c) There exists a constant 2 depending only on ‖ 0 ‖ 1 ( ) , and , such that

Proof of Main Result
From the conservation law (2), we have Differentiating the first equation of problem (4) with respect to and writing / = , we obtain Lemma 3. Let 0 < < 1, > 0, and , ∈ , < . Then there exists a positive constant 1 depending only on ‖ 0 ‖ 1 ( ) , , , , , and , such that the space higher integrability estimate holds where = ( , ) is the unique solution of problem (3).