Gevrey regularity and analyticity for the modified Camassa–Holm equation

This paper deals with the analyticity and Gevrey regularity for the Cauchy problem of a modified Camassa–Holm (mCH) equation in Sobolev–Gevrey space Gr,sδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{r,s}^\delta$$\end{document} with s>12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>\frac{1}{2}$$\end{document}, r≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\ge 1$$\end{document} and 0<δ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\delta <1$$\end{document}, and a lower bound of lifespan and continuity of the data-to-solution map was also obtained.


3
The original Camassa-Holm equation was obtained by Fuchssteiner and Fokas by using the method of recursion operators [14], and later derived as a model for unidirectional propagation of shallow water over a flat bottom by Camassa and Holm [1]. Analogous to the famous KdV equation, the Camassa-Holm equation (1.2) also has a bi-Hamiltonian structure [1,14], which is completely integrable not only in the sense of the existence of a Lax pair [1], but also (by means of inverse scattering and inverse spectral theory) as an infinitedimensional Hamiltonian flow that can be linearised in suitable action-angle variables, cf. (see [3,4,8,11,12]). The orbital stability of solitary waves and the stability of the peakons (k = 0) for the Camassa-Holm equation are investigated by Constantin and Strauss [9,10]. The advantage of the Comassa-Holm equation in comparison with the famous KdV equation lies in the fact that the Camassa-Holm models the peculiar wave breaking phenomena, that is, the solution remains bounded but its slope becomes unbounded in finite time (cf. [2,7]). In addition to wave breaking, one of the most interesting aspects of the equation is the existence of peakon solutions, namely, the travelling wave solutions of greatest height of the governing equations for water waves have a peak at their crest (cf. [5,6]). The latest progress about the Camassa-Holm type equations can be refer to [20] and references therein.
Recently, Luo, Qiao and Yin [19] established the local well-posedness for the Cauchy problem of the Equ. (1.1) in nonhomogeneous Besov spaces by Littlewood-Paley theory and transport equation theory. In this paper, we show the Geverey and analytic regularity of the Cauchy problem for mCH system. We first introduce the Sobolev-Gevrey spaces (cf. [13]).
. For 0 < r < 1 , it is called ultra-analytic function. If r = 1 , it is usual analytic function and is called the radius of analyticity. If r > 1 , it is the Gevrey class function.
It is well known that the solutions to the Camassa-Holm type systems are analytic in both space and time variables for a short time (see [17]). However, it gives no estimate about the size of the analytic lifespan . Also, it dose not supply information about the evolution of the uniform radius of analyticity. Recently, Barostichi et al. solved these important problems for Comassa-Holm and other nonlocal equations and systems by using an Ovsyannikov type theorem for an autonomous abstract Cauchy problem in a scale of decreasing Banach spaces, and they also studied the stability of their solution map. By taking the advantage of this idea, Luo and Yin [18] proved the Gevrey regularity and analyticity of these systems be a generalized Journal of Nonlinear Mathematical Physics (2022) 29:493-503 Ovsyannikov theorem. Our motivation to solve this problem follow the line of [17,18].
Thus, for all 0 < < 1 , there exists a T 0 > 0 such that the mCH system has a unique solution which is holomor- . Moreover With the existence and uniqueness at hand, we next show the continuity of the data-to-solution map in G 1 r,s (ℝ) , which means that: Our last result about the continuity of the data-to-solution map was stated as follows:

Remark 1
In the period case, the Sobolev-Gevrey norm can be defined as follows in view of the same argument as in Theorems 1.1 and 1.2, we can get the similar Gevrey regularity and analytic for the mCH (1.1).
The plan of this paper is organized as follows. In the next section, we recall some properties about Sobolev-Gevrey spaces. In the last section, we prove the analyticity and Gevrey regularity of the solutions to the modified Camassa-Holm equation, and prove Theorems 1.1-1.2.

3 2 Preliminaries
In this section, we will investigate the analytic solutions to mCH system (1.1) by nonlinear Cauchy-Kowalevski theorem, and the Cauchy-Kowalevski theorem is read as follows: Theorem 2.1 (See Theorem 3.1 in [18,22]) Let X , ∥ ⋅ ∥ 0< ≤1 be a scale of decreasing Banach spaces, such that for any 0

Consider the Cauchy problem
Let T, R > 0 and r ≥ 1 . For given 0 ∈ X 1 , suppose that F satisfies the following conditions: (1) If for any 0 < ′ < < 1 , the function t ↦ (t) is holomorphic on | t |< T and continuous on | t |≤ T with values in X and then t ↦ F(t, (t)) is a holomorphic function on | t |< T with values in X .
G r,s (ℝ) has the following three properties: [18]) Let s > 1∕2 , r ≥ 1 and > 0 , then G r,s (ℝ) is an algebra, and for any 1 ∈ G r,s−1 (ℝ) , 2 ∈ G r,s (ℝ) , there is a constant C s that is only dependent of s such that Moreover, for any 1 , 2 ∈ G r,s (ℝ) , there is a constant C ′ s that is only dependent of s such that

Proof of Theorem 1.1 and 1.2
In this section, we will prove Theorem 1.
According to the same token as the above, we can get that So we see that F( ) satisfies the condition (3) According to the above inequality, we have proved that F( ) satisfies the condition (2) of Theorem 2.1 with L = 3C s (e −r r r + 2) ∥ 0 ∥ G r,s (ℝ) +R + 1 ∥ 0 ∥ G r,s (ℝ) +R + 1 3 , and that . ◻ Next, we will prove the Theorem 1.2 as follows.
Proof of Theorem 1.2 Applying the proof of Theorem 1.1, for the given r ≥ 1 , s > 1∕2 , we can know that the lifespan of the corresponding solution to the mCH Cauchy problem n and ∞ are given by (3.13) (3.14) (3.15) T ≐ 1 As in proof Theorem 1.1, we see that T n and T ∞ are the existence time corresponding to ∥ n 0 ∥ G 1 r,s (ℝ) and ∥ ∞ 0 ∥ G 1 r,s (ℝ) respectively. Which implies that for any n ≥ N where F is given in (3.8). On the basis of the above equation, we affirm that for any In consideration of (3.11), we have using the Lemma 3.7 in [22] with a = T , we derive that According to the definition of L and T (see (3.12) and (3.15)), we can obtain that 2 2r+3 LT < 1 2 . So we get which leads to (3.16) n (t, x) = n 0 (x) + � t 0 F( n (t, ))d , 0 ≤ t ≤ D r T(1 − ) r 2 r+1 ,  (3.21) (3.23)