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1.   Introduction

In this paper, we are concerned with the Cauchy problem for the generalized Camassa-Holm equation

which is introduced by S. Hakkaev and K. Kirchev ^[14,15]. In Eq (1.1), $u = u\; (t, x)$ stands for the fluid velocity at time $t > 0$ in the spatial direction. Equation (1.1) admits following conservative laws

and

An important feature of Eq (1.1) is the existence of traveling solitary waves, interacting like solitons, also called "peakons"

Since then, Eq (1.1) attracted lots of the attentions in the last few years. The well-posedness of the solutions for Eq (1.1) is obtained by parabolic regularization method ^[14], Kato's semigroup approach ^[20] and by classical Friedrichs's regularization method ^[21], respectively. In ^[17], Lai and Wu obtained a sufficient condition for the existence of weak solutions of Eq (1.1) in lower order Sobolev space $H^{s}(\mathbb{R})$ with $1 < s\leq3/2$. Yan ^[23] proved that Eq (1.1) does not depend uniformly continuously on the initial data in $H^{s}(\mathbb{R})$ with $s < 3/2$ and that the Cauchy problem for the generalized Camassa-Holm equation is locally well-posedness in $B_{2, 1}^{3/2}$. In ^[24], Zhou focused on the persistence property in weighted $L^{p}$ spaces.

When $k = 1$, Eq (1.1) reduced to the well-known Camassa-Holm equation

which was derived by Camassa and Holm ^[1] and by Fokas and Fuchssteiner ^[11]. It describes the motion of shallow water waves and possesses soliton solutions, a Lax pair, a bi-Hamiltonian structures and infinitely many conserved integrals ^[1,32], and it can be solved by the Inverse Scattering Method. The dynamic properties related the equation can be found in ^{[3,4,5,10,12,13,14,15,16,18,19,20,22,25]} and the references therein. It is well-known that a major interest in water waves is the existence of breaking waves (solutions that remain bounded but whose slope becomes unbounded in finite time^[7]). Comparing with KdV equation, another important feature of Camassa-Holm equation is that it possesses breaking wave^[6,7,8,9].

To our best knowledge, blow-up, analyticity and analytical solutions have not been investigated yet for the problems (1.1) and (1.2). Inspired by the ideas from ^[7], the objective of this paper is to investigate the blow-up phenomenon, analyticity and analytical solutions for the problems (1.1) and (1.2). In our blow-up phenomenon analysis, the quantity $\int_{\mathbb{R}}((u^{k})_{x})^{3}dx$ plays a key role. Taking advantage of complicated calculation, we obtain the Riccati inequality of quantity $\int_{\mathbb{R}}((u^{k})_{x})^{3}dx$ to arrive at a new blow-up result. In addition, we present some analytical solutions for the problems (1.1) and (1.2). Finally, we prove the analyticity. The results we obtained complements earlier results in this direction.

Notations. The space of all infinitely differentiable functions $\phi(t, x)$ with compact support in $[0, +\infty)\times \mathbb{R}$ is denoted by $C^\infty_0$. Let $L^p = L^p(\mathbb{R})(1\leq p < +\infty)$ be the space of all measurable functions $h$ such that $\parallel h\parallel^{P}_{L^{P}} = \int_{\mathbb{R}}|h(t, x)|^{p} dx < \infty$. We define $L^\infty = L^{\infty}(\mathbb{R})$ with the standard norm $\parallel h\parallel_{L^{\infty}} = {\rm{inf}}_{m(e) = 0} {\rm{sup}}_{x\in R\setminus e}| h(t, x)|$. For any real number $s$, $H^{s} = H^{s}(\mathbb{R})$ denotes the Sobolev space with the norm defined by

where $\hat{h}(t, \xi) = \int_\mathbb{R}e^{-ix\xi}h(t, x)dx$.

We denote by $\ast$ the convolution, using the green function $G(x) = \frac{1}{2}e^{-\mid x\mid}$, we have $(1-\partial_{x}^{2})^{-1}f = G(x)*f$ for all $f\in L^{2}$, and $p*(u-u_{xx}) = u$. For $T > 0$ and nonnegative number $s$, $C([0, T); H^{s}(\mathbb{R}))$ denotes the Frechet space of all continuous $H^{s}$-valued functions on $[0, T)$. For simplicity, throughout this article, we let $c$ denote any positive constant

The Cauchy problems (1.1) and (1.2) is equivalent to

which is also equivalent to

The rest sections are organized as follows. In the second section, we give a blow up criterion and a new blow up phenomenon. Existence of weak solution (CH-type peakon) and analytical solutions are studied in third section. In the forth section, we proved analyticity of strong solutions..

2.   Blow-up criterion and blow-up phenomenon

We firstly give some useful Lemmas.

Lemma 2.1. Given $u(x, 0) = u_{0}\in H^{s}(\mathbb{R}), s > 3/2$, then there exist a maximal $T = T(u_{0})$ and a unique solution $u$ to the problems (1.1) and (1.2) such that

Moreover, the solution depends continuously on the initial data, i.e., the mapping $u_{0}\rightarrow u(\cdot, u_{0}):H^{s}\rightarrow C([0, T); H^{s}(\mathbb{R}))\bigcap C^{1}([0, T); H^{s-1}(\mathbb{R}))$ is continuous.

Proof. Using the Kato's theorem^[27], we can prove the above theorem. Because there exist some similarities, here we omit the proof of Lemma 2.1, a detailed proof can be found in ^[26].

Lemma 2.2. (^[2]) Let $f\in C^{1}(\mathbb{R})$, $a > 0$, $b > 0$ and $f(0) > \sqrt{\frac{b}{a}}$. If $f'(t)\geq af^{2}(t)-b$, then

Lemma 2.3. (see ^[23]) Let $u_{0}\in B_{2, 1}^{3/2}$ and $u$ be the corresponding solution to (1.1). Assume that $T$ is the maximal time of existence of the solution to the problems (1.1) and (1.2). If $T < \infty$, then

Remark 1. For $s > \frac{3}{2}$, it is well known that $H^{s}\hookrightarrow B_{2, 1}^{3/2}$, so we have the following result:

Let $u_{0}\in H^{s}$ with $s > \frac{3}{2}$ and $u$ be the corresponding solution to (1.1). Assume that $T$ is the maximal time of existence of the solution to the problems (1.1) and (1.2). If $T < \infty$, then

Lemma 2.4. Let $u_{0}\in H^{s}(\mathbb{R})$ with $s > \frac{3}{2}$. Let $T > 0$ be the maximum existence time of the solution $u$ to the problems (1.1) and (1.2) with the initial data $u_{0}$. Then the corresponding solution $u$ blows up in finite time if and only if

Proof. Applying Lemma 2.1 and a simple density argument, it suffices to consider the case $s = 3$. Let $T > 0$ be the maximal time of existence of solution $u$ to the problems (1.1) and (1.2) with initial data $u_{0}\in H^{3}(\mathbb{R})$. From Lemma 2.1 we know that $u\in C([0, T); H^{3}(\mathbb{R}))\bigcap C^{1}([0, T); H^{2}(\mathbb{R}))$. Due to $y = u-u_{xx}$, by direct computation, one has

So,

Multiplying equation (1.10) by $2y$ and integrating by parts and using the interpolation $\parallel u_{x}\parallel^{2}_{L^{\infty}}\leq C \parallel u\parallel_{H^{1}}\parallel y\parallel_{L^{2}}$, we obtain

where $C_{1} = C_{1}(\parallel u_{0}\parallel_{H^{1}})$.

If there exists a constant $M > 0$ such that $\parallel u^{k-1}u_{x}\parallel_{L^{\infty}} < M$, from (2.6) we deduce that

By virtue of Gronwall's inequality, one has

On the other hand, due to $u = p\ast y$ and $u_{x} = p_{x}\ast y$, then

This completes the proof of Lemma 2.4.

Now, we present the blow-up phenomenon.

Theorem 2.5. Let $u_{0}\in H^{s}(\mathbb{R})$ for $s > \frac{3}{2}$. Suppose that $u(t, x)$ be corresponding solution of problems (1.1) and (1.2) with the initial datum $u_{0}$. If the slope of $u^{k}_{0}$ satisfies

where $K = \frac{3k(6k^{2}+7k-2)}{4(k+1)}\parallel u_{0}\parallel^{2k}_{H^{1}}$ and $K_{1} = \frac{1}{2ck\parallel u_{0}\parallel^{k}_{H^{1}}}$. Then there exists the lifespan $T < \infty$ such that the corresponding solution $u(t, x)$ blows up in finite time $T$ with

Proof. Defining $g(t) = u^{k}(t, x)$, $h(t) = \int_{\mathbb{R}}g^{3}_{x}dx$, it follows that

where $Q = -\partial_{x}(1-\partial^{2}_{x})^{-1}(\frac{k(k+2)}{2(k+1)}u^{k+1}+\frac{k}{2}u^{k-1}u^{2}_{x})$.

Differentiating the above Eq (2.11) with respect $x$ yields

Multiplying $3g^{2}_{x}$ both sides of (2.12) and integrating with respect $x$ over $\mathbb{R}$, one has

Using the Hölder's inequality, Yong's inequality and (1.4), from (2.13) we get

and

Combining the above inequalities (2.14)–(2.17), we obtain

Choosing $\epsilon = \frac{2(k+1)}{3k(6k^{2}+7k-2)}$, which results in

in which $K^{2} = \frac{9k^{2}(6k^{2}+7k-2)^{2}}{16(k+1)^{2}}\parallel u_{0}\parallel^{4k}_{H^{1}}$. Using the Hölder's inequality, we get

Combining (2.19) and (2.20), we have

where $K^{2}_{1} = \frac{1}{4ck^{2}\parallel u_{0}\parallel^{2k}_{H^{1}}}$.

It is observed from assumption of Theorem that $h(0) < -\frac{K}{K_{1}}$, the continuity argument ensures that $h(t) < h(0)$. Lemma 2.1 ($a = K^{2}_{1}$ and $b = K^{2}$) implies that $h(t)\rightarrow -\infty$ as $t\rightarrow T = \frac{1}{2K_{1}K}\log\frac{K_{1}h(0)-K}{K_{1}h(0)+K}$.

On the other hand, by using the fact that

Lemma 2.4 implies that the Theorem 2.5 is true. This completes the proof of Theorem 2.5.

3.   Analytical solutions

The solitons do not belong to the spaces $H^{s}(\mathbb{R})$ with $s > \frac{3}{2}$ ^[28,29], so it motivates us to carry out the study of analytical solutions to problems (1.1) and (1.2).

Definition 3.1. Given initial data $u_{0}\in H^{s}$, $s > \frac{3}{2}$, the function $u$ is said to be a weak solution to the initial-value problems (1.8) and (1.9) if it satisfies the following identity

for any smooth test function $\varphi(t, x)\in C^{\infty}_{c}([0, T)\times \mathbb{R})$. If $u$ is a weak solution on $[0, T)$ for every $T > 0$, then it is called a global weak solution.

Theorem 3.2 The peakon function of the form

is a global weak solution to problems (1.1) and (1.2) in the sense of Definition 3.1. Assumed that the functions $p(t)$ and $q(t)$ satisfy

and

where $'$ denotes differentiation.

Proof. We firstly claim that

is a peakon solution of (1.1) and

Hence, using (3.1), (3.4) and integration by parts, we derive that

On the other hand, using (3.4), we obtain

Note that $G_{x} = -\frac{1}{2}sign(x)e^{-|x|}$. For $x > q(t)$, directly calculate

We directly compute $I_{1}$ as follows

In a similar procedure,

and

Substituting (3.8)–(3.10) into (3.7), we deduce that for $x > q(t)$

where $\Omega = -\frac{1}{2}\frac{k(k+2)}{k+1}p^{k+1}(t)$.

For $x < q(t)$,

We directly compute $\Delta_{1}$ as follows

In a similar procedure,

and

Therefore, from (3.8)–(3.10), we deduce that for $x < q(t)$

where $\Theta = \frac{1}{2}\frac{k(k+2)}{k+1}p^{k+1}(t)$.

Recalling $u = p(t)e^{-|x-q(t)|}$, we have

To ensure that $u = p(t)e^{-|x-q(t)|}$ is a global weak solution of (1.1) in the sense of Definition 3.1, we infer that

and

hold.

It completes the proof of Theorem 3.2.

Remark 2. Solving Eqs (3.17) and (3.18), we get

Therefore, we conclude that peakon solution for problems (1.1) and (1.2)

Remark 3. For $x > q(t)$, the solution of problems (1.1) and (1.2) is of following form

where $p(t)$ and $q(t)$ satisfy

For $x < q(t)$, the solution of problems (1.1) and (1.2) is of following form

where $p(t)$ and $q(t)$ satisfy

Example. For $x > q(t)$, letting $q(t) = \sqrt{t}+c, c > 0$, from (3.17) we derive that

(3.25) implies that

Hence, we obtain from (3.3) the solution of (1.1) for $x > q(t)$.

For $x < q(t)$, letting $q(t) = \sqrt{t}+c, c > 0$, from (3.18) we derive that

(3.28) implies that

Therefore, we obtain from (3.3) the solution of (1.1) for $x < q(t)$.

4.   Analyticity of solution

In this section, we focus on the analyticity of the Cauchy problems (1.1) and (1.2) based on a contraction type argument in a suitably chosen scale of the Banach spaces. In order to state the main result, we will need a suitable scale of the Banach spaces as follows. For any $s > 0$, we set

where $H^{2}(\mathbb{R})$ is the Sobolev space of order two on the real line and $N_{0}$ is the set nonnegative integers. One can easily verify that $E_{s}$ equipped with the norm $|||\cdot|||_{s}$ is a Banach space and that, for any $0 < s' < s$, $E_{s}$ is continuously embedded in $E_{s'}$ with

Another simple consequence of the definition is that any $u$ in $E_{s}$ is a real analytic function on $\mathbb{R}$. Our main theorem is stated as follows.

Theorem 4.1. If the initial data $u_{0}$ is analytic and belongs to a space $E_{s_{0}}$, for some $0 < s_{0}\leq1$, then there exist an $\varepsilon > 0$ and a unique solution $u(t, x)$ to the Cauchy problems (1.1) and (1.2) that is analytic on $(-\varepsilon, \varepsilon)\times \mathbb{R}$.

For the proof of Theorem 4.1, we need the following lemmas

Lemma 4.2. (^[30]) Let $0 < s < 1$. There is a constant $C > 0$, independent of $s$, such that for any $u$ and $v$ in $E_{s}$ we have

Lemma 4.3. (^[30]) There is a constant $C > 0$ such that for any $0 < s' < s < 1$, we have $|||\partial_{x}u|||_{s'}\leq \frac{C}{s-s'}$, and $|||(1-\partial^{2}_{x})^{-1}u|||_{s}\leq|||u|||_{s}$, $|||\partial_{x}(1-\partial^{2}_{x})^{-1}u|||_{s}\leq|||u|||_{s}.$

Lemma 4.4. (^[31]) Let $\{X_{s}\}_{0 < s < 1}$ be a scale of decreasing Banach spaces, namely for any $s' < s$ we have $X_{s}\subset X_{s'}$ and $|||\cdot|||_{s'}\leq|||\cdot|||_{s}$. Consider the Cauchy problem

Let $T$, $R$ and $C$ be positive constants and assume that $F$ satisfies the following conditions:

$(1)$ If for $0 < s' < s < 1$ the function $t\mapsto u(t)$ is holomorphic in $|t| < T$ and continuous on $|t|\leq T$ with values in $X_{s}$ and

then $t\mapsto F(t, u(t))$ is a holomorphic function on $|t| < T$ with values in $X_{s'}$.

$(2)$ For any $0 < s' < s < 1$ and only $u, v\in X_{s}$ with $|||u|||_{s} < R$, $|||v|||_{s} < R$,

$(3)$ There exists $M > 0$ such that for any $0 < s < 1$,

then there exist a $T_{0}\in (0, T)$ and a unique function $u(t)$, which for every $s\in (0, 1)$ is holomorphic in $|t| < (1-s)T_{0}$ with values in $X_{s}$, and is a solution to the Cauchy problems (1.1) and (1.2).

Let $u_{1} = u$ and $u_{2} = u_{x}$, then the problems (1.1) and (1.2) can be written as a system for $u_{1}$ and $u_{2}$.

To apply Lemma 4.4 to prove Theorem 4.1, we rewrite the system (4.3)-(3.28) as

where $U = (u_{1}, u_{2})$ and $F(t, U) = F(u_{1}, u_{2}) = (F_{1}(u_{1}, u_{2}), F_{2}(u_{1}, u_{2}))$.

Proof of Theorem 4.1. Theorem 4.1 is a straightforward consequence of the Cauchy-Kowalevski theorem ^[31]. We only need verify the conditions (1)–(3) in the statement of the abstract Cauchy-Kowalevski theorem (see Lemma 4.4) for both $F_{1}(u_{1}, u_{2})$ and $F_{2}(u_{1}, u_{2})$ in the systems (4.3)–(3.28), since neither $F_{1}$ nor $F_{2}$ depends on $t$ explicitly. For $0 < s' < s < 1$, we derive from Lemmas 4.2 and 4.3 that

where the constant $C$ depends only on $R$, so condition $(1)$ holds.

Notice that to verify the second condition it is sufficient to estimate

Using Lemmas 4.2 and 4.3, we get the following estimates

and

Therefore, we arrive at

where the constant $C$ depends only on $R$ and $k$. The conditions (1)–(3) above are easily verified once our system is transformed into a new system with zero initial data as (4.1) and (4.2). So, we have completed the proof of Theorem 4.1.

5.   Conclusions

In this paper, we focus on several dynamic properties of the Cauchy problems (1.1) and (1.2). We first establish a new blow-up criterion and a blow-up phenomenon for the problem, then we study analytical solutions for the equation by using a new method, here, we present two analytical solutions for the problems (1.1) and (1.2) for the first time. Finally, we study the analyticity in a suitable scale of the Banach spaces. The properties of the problems (1.1) and (1.2) not only present fundamental importance from mathematical point of view but also are of great physical interest.

Acknowledgments

This research was funded by the Guizhou Province Science and Technology Basic Project (Grant No. QianKeHe Basic [2020]1Y011).

Conflict of interest

There is no conflict of interest.