Asymptotic Stability of Singular Solution for Camassa-Holm Equation

Yuetian Gao

School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, China.

DOI: 10.4236/jamp.2021.97102                

Abstract

The aim of this paper is to study singular dynamics of solutions of Camassa-Holm equation. Based on the semigroup theory of linear operators and Banach contraction mapping principle, we prove the asymptotic stability of the explicit singular solution of Camassa-Holm equation.

Keywords

Asymptotic Stability, Camassa-Holm Equation, Explicit Solution, Semigroup Theory, Banach Contraction Mapping Principle

1. Introduction and Main Results

1.1. Introduction

Consider the well-known Camassa-Holm equation as follows (see [1] ):

$m_t+c_0{u}_x+u{m}_x+2m{u}_x=0,$ (1.1)

where $\left(t\mathrm{,}x\right)\in {ℝ}^{+}\times ℝ$, $u=u\left(t\mathrm{,}x\right)$ is the velocity of fluid, m is the momentum given by

$m=m\left(t,x\right)=u\left(t,x\right)-{\alpha }^2{u}_{xx}\left(t,x\right),$

$c_0\in ℝ$ is the critical speed and $\alpha \in ℝ$ relates to the length scale. Thus,

$u_t-{\alpha }^2{u}_{txx}+c_0{u}_x+3u{u}_x={\alpha }^2\left(2u_x{u}_{xx}+u{u}_{xxx}\right).$ (1.2)

Given the initial value as $u\left(\mathrm{0,}x\right)=u_0\left(x\right)$ for $x\in ℝ$.

The Camassa-Holm equation describes unidirectional propagation of surface water waves in shallow water area. For the global well-posedness and stability of solutions, we recommend that the reader refers to [2] - [9], etc. For the wave breaking analysis, we refer the reader to [6] [10] - [15], etc. When $c_0=0$ and $\alpha =1$, the Camassa-Holm equation becomes to the classical Camassa-Holm equation, which admits a bi-Hamiltonian structure [1] [5]. Moreover, the explicit peakon solution and its stability have been established in [12] [16] [17] [18] [19], etc.

Since it is rare to see the explicit stable blowup solutions of Camassa-Holm equation, in this paper, we study the stability of the explicit solution of (1.2) as follows (see [20] ):

$\bar{u}\left(t\mathrm{,}x\right)=-\frac{1}{3}\left(c_0+\frac{x}{T-t}+\frac{1}{T-t}\right)\mathrm{,}$ (1.3)

where $T>0$ is a constant.

1.2. Main Results

Now, we state our main result of this paper.

Theorem 1.1. Let $s>2$ be an integer and $\delta$ is a sufficiently small constant. Then the explicit solution (1.3) of the Camassa-Holm Equation (1.2) is asymptotic stable, i.e., if the initial data $u_0\left(x\right)$ satisfies

${\Vert u_0\left(x\right)+\frac{1}{3}\left(c_0+\frac{x}{T}+\frac{1}{T}\right)\Vert }_{{ℍ}^{s+1}\left(ℝ\right)}\le \delta \mathrm{,}$

then there is a solution $u\left(t\mathrm{,}x\right)$ of (1.2) satisfying

${\Vert u\left(t\mathrm{,}x\right)-\bar{u}\left(t\mathrm{,}x\right)\Vert }_{{ℍ}^s\left(ℝ\right)}\le \frac{\tilde{C}\left(T-t\right)}{{\alpha }^2\left(1+C\ln \left(T-t\right)\right)}\mathrm{,}\left(t\mathrm{,}x\right)\in \left(\mathrm{0,}T\right)\times ℝ\mathrm{,}$

where C and $\tilde{C}$ are positive constants that depend on s.

1.3. Notations

Denote ${\mathbb{L}}^2\left(ℝ\right)={\mathbb{L}}^2$ and ${ℍ}^s\left(ℝ\right)={ℍ}^s$ by the Lebesgue spaces and Sobolev spaces with norms ${\Vert \cdot \Vert }_{{\mathbb{L}}^2}$ and ${\Vert \cdot \Vert }_{{ℍ}^s}$, respectively. * denotes the convolution. $\left[A\mathrm{,}B\right]$ stands for the commutator.

2. Proof of Theorem 1.1

Let

$u\left(t,x\right)=v\left(t,x\right)+\bar{u}\left(t,x\right),$ (2.1)

be the solution of (1.2), where $\bar{u}\left(t\mathrm{,}x\right)=-\frac{1}{3}\left(c_0+\frac{x}{T-t}+\frac{1}{T-t}\right)$ is the explicit solution. Substituting (2.1) into (1.2), we get

$\begin{array}{l}{v}_t-{\alpha }^2{v}_{txx}+\left[\frac{{\alpha }^2}{3}\left(c_0+\frac{x}{T-t}+\frac{1}{T-t}\right)\right]v_{xxx}+\frac{2{\alpha }^2}{3\left(T-t\right)}{v}_{xx}\\ -\left(\frac{x}{T-t}+\frac{1}{T-t}\right)v_x-\frac{1}{T-t}v+3v{v}_x\\ ={\alpha }^2\left(2v_x{v}_{xx}+v{v}_{xxx}\right)\mathrm{,}\forall \left(t\mathrm{,}x\right)\in \left(\mathrm{0,}T\right)\times ℝ\end{array}$ (2.2)

with the initial condition $v\left(\mathrm{0,}x\right)=v_0\left(x\right)=u_0\left(x\right)+\frac{1}{3}\left(c_0+\frac{x}{T}+\frac{1}{T}\right)$ for $x\in ℝ$.

For the singular coefficients in (2.2), let $v\left(t\mathrm{,}x\right)=\psi \left(\tau \mathrm{,}\rho \right)$ by $\tau =-\ln \left(T-t\right)$ and $\rho =\frac{x}{T-t}$, then (2.2) becomes to

$\begin{array}{l}{\psi }_{\tau }+\rho {\psi }_{\rho }-{\alpha }^2{\text{e}}^{2\tau }\left({\psi }_{\tau \rho \rho }+2{\psi }_{\rho \rho }+\rho {\psi }_{\rho \rho }\right)+{\text{e}}^{2\tau }\left[\frac{{\alpha }^2}{3}\left(c_0+\rho +{\text{e}}^{\tau }\right)\right]{\psi }_{\rho \rho \rho }\\ +\frac{2{\alpha }^2}{3}{\text{e}}^{2\tau }{\psi }_{\rho \rho }-\left(\rho +{\text{e}}^{\tau }\right){\psi }_{\rho }-\psi +3\psi {\psi }_{\rho }={\alpha }^2{\text{e}}^{2\tau }\left(2{\psi }_{\rho }{\psi }_{\rho \rho }+\psi {\psi }_{\rho \rho \rho }\right)\mathrm{.}\end{array}$ (2.3)

Let $\kappa ={\text{e}}^{-\tau }\rho$ and $\bar{v}\left(\tau \mathrm{,}\kappa \right)={\text{e}}^{-\tau }\psi \left(\tau \mathrm{,}\rho \right)$. Then (2.3) becomes to

$\begin{array}{l}{\bar{v}}_{\tau }-{\alpha }^2{\bar{v}}_{\tau \kappa \kappa }-\frac{{\alpha }^2}{3}{\bar{v}}_{\kappa \kappa }+{\text{e}}^{-\tau }\left[\gamma +\frac{{\alpha }^2}{3}\left(c_0+\kappa {\text{e}}^{\tau }+{\text{e}}^{\tau }\right)\right]{\bar{v}}_{\kappa \kappa \kappa }\\ -\left(\kappa +1\right){\bar{v}}_{\kappa }+3\bar{v}{\bar{v}}_{\kappa }={\alpha }^2\left(2{\bar{v}}_{\kappa }{\bar{v}}_{\kappa \kappa }+\bar{v}{\bar{v}}_{\kappa \kappa \kappa }\right)\mathrm{.}\end{array}$ (2.4)

Let the operator $\mathcal{A}={\left(1-{\alpha }^2{\partial }_{\kappa \kappa }\right)}^{\frac{1}{2}}$. Since $1-{\alpha }^2{\partial }_{\kappa \kappa }$ admits a fundamental solution $\wp \left(x\right)=\frac{1}{2\alpha }{\text{e}}^{-\left|\frac{\kappa }{\alpha }\right|}$, we have ${\mathcal{A}}^{-2}\bar{v}=\wp \left(\kappa \right)\ast \bar{v}$ for all $\bar{v}\in {\mathbb{L}}^2$. Let $w\left(\tau \mathrm{,}\kappa \right)=\bar{v}\left(\tau \mathrm{,}\kappa \right)-{\alpha }^2{\bar{v}}_{\kappa \kappa }\left(\tau \mathrm{,}\kappa \right)$, then $\bar{v}\left(\tau \mathrm{,}\kappa \right)=\wp \ast w$, where $\kappa \in ℝ$. Furthermore, we have ${\left(\rho \ast w\right)}_{\kappa \kappa }={\alpha }^{-2}\left(\rho \ast w-w\right)$, ${\bar{v}}_{\kappa }={\left(\wp \ast w\right)}_{\kappa }$ and ${\bar{v}}_{\kappa \kappa \kappa }={\alpha }^{-2}\left({\left(\wp \ast w\right)}_{\kappa }-w_{\kappa }\right)$. Then (2.3) can be rewritten as

$\begin{array}{l}{w}_{\tau }+\frac{1}{3}w-{\text{e}}^{-\tau }\left[\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right]w_{\kappa }-\frac{1}{3}\wp \ast w\\ +\left\{{\text{e}}^{-\tau }\left[\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right]-\left(\kappa +1\right)\right\}{\left(\wp \ast w\right)}_{\kappa }+3\left(\wp \ast w\right){\left(\wp \ast w\right)}_{\kappa }\\ =2{\left(\wp \ast w\right)}_{\kappa }\left(\wp \ast w-w\right)+\left(\wp \ast w\right)\left[{\left(\wp \ast w\right)}_{\kappa }-w_{\kappa }\right]\end{array}$ (2.5)

with the initial data

$w_0\left(\kappa \right)=u_0\left(x\right)-{\alpha }^2{u^{″}}_0\left(x\right)+\frac{1}{3}\left(\frac{x}{T}+\frac{1}{T}+c_0\right)\mathrm{,}$ (2.6)

and the boundary condition

$\underset{\left|\kappa \right|\to +\infty }{\lim }w\left(\tau \mathrm{,}\kappa \right)=\mathrm{0,}\underset{\left|\kappa \right|\to +\infty }{\lim }{w}_{\kappa }\left(\tau \mathrm{,}\kappa \right)=0.$ (2.7)

Before making a priori estimate of the solutions to problems (2.5)-(2.7). We recall the following commutator estimate.

Lemma 2.1 ( [21] ). Let $s>0$. Then it holds

${\Vert \left[{\mathcal{A}}^s\mathrm{,}u\right]v\Vert }_{{\mathbb{L}}^2}\le C\left({\Vert {\partial }_{x}u\Vert }_{{\mathbb{L}}^{\infty }}{\Vert {\mathcal{A}}^{s-1}v\Vert }_{{\mathbb{L}}^2}+{\Vert {\mathcal{A}}^{s}u\Vert }_{{\mathbb{L}}^2}{\Vert v\Vert }_{{\mathbb{L}}^{\infty }}\right)\mathrm{,}$ (2.8)

where C is a positive constant that depends on s.

Now, we derive a priori estimate of the solutions for (2.5).

Lemma 2.2. Let $s>2$ and $\alpha \ne 0$. Assume that w be a solution of (2.5), then

${\Vert w\Vert }_{{ℍ}^s}\le \frac{1}{{\Vert w_0\Vert }_{{ℍ}^s}^{-1}-C\tau }\mathrm{,}$ (2.9)

where C is a positive constant depending upon s.

Proof. Applying ${\mathcal{A}}^s$ to both sides of (2.5) and taking the ${\mathbb{L}}^2$ -inner product with ${\mathcal{A}}^{s}w$, we get

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}\tau }{\Vert w\Vert }_{{ℍ}^s}^2+\frac{1}{3}{\Vert w\Vert }_{{ℍ}^s}^2-\frac{1}{3}{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left(\wp \ast w\right)\text{d}\kappa \\ -{\text{e}}^{-\tau }{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left[\left(\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right)w_{\kappa }\right]\text{d}\kappa \\ +{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left\{\left[{\text{e}}^{-\tau }\left(\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right)-\left(\kappa +1\right)\right]{\left(\wp \ast w\right)}_{\kappa }\right\}\text{d}\kappa \\ +3{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left[\left(\wp \ast w\right){\left(\wp \ast w\right)}_{\kappa }\right]\text{d}\kappa \\ =2{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left[{\left(\wp \ast w\right)}_{\kappa }\left(\wp \ast w-w\right)\right]\text{d}\kappa \\ +{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left[\left(\wp \ast w\right)\left({\left(\wp \ast w\right)}_{\kappa }-w_{\kappa }\right)\right]\text{d}\kappa \mathrm{.}\end{array}$ (2.10)

Next, we estimate each of terms in (2.10).

$-\frac{1}{3}{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left(\wp \ast w\right)\text{d}\kappa =-\frac{1}{3}{\Vert w\Vert }_{{ℍ}^{s-1}}^2\mathrm{,}$ (2.11)

$\begin{array}{l}-{\text{e}}^{-\tau }{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left[\left(\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right)w_{\kappa }\right]\text{d}\kappa \\ ={\text{e}}^{-\tau }{\displaystyle {\int }_{ℝ}}{\left[\left(\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right){\mathcal{A}}^{2s}w\right]}_{\kappa }w\text{d}\kappa \\ =\frac{1}{3}{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{2s}w\cdot w\text{d}\kappa -\frac{1}{2}\times \frac{1}{3}{\displaystyle {\int }_{ℝ}}{\left({\mathcal{A}}^{s}w\right)}^2\text{d}\kappa =\frac{1}{6}{\Vert w\Vert }_{{ℍ}^s}^2,\end{array}$ (2.12)

$\begin{array}{l}{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left\{\left[{\text{e}}^{-\tau }\left(\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right)-\left(\kappa +1\right)\right]{\left(\wp \ast w\right)}_{\kappa }\right\}\text{d}\kappa \\ =-{\displaystyle {\int }_{ℝ}}{\left\{{\mathcal{A}}^{2s}w\left[{\text{e}}^{-\tau }\left(\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right)-\left(\kappa +1\right)\right]\right\}}_{\kappa }\left(\wp \ast w\right)\text{d}\kappa \\ =-{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{2s}{w}_{\kappa }\left[{\text{e}}^{-\tau }\left(\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right)-\left(\kappa +1\right)\right]\left(\wp \ast w\right)\text{d}\kappa \\ -\left(\frac{1}{3}-1\right){\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{2s}w\left(\wp \ast w\right)\text{d}\kappa \\ =\frac{1}{2}\times \left(\frac{1}{3}-1\right){\displaystyle {\int }_{ℝ}}{\left({\mathcal{A}}^{s-1}w\right)}^2\text{d}\kappa +\frac{2}{3}{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{2s}w\left(\wp \ast w\right)\text{d}\kappa =\frac{1}{3}{\Vert w\Vert }_{{ℍ}^{s-1}}^2,\end{array}$ (2.13)

$\begin{array}{l}3{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left(\left(\wp \ast w\right){\left(\wp \ast w\right)}_{\kappa }\right)\text{d}\kappa \\ =-\frac{3}{2}{\displaystyle {\int }_{ℝ}}{w}_{\kappa }{\mathcal{A}}^{2s}\left({\left(\wp \ast w\right)}^2\right)\text{d}\kappa \le \frac{3}{2}{\Vert w_{\kappa }\Vert }_{{\mathbb{L}}^{\infty }}{\Vert w\Vert }_{{ℍ}^{s-1}}^2\le \frac{3}{2}{\Vert w\Vert }_{{ℍ}^{s-1}}^3\mathrm{,}\end{array}$ (2.14)

In addition, using (2.8), we have

$\begin{array}{l}2\left|{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left({\left(\wp \ast w\right)}_{\kappa }\left(\wp \ast w-w\right)\right)\text{d}\kappa \right|\\ =2\left|{\displaystyle {\int }_{ℝ}}\left[{\mathcal{A}}^s,\left(\wp \ast w-w\right)\right]{\left(\wp \ast w\right)}_{\kappa }{\mathcal{A}}^{s}w\text{d}\kappa \right|\\ +2\left|{\displaystyle {\int }_{ℝ}}\left(\wp \ast w-w\right){\mathcal{A}}^s{\left(\wp \ast w\right)}_{\kappa }{\mathcal{A}}^{s}w\text{d}\kappa \right|\\ \le C\left({\Vert \wp \ast w-w\Vert }_{{\mathbb{L}}^{\infty }}{\Vert {\mathcal{A}}^{s-1}{\left(\wp \ast w\right)}_{\kappa }\Vert }_{{\mathbb{L}}^2}+{\Vert {\mathcal{A}}^s\left(\wp \ast w-w\right)\Vert }_{{\mathbb{L}}^2}{\Vert {\left(\wp \ast w\right)}_{\kappa }\Vert }_{{\mathbb{L}}^{\infty }}\right){\Vert w\Vert }_{{ℍ}^s}\\ +2\left({\Vert \left(\wp \ast w-w\right)\Vert }_{{\mathbb{L}}^{\infty }}+{\Vert {\left(\wp \ast w-w\right)}_{\kappa }\Vert }_{{\mathbb{L}}^{\infty }}\right){\Vert w\Vert }_{{ℍ}^2}^2\\ \le C{\Vert w\Vert }_{{ℍ}^s}^3,\end{array}$ (2.15)

similarly,

$\left|{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}w{\mathcal{A}}^s\left[\left(\wp \ast w\right)\left({\left(\wp \ast w\right)}_{\kappa }-w_{\kappa }\right)\right]\text{d}\kappa \right|\le C{\Vert w\Vert }_{{ℍ}^s}^3,$ (2.16)

where C is a positive constant depending upon s.

Substituting (2.11)-(2.16) into (2.10), we get $\frac{1}{2}\frac{\text{d}}{\text{d}\tau }{\Vert w\Vert }_{{ℍ}^s}^2\le C{\Vert w\Vert }_{{ℍ}^s}^3$, and then $-\frac{\text{d}}{\text{d}\tau }{\Vert w\Vert }_{{ℍ}^s}^{-1}\le C$. Integrating this inequality above with respect to $\tau$ from 0 to $\tau$, we get

${\Vert w\Vert }_{{ℍ}^s}\le \frac{1}{{\Vert w_0\Vert }_{{ℍ}^s}^{-1}-C\tau }\mathrm{.}$ (2.17)

This completes the proof of Lemma 2.2. o

Proof of Theorem 1.1. Now, we study the well-posedness for (2.5)-(2.7). Define the linear operator L as

$\begin{array}{c}L\left[w\right]=-\frac{1}{3}w+\frac{1}{3}\wp \ast w+{\text{e}}^{-\tau }\left[\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right]w_{\kappa }\\ -\left[{\text{e}}^{-\tau }\left(\frac{1}{3}\left(c_0+{\text{e}}^{\tau }\kappa +{\text{e}}^{\tau }\right)\right)-\left(\kappa +1\right)\right]{\left(\wp \ast w\right)}_{\kappa },\end{array}$ (2.18)

then (2.5) becomes to

$w_t=L\left[w\right]+f\left(w\right)\mathrm{,}$ (2.19)

where f is the nonlinear terms:

$\begin{array}{c}f\left(w\right)=-3\left(\wp \ast w\right){\left(\wp \ast w\right)}_{\kappa }+2\left(\wp \ast w\right)\left(\wp \ast w-w\right)\\ +\left(\wp \ast w\right)\left[{\left(\wp \ast w\right)}_{\kappa }-w_{\kappa }\right]\mathrm{.}\end{array}$ (2.20)

Lemma 2.3. Let $s>2$. Then

· $L\left[w\right]\in {ℍ}^s$ for $\forall w\in \mathcal{D}\left(L\right)$.

· L is a closed and densely defined linear operator in ${ℍ}^s$.

Proof. It is a direct verification by the definition of L. o

Lemma 2.4. Let $s>2$. Then L is a dissipative operator in ${ℍ}^s$ , i.e., ${\left(L\left[w\right]\mathrm{,}w\right)}_s\le 0$.

Proof. Using (2.11)-(2.14), a direct calculation shows that

$\begin{array}{c}{\displaystyle {\int }_{ℝ}}\left({\mathcal{A}}^{s}L\left[w\right]\right){\mathcal{A}}^{s}w\text{d}\kappa =-\frac{1}{3}{\Vert w\Vert }_{{ℍ}^s}^2+\frac{1}{3}{\Vert w\Vert }_{{ℍ}^{s-1}}^2-\frac{1}{6}{\Vert w\Vert }_{{ℍ}^s}^2-\frac{1}{3}{\Vert w\Vert }_{{ℍ}^{s-1}}^2\\ =-\frac{1}{2}{\Vert w\Vert }_{{ℍ}^s}^2\le 0.\end{array}$ (2.21)

This completes the proof. o

Lemma 2.5 (Young inequality with $\epsilon$, see [22] ). Let $a,b>0$ and $\epsilon >0$. If $p\mathrm{,}q\in \left(\mathrm{1,}\infty \right)$ satisfy $\frac{1}{p}+\frac{1}{q}=1$. Then

$ab\le \epsilon a^p+C\left(\epsilon \right)b^q\mathrm{,}$ (2.22)

where $C\left(\epsilon \right)={\left(\epsilon p\right)}^{-\frac{q}{p}}{q}^{-1}$.

Lemma 2.6. Let $s>2$. Then the operator L is invertible in ${ℍ}^s$. Furthermore, it generates a ${ℂ}_0$ -semigroup ${\left(S\left(t\right)\right)}_{\tau \ge 0}$ in ${ℍ}^s$.

Proof. Firstly, we show that the existence of $L^{-1}$. Indeed, we need to prove L is injective and surjective. On the one hand, let $w\in \mathcal{D}\left(L\right)$ such that $L\left[w\right]=0$, then

${\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}L\left[w\right]{\mathcal{A}}^{s}w\text{d}\kappa =-\frac{1}{2}{\Vert w\Vert }_{{ℍ}^s}^2=0.$ (2.23)

This combining with the boundary condition (2.7) gives that $w=0$. So the operator L is injective. On the other hand, for all $g\in {ℍ}^1$, put

$L\left[w\right]=g\mathrm{.}$ (2.24)

Applying ${\mathcal{A}}^s$ to (2.24) and multiplying the result by ${\mathcal{A}}^{s}w$, and then integrating over $ℝ$, we get

${\Vert w\Vert }_{{ℍ}^s}^2=-2{\displaystyle {\int }_{ℝ}}{\mathcal{A}}^{s}g{\mathcal{A}}^{s}w\text{d}\kappa \mathrm{.}$ (2.25)

It follows from the Young inequality with $\epsilon$ in Lemma 2.5 that

${\Vert w\Vert }_{{ℍ}^s}\le C{\Vert g\Vert }_{{ℍ}^s}\mathrm{.}$ (2.26)

Note that $s>2$, then by the standard theory of elliptic equations (see [22] ), there exists a unique weak solution $w\in {ℍ}^1$, moreover, we have $w\in {ℍ}^{s+1}$ if $g\in {ℍ}^s$. Thus, the operator L is surjective. Secondly, by the Lumer-Phillips theorem (see [23] ), the operator L generates a ${ℂ}_0$ -semigroup ${\left(S\left(t\right)\right)}_{\tau \ge 0}$ in ${ℍ}^s$. This completes the proof. o

As a consequence, we have

Proposition 2.7. Let $s>2$. Then the Cauchy problem

$(\begin{array}{l}\frac{\text{d}}{\text{d}\tau }w=Lw,\\ w\left(0\right)=w_0\end{array}$ (2.27)

with zero boundary condition exists a unique solution $w\left(\tau \right)=S\left(\tau \right)w_0$, where $w_0$ is the initial data defined in (2.6).

Using the Duhamel’s principle, the solutions of (2.19) satisfies the integral equation:

$w\left(\tau \right)=S\left(\tau \right)w_0+{\displaystyle {\int }_0^{\tau }}S\left(\tau -s\right)f\left(w\left(s\right)\right)\text{d}s\mathrm{.}$ (2.28)

To show this integral equation exists a solution, we define the solution space as

$B_{\delta }=\left\{w\in {ℍ}^s\mathrm{<mo>\; :\; </mo>}{\Vert w\Vert }_{{ℍ}^s}<\delta \ll 1\right\}\mathrm{,}$ (2.29)

and the map $\mathcal{T}$ as

$\mathcal{T}w\left(\tau \right)=S\left(\tau \right)w_0+{\displaystyle {\int }_0^{\tau }}S\left(\tau -s\right)f\left(w\left(s\right)\right)\text{d}s\mathrm{.}$ (2.30)

We need to prove that $\mathcal{T}$ has a fixed point in the space $B_{\delta }$.

Lemma 2.8 ( [21] ). Let $s>2$. Then $B_{\delta }$ is an algebra, and

${\Vert uv\Vert }_{{ℍ}^s}\le C\left({\Vert u\Vert }_{{\mathbb{L}}^{\infty }}{\Vert v\Vert }_{{ℍ}^s}+{\Vert u\Vert }_{{ℍ}^s}{\Vert v\Vert }_{{\mathbb{L}}^{\infty }}\right)\mathrm{,}$ (2.31)

where C is a positive constant depending upon s.

Lemma 2.9. Let $s>2$ be an integer. Assume that ${\Vert w_0\Vert }_{{ℍ}^{s+1}}<\delta$ for some sufficiently small $\delta >0$. Then $\mathcal{T}$ is a self-mapping on $B_{\delta }$. Moreover, $\mathcal{T}$ is a contraction mapping.

Proof. By Lemma 2.8, we have

$\begin{array}{c}{\Vert f\left(w\right)\Vert }_{{ℍ}^s}\le 3{\Vert \left(\wp \ast w\right){\left(\wp \ast w\right)}_{\kappa }\Vert }_{{ℍ}^s}+2{\Vert {\left(\wp \ast w\right)}_{\kappa }\left(\wp \ast w-w\right)\Vert }_{{ℍ}^s}\\ +{\Vert \left(\wp \ast w\right)\left({\left(\wp \ast w\right)}_{\kappa }-w_{\kappa }\right)\Vert }_{{ℍ}^s}\\ \le C_1({\Vert \wp \ast w\Vert }_{{ℍ}^s}{\Vert {\left(\wp \ast w\right)}_{\kappa }\Vert }_{{\mathbb{L}}^{\infty }}+{\Vert {\left(\wp \ast w\right)}_{\kappa }\Vert }_{{\mathbb{L}}^{\infty }}{\Vert \left(\wp \ast w-w\right)\Vert }_{{ℍ}^s}\\ +{\Vert \wp \ast w\Vert }_{{ℍ}^s}{\Vert {\left(\wp \ast w\right)}_{\kappa }-w_{\kappa }\Vert }_{{\mathbb{L}}^{\infty }}),\end{array}$ (2.32)

where $C_1$ is a positive constant.

Note that ${ℍ}^s\subset {\mathbb{L}}^{\infty }$ and $w={\mathcal{A}}^2\left(\wp \left(\kappa \right)\ast \bar{v}\right)$, then using Lemma 2.2, we have

${\Vert f\left(w\right)\Vert }_{{ℍ}^s}\le C_1{\Vert w\Vert }_{{ℍ}^s}^2<\frac{C_1}{{\Vert w\Vert }_{{ℍ}^s}^{-1}-C\tau }<\frac{C_1}{{\delta }^{-1}-C\tau }<\delta$ (2.33)

for sufficiently small $\delta$. Thus, $\mathcal{T}$ is a self-mapping on $B_{\delta }$.

To show $\mathcal{T}$ is a contraction mapping, we choose $w\mathrm{,}\bar{w}\in B_{\delta }$, by Lemma 2.8 and a direct calculation show that

$\begin{array}{l}{\Vert f\left(w\right)-f\left(\bar{w}\right)\Vert }_{{ℍ}^s}\\ =\Vert -3\left(\wp \ast w\right){\left(\wp \ast w\right)}_{\kappa }+2{\left(\wp \ast w\right)}_{\kappa }\left(\wp \ast w\right)\\ -2{\left(\wp \ast w\right)}_{\kappa }w+\left(\wp \ast w\right)\left({\left(\wp \ast w\right)}_{\kappa }-w_{\kappa }\right)\\ +3\left(\wp \ast \bar{w}\right){\left(\wp \ast \bar{w}\right)}_{\kappa }-2{\left(\wp \ast \bar{w}\right)}_{\kappa }\left(\wp \ast \bar{w}\right)\\ +2{\left(\wp \ast \bar{w}\right)}_{\kappa }\bar{w}-{\left(\wp \ast \bar{w}\right)\left({\left(\wp \ast \bar{w}\right)}_{\kappa }-{\bar{w}}_{\kappa }\right)\Vert }_{{ℍ}^s}\end{array}$

$\begin{array}{l}\le \Vert 3\left\{\left(\wp \ast \bar{w}\right){\left[\wp \ast \left(\bar{w}-w\right)\right]}_{\kappa }+{\left(\wp \ast \bar{w}\right)}_{\kappa }\left[\wp \ast \left(\bar{w}-w\right)\right]\right\}\\ +2\left\{{\left(\wp \ast w\right)}_{\kappa }\left[\wp \ast \left(\bar{w}-w\right)\right]+\left(\wp \ast \bar{w}\right){\left[\wp \ast \left(\bar{w}-w\right)\right]}_{\kappa }\right\}\\ +3\left\{\left(\wp \ast \bar{w}\right){\left[\wp \ast \left(\bar{w}-w\right)\right]}_{\kappa }+{\left(\wp \ast \bar{w}\right)}_{\kappa }\left[\wp \ast \left(\bar{w}-w\right)\right]\right\}\\ +\left\{\left(\wp \ast \bar{w}\right){\left[\wp \ast \left(\bar{w}-w\right)\right]}_{\kappa }+{\left(\wp \ast \bar{w}\right)}_{\kappa }\left[\wp \ast \left(\bar{w}-w\right)\right]\right\}\\ +{\left\{\left[\wp \ast \left(\bar{w}-w\right)\right]{\bar{w}}_{\kappa }+\left(\wp \ast w\right){\left(\bar{w}-w\right)}_{\kappa }\right\}\Vert }_{{ℍ}^s}\\ \le C\delta {\Vert w-\bar{w}\Vert }_{{ℍ}^s}.\end{array}$ (2.34)

Thus,

${\Vert \mathcal{T}w\left(\tau \right)-\mathcal{T}\bar{w}\left(\tau \right)\Vert }_{{ℍ}^s}\le C\delta {\Vert w-\bar{w}\Vert }_{{ℍ}^s}\mathrm{.}$ (2.35)

Since $\delta >0$ is sufficiently small, $\mathcal{T}$ is a contraction mapping. o

Thus, we have the following existence results.

Proposition 2.10. Let $s>2$ be a fixed integer and $\delta >0$ is a sufficiently small constant. Then

· if ${\Vert w_0\Vert }_{{ℍ}^{s+1}}<\delta$, there exists a unique solution $w\in B_{\delta }$ to (2.5) with the initial data (2.6) and the boundary condition (2.7).

· there exists a global solution $\psi \left(\tau \mathrm{,}\rho \right)\in {ℍ}^s$ to (2.3) with the initial data (2.6) and the boundary condition (2.7). Moreover, if the initial data ${\psi }_0$ satisfies ${\Vert {\psi }_0\Vert }_{{ℍ}^{s+1}}<\delta$, then

${\Vert \psi \Vert }_{{ℍ}^s}\le \frac{\tilde{C}}{{\alpha }^2{\text{e}}^{\tau }\left(1-C\tau \right)}\mathrm{.}$ (2.36)

Here C and $\tilde{C}$ are two positive constants that depend on s.

Proof. By Lemma 2.9 and the Banach fixed point theorem, the map $\mathcal{T}$ has a fixed point in $B_{\delta }$, which is a solution of Equation (2.5). Thus, there exists a global solution of (2.3) as

$\psi \left(\tau \mathrm{,}\rho \right)={\text{e}}^{\tau }\bar{v}\left(\tau \mathrm{,}{\text{e}}^{-\tau }\rho \right)={\text{e}}^{\tau }\left(\left(\wp \ast w\right)\left(\tau \mathrm{,}{\text{e}}^{-\tau }\rho \right)\right)\mathrm{.}$ (2.37)

Furthermore, we have

$v_{\rho \rho }={\psi }_{\rho \rho }={\left(\wp \ast w\right)}_{\kappa \kappa }{\text{e}}^{-\tau }={\alpha }^{-2}{\text{e}}^{-\tau }\left(\wp \ast w-w\right).$ (2.38)

Thus, by Lemma 2.2, we get

$\begin{array}{c}{\Vert {\psi }_{\rho \rho }\Vert }_{{ℍ}^{s-2}}\le {\alpha }^{-2}{\text{e}}^{-\tau }{\Vert \wp \ast w-w\Vert }_{{ℍ}^{s-2}}\le \tilde{C}{\alpha }^{-2}{\text{e}}^{-\tau }{\Vert w\Vert }_{{ℍ}^{s-2}}\\ \le \frac{\tilde{C}}{{\alpha }^2{\text{e}}^{\tau }\left({\Vert w_0\Vert }_{{ℍ}^s}^{-1}-C\tau \right)}\le \frac{\tilde{C}}{{\alpha }^2{\text{e}}^{\tau }\left(1-C\tau \right)}\mathrm{,}\end{array}$ (2.39)

where we have used $\delta <1$ in the last inequality. This completes the proof. o

As a consequence, we obtain that the global well-posedness of the initial value problem (2.2). This implies that the asymptotic stability of the explicit singular solution (1.3) for the Camassa-Holm Equation (1.2). Hence, we complete the proof of Theorem 1.1.

3. Conclusion

In this paper, the Semigroup theory of linear operators has been used to study the asymptotic stability of the explicit blowup solution of Camassa-Holm equation. This result shows that the explicit solution is a meaningful physical solution. However, this explicit solution does not depend on the wavelength (i.e., it does not depend on $\alpha$ ). Thus, further studies are needed to construct the explicit solutions that depend on $\alpha$, and then prove their stability.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Camassa, R. and Holm, D.D. (1993) An Integrable Shallow Water Equation with Peaked Solitons. Physical Review Letters, 71, 1661-1664. https://doi.org/10.1103/PhysRevLett.71.1661
[2]	Bressan, A. and Constantin, A. (2007) Global Conservative Solutions of the Camassa-Holm Equation. Archive for Rational Mechanics and Analysis, 183, 215-239. https://doi.org/10.1007/s00205-006-0010-z
[3]	Bressan, A. and Constantin, A. (2007) Global Dissipative Solutions of the Camassa-Holm Equation. Analysis and Applications, 5, 1-27. https://doi.org/10.1142/S0219530507000857
[4]	Constantin, A. (1997) On the Cauchy Problem for the Periodic Camassa-Holm Equation. Journal of Differential Equations, 141, 218-235. https://doi.org/10.1006/jdeq.1997.3333
[5]	Constantin, A. (1997) The Hamiltonian Structure of the Camassa-Holm Equation. Expositiones Mathematicae, 15, 53-85.
[6]	Constantin, A. and Escher, J. (1998) Global Existence and Blow-up for a Shallow Water Equation. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 26, 303-328.
[7]	Constantin, A. (2001) On the Scattering Problem for the Camassa-Holm Equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 457, 953-970. https://doi.org/10.1098/rspa.2000.0701
[8]	Constantin, A. and Molinet, L. (2000) Global Weak Solutions for a Shallow Water Equation. Communications in Mathematical Physics, 211, 45-61. https://doi.org/10.1007/s002200050801
[9]	Constantin, A. and Lannes, D. (2009) The Hydrodynamical Relevance of the Camassa-Holm and Degasperis-Procesi Equations. Archive for Rational Mechanics and Analysis, 192, 165-186. https://doi.org/10.1007/s00205-008-0128-2
[10]	Bendahmane, M., Coclite, G.M. and Karlsen, K.H. (2006) H1-Perturbations of Smooth Solutions for a Weakly Dissipative Hyperelastic-Rod Wave Equation. Mediterranean Journal of Mathematics, 3, 419-432. https://doi.org/10.1007/s00009-006-0088-4
[11]	Brandolese, L. (2014) Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations. Communications in Mathematical Physics, 330, 401-414. https://doi.org/10.1007/s00220-014-1958-4
[12]	Constantin, A. (2000) Existence of Permanent and Breaking Waves for a Shallow Water Equation: A Geometric Approach. Annales de l’Institut Fourier, 50, 321-362. https://doi.org/10.5802/aif.1757
[13]	Constantin, A. and Escher, J. (1998) Wave Breaking for Nonlinear Nonlocal Shallow Water Equations. Acta Mathematica, 181, 229-243. https://doi.org/10.1007/BF02392586
[14]	Liu, Y. (2006) Global Existence and Blow-Up Solutions for a Nonlinear Shallow Water Equation. Mathematische Annalen, 335, 717-735. https://doi.org/10.1007/s00208-006-0768-1
[15]	McKean, H.P. (2004) Breakdown of the Camassa-Holm Equation. Communications on Pure and Applied Mathematics, 57, 416-418. https://doi.org/10.1002/cpa.20003
[16]	Constantin, A. and Strauss, W. (2002) Stability of the Camassa-Holm Solitons. Journal of Nonlinear Science, 12, 415-422. https://doi.org/10.1007/s00332-002-0517-x
[17]	Constantin, A. and Molinet, L. (2001) Orbital Stability of Solitary Waves for a Shallow Water Equation. Physica D: Nonlinear Phenomena, 157, 75-89. https://doi.org/10.1016/S0167-2789(01)00298-6
[18]	Constantin, A. and Strauss, W. (2000) Stability of a Class of Solitary Waves in Compressible Elastic Rods. Physics Letters A, 270, 140-148. https://doi.org/10.1016/S0375-9601(00)00255-3
[19]	Hakkaev, S. and Kirchev, K. (2005) Local Well-Posedness and Orbital Stability of Solitary Wave Solutions for the Generalized Camassa-Holm Equation. Communications in Partial Differential Equations, 30, 761-781. https://doi.org/10.1081/PDE-200059284
[20]	Gao, Y. and Chen, J. (2021) Asymptotic Stability of Singular Waves for Dullin-Gottwald-Holm Equation. (Submitted)
[21]	Kato, T. (1975) Quasi-Linear Equations of Evolution, with Applications to Partial Differential Equations. In: Everitt, W.N., Ed., Spectral Theory and Differential Equations, Vol. 448, Springer, Berlin, Heidelberg, 25-70. https://doi.org/10.1007/BFb0067080
[22]	Evans, L.C. (1998) Partial Differential Equations (Graduate Studies in Mathematics, Vol. 19). American Mathematical Society, Providence, xviii+662 p.
[23]	Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. Vol. 44. Springer-Verlag, New York, viii+279 p. https://doi.org/10.1007/978-1-4612-5561-1