Asymptotic Stability of Singular Solution for Camassa-Holm Equation

The aim of this paper is to study singular dynamics of solutions of Camas-sa-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.

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]): where 0 T > is a constant.

Main Results
Now, we state our main result of this paper.
where C and C  are positive constants that depend on s. [ ] , A B stands for the commutator.

Proof of Theorem 1.1
be the solution of (1.2), where ( ) with the initial condition ( ) ( ) ( ) Let the operator ( ) and the boundary condition Before making a priori estimate of the solutions to problems (2.5)-(2.7). We recall the following commutator estimate.
where C is a positive constant that depends on s. Now, we derive a priori estimate of the solutions for (2.5). where C is a positive constant depending upon s.

Lemma 2.2. Let
Proof. Applying s  to both sides of (2.5) and taking the 2  -inner product Next, we estimate each of terms in (2.10).
( ) where f is the nonlinear terms:  Note that 2 s > , then by the standard theory of elliptic equations (see [22]), there exists a unique weak solution 1 w ∈  , moreover, we have Thus, the operator L is surjective. Secondly, by the Lumer-Phillips theorem (see [23]), the operator L generates a 0  -semigroup Using the Duhamel's principle, the solutions of (2.19) satisfies the integral equation:   where we have used 1 δ < in the last inequality. This completes the proof.  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.

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 α ). Thus, further studies are needed to construct the explicit solutions that depend on α , and then prove their stability.

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