Let

u ( t , x ) = v ( t , x ) + u ¯ ( t , x ) , (2.1)

be the solution of (1.2), where u ¯ ( t , x ) = − 1 3 ( c 0 + x T − t + 1 T − t ) is the explicit solution. Substituting (2.1) into (1.2), we get

v t − α 2 v t x x + [ α 2 3 ( c 0 + x T − t + 1 T − t ) ] v x x x + 2 α 2 3 ( T − t ) v x x − ( x T − t + 1 T − t ) v x − 1 T − t v + 3 v v x = α 2 ( 2 v x v x x + v v x x x ) ,   ∀ ( t , x ) ∈ ( 0, T ) × ℝ (2.2)

with the initial condition v ( 0, x ) = v 0 ( x ) = u 0 ( x ) + 1 3 ( c 0 + x T + 1 T ) for x ∈ ℝ .

For the singular coefficients in (2.2), let v ( t , x ) = ψ ( τ , ρ ) by τ = − ln ( T − t ) and ρ = x T − t , then (2.2) becomes to

ψ τ + ρ ψ ρ − α 2 e 2 τ ( ψ τ ρ ρ + 2 ψ ρ ρ + ρ ψ ρ ρ ) + e 2 τ [ α 2 3 ( c 0 + ρ + e τ ) ] ψ ρ ρ ρ + 2 α 2 3 e 2 τ ψ ρ ρ − ( ρ + e τ ) ψ ρ − ψ + 3 ψ ψ ρ = α 2 e 2 τ ( 2 ψ ρ ψ ρ ρ + ψ ψ ρ ρ ρ ) . (2.3)

Let κ = e − τ ρ and v ¯ ( τ , κ ) = e − τ ψ ( τ , ρ ) . Then (2.3) becomes to

v ¯ τ − α 2 v ¯ τ κ κ − α 2 3 v ¯ κ κ + e − τ [ γ + α 2 3 ( c 0 + κ e τ + e τ ) ] v ¯ κ κ κ − ( κ + 1 ) v ¯ κ + 3 v ¯ v ¯ κ = α 2 ( 2 v ¯ κ v ¯ κ κ + v ¯ v ¯ κ κ κ ) . (2.4)

Let the operator A = ( 1 − α 2 ∂ κ κ ) 1 2 . Since 1 − α 2 ∂ κ κ admits a fundamental solution ℘ ( x ) = 1 2 α e − | κ α | , we have A − 2 v ¯ = ℘ ( κ ) ∗ v ¯ for all v ¯ ∈ L 2 . Let w ( τ , κ ) = v ¯ ( τ , κ ) − α 2 v ¯ κ κ ( τ , κ ) , then v ¯ ( τ , κ ) = ℘ ∗ w , where κ ∈ ℝ . Furthermore, we have ( ρ ∗ w ) κ κ = α − 2 ( ρ ∗ w − w ) , v ¯ κ = ( ℘ ∗ w ) κ and v ¯ κ κ κ = α − 2 ( ( ℘ ∗ w ) κ − w κ ) . Then (2.3) can be rewritten as

w τ + 1 3 w − e − τ [ 1 3 ( c 0 + e τ κ + e τ ) ] w κ − 1 3 ℘ ∗ w + { e − τ [ 1 3 ( c 0 + e τ κ + e τ ) ] − ( κ + 1 ) } ( ℘ ∗ w ) κ + 3 ( ℘ ∗ w ) ( ℘ ∗ w ) κ = 2 ( ℘ ∗ w ) κ ( ℘ ∗ w − w ) + ( ℘ ∗ w ) [ ( ℘ ∗ w ) κ − w κ ] (2.5)

with the initial data

w 0 ( κ ) = u 0 ( x ) − α 2 u ″ 0 ( x ) + 1 3 ( x T + 1 T + c 0 ) , (2.6)

and the boundary condition

lim | κ | → + ∞ w ( τ , κ ) = 0,   lim | κ | → + ∞ w κ ( τ , κ ) = 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

‖ [ A s , u ] v ‖ L 2 ≤ C ( ‖ ∂ x u ‖ L ∞ ‖ A s − 1 v ‖ L 2 + ‖ A s u ‖ L 2 ‖ v ‖ L ∞ ) , (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 α ≠ 0 . Assume that w be a solution of (2.5), then

‖ w ‖ ℍ s ≤ 1 ‖ w 0 ‖ ℍ s − 1 − C τ , (2.9)

where C is a positive constant depending upon s.

Proof. Applying A s to both sides of (2.5) and taking the L 2 -inner product with A s w , we get

1 2 d d τ ‖ w ‖ ℍ s 2 + 1 3 ‖ w ‖ ℍ s 2 − 1 3 ∫ ℝ     A s w A s ( ℘ ∗ w ) d κ   − e − τ ∫ ℝ A s w A s [ ( 1 3 ( c 0 + e τ κ + e τ ) ) w κ ] d κ   + ∫ ℝ     A s w A s { [ e − τ ( 1 3 ( c 0 + e τ κ + e τ ) ) − ( κ + 1 ) ] ( ℘ ∗ w ) κ } d κ   + 3 ∫ ℝ     A s w A s [ ( ℘ ∗ w ) ( ℘ ∗ w ) κ ] d κ = 2 ∫ ℝ     A s w A s [ ( ℘ ∗ w ) κ ( ℘ ∗ w − w ) ] d κ   + ∫ ℝ     A s w A s [ ( ℘ ∗ w ) ( ( ℘ ∗ w ) κ − w κ ) ] d κ . (2.10)

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

− 1 3 ∫ ℝ     A s w A s ( ℘ ∗ w ) d κ = − 1 3 ‖ w ‖ ℍ s − 1 2 , (2.11)

− e − τ ∫ ℝ     A s w A s [ ( 1 3 ( c 0 + e τ κ + e τ ) ) w κ ] d κ = e − τ ∫ ℝ [ ( 1 3 ( c 0 + e τ κ + e τ ) ) A 2 s w ] κ w d κ = 1 3 ∫ ℝ     A 2 s w ⋅ w d κ − 1 2 × 1 3 ∫ ℝ ( A s w ) 2 d κ = 1 6 ‖ w ‖ ℍ s 2 , (2.12)

∫ ℝ     A s w A s { [ e − τ ( 1 3 ( c 0 + e τ κ + e τ ) ) − ( κ + 1 ) ] ( ℘ ∗ w ) κ } d κ = − ∫ ℝ { A 2 s w [ e − τ ( 1 3 ( c 0 + e τ κ + e τ ) ) − ( κ + 1 ) ] } κ ( ℘ ∗ w ) d κ = − ∫ ℝ     A 2 s w κ [ e − τ ( 1 3 ( c 0 + e τ κ + e τ ) ) − ( κ + 1 ) ] ( ℘ ∗ w ) d κ         − ( 1 3 − 1 ) ∫ ℝ     A 2 s w ( ℘ ∗ w ) d κ = 1 2 × ( 1 3 − 1 ) ∫ ℝ ( A s − 1 w ) 2 d κ + 2 3 ∫ ℝ     A 2 s w ( ℘ ∗ w ) d κ = 1 3 ‖ w ‖ ℍ s − 1 2 , (2.13)

3 ∫ ℝ     A s w A s ( ( ℘ ∗ w ) ( ℘ ∗ w ) κ ) d κ = − 3 2 ∫ ℝ     w κ A 2 s ( ( ℘ ∗ w ) 2 ) d κ ≤ 3 2 ‖ w κ ‖ L ∞ ‖ w ‖ ℍ s − 1 2 ≤ 3 2 ‖ w ‖ ℍ s − 1 3 , (2.14)

In addition, using (2.8), we have

2 | ∫ ℝ     A s w A s ( ( ℘ ∗ w ) κ ( ℘ ∗ w − w ) ) d κ | = 2 | ∫ ℝ [ A s , ( ℘ ∗ w − w ) ] ( ℘ ∗ w ) κ A s w d κ |     + 2 | ∫ ℝ ( ℘ ∗ w − w ) A s ( ℘ ∗ w ) κ A s w d κ | ≤ C ( ‖ ℘ ∗ w − w ‖ L ∞ ‖ A s − 1 ( ℘ ∗ w ) κ ‖ L 2 + ‖ A s ( ℘ ∗ w − w ) ‖ L 2 ‖ ( ℘ ∗ w ) κ ‖ L ∞ ) ‖ w ‖ ℍ s     + 2 ( ‖ ( ℘ ∗ w − w ) ‖ L ∞ + ‖ ( ℘ ∗ w − w ) κ ‖ L ∞ ) ‖ w ‖ ℍ 2 2 ≤ C ‖ w ‖ ℍ s 3 , (2.15)

similarly,

| ∫ ℝ     A s w A s [ ( ℘ ∗ w ) ( ( ℘ ∗ w ) κ − w κ ) ] d κ | ≤ C ‖ w ‖ ℍ s 3 , (2.16)

where C is a positive constant depending upon s.

Substituting (2.11)-(2.16) into (2.10), we get 1 2 d d τ ‖ w ‖ ℍ s 2 ≤ C ‖ w ‖ ℍ s 3 , and then − d d τ ‖ w ‖ ℍ s − 1 ≤ C . Integrating this inequality above with respect to τ from 0 to τ , we get

‖ w ‖ ℍ s ≤ 1 ‖ w 0 ‖ ℍ s − 1 − C τ . (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

L [ w ] = − 1 3 w + 1 3 ℘ ∗ w + e − τ [ 1 3 ( c 0 + e τ κ + e τ ) ] w κ       − [ e − τ ( 1 3 ( c 0 + e τ κ + e τ ) ) − ( κ + 1 ) ] ( ℘ ∗ w ) κ , (2.18)

then (2.5) becomes to

w t = L [ w ] + f ( w ) , (2.19)

where f is the nonlinear terms:

f ( w ) = − 3 ( ℘ ∗ w ) ( ℘ ∗ w ) κ + 2 ( ℘ ∗ w ) ( ℘ ∗ w − w )       + ( ℘ ∗ w ) [ ( ℘ ∗ w ) κ − w κ ] . (2.20)

Lemma 2.3. Let s > 2 . Then

· L [ w ] ∈ ℍ s for ∀ w ∈ D ( L ) .

· 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., ( L [ w ] , w ) s ≤ 0 .

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

∫ ℝ ( A s L [ w ] ) A s w d κ = − 1 3 ‖ w ‖ ℍ s 2 + 1 3 ‖ w ‖ ℍ s − 1 2 − 1 6 ‖ w ‖ ℍ s 2 − 1 3 ‖ w ‖ ℍ s − 1 2 = − 1 2 ‖ w ‖ ℍ s 2 ≤ 0. (2.21)

This completes the proof. o

Lemma 2.5 (Young inequality with ε , see [22] ). Let a , b > 0 and ε > 0 . If p , q ∈ ( 1, ∞ ) satisfy 1 p + 1 q = 1 . Then

a   b ≤ ε   a p + C ( ε ) b q , (2.22)

where C ( ε ) = ( ε p ) − 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 ( S ( t ) ) τ ≥ 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 ∈ D ( L ) such that L [ w ] = 0 , then

∫ ℝ     A s L [ w ] A s w d κ = − 1 2 ‖ w ‖ 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 ∈ ℍ 1 , put

L [ w ] = g . (2.24)

Applying A s to (2.24) and multiplying the result by A s w , and then integrating over ℝ , we get

‖ w ‖ ℍ s 2 = − 2 ∫ ℝ     A s g A s w d κ . (2.25)

It follows from the Young inequality with ε in Lemma 2.5 that

‖ w ‖ ℍ s ≤ C ‖ g ‖ ℍ s . (2.26)

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

As a consequence, we have

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

( d d τ w = L w , w ( 0 ) = w 0 (2.27)

with zero boundary condition exists a unique solution w ( τ ) = S ( τ ) 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 ( τ ) = S ( τ ) w 0 + ∫ 0 τ     S ( τ − s ) f ( w ( s ) ) d s . (2.28)

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

B δ = { w ∈ ℍ s : ‖ w ‖ ℍ s < δ ≪ 1 } , (2.29)

and the map T as

T w ( τ ) = S ( τ ) w 0 + ∫ 0 τ     S ( τ − s ) f ( w ( s ) ) d s . (2.30)

We need to prove that T has a fixed point in the space B δ .

Lemma 2.8 ( [21] ). Let s > 2 . Then B δ is an algebra, and

‖ u v ‖ ℍ s ≤ C ( ‖ u ‖ L ∞ ‖ v ‖ ℍ s + ‖ u ‖ ℍ s ‖ v ‖ L ∞ ) , (2.31)

where C is a positive constant depending upon s.

Lemma 2.9. Let s > 2 be an integer. Assume that ‖ w 0 ‖ ℍ s + 1 < δ for some sufficiently small δ > 0 . Then T is a self-mapping on B δ . Moreover, T is a contraction mapping.

Proof. By Lemma 2.8, we have

‖ f ( w ) ‖ ℍ s ≤ 3 ‖ ( ℘ ∗ w ) ( ℘ ∗ w ) κ ‖ ℍ s + 2 ‖ ( ℘ ∗ w ) κ ( ℘ ∗ w − w ) ‖ ℍ s   + ‖ ( ℘ ∗ w ) ( ( ℘ ∗ w ) κ − w κ ) ‖ ℍ s ≤ C 1 ( ‖ ℘ ∗ w ‖ ℍ s ‖ ( ℘ ∗ w ) κ ‖ L ∞ + ‖ ( ℘ ∗ w ) κ ‖ L ∞ ‖ ( ℘ ∗ w − w ) ‖ ℍ s   + ‖ ℘ ∗ w ‖ ℍ s ‖ ( ℘ ∗ w ) κ − w κ ‖ L ∞ ) , (2.32)

where C 1 is a positive constant.

Note that ℍ s ⊂ L ∞ and w = A 2 ( ℘ ( κ ) ∗ v ¯ ) , then using Lemma 2.2, we have

‖ f ( w ) ‖ ℍ s ≤ C 1 ‖ w ‖ ℍ s 2 < C 1 ‖ w ‖ ℍ s − 1 − C τ < C 1 δ − 1 − C τ < δ (2.33)

for sufficiently small δ . Thus, T is a self-mapping on B δ .

To show T is a contraction mapping, we choose w , w ¯ ∈ B δ , by Lemma 2.8 and a direct calculation show that

‖ f ( w ) − f ( w ¯ ) ‖ ℍ s = ‖ − 3 ( ℘ ∗ w ) ( ℘ ∗ w ) κ + 2 ( ℘ ∗ w ) κ ( ℘ ∗ w )       − 2 ( ℘ ∗ w ) κ w + ( ℘ ∗ w ) ( ( ℘ ∗ w ) κ − w κ )       + 3 ( ℘ ∗ w ¯ ) ( ℘ ∗ w ¯ ) κ − 2 ( ℘ ∗ w ¯ ) κ ( ℘ ∗ w ¯ )       + 2 ( ℘ ∗ w ¯ ) κ w ¯ − ( ℘ ∗ w ¯ ) ( ( ℘ ∗ w ¯ ) κ − w ¯ κ ) ‖ ℍ s

≤ ‖ 3 { ( ℘ ∗ w ¯ ) [ ℘ ∗ ( w ¯ − w ) ] κ + ( ℘ ∗ w ¯ ) κ [ ℘ ∗ ( w ¯ − w ) ] }       + 2 { ( ℘ ∗ w ) κ [ ℘ ∗ ( w ¯ − w ) ] + ( ℘ ∗ w ¯ ) [ ℘ ∗ ( w ¯ − w ) ] κ }       + 3 { ( ℘ ∗ w ¯ ) [ ℘ ∗ ( w ¯ − w ) ] κ + ( ℘ ∗ w ¯ ) κ [ ℘ ∗ ( w ¯ − w ) ] }       + { ( ℘ ∗ w ¯ ) [ ℘ ∗ ( w ¯ − w ) ] κ + ( ℘ ∗ w ¯ ) κ [ ℘ ∗ ( w ¯ − w ) ] }       + { [ ℘ ∗ ( w ¯ − w ) ] w ¯ κ + ( ℘ ∗ w ) ( w ¯ − w ) κ } ‖ ℍ s ≤ C δ ‖ w − w ¯ ‖ ℍ s . (2.34)

Thus,

‖ T w ( τ ) − T w ¯ ( τ ) ‖ ℍ s ≤ C δ ‖ w − w ¯ ‖ ℍ s . (2.35)

Since δ > 0 is sufficiently small, T is a contraction mapping. o

Thus, we have the following existence results.

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

· if ‖ w 0 ‖ ℍ s + 1 < δ , there exists a unique solution w ∈ B δ to (2.5) with the initial data (2.6) and the boundary condition (2.7).

· there exists a global solution ψ ( τ , ρ ) ∈ ℍ s to (2.3) with the initial data (2.6) and the boundary condition (2.7). Moreover, if the initial data ψ 0 satisfies ‖ ψ 0 ‖ ℍ s + 1 < δ , then

‖ ψ ‖ ℍ s ≤ C ˜ α 2 e τ ( 1 − C τ ) . (2.36)

Here C and C ˜ are two positive constants that depend on s.

Proof. By Lemma 2.9 and the Banach fixed point theorem, the map T has a fixed point in B δ , which is a solution of Equation (2.5). Thus, there exists a global solution of (2.3) as

ψ ( τ , ρ ) = e τ v ¯ ( τ , e − τ ρ ) = e τ ( ( ℘ ∗ w ) ( τ , e − τ ρ ) ) . (2.37)

Furthermore, we have

v ρ ρ = ψ ρ ρ = ( ℘ ∗ w ) κ κ e − τ = α − 2 e − τ ( ℘ ∗ w − w ) . (2.38)

Thus, by Lemma 2.2, we get

‖ ψ ρ ρ ‖ ℍ s − 2 ≤ α − 2 e − τ ‖ ℘ ∗ w − w ‖ ℍ s − 2 ≤ C ˜ α − 2 e − τ ‖ w ‖ ℍ s − 2 ≤ C ˜ α 2 e τ ( ‖ w 0 ‖ ℍ s − 1 − C τ ) ≤ C ˜ α 2 e τ ( 1 − C τ ) , (2.39)

where we have used δ < 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.

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.

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

Gao, Y.T. (2021) Asymptotic Stability of Singular Solution for Camassa-Holm Equation. Journal of Applied Mathematics and Physics, 9, 1505-1514. https://doi.org/10.4236/jamp.2021.97102