On the Study of Local Solutions for a Generalized Camassa-Holm Equation

and Applied Analysis 3 Theorem 2.1. Suppose that the initial function u0 x belongs to the Sobolev space H R with s > 3/2 and λ is a constant. Then, there is a T > 0, which depends on ‖u0‖Hs , such that problem 2.2 has a unique solution u t, x satisfying u t, x ∈ C 0, T ;H R ⋂ C1 ( 0, T ;Hs−1 R ) . 2.3 3. Local Well-Posedness In order to prove Theorem 2.1, we consider the associated regularized problem ut − utxx εutxxxx − k m 1 ( u 1 ) x − m 3 m 2 ( u 2 ) x 1 m 2 ∂x ( u 2 ) − m 1 ∂x ( uux ) uuxuxx λ u − uxx , u 0, x u0 x , 3.1 where the parameter ε satisfies 0 < ε < 1/4. Lemma 3.1. For s ≥ 1 and f x ∈ H R and letting k1 > 0 be an integer such that k1 ≤ s − 1, f, f ′, . . . , f1 are uniformly continuous bounded functions that converge to 0 at x ±∞. The proof of Lemma 3.1 was stated on page 559 by Bona and Smith 18 . Lemma 3.2. If u t, x ∈ H s > 7/2 is a solution to problem 3.1 , it holds that


Introduction
In recent years, extensive research has been carried out worldwide to study highly nonlinear equations including the Camassa-Holm CH equation and its various generalizations 1-6 .It is shown in 7-9 that the inverse spectral or scattering approach is a powerful technique to handle the Camassa-Holm equation and analyze its dynamics.It is pointed out in 10-12 that the CH equation gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group, and this geometric illustration leads to a proof that the Least Action Principle holds.Li and Olver 13 established the local well-posedness to the CH model in the Sobolev space H s R with s > 3/2 and gave conditions on the initial data that lead to finite time blow-up of certain solutions.Constantin and Escher 14 proved that the blow-up occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time.Hakkaev and Kirchev 15 investigated a generalized form of the Camassa-Holm equation with high order nonlinear terms and obtained the orbit stability of the traveling wave solutions under certain assumptions.Lai and Wu 16 discussed a generalized Camassa-Holm model and acquired its local existence and uniqueness.Recently, Li et al. 17  where m ≥ 0 is a natural number and k ≥ 0. The authors in 17 assume that the initial value satisfies the sign condition and establish the global existence of solutions for 1.1 .
In this paper, we will study the following generalization of 1.1 : where m ≥ 0 is a natural number, k ≥ 0, and λ is a constant.The objective of this paper is to study the local well-posedness of 1.2 .Its local wellposedness of strong solutions in the Sobolev space H s R with s > 3/2 is investigated by using the pseudoparabolic regularization method.Comparing with the work by Li et al. 17 , 1.2 considered in this paper possesses a conservation law different to that in 17 see Lemma 3.2 in Section 3 .Also 1.2 contains a dissipative term λ u − u xx , which causes difficulty to establish its local and global existence in the Sobolev space.It should be mentioned that the existence and uniqueness of local strong solutions for the generalized nonlinear Camassa-Holm models like 1.2 have never been investigated in the literatures.
The organization of this work is as follows.The main result is given in Section 2. Section 3 establishes several lemmas, and the last section gives the proof of the main result.

Main Result
Firstly, we introduce several notations.
For any real number s, H s H s R denotes the Sobolev space with the norm defined by where h t, ξ R e −ixξ h t, x dx.For T > 0 and nonnegative number s, C 0, T ; H s R denotes the Frechet space of all continuous H s -valued functions on 0, T .We set Λ 1 − ∂ 2 x 1/2 .For simplicity, throughout this paper, we let c denote any positive constant that is independent of parameter ε.
We consider the Cauchy problem of 1.2

2.2
Now, we give our main results for problem of 2.2 .
Abstract and Applied Analysis 3 Theorem 2.1.Suppose that the initial function u 0 x belongs to the Sobolev space H s R with s > 3/2 and λ is a constant.Then, there is a T > 0, which depends on u 0 H s , such that problem 2.2 has a unique solution u t, x satisfying

Local Well-Posedness
In order to prove Theorem 2.1, we consider the associated regularized problem where the parameter ε satisfies 0 < ε < 1/4.
Lemma 3.1.For s ≥ 1 and f x ∈ H s R and letting k 1 > 0 be an integer such that k 1 ≤ s − 1, f, f , . . ., f k 1 are uniformly continuous bounded functions that converge to 0 at x ±∞.
The proof of Lemma 3.

3.3
Direct calculation and integration by parts give rise to 1 2 in which we have used 3.3 .From 3.4 , we obtain the conservation law 3.2 .
Lemma 3.3.Let s ≥ 7/2.The function u t, x is a solution of problem 3.1 and the initial value u 0 x ∈ H s .Then, the following inequality holds:

3.6
For q ∈ 0, s − 1 , there is a constant c independent of ε such that The proof of this lemma is similar to that of Lemma 3.5 in 17 .Here we omit it.
Lemma 3.4.Let r and q be real numbers such that −r < q ≤ r.Then,

3.8
This lemma can be found in 19 or 20 .
x −1 , we know that D : H s → H s 4 is a bounded linear operator.Applying the operator D on both sides of the first equation of system 3.1 and then integrating the resultant equation with respect to t over the interval 0, t , we get 3.9 Suppose that both u and v are in the closed ball B M 0 0 of radius M 0 about the zero function in C 0, T ; H s R and A is the operator in the right-hand side of 3.9 .For any fixed t ∈ 0, T , we obtain

3.10
where C 1 may depend on ε.The algebraic property of

3.11
Using the first inequality of Lemma 3.4 gives rise to where C may depend on ε.From 3.11 -3.12 , we obtain and C 2 is independent of 0 < t < T. Choosing T sufficiently small such that θ < 1, we know that A is a contraction.Similarly, it follows from 3.10 that 3.14 Choosing T sufficiently small such that θM 0 u 0 H s < M 0 , we deduce that A maps B M 0 0 to itself.It follows from the contraction-mapping principle that the mapping A has a unique fixed point u in B M 0 0 .It completes the proof.
From the above and Lemma 3.2, we have Therefore, which together with Lemma 3.3 completes the proof of the global existence.
Lemma 3.6.For s > 0, u 0 ∈ H s , it holds that where c is a constant independent of ε.
The proof of Lemma 3.6 can be found in 16 .

3.23
where c 0 is independent of ε and t.
Lemma 3.8.Suppose u 0 x ∈ H s R with s ≥ 1 such that u 0x L ∞ < ∞.Let u ε0 be defined as in system 3.17 .Then, there exist two positive constants T and c, which are independent of ε, such that the solution u ε of problem 3.17 satisfies u εx L ∞ ≤ c for any t ∈ 0, T .
Here we omit the proof of Lemma 3.8 since it is similar to Lemma 3.9 presented in 17 .
Lemma 3.9 see Li and Olver 13 .If u and f are functions in

3.24
Lemma 3.10 see Lai and Wu 16 .For u, v ∈ H s R with s > 3/2, w u − v, q > 1/2, and a natural number n, it holds that

3.25
Lemma 3.11 see Lai and Wu 16 .If 1/2 < q < min{1, s−1} and s > 3/2, then for any functions w, f defined on R, it holds that R Λ q wΛ q−2 wf x dx ≤ c w 2 H q f H q , 3.26 R Λ q wΛ q−2 w x f x x dx ≤ c w 2 H q f H s .

3.27
Lemma 3.12.For problem 3.17 , s > 3/2, and u 0 ∈ H s R , there exist two positive constants c and M, which are independent of ε, such that the following inequalities hold for any sufficiently small ε and t ∈ 0, T :

3.28
Slightly modifying the methods presented in 16 can complete the proof of Lemma 3.12.
Our next step is to demonstrate that u ε is a Cauchy sequence.Let u ε and u δ be solutions of problem 3.17 , corresponding to the parameters ε and δ, respectively, with 0 < ε < δ < 1/4, and let w u ε − u δ .Then, w satisfies the problem

3.30
Lemma 3.13.For s > 3/2, u 0 ∈ H s R , there exists T > 0 such that the solution u ε of 3.17 is a Cauchy sequence in C 0, T ; Proof.For q with 1/2 < q < min{1, s− 1}, multiplying both sides of 3.29 by Λ q wΛ q and then integrating with respect to x give rise to 1 2

3.31
It follows from the Schwarz inequality that

3.32
Using the first inequality in Lemma 3.9, we have

3.33
where For the last three terms in 3.32 , using Lemmas 3.4 and 3.12, 1/2 < q < min{1, s − 1}, s > 3/2, the algebra property of H s 0 with s 0 > 1/2, and 3.23 , we have 3.36 Using 3.26 , we derive that the inequality 3.37 holds for some constant c, where g m 1 Using the algebra property of H q with q > 1/2, q 1 < s and Lemma 3.11, we have g m H q 1 ≤ c for t ∈ 0, T .Then, it follows from 3.28 and 3.33 -3.37 that there is a constant c depending on T such that the estimate 3.38 holds for any t ∈ 0, T , where γ 1 if s ≥ 3 q and γ 1 s − q /4 if s < 3 q.Integrating 3.38 with respect to t, one obtains the estimate

3.39
Applying the Gronwall inequality and using 3.20 and 3.22 yield u H q ≤ cδ s−q /4 e ct δ γ e ct − 1 3.40 for any t ∈ 0, T .Multiplying both sides of 3.29 by Λ s wΛ s and integrating the resultant equation with respect to x, one obtains 1 2

3.41
From Lemma 3.12, we have

3.42
From Lemma 3.10, it holds that

3.43
Using the Cauchy-Schwartz inequality and the algebra property of H s 0 with s 0 > 1/2, for s > 3/2, we have

3.48
It follows from the Gronwall inequality and 3.48 that

3.49
where c 1 is independent of ε and δ.Then, 3.22 and the above inequality show that Next, we consider the convergence of the sequence {u εt }.Multiplying both sides of 3.29 by Λ s−1 w t Λ s−1 and integrating the resultant equation with respect to x, we obtain

3.51
It follows from inequalities 3.28 and the Schwartz inequality that there is a constant c depending on T and m such that

3.54
It follows from 3.40 and 3.50 that w t → 0 as ε, δ → 0 in the H s−1 norm.This implies that u ε is a Cauchy sequence in the spaces C 0, T ; H s R and C 0, T ; H s−1 R , respectively.The proof is completed.

Proof of the Main Result
We consider the problem

4.1
Letting u t, x be the limit of the sequence u ε and taking the limit in problem 4.1 as ε → 0, from Lemma 3.13, we know that u is a solution of the problem

4.3
For any 1/2 < q < min{1, s − 1}, applying the operator Λ q wΛ q to both sides of 4.3 and integrating the resultant equation with respect to x, we obtain the equality 1 2 4.4 By the similar estimates presented in Lemma 3.13, we have d dt w 2 H q ≤ c w 2 H q .4.5 Using the Gronwall inequality leads to the conclusion that w H q ≤ 0 × e ct 0 4.6 for t ∈ 0, T .This completes the proof.
investigated the generalized Camassa-Holm equationu t − u txx ku m u x m 3 u m 1 u x m 2 u m u x u xx u m 1 u xxx , 1.1

1 −∂ x u m 1
hence u is a solution of problem 4.2 in the sense of distribution.In particular, if s ≥ 4, u is also a classical solution.Let u and v be two solutions of 4.2 corresponding to the same initial value u 0 such that u, v ∈ C 0, T ; H s R .Then, w u−v satisfies the Cauchy problem∂ x w − ∂ x u m 1 − v m 1 ∂ x v u m u x u xx − v m v x v xx λw,w 0, x 0.
1 was stated on page 559 by Bona and Smith 18 .