The Well-Posedness of Solutions for a Generalized Shallow Water Wave Equation

and Applied Analysis 3 where ĥ t, ξ ∫ R e −ixξh t, x dx. Here, we note that the norms ‖ · ‖pLp , ‖ · ‖L∞ , and ‖ · ‖Hs depend on variable t. For T > 0 and nonnegative number s, letC 0, T ;H R denote the space of functions u : 0, T ×R → Rwith the properties that u t, · ∈ H R for each t ∈ 0, T , and the mapping u : 0, T → H R is continuous and bounded. For simplicity, throughout this paper, we let c denote any positive constant which is independent of parameter ε and set Λ 1 − ∂x . In order to study the existence of solutions for 1.5 , we consider its Cauchy problem in the form ut − utxx −∂x ( 2ku m 2 u2 ) auxuxx buuxxx βux, u 0, x u0 x , 2.2 where b > 0, a, k,m, β, and n are arbitrary constants. Now, we give the theorem to describe the local well-posedness of solutions for problem 2.2 . Theorem 2.1. Let u0 x ∈ H R with s > 3/2, then the Cauchy problem 2.2 has a unique solution u t, x ∈ C 0, T ;H R C1 0, T ;Hs−1 R where T > 0 depends on ‖u0‖Hs R . For a real number swith s > 0, suppose that the function u0 x is inH R , and let uε0 be the convolution uε0 φε u0 of the function φε x ε−1/4φ ε−1/4x and u0 such that the Fourier transform φ̂ of φ satisfies φ̂ ∈ C∞ 0 , φ̂ ξ ≥ 0, and φ̂ ξ 1 for any ξ ∈ −1, 1 . Thus one has uε0 x ∈ C∞. It follows from Theorem 2.1 that for each ε satisfying 0 < ε < 1/4, the Cauchy problem ut − utxx −∂x ( 2ku m 2 u2 ) auxuxx buuxxx βux, u 0, x uε0 x , x ∈ R 2.3 has a unique solution uε t, x ∈ C∞ 0, Tε ;H∞ , in which Tε may depend on ε. However, one will show that under certain assumptions, there exist two constants c and T > 0, both independent of ε, such that the solution of problem 2.3 satisfies ‖uεx‖L∞ ≤ c for any t ∈ 0, T , and there exists a weak solution u t, x ∈ L2 0, T ,H for problem 2.2 . These results are summarized in the following two theorems. Theorem 2.2. If u0 x ∈ H R with s ∈ 1, 3/2 such that ‖u0x‖L∞ < ∞, let uε0 be defined as in system 2.3 , then there exist two constants c and T > 0, which are independent of ε, such that the solution uε of problem 2.3 satisfies ‖uεx‖L∞ ≤ c for any t ∈ 0, T . Theorem 2.3. Suppose that u0 x ∈ H with 1 ≤ s ≤ 3/2 and ‖u0x‖L∞ < ∞, then there exists a T > 0 such that problem 2.2 has a weak solution u t, x ∈ L2 0, T ,H R in the sense of distribution and ux ∈ L∞ 0, T × R . 4 Abstract and Applied Analysis 3. Proof of Theorem 2.1 Consider the abstract quasilinear evolution equation dv dt A v v f v , t ≥ 0, v 0 v0. 3.1 Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and let Q : Y → X be a topological isomorphism. Let L Y,X be the space of all bounded linear operators from Y to X. If X Y , we denote this space by L X . We state the following conditions in which ρ1, ρ2, ρ3, and ρ4 are constants depending only on max{‖y‖Y , ‖z‖Y}: I A y ∈ L Y,X for y ∈ X with ∥ ∥A ( y ) −A z w∥∥X ≤ ρ1 ∥ ∥y − z∥∥X‖w‖Y , y, z, w ∈ Y, 3.2 and A y ∈ G X, 1, β i.e., A y is quasi-m-accretive , uniformly on bounded sets in Y . II QA y Q−1 A y B y , where B y ∈ L X is bounded, uniformly on bounded sets in Y . Moreover, ∥B ( y ) − B z w∥∥X ≤ ρ2 ∥y − z∥∥Y‖w‖X, y, z ∈ Y, w ∈ X. 3.3 III f : Y → Y extends to a map from X into X, is bounded on bounded sets in Y , and satisfies ∥f ( y ) − f z ∥∥Y ≤ ρ3 ∥y − z∥∥Y , y, z ∈ Y, ∥f ( y ) − f z ∥∥X ≤ ρ4 ∥y − z∥∥X, y, z ∈ Y. 3.4 Kato Theorem (see [23]) Assume that I , II , and III hold. If v0 ∈ Y , there is a maximal T > 0 depending only on ‖v0‖Y and a unique solution v to problem 3.1 such that v v ·, v0 ∈ C 0, T ;Y ⋂ C1 0, T ;X . 3.5 Moreover, the map v0 → v ·, v0 is a continuous map from Y to the space C 0, T ;Y ⋂ C1 0, T ;X . 3.6 In fact, problem 2.2 can be written as ut − utxx − [ 2ku m 2 u2 ] x b 2 ∂x u 2 − 3b − a 2 ∂x ( ux ) βux, u 0, x u0 x , 3.7 Abstract and Applied Analysis 5 which is equivalent toand Applied Analysis 5 which is equivalent to ut buux −Λ−2 [( 2ku m 2 u2 ) x buux − 3b − a 2 ∂x ( ux ) βux ] , u 0, x u0 x . 3.8 We set A u bu∂x with constant b > 0, Y H R , X Hs−1 R , Λ 1 − ∂x , f u −Λ−2 2ku m/2 u2 x bΛ−2 uux − 3b − a /2 Λ−2∂x ux βΛux, and Q Λ. We know that Q is an isomorphism of H onto Hs−1. In order to prove Theorem 2.1, we only need to check that A u and f u satisfy assumptions I – III . Lemma 3.1. The operator A u u∂x with u ∈ H R , s > 3/2 belongs to G Hs−1, 1, β . Lemma 3.2. Let A u bu∂x with u ∈ H and s > 3/2, then A u ∈ L Hs,Hs−1 for all u ∈ H. Moreover, ‖ A u −A z w‖Hs−1 ≤ ρ1‖u − z‖Hs−1‖w‖Hs , u, z,w ∈ H R . 3.9 Lemma 3.3. For s > 3/2, u, z ∈ H, and w ∈ Hs−1, it holds that B u Λ, u∂x Λ−1 ∈ L Hs−1 for u ∈ H and ‖ B u − B z w‖Hs−1 ≤ ρ2‖u − z‖Hs‖w‖Hs−1 . 3.10 Proofs of the above Lemmas 3.1–3.3 can be found in 24 or 25 . Lemma 3.4 see 23 . Let r and q be real numbers such that −r < q ≤ r, then ‖uv‖Hq ≤ c‖u‖Hr‖v‖Hq , if r > 1 2 , ‖uv‖Hr q−1/2 ≤ c‖u‖Hr‖v‖Hq , if r < 1 2 . 3.11 Lemma 3.5. Let u, z ∈ H with s > 3/2 and f u −Λ−2 2ku m/2 u2 x bΛ−2 uux − 3b − a /2 Λ−2∂x ux βΛ −2 ux , then f is bounded inH s and satisfies ∥f u − f z ∥∥Hs ≤ ρ3‖u − z‖Hs, ∥f u − f z ∥∥Hs−1 ≤ ρ4‖u − z‖Hs−1 . 3.12 6 Abstract and Applied Analysis Proof. Using the algebra property of the space H0 with s0 > 1/2 and s − 1 > 1/2, we have ∥ ∥f u − f z ∥∥Hs ≤ ∥ ∥ ∥ ∥Λ −2 (( 2ku m 2 u2 ) x − ( 2kz m 2 z2 ) x )∥ ∥ ∥ Hs ∥ ∥∥bΛ−2 uux − zzx ∥ ∥∥ Hs ∥ ∥ ∥ ∥ 3b − a 2 Λ−2∂x ( ux − zx )∥ ∥ ∥ Hs ∥∥ ∥βΛ−2 ux − zx ∥∥ ∥ Hs ≤ c ⎛ ⎝‖u − z‖Hs−1 1 ‖u‖Hs−1 ‖z‖Hs−1 ∥ ∥ ∥Λ−2 ( u2 − z2 ) x ∥ ∥ ∥ Hs ∥ ∥ ∥ux − zx ∥ ∥ ∥ Hs−1 ‖ux − zx‖Hs−1 n−1 ∑ j 0 ‖ux‖ Hs−1‖zx‖ j Hs−1 ⎞ ⎠ ≤ c ⎛ ⎝‖u − z‖Hs 1 ‖u‖Hs ‖z‖Hs ‖ u − z u z ‖Hs−1 ∥∥∥u2x − zx ∥∥ Hs−1 ‖ux − zx‖Hs−1 n−1 ∑ j 0 ‖u‖n−j Hs ‖z‖ j Hs ⎞ ⎠ ≤ c‖u − z‖Hs ⎛ ⎝1 ‖u‖Hs ‖z‖Hs n−1 ∑ j 0 ‖u‖n−j Hs ‖z‖ j Hs ⎞ ⎠ ≤ cρ3‖u − z‖Hs, 3.13 from which we obtain 3.12 . Applying Lemma 3.4, uux 1/2 u2 x, s > 3/2, we get ∥f u − f z ∥∥Hs−1 ≤ c ∥∥ ∥2ku m 2 u2 − ( 2kz m 2 z2 ∥∥∥ Hs−2 ∥ ∥u − z2 ∥∥∥ Hs−2 ‖ ux − zx ux zx ‖Hs−2 ‖ux − zx‖Hs−1 ) ≤ c‖u − z‖Hs−2 1 ‖u‖Hs−1 ‖z‖Hs−1 c‖ux − zx‖Hs−2 ‖ux‖Hs−1 ‖zx‖Hs−1 c‖ux − zx‖Hs−2 n−1 ∑ j 0 ‖ux‖ Hs−1‖zx‖ j Hs−1 ≤ c‖u − z‖Hs−1 ⎛ ⎝1 ‖u‖Hs ‖z‖Hs n−1 ∑ j 0 ‖u‖n−j Hs ‖z‖ j Hs ⎞ ⎠, 3.14 which completes the proof of 3.12 . Abstract and Applied Analysis 7 Proof of Theorem 2.1. Using the Kato theorem, Lemmas 3.1, 3.2, 3.3, and 3.5, we know that system 3.11 or problem 2.2 has a unique solution u t, x ∈ C 0, T ;H R ⋂ C1 ( 0, T ;Hs−1 ) . 3.15and Applied Analysis 7 Proof of Theorem 2.1. Using the Kato theorem, Lemmas 3.1, 3.2, 3.3, and 3.5, we know that system 3.11 or problem 2.2 has a unique solution u t, x ∈ C 0, T ;H R ⋂ C1 ( 0, T ;Hs−1 ) . 3.15 4. Proofs of Theorems 2.2 and 2.3 Before establishing the proofs of Theorems 2.2 and 2.3, we give several lemmas. Lemma 4.1 Kato and Ponce 26 . If r ≥ 0, thenH L∞ is an algebra. Moreover, ‖uv‖Hr ≤ c ‖u‖L∞‖v‖Hr ‖u‖Hr‖v‖L∞ , 4.1 where c is a constant depending only on r. Lemma 4.2 Kato and Ponce 26 . Let r > 0. If u ∈ H W1,∞ and v ∈ Hr−1L∞, then ‖ Λ , u v‖L2 ≤ c ( ‖∂xu‖L∞ ∥∥∥Λr−1v ∥∥∥ L2 ‖Λu‖L2‖v‖L∞ ) , 4.2 where Λ , u v Λ uv − uΛv. Using the first equation of problem 2.2 gives rise to d dt [∫ R ( u2 ux ) dx ] a − 2b ∫


Introduction
Equations 1.3 and 1.4 are bi-Hamiltonian and arise in the modeling of shallow water waves.These two equations pertain to waves of medium amplitude cf. the discussions in 1, 4 and accommodate wave-breaking phenomena.Moreover, the Camassa-Holm and Degasperis-Procesi models admit peaked periodic as well as solitary traveling waves capturing the main feature of the exact traveling wave solutions of the greatest height of the governing equations for water waves cf. 5, 6 .For other dynamic properties about 1.3 and 1.4 , the reader is referred to 7-20 .
Recently, Lai and Wu 21 investigate 1.2 in the case where k 0, m a b, a > 0, and b > 0. The well-posedness of global solutions is established in 21 in Sobolev space H s R with s > 3/2 under certain assumptions on the initial value.The local strong and weak solutions for 1.2 are discussed in 22 in the case where b > 0, a, k, and m are arbitrary constants.
Motivated by the desire to extend the work in 22 , we investigate the following generalized model of 1.2 : where b > 0, a, k, m, and β are arbitrary constants, and n is a positive integer.The aim of this paper is to investigate 1.5 .Since a, b, m, and β are arbitrary constants, we do not have the result that the H 1 norm of the solution of 1.5 remains constant.We will apply the Kato theorem 23 to prove the existence and uniqueness of local solutions for 1.5 in the space C 0, T , H s R C 1 0, T , H s−1 R s > 3/2 provided that the initial value u 0 x belongs to H s R s > 3/2 .Moreover, it is shown that there exists a weak solution of 1.5 in lower-order Sobolev space The structure of this paper is as follows.The main results are given in Section 2. The existence and uniqueness of the local strong solution for the Cauchy problem 1.5 are proved in Section 3. The existence of weak solutions is established in Section 4.

Main Results
Firstly, we give some notations.
The space of all infinitely differentiable functions φ t, x with compact support in 0, ∞ × R is denoted by C ∞ 0 .We let L p L p R 1 ≤ p < ∞ be the space of all measurable functions h such that For any real number s, we let H s H s R denote the Sobolev space with the norm defined by For T > 0 and nonnegative number s, let C 0, T ; H s R denote the space of functions u : 0, T × R → R with the properties that u t, • ∈ H s R for each t ∈ 0, T , and the mapping u : 0, T → H s R is continuous and bounded.
For simplicity, throughout this paper, we let c denote any positive constant which is independent of parameter ε and set In order to study the existence of solutions for 1.
For a real number s with s > 0, suppose that the function u 0 x is in H s R , and let u ε0 be the convolution u ε0 φ ε u 0 of the function φ ε x ε −1/4 φ ε −1/4 x and u 0 such that the Fourier transform φ of φ satisfies φ ∈ C ∞ 0 , φ ξ ≥ 0, and φ ξ 1 for any ξ ∈ −1, 1 .Thus one has u ε0 x ∈ C ∞ .It follows from Theorem 2.1 that for each ε satisfying 0 < ε < 1/4, the Cauchy problem in which T ε may depend on ε.However, one will show that under certain assumptions, there exist two constants c and T > 0, both independent of ε, such that the solution of problem 2.3 satisfies u εx L ∞ ≤ c for any t ∈ 0, T , and there exists a weak solution u t, x ∈ L 2 0, T , H s for problem 2.2 .These results are summarized in the following two theorems.
let u ε0 be defined as in system 2.3 , then there exist two constants c and T > 0, which are independent of ε, such that the solution u ε of problem 2.3 satisfies u εx L ∞ ≤ c for any t ∈ 0, T .
Theorem 2.3.Suppose that u 0 x ∈ H s with 1 ≤ s ≤ 3/2 and u 0x L ∞ < ∞, then there exists a T > 0 such that problem 2.2 has a weak solution u t, x ∈ L 2 0, T , H s R in the sense of distribution and u x ∈ L ∞ 0, T × R .

Proof of Theorem 2.1
Consider the abstract quasilinear evolution equation Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and let Q : Y → X be a topological isomorphism.Let L Y, X be the space of all bounded linear operators from Y to X.If X Y , we denote this space by L X .We state the following conditions in which ρ 1 , ρ 2 , ρ 3 , and ρ 4 are constants depending only on max{ y Y , z Y }: III f : Y → Y extends to a map from X into X, is bounded on bounded sets in Y , and satisfies

3.4
Kato Theorem (see [23]) Assume that I , II , and III hold.If v 0 ∈ Y , there is a maximal T > 0 depending only on v 0 Y and a unique solution v to problem 3.1 such that In fact, problem 2.2 can be written as which is equivalent to

3.8
We set x , and Q Λ s .We know that Q is an isomorphism of H s onto H s−1 .In order to prove Theorem 2.1, we only need to check that A u and f u satisfy assumptions I -III . 3.10 Proofs of the above Lemmas 3.1-3.3can be found in 24 or 25 .
Lemma 3.4 see 23 .Let r and q be real numbers such that −r < q ≤ r, then x , then f is bounded in H s and satisfies

3.12
Proof.Using the algebra property of the space H s 0 with s 0 > 1/2 and s − 1 > 1/2, we have 3.13 from which we obtain 3.12 .Applying Lemma 3.4, uu x 1/2 u 2 x , s > 3/2, we get which completes the proof of 3.12 .

Proof of Theorem
where c is a constant depending only on r.
Using the first equation of problem 2.2 gives rise to Lemma 4.3.Let s ≥ 3/2, and the function u t, x is a solution of the problem 2.2 and the initial data where c 0 1/2 max |a − 2b|, |β| .For q ∈ 0, s − 1 , there is a constant c depending only on q such that R If q ∈ 0, s − 1 , there is a constant c depending only on q such that x dx and 4.4 derives 4.5 .We write 1.5 in the equivalent form 4.8 Applying ∂ 2 x −Λ 2 1 and the Parseval's equality gives rise to For q ∈ 0, s − 1 , applying Λ q u Λ q on both sides of 4.8 , noting the above equality, and integrating the new equation with respect to x by parts, we obtain the equation 1 2 Λ q uΛ q u n x dx.

4.10
We will estimate each of the terms on the right-hand side of 4.10 .For the first and the fourth terms, using integration by parts, the Cauchy-Schwartz inequality, and Lemmas 4.1-4.2,we have where c only depends on q.Using the above estimate to the second term yields Abstract and Applied Analysis 9 For the third term, using Lemma 4.1 gives rise to

4.13
For the last term, using Lemma 4.1 repeatedly, we get It follows from 4.10 -4.14 that which results in 4.6 .Applying the operator 1 − ∂ 2 x −1 on both sides of 4.8 yields the equation

4.16
Multiplying both sides of 4.16 by Λ q u t Λ q for q ∈ 0, s − 1 and integrating the resultant equation by parts give rise to

4.17
On the right-hand side of 4.17 , we have in which we have used Lemma 4.1.As

4.22
Using the Cauchy-Schwartz inequality and Lemma 4.1 yields Applying 4.18 -4.23 to 4.17 yields the inequality for a constant c > 0.
The proof of this lemma can be found in 21 .Applying Lemmas 4.3 and 4.4, we can now state the following lemma, which plays an important role in proving existence of weak solutions.Lemma 4.5.For s ≥ 1 and u 0 ∈ H s R , there exists a constant c independent of ε, such that the solution u ε of problem 2.3 satisfies where c 0 1/2 max |a − 2b|, |β| .
Proof.The proof can be directly obtained from Lemma 4.4 and inequality 4.5 .
Proof of Theorem 2.2.Using notation u u ε and differentiating 4.16 with respect to x give rise to

4.29
Letting p > 0 be an integer and multiplying 4.27 by u x 2p 1 and then integrating the resulting equation with respect to x yield the equality

4.30
Applying the H ölder's inequality, we get where integrating 4.32 with respect to t and taking the limit as p → ∞ result in the estimate

4.38
It follows from the contraction mapping principle that there is a T > 0 such that the equation has a unique solution W ∈ C 0, T .From 4.39 , we know that the variable T only depends on c and u 0x L ∞ .Using the theorem presented on page 51 in 16 or Theorem 2 in Section 1.1 in 27 derives that there are constants T > 0 and c > 0 independent of ε such that u εx L ∞ ≤ W t for arbitrary t ∈ 0, T , which leads to the conclusion of Theorem 2.
Remark 4.6.Under the assumptions of Theorem 2.2, there exist two constants T and c, both independent of ε, such that the solution u ε of problem 2.3 satisfies u εx L ∞ ≤ c for any t ∈ 0, T .This states that in Lemma 4.5, there exists a T independent of ε such that 4.26 holds.Using Theorem 2.2, Lemma 4.5, 4.6 , 4.7 , notation u ε u, and Gronwall's inequality results in the inequalities

4.40
where q ∈ 0, s , r ∈ 0, s − 1 , and t ∈ 0, T .It follows from Aubin's compactness theo-rem that there is a subsequence of {u ε }, denoted by {u ε n 1 }, such that {u ε n 1 } and their temporal derivatives {u ε n 1 t } are weakly convergent to a function u t, x and its derivative u t in L 2 0, T , H s and L 2 0, T , H s−1 , respectively.Moreover, for any real number R 1 > 0, {u ε n 1 } is convergent to the function u strongly in the space L 2 0, T , H q −R 1 , R 1 for q ∈ 0, s , and {u ε n 1 t } converges to u t strongly in the space L 2 0, T , H r −R 1 , R 1 for r ∈ 0, s − 1 .
Proof of Theorem 2.3.From Theorem 2.2, we know that {u ε n 1 x } ε n 1 → 0 is bounded in the space L ∞ .Thus, the sequences{u ε n 1 }, {u ε n 1 x }, {u 2 ε n 1 x }, and {u n ε n 1 x } are weakly convergent to u, u x , u 2 x , and u n x in L 2 0, T , H r −R 1 , R 1 for any r ∈ 0, s − 1 , separately.Hence, u satisfies the equation with u 0, x u 0 x and g ∈ C ∞ 0 .Since X L 1 0, T × R is a separable Banach space and {u ε n 1 x } is a bounded sequence in the dual space X * L ∞ 0, T × R of X, there exists a subsequence of {u ε n 1 x }, still denoted by {u ε n 1 x }, weakly star convergent to a function v in L ∞ 0, T × R .As {u ε n 1 x } weakly converges to u x in L 2 0, T × R , it results that u x v almost everywhere.Thus, we obtain u x ∈ L ∞ 0, T × R .

Lemma 4 . 2
Kato and Ponce 26 .Let r > 0 and μ satisfy certain conditions.Under several restrictions on the coefficients of model 1.1 , the large time well-posedness was established on a time scale O |ρ| −1 provided that the initial value u 0 belongs to H s R with s > 5/2, and the wave-breaking phenomena were also discussed in 1 .As stated in 1 , using suitable mathematical transformations, one can turn 1.1 into the formu t −u txx 2ku x muu x au x u xx buu xxx , 1.2 where a, b, k, and m are constants.Obviously, 1.2 is a generalization of both the Camassa-Holm equation 2 u t − u txx 2ku x 3uu x 2u x u xx uu xxx , k is a constant 1.3 and the Degasperis-Procesi model 3 u t − u xxt 2ku x 4uu x 3u x u xx uu xxx .
, • L ∞ , and • H s depend on variable t.
R e −ixξ h t, x dx.Here, we note that the norms • p L p 2.1.Using the Kato theorem, Lemmas 3.1, 3.2, 3.3, and 3.5, we know that system 3.11 or problem 2.2 has a unique solution Before establishing the proofs of Theorems 2.2 and 2.3, we give several lemmas.Kato and Ponce 26 .If r ≥ 0, then H r L ∞ is an algebra.Moreover,