The local strong and weak solutions to a generalized Novikov equation

A nonlinear partial differential equation, which includes the Novikov equation as a special case, is investigated. The well-posedness of local strong solutions for the equation in the Sobolev space Hs(R) with s>32 is established. Although the H1-norm of the solutions to the nonlinear model does not remain constant, the existence of its local weak solutions in the lower order Sobolev space Hs(R) with 1≤s≤32 is established under the assumptions u0∈Hs and ∥u0x∥L∞<∞. MSC:35Q35, 35Q51.


Introduction
The Novikov equation with cubic nonlinearities takes the form u t − u txx 4u 2 u x 3uu x u xx u 2 u xxx , 1.1 which was derived by Vladimir Novikov in a symmetry classification of nonlocal partial differential equations 1 . Using the perturbed symmetry approach, Novikov was able to isolate 1.1 and investigate its symmetries. A scalar Lax pair for it was discovered in 1, 2 and was shown to be related to a negative flow in the Sawada-Kotera hierarchy. Many conserved quantities were found as well as a bi-Hamiltonian structure. The scattering theory was employed by Hone et al. 3 to find nonsmooth explicit soliton solutions with multiple peaks for 1.1 . This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations see 4-10 . Ni and Zhou 11 proved that the Novikov equation associated with initial value is locally well-posedness in Sobolev space H s with s > 3/2 by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for 1.1 were established. It is shown in 12 that the local well-posedness for the periodic Cauchy problem of the Novikov equation in Sobolev space H s with s > 5/2 .
The orbit invariants are employed to get the existence of periodic global strong solution if the Sobolev index s ≥ 3 and a sign condition holds. For analytic initial data, the existence and uniqueness of analytic solutions for 1.1 are also obtained in 12 . In this paper, motivated by the work in 7, 13 , we study the following generalized Novikov equation: The main tasks of this work are two-fold. Firstly, by using the Kato theorem for abstract differential equations, we establish the local existence and uniqueness of solutions for the 1.2 with any β and arbitrary positive integer N in space C 0, T , H s R C 1 0, T , H s−1 R with s > 3/2 . Secondly, it is shown that there exist local weak solutions in lowerorder Sobolev space H s R with 1 ≤ s ≤ 3/2 . The ideas of proving the second result come from those presented in Li and Olver 8 .

Main Results
Firstly, some notations are presented as follows.
The space of all infinitely differentiable functions φ t, x with compact support in 0, ∞ × R is denoted by Abstract and Applied Analysis 3 which is equivalent to

2.3
Now, we state our main results.

Local Well-Posedness
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 on max{ y Y , z Y }.
and A y ∈ G X, 1, β i.e., A y is quasi-m-accretive , uniformly on bounded sets in Y . 3.3 III f : Y → Y extends to a map from X into X, is bounded on bounded sets in Y , and satisfies
and Q Λ. In order to prove Theorem 2.1, we only need to check that A u and f u satisfy assumptions I -III .

Lemma 3.3 Ni and Zhou 11 . For
Lemma 3.4. Let r and q be real numbers such that −r < q ≤ r. Then

3.10
The above first two inequalities of this lemma can be found in 14, 15 , and the third inequality can be found in 7 .

Lemma 3.5. Letting u, z ∈ H s with s > 3/2 , then f u is bounded on bounded sets in H s and satisfies
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
It follows from Lemma 3.4 and s − 1 > 1/2 that 3.14 6 Abstract and Applied Analysis From 3.7 , and 3.13 , we know that 3.11 is valid, while inequality 3.12 follows from 3.14 .
Proof of Theorem 2.1. Using the Kato Theorem, Lemmas 3.1-3.3 and 3.5, we know that system 2.2 or problem 2.3 has a unique solution

Existence of Weak Solutions
For s ≥ 2, using the first equation of system 1.3 derives

4.6
Abstract and Applied Analysis 7 For q ∈ 0, s − 1 , there is a constant c such that Proof. The identity u 2 H 1 R u 2 u 2 x dx, 4.2 , and the Gronwall inequality result in 4.5 . Using ∂ 2 x −Λ 2 1 and the Parseval equality gives rise to For q ∈ 0, s − 1 , applying Λ q u Λ q to both sides of the first equation of system 2.2 and integrating with respect to x by parts, we have the identity

4.9
We will estimate the terms on the right-hand side of 4.9 separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2, we have

4.10
Using the above estimate to the second term yields For the third term, using the Cauchy-Schwartz inequality and Lemma 4.2, we obtain

4.12
in which we have used uu x H q ≤ c u 2 x H q ≤ c u L ∞ u H q 1 .

Abstract and Applied Analysis
For the fouth term in 4.9 , using u u 2

4.13
For K 1 , it follows from 4.12 that For K 2 , applying Lemma 4.2 derives

4.15
For the last term in 4.9 , using Lemma 4.1 repeatedly results in It follows from 4.10 -4.16 that there exists a constant c such that

4.17
Integrating both sides of the above inequality with respect to t results in inequality 4.6 .
To estimate the norm of u t , we apply the operator 1 − ∂ 2 x −1 to both sides of the first equation of system 2.2 to obtain the equation Applying Λ q u t Λ q to both sides of 4.18 for q ∈ 0, s − 1 gives rise to

4.19
Abstract and Applied Analysis 9 For the right-hand of 4.19 , we have

4.22
Using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2 yields for a constant c > 0. This completes the proof of Lemma 4.3.

Abstract and Applied Analysis
Defining
It follows from Theorem 2.1 that for each ε the Cauchy problem

Lemma 4.4.
Under the assumptions of problem 4.27 , the following estimates hold for any ε with 0 < ε < 1/4 and s > 0:

4.28
where c 1 is a constant independent of ε.
The proof of this Lemma can be found in Lai and Wu 7 .

Lemma 4.5.
If u 0 x ∈ H s R with s ∈ 1, 3/2 such that u 0x L ∞ < ∞, and u ε0 is defined as in system 4.27 . Then there exist two positive constants T and c, which are independent of ε, such that the solution u ε of problem 4.27 satisfies u εx L ∞ ≤ c for any t ∈ 0, T .
Proof. Using notation u u ε and differentiating both sides of the first equation of problem 4.27 with respect to x give rise to

4.29
Integrating by parts leads to Multiplying the above equation by u x 2p 1 and then integrating the resulting equation with respect to x yield the equality

4.32
Applying the Hölder's inequality yields Since f L p → f L ∞ as p → ∞ for any f ∈ L ∞ L 2 , integrating both sides of the inequality 4.34 with respect to t and taking the limit as p → ∞ result in the estimate Using the algebra property of H s 0 R with s 0 > 1/2 yields u ε H 1/2 means that there exists a sufficiently small δ > 0 such that u ε H 1/2 u ε H 1/2 δ : Using u L ∞ ≤ u H 1 , from 4.36 and 4.38 , it has From Lemma 4.4, 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 . Using the theorem presented at page 51 in 8 yields that there are constants T > 0 and c > 0 independent of ε such that u x L ∞ ≤ W t for arbitrary t ∈ 0, T , which leads to the conclusion of Lemma 4.5.
Using Lemmas 4.3 and 4.5, notation u ε u, and Gronwall's inequality results in the inequalities u ε H q ≤ C T e C T , u εt H r ≤ C T e C T ,

4.41
where q ∈ 0, s , r ∈ 0, s − 1 , and C T depends on T . It follows from Aubin's compactness theorem that there is a subsequence of {u ε }, denoted by {u ε n }, such that {u ε n } and their temporal derivatives {u ε n 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 } 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 t } converges to u t strongly in the space L 2 0, T , H r −R 1 , R 1 for r ∈ 0, s − 1 . Thus, we can prove the existence of a weak solution to 1.2 .
Proof of Theorem 2.2. From Lemma 4.5, we know that {u ε n x } ε n → 0 is bounded in the space L ∞ . Thus, the sequences {u ε n } and {u N ε n x } are weakly convergent to u and u N x in L 2 0, T , H r −R, R for any r ∈ 0, s − 1 , respectively. Therefore, 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 x } is a bounded sequence in the dual space X * L ∞ 0, T × R of X, there exists a subsequence of {u ε n x }, still denoted by {u ε n x }, weakly star convergent to a function v in L ∞ 0, T × R . It derives from the {u ε n x } weakly convergent to u x in L 2 0, T × R that u x v almost everywhere. Thus, we obtain u x ∈ L ∞ 0, T × R .