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 with is established. Although the -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 with is established under the assumptions and .
MSC: 35Q35, 35Q51.
Keywords:local strong solution; local weak solution; generalized Novikov equation
Novikov  derived the integrable equation with cubic nonlinearities
which has been investigated by many scholars. Grayshan  studied both the periodic and the non-periodic Cauchy problem for Eq. (1) and discussed continuity results for the data-to-solution map in the Sobolev spaces. A Galerkin-type approximation method was used in Himonas and Holliman’s paper  to establish the well-posedness of Eq. (1) in the Sobolev space with on both the line and the circle. Hone et al. applied the scattering theory to find non-smooth explicit soliton solutions with multiple peaks for Eq. (1). This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations (see [5-7]). A matrix Lax pair for Eq. (1) was acquired in [8,9] and was shown to be related to a negative flow in the Sawada-Kotera hierarchy. Sufficient conditions on the initial data to guarantee the formation of singularities in finite time for Eq. (1) were given in Jiang and Li . Mi and Mu  obtained many dynamic results for a modified Novikov equation with a peak solution. It is shown in Ni and Zhou  that the Novikov equation associated with the initial value is locally well-posed in Sobolev space with by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for Eq. (1) are established in . Tiglay  proved the local well-posedness for the periodic Cauchy problem of the Novikov equation in Sobolev space with . The orbit invariants are used to show the existence of a periodic global strong solution if the Sobolev index and a sign condition holds. For analytic initial data, the existence and uniqueness of analytic solutions for Eq. (1) are obtained in . Using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, Yan et al. proved that Eq. (1) is locally well-posed in the Besov space under certain assumptions. For other methods to handle the Novikov equation and the related partial differential equations, the reader is referred to [15-24] and the references therein.
We note that the coefficients of the terms , and in the Novikov equation (1) are 4, 3 and 1, respectively. Namely, . This guarantees that the conservation law of Eq. (1) holds
which takes a key role in obtaining various dynamic properties in the previous works.
where m, a and are arbitrary constants. Clearly, letting , and , Eq. (2) becomes the Novikov equation (1).
The objective of this paper is to investigate Eq. (2). Since m, a and are arbitrary constants, we do not have the result that the norm of the solution of Eq. (2) remains constant. We will apply the Kato theorem for abstract differential equations to prove the existence and uniqueness of local solutions for Eq. (2) subject to the initial value ( ). In addition, the existence of local weak solutions for Eq. (2) is established in the lower-order Sobolev space with under the assumptions and .
The rest of this paper is organized as follows. The main results are given in Section 2. The proof of a local well-posedness result is established in Section 3, while the existence of local weak solutions is proved in Section 4.
2 Main results
Firstly, we state some notations.
The space of all infinitely differentiable functions with compact support in is denoted by . ( ) is the space of all measurable functions h such that . We define with the standard norm . For any real number s, denotes the Sobolev space with the norm defined by
For and nonnegative number s, denotes the Frechet space of all continuous -valued functions on . We set . For simplicity, throughout this article, we let c denote any positive constant which is independent of parameter ε.
and setting with and , we know that for any , (see ).
We consider the Cauchy problem for Eq. (2)
which is equivalent to
Now, we give our main results for problem (3).
Theorem 1Let with . Then the Cauchy problem (3) has a unique solution , where depends on .
It follows from Theorem 1 that for each ε satisfying , the Cauchy problem
has a unique solution , in which may depend on ε. However, we shall show that under certain assumptions, there exist two constants c and , both independent of ε, such that the solution of problem (5) satisfies for any and there exists a weak solution for problem (3). These results are summarized in the following two theorems.
Theorem 2If with such that . Let be defined as in system (5). Then there exist two constantscand , which are independent ofε, such that the solution of problem (5) satisfies for any .
Theorem 3Suppose that with and . Then there exists a such that problem (3) has a weak solution in the sense of distribution and .
3 Proof of Theorem 1
Consider the abstract quasi-linear evolution equation
Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and let be a topological isomorphism. Let be the space of all bounded linear operators from Y to X. If , we denote this space by . We state the following conditions in which , , and are constants depending only on .
(I) for with
and (i.e., is quasi-m-accretive), uniformly on bounded sets in Y.
(II) , where is bounded, uniformly on bounded sets in Y. Moreover,
(III) extends to a map from X into X, is bounded on bounded sets in Y, and satisfies
Kato theorem (see )
Assume that (I), (II) and (III) hold. If , there is a maximal depending only on and a unique solutionvto problem (6) such that
Moreover, the map is a continuous map fromYto the space
We set with constant , , , , and . We know that Q is an isomorphism of onto . In order to prove Theorem 1, we only need to check that and satisfy assumptions (I)-(III).
Lemma 3.1The operator with , belongs to .
Lemma 3.2Let with and . Then for all . Moreover,
Lemma 3.3For , and , it holds that for and
The above three lemmas can be found in Ni and Zhou .
Lemma 3.4Letrandqbe real numbers such that . Then
Lemma 3.5Let with and . Thenfis bounded on bounded sets in and satisfies
Proof Using the algebra property of the space with , we get
which completes the proof of (9). Using and the first inequality in Lemma 3.4, we have
Using (12) and (13) yields
which completes the proof of inequality (10). □
Proof of Theorem 1 Using the Kato theorem, Lemmas 3.1, 3.2, 3.3 and Lemma 3.5, we know that system (3) or problem (4) has a unique solution
4 Proofs of Theorems 2 and 3
Using the first equation of system (3) derives
from which we have the conservation law
Lemma 4.1 (Kato and Ponce )
If , then is an algebra. Moreover,
wherecis a constant depending only onr.
Lemma 4.2 (Kato and Ponce )
Let . If and , then
Lemma 4.3Let and the function is a solution of problem (3) and the initial data . Then the following results hold:
For , there is a constant c only depending on m, a and b such that
For , there is a constant c only depending on m, a and b such that
Proof Using , the Gronwall inequality and (15) derives (16).
Using and the Parseval equality gives rise to
For , applying to both sides of the first equation of system (3) and integrating with respect to x by parts, we have the identity
We will estimate the terms on the right-hand side of (19) separately. For the first term, by using the Cauchy-Schwarz inequality and Lemmas 4.1 and 4.2, we have
Using the above estimate to the second term yields
Using the Cauchy-Schwarz inequality and Lemma 4.1, we obtain
For the last term in (19), using results in
For , it follows from (22) that
For , applying Lemma 4.1 derives
It follows from (20)-(25) that there exists a constant c such that
Integrating both sides of the above inequality with respect to t results in inequality (17).
To estimate the norm of , we apply the operator to both sides of the first equation of system (3) to obtain the equation
Applying to both sides of Eq. (27) for gives rise to
For the right-hand of Eq. (28), we have
using Lemma 4.1, and , we have
Using the Cauchy-Schwarz inequality and Lemma 4.1 yields
Applying (29)-(33) into (28) yields the inequality
for a constant . This completes the proof of Lemma 4.3. □
Lemma 4.4 ()
For , and , the following estimates hold for anyεwith
wherecis a constant independent ofε.
Proof of Theorem 2 Using notation and differentiating both sides of the first equation of problem (5) or Eq. (27) with respect to x give rise to
Letting be an integer and multiplying the above equation by and then integrating the resulting equation with respect to x yield the equality
Applying the Hölder’s inequality yields
Since as for any , integrating both sides of the inequality (42) with respect to t and taking the limit as result in the estimate
where c only depends on m, a, b.
Using the algebraic property of with and the inequality (16) yields
where c is a constant independent of ε. From (45), we have
It follows from (43) and (46) that
It follows from the contraction mapping principle that there is a such that the equation
has a unique solution . Using the theorem presented on p.51 in Li and Olver  yields that there are constants and , which are independent of ε, such that for arbitrary , which leads to the conclusion of Theorem 2. □
Using Theorem 2, (17), (18) and (44), notation and Gronwall’s inequality results in the inequalities
where , and . It follows from Aubin’s compactness theorem that there is a subsequence of , denoted by , such that and their temporal derivatives are weakly convergent to a function and its derivative in and , respectively. Moreover, for any real number , is convergent to the function u strongly in the space for and converges to strongly in the space for .
Proof of Theorem 3 From Theorem 2, we know that ( ) is bounded in the space . Thus, the sequences , , and are weakly convergent to u, , and in for any , separately. Hence, u satisfies the equation
with and . Since is a separable Banach space and is a bounded sequence in the dual space of X, there exists a subsequence of , still denoted by , weakly star convergent to a function v in . As weakly converges to in , it results that almost everywhere. Thus, we obtain . □
The authors declare that they have no competing interests.
The article is a joint work of two authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Thanks are given to referees whose comments and suggestions are very helpful to revise our paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).
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