The local strong and weak solutions to a generalized Novikov equation
© Lai and Wu; licensee Springer. 2013
Received: 8 December 2012
Accepted: 3 May 2013
Published: 20 May 2013
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 .
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.
which takes a key role in obtaining various dynamic properties in the previous works.
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.
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 ).
Now, we give our main results for problem (3).
Theorem 1 Let with . Then the Cauchy problem (3) has a unique solution , where depends on .
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 2 If with such that . Let be defined as in system (5). Then there exist two constants c and , which are independent of ε, such that the solution of problem (5) satisfies for any .
Theorem 3 Suppose 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
- (I)for with
- (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 )
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 3.1 The operator with , belongs to .
The above three lemmas can be found in Ni and Zhou .
which completes the proof of inequality (10). □
4 Proofs of Theorems 2 and 3
Lemma 4.1 (Kato and Ponce )
where c is a constant depending only on r.
Lemma 4.2 (Kato and Ponce )
Proof Using , the Gronwall inequality and (15) derives (16).
Integrating both sides of the above inequality with respect to t results in inequality (17).
for a constant . This completes the proof of Lemma 4.3. □
Lemma 4.4 ()
where c is a constant independent of ε.
where c only depends on m, a, b.
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. □
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 .
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 . □
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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