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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 158126, 14 pages
http://dx.doi.org/10.1155/2012/158126
Research Article

The Local Strong and Weak Solutions for a Generalized Novikov Equation

Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 8 December 2011; Accepted 13 January 2012

Academic Editor: Shaoyong Lai

Copyright © 2012 Meng Wu and Yue Zhong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Kato theorem for abstract differential equations is applied to establish the local well-posedness of the strong solution for a nonlinear generalized Novikov equation in space with . The existence of weak solutions for the equation in lower-order Sobolev space with is acquired.

1. Introduction

The Novikov equation with cubic nonlinearities takes the form 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 [410]). Ni and Zhou [11] proved that the Novikov equation associated with initial value is locally well-posedness in Sobolev space with 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 with . The orbit invariants are employed to get the existence of 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 (1.1) are also obtained in [12].

In this paper, motivated by the work in [7, 13], we study the following generalized Novikov equation: where is a natural number. In fact, (1.1) has the property

Due to the term appearing in (1.2), the conservation law (1.3) for (1.2) is not valid. This brings us a difficulty to obtain the bounded estimates for the solution of (1.2). However, we will overcome this difficulty to investigate the local existence and uniqueness of the solution to (1.2) subject to initial value with . Meanwhile, a sufficient condition is presented to guarantee the existence of local weak solution for (1.2).

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 and arbitrary positive integer in space with . Secondly, it is shown that there exist local weak solutions in lower-order Sobolev space with . The ideas of proving the second result come from those presented in Li and Olver [8].

2. Main Results

Firstly, some notations are presented as follows.

The space of all infinitely differentiable functions with compact support in is denoted by . is the space of all measurable functions such that . We define with the standard norm . For any real number , denotes the Sobolev space with the norm defined by where .

For and nonnegative number , denotes the Frechet space of all continuous -valued functions on . We set .

The Cauchy problem for (1.2) is written in the form which is equivalent to

Now, we state our main results.

Theorem 2.1. Let with . Then problem (2.2) or problem (2.3) has a unique solution , where depends on .

Theorem 2.2. Suppose that with and . Then there exists a such that (1.2) subject to initial value has a weak solution in the sense of distribution and .

3. Local Well-Posedness

Consider the abstract quasilinear evolution equation Let and be Hilbert spaces such that is continuously and densely embedded in , and let be a topological isomorphism. Let be the space of all bounded linear operators from to . If , we denote this space by . We state the following conditions in which , , , and are constants depending on .(I) for with and (i.e., is quasi-m-accretive), uniformly on bounded sets in .(II), where is bounded, uniformly on bounded sets in . Moreover,(III) extends to a map from into , is bounded on bounded sets in , and satisfies

Kato Theorem (see [14])
Assume that (I), (II), and (III) hold. If , there is a maximal depending only on and a unique solution to problem (3.1) such that Moreover, the map is a continuous map from to the space For problem (2.3), we set , , , , and . In order to prove Theorem 2.1, we only need to check that and satisfy assumptions (I)–(III).

Lemma 3.1 (Ni and Zhou [11]). The operator with , belongs to .

Lemma 3.2 (Ni and Zhou [11]). Let with and . Then for all . Moreover,

Lemma 3.3 (Ni and Zhou [11]). For and , it holds that for and

Lemma 3.4. Let and be real numbers such that . Then

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 with , then is bounded on bounded sets in and satisfies

Proof. Using the algebra property of the space with and , we have It follows from Lemma 3.4 and that 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 3.13.3 and 3.5, we know that system (2.2) or problem (2.3) has a unique solution

4. Existence of Weak Solutions

For , using the first equation of system (1.3) derives from which we have the conservation law

Lemma 4.1 (Kato and Ponce [15]). If , then is an algebra. Moreover where is a constant depending only on .

Lemma 4.2 (Kato and Ponce [15]). Letting . If and , then

Lemma 4.3. Let and the function is a solution of problem (2.2) and the initial data . Then the following results hold:
For , there is a constant such that
For , there is a constant such that

Proof. The identity , (4.2), and the Gronwall inequality result in (4.5).
Using and the Parseval equality gives rise to
For , applying to both sides of the first equation of system (2.2) and integrating with respect to by parts, we have the identity 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 Using the above estimate to the second term yields For the third term, using the Cauchy-Schwartz inequality and Lemma 4.2, we obtain in which we have used .
For the fouth term in (4.9), using results in For , it follows from (4.12) that For , applying Lemma 4.2 derives
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 such that Integrating both sides of the above inequality with respect to results in inequality (4.6).
To estimate the norm of , we apply the operator to both sides of the first equation of system (2.2) to obtain the equation Applying to both sides of (4.18) for gives rise to For the right-hand of (4.19), we have Since using Lemma 4.2, , and , we have Using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2 yields in which we have used inequality (4.15).
Applying (4.20)–(4.24) into (4.19) yields the inequality for a constant . This completes the proof of Lemma 4.3.

Defining and setting with and , we know that for any and .

It follows from Theorem 2.1 that for each the Cauchy problem has a unique solution .

Lemma 4.4. Under the assumptions of problem (4.27), the following estimates hold for any with and : where is a constant independent of .

The proof of this Lemma can be found in Lai and Wu [7].

Lemma 4.5. If with such that , and is defined as in system (4.27). Then there exist two positive constants and , which are independent of , such that the solution of problem (4.27) satisfies for any .

Proof. Using notation and differentiating both sides of the first equation of problem (4.27) with respect to give rise to Integrating by parts leads to from which we obtain Multiplying the above equation by and then integrating the resulting equation with respect to yield the equality Applying the Hölder's inequality yields or where Since as for any , integrating both sides of the inequality (4.34) with respect to and taking the limit as result in the estimate Using the algebra property of with yields ( means that there exists a sufficiently small such that ): in which Lemma 3.4 is used. From Lemma 4.3, we get Using , from (4.36) and (4.38), it has
From Lemma 4.4, it follows from the contraction mapping principle that there is a such that the equation has a unique solution . Using the theorem presented at page 51 in [8] yields that there are constants and independent of such that for arbitrary , which leads to the conclusion of Lemma 4.5.
Using Lemmas 4.3 and 4.5, notation , and Gronwall's inequality results in the inequalities where , , and depends on . 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 strongly in the space for and converges to strongly in the space for . 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 is bounded in the space . Thus, the sequences and are weakly convergent to and in , for any , respectively. Therefore, satisfies the equation with and . Since is a separable Banach space and is a bounded sequence in the dual space of , there exists a subsequence of , still denoted by , weakly star convergent to a function in . It derives from the weakly convergent to in that almost everywhere. Thus, we obtain .

Acknowledgment

This work is supported by the Applied and Basic Project of Sichuan Province (2012JY0020).

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