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# The Cauchy problem for the seventh-order dispersive equation in Sobolev space

Hongjun Wang1* and Yan Zheng2

Author Affiliations

1 College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, 453007, P.R. China

2 Henan Vocational College of Agriculture, Zhengzhou, Henan, 451450, P.R. China

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Boundary Value Problems 2014, 2014:122  doi:10.1186/1687-2770-2014-122

 Received: 5 March 2014 Accepted: 6 May 2014 Published: 20 May 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

### Abstract

This paper is devoted to the Cauchy problem for the higher-order dispersive equation , . The local well-posedness of the associated Cauchy problem is established in Sobolev space with with the aid of the Fourier restriction norm method.

MSC: 35K30.

##### Keywords:
Cauchy problem; well-posedness; Sobolev spaces

### 1 Introduction

In this paper, we are concerned with the Cauchy problem for the following seventh-order dispersive equation:

(1.1)

(1.2)

Kenig et al.[1] established that

(1.3)

(1.4)

is locally well-posed in some weighted Sobolev spaces for small initial data and for arbitrary initial data. Recently, Pilod [2] studied the following higher-order nonlinear dispersive equation:

(1.5)

where and u is a real- (or complex-) valued function and proved it is locally well-posed in weighted Besov and Sobolev spaces for small initial data and proved ill-posedness results when for some in the sense that (1.5) cannot have its flow map at the origin in . Very recently, Guo et al.[3] studied the Cauchy problem for

(1.6)

and he proved that it is locally well-posed in with with the aid of a short time Bourgain space.

In this paper, inspired by [1-5], by using the Fourier restriction norm method, we establish that (1.1)-(1.2) is locally well-posed in Sobolev space with .

Now we give some notations and definitions. Throughout this paper, we always assume that ψ is a smooth function, , satisfying , when , and , (),

for any , and denotes the Fourier transformation of u with respect to its all variables. denotes the Fourier inverse transformation of u with respect to its all variables. denotes the Fourier transformation of u with respect to its space variable. denotes the Fourier inverse transformation of u with respect to its space variable. is the Schwarz space and is its dual space. is the Sobolev space with norm . For any , is the Bourgain space with phase function . That is, a function in belongs to iff

For any given interval L, is the space of the restriction of all functions in on , and for its norm is

When , is abbreviated as .

The main result of this paper is as follows.

Theorem 1.1Assume thatwith. Then the Cauchy problem for (1.1) is locally well-posed.

The remainder of paper is arranged as follows. In Section 2, we make some preliminaries. In Section 3, we give an important bilinear estimate. In Section 4, we establish Theorem 1.1.

### 2 Preliminaries

Lemma 2.1Let. Then

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

Proof For the proof of (2.1)-(2.5), we refer the readers to Lemma 2.1 of [5].

We have completed the proof of Lemma 2.1. □

Lemma 2.2Assume that. Then

(2.6)

where

Lemma 2.2 is the case of of Lemma 3.1 of [5].

Lemma 2.3For any, and, for, we have

(2.7)

For, we have

(2.8)

Lemma 2.3 can be found as Lemma 2.4 of [6].

### 3 Bilinear estimates

In this section, we will give an important bilinear estimate.

We give an important relation before proving the bilinear estimate.

where

(3.1)

Lemma 3.1Let, , where, . Then

(3.2)

Proof Let

To establish (3.2), it is sufficient to derive the following inequality:

(3.3)

where

(3.4)

Without loss of generality, we assume that , (). To derive (3.3), it suffices to prove that

(3.5)

By using the symmetry between and , without loss of generality, we can assume that . Obviously,

where

We will denote the integrals in (3.5) corresponding to () by (), respectively. Let , , .

(1) Subregion. Since , we have , which yields

Then, by the Plancherel identity, the Hölder inequality, and , we derive

(2) Subregion. In this subregion, obviously, .

It is easily checked that

Consequently, by the Cauchy-Schwarz inequality and Lemma 2.2, we have

(3) Subregion. In this subregion, we derive . Thus,

(i) Case . By (3.1), we derive

If , then

If , then

This case can be proved similarly to .

(ii) Case . Since , we have

If , we have

consequently, by using the Cauchy-Schwarz inequality and (2.5) and (2.4), we have

If , since , we have

This case can be proved similarly to the above case.

(iii) Case . This case is similar to (ii) case .

(4) Subregion. In this subregion, , and it is easy to obtain

(i) Case . By using, , when , we have

This case can be proved similarly to Subregion. When , we have

If , since , then

By using the Cauchy-Schwarz inequality, we have

(ii) Case . Since , by using , we have

This case can be proved similarly to .

(iii) Case .

This case can be proved similarly to .

(5) Subregion. In this region , thus, we have

(i) If , by using (3.1) and , we have

By using the Plancherel identity, the Hölder inequality, and as well as (2.5), we have

(ii) If , then . By using (3.1), we have

By using the Plancherel identity, the Hölder inequality, (2.5) and , we have

(iii) If .

This case can be proved similarly to the case .

(6) Subregion. In this region, we have .

This case can be proved similarly to the Subregion.

We have completed the proof of Lemma 3.1. □

### 4 Proof of Theorem 1.1

The system (1.1)-(1.2) is equivalent to the following integral equation:

(4.1)

We define

(4.2)

Combining Lemmas 2.3 and 3.1 with the fixed point theorem, we easily obtain Theorem 1.1.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors read and approved the final manuscript.

### Acknowledgements

We would like to thank reviewers for a careful reading and valuable comments on the original draft. The first author is supported by Foundation and Frontier of Henan Province under grant Nos. 122300410414, 132300410432.

### References

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4. Bourgain, J: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: the KdV equation. Geom. Funct. Anal.. 3, 209–262 (1993)

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