The Cauchy problem for the seventh-order dispersive equation in Sobolev space
© Wang and Zheng; licensee Springer. 2014
Received: 5 March 2014
Accepted: 6 May 2014
Published: 20 May 2014
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.
KeywordsCauchy problem well-posedness Sobolev spaces
and he proved that it is locally well-posed in with with the aid of a short time Bourgain space.
When , is abbreviated as .
The main result of this paper is as follows.
Theorem 1.1 Assume that with . 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.
Proof For the proof of (2.1)-(2.5), we refer the readers to Lemma 2.1 of .
We have completed the proof of Lemma 2.1. □
Lemma 2.2 is the case of of Lemma 3.1 of .
Lemma 2.3 can be found as Lemma 2.4 of .
3 Bilinear estimates
In this section, we will give an important bilinear estimate.
- (1)Subregion . Since , we have , which yields
Subregion . In this subregion, obviously, .
- (3)Subregion . In this subregion, we derive . Thus,
- (i)Case . By (3.1), we derive
- (ii)Case . Since , we have
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
- (ii)Case . Since , by using , we have
- (5)Subregion . In this region , thus, we have
- (i)If , by using (3.1) and , we have
- (ii)If , then . By using (3.1), we have
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
Combining Lemmas 2.3 and 3.1 with the fixed point theorem, we easily obtain Theorem 1.1.
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.
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