Keywords:Cauchy problem; well-posedness; Sobolev spaces
In this paper, we are concerned with the Cauchy problem for the following seventh-order dispersive equation:
Kenig et al. established that
is locally well-posed in some weighted Sobolev spaces for small initial data and for arbitrary initial data. Recently, Pilod  studied the following higher-order nonlinear dispersive equation:
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. studied the Cauchy problem for
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
The main result of this paper is as follows.
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.
We give an important relation before proving the bilinear estimate.
To establish (3.2), it is sufficient to derive the following inequality:
It is easily checked that
Consequently, by the Cauchy-Schwarz inequality and Lemma 2.2, we have
consequently, by using the Cauchy-Schwarz inequality and (2.5) and (2.4), we have
This case can be proved similarly to the above case.
By using the Cauchy-Schwarz inequality, we have
We have completed the proof of Lemma 3.1. □
4 Proof of Theorem 1.1
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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.