Abstract
Keywords:
Cauchy problem; wellposedness; Sobolev spaces1 Introduction
In this paper, we are concerned with the Cauchy problem for the following seventhorder dispersive equation:
Kenig et al.[1] established that
is locally wellposed in some weighted Sobolev spaces for small initial data and for arbitrary initial data. Recently, Pilod [2] studied the following higherorder nonlinear dispersive equation:
where and u is a real (or complex) valued function and proved it is locally wellposed in weighted Besov and Sobolev spaces for small initial data and proved illposedness 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
and he proved that it is locally wellposed in with with the aid of a short time Bourgain space.
In this paper, inspired by [15], by using the Fourier restriction norm method, we establish that (1.1)(1.2) is locally wellposed 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
The main result of this paper is as follows.
Theorem 1.1Assume thatwith. Then the Cauchy problem for (1.1) is locally wellposed.
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
3 Bilinear estimates
In this section, we will give an important bilinear estimate.
We give an important relation before proving the bilinear estimate.
where
Proof Let
To establish (3.2), it is sufficient to derive the following inequality:
where
Without loss of generality, we assume that , (). To derive (3.3), it suffices to prove that
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 CauchySchwarz inequality and Lemma 2.2, we have
(3) Subregion. In this subregion, we derive . Thus,
(i) Case . By (3.1), we derive
This case can be proved similarly to .
consequently, by using the CauchySchwarz inequality and (2.5) and (2.4), 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
By using the CauchySchwarz inequality, we have
(ii) Case . Since , by using , we have
This case can be proved similarly to .
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
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
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.
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