Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article
Special Issue

Nonlinear Analysis: Algorithm, Convergence, and Applications 2014

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Research Article | Open Access

Volume 2014 |Article ID 404781 | 8 pages | https://doi.org/10.1155/2014/404781

The Cauchy Problem for a Fifth-Order Dispersive Equation

Academic Editor: Changsen Yang
Received21 Jan 2014
Accepted08 Feb 2014
Published27 Mar 2014

Abstract

This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev space with . We also establish the ill-posedness for the initial data in with . Thus, the regularity requirement for the fifth-order dispersive equations is sharp.

1. Introduction

In this paper, we consider the Cauchy problem and existence of solitary waves of the following fifth-order dispersive equation: Obviously, (1) can be seen as the higher modification of the following dispersive equation:

We recall that where which has been studied in [1], is the generalized Camassa-Holm equation. In fact, Hakkaev and Kirchev [1] studied the local well-posedness and orbital stability and instability of (4) with the aid of the pseudoparabolic regularization and spectral analysis. Obviously, when , (4) is the well-known Camassa-Holm equation; when and , (4) becomes (3). Some people consider the Cauchy problem for the higher modification of the nonlocal form of Camassa-Holm equation; we refer the readers to [25].

By acting on both sides of (1), we obtain the following equivalent form: Obviously, (6) is a modification of mKdV equation which has been intensively studied; we refer the readers to [611]. In [8], the authors proved that (7) is locally well-posed for the initial data in with . The regularity requirement for the modified KdV equation is sharp; see [9]. In [11], by using the -method which can be seen in [1114] and Mirua transformation which can be seen in [15], the authors proved that the modified KdV equation is globally well-posed for the initial data in with . In [16], by using the dyadic bilinear estimates and resolution spaces which can be seen in [17, 18], the author proved that the modified KdV equation is globally well-posed for the initial data in with .

In this paper, by using the Fourier restriction norm method introduced in [19], we prove that (1) and (2) are locally well-posed for the initial data in with . When , , we prove that the Cauchy problem for (1) is globally well-posed in . By using the general well-posedness principle proposed by [20], we establish the ill-posedness for the initial data in with . Thus, the requirement for regularity is sharp.

Before stating the main results, we introduce some notations and definitions. We use when , where ; we use when , where is some positive number which is larger than 2. for any , and is the Fourier transform of with respect to its all variables. is the Fourier transform of with respect to its space variables. Denote . () is the Schwartz space and is its dual space. is the usual Sobolev space with norm . For any , is the Bourgain space with phase function . That is, a function in belongs to if and only if For any given interval , is the space of the restriction of all functions in on , and for its norm is We write for when . Let .

The main results of this paper are as follows.

Theorem 1. The Cauchy problem for (1) is locally well-posed for the initial data in with .

Theorem 2. Let . Then (1) and (2) are ill-posed in in the sense that the solution map of the Cauchy problem for (1) is not Lipschitz continuous at zero. More precisely, for any , the solution map is not Lipschitz continuous at zero.

The rest of this paper is arranged as follows. In Section 2, we make some preliminaries. In Section 3, we establish two crucial trilinear estimates. In Section 4, we prove Theorem 1. In Section 5, we prove Theorem 2.

2. Preliminaries

Lemma 3. Let . Then

Proof. From [21], we have From Plancherel’s identity, we have Interpolating (15) with (16) yields (14). Equations (11)–(13) can be seen in [21, 22].

Lemma 4. For , , and , we have and for , we have

Equations (17) and (18) can be seen in [6, 23].

Lemma 5. Let . Then

Lemma 5 can be found in [11].

Lemma 6. Let denote the Riesz potential of order and , . Then the following estimate holds true: where

In particular, when and , we have

Lemma 6 can be seen in [6].

3. Trilinear Estimates

In this section, we will prove two crucial trilinear estimates.

Lemma 7. Let , , , . Then

Proof. Let By duality and Plancherel’s identity, to derive (23), it suffices to prove where Without loss of generality, we may assume that and . By the symmetry among , , , without loss of generality, we may assume that . Obviously, , where In this lemma, integrals over the subregion ’s are, respectively, denoted as (, ). Consider
(1) Subregion . In this subregion, since and , we have By using Cauchy Schwartz’s inequality and Plancherel’s identity as well as Hölder’s inequality, (11), and , we derive
(2) Subregion . In this subregion, and , since , we have By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (22) of Lemma 6, (11), , and , since , we obtain
(3) Subregion . In this subregion, since and , we have This case can be treated similarly to .
(4) Subregion . In this subregion and .
When , since and , we have This case can be treated similarly to .
When , we consider the case and the case .
When , since , we have This case can be treated similarly to .
When , From Lemma 5, we have which yields ; thus one of the following four cases must occur: When (37) is valid, since we have when , since and , we have when , since and , we have By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), , we obtain When (38) is valid, since we have when , we have when , since and , we have By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), , and , we obtain Cases (39) and (40) can be treated similarly to case (38).
(5) Subregion . In this subregion, since , which yields Thus, applying Cauchy-Schwartz’s inequality and Plancherel’s identity, (12) and (13), we have
(6) Subregion . In this subregion, since , we have and and . From Lemma 5, we have which yields ; thus one of the following four cases must occur: When , we have this case can be handled similarly to .
When , we consider (54), (55), (56), and (57), respectively.
When (37) is valid, since , and as well as , we have By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), , we obtain When (55) holds, since , and as well as , we have By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), , and , we obtain Cases (56) and (57) can be treated similarly to case (55).
Putting the estimates of (, ) together, we have (25).
Thus, we complete the proof of Lemma 7.

Lemma 8. Let , , , . Then

Lemma 8 can be proved similarly to Lemma 7.

4. The Proof of Theorem 1

In order to prove Theorem 1, firstly, for and , , we define by where By using Lemmas 4, 7, and 8, we have where , , of (66) concord with , , and of Lemmas 7 and 8. Let where . From (66) and (67), we know that is a mapping from the closed ball into itself. Similarly, we have thus, is a contraction mapping from the closed ball into itself; by using Banach fixed point Theorem, we have .

The rest of local well-posedness of Theorem 1 follows from a standard proof.

5. The Proof of Theorem 2

In this section, we prove Theorem 2.

By contradiction, we assume that the solution map of (1) and (2) is Lipschitz continuous at zero with . From the general well-posedness principle of [20], we must have where We consider the initial data where and . Thus, we have which can be seen in [24], where denotes the characteristic function of a set . It is easy to check that . Let and and . We have and thus where where Since , we define resulting from Lemma 5 and . To estimate , we consider the following three cases. Case 1: .Case 2: .Case 3: , , or , , or , , or , .The integrals in (76) corresponding to Cases 1, 2, and 3 are denoted as , , , respectively.

Case 1. In this case, from Lemma 6, we have and . Since , we have

Case 2. In this case, from Lemma 6, we have and . Since , we have