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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


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/122


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

© 2014 Wang and Zheng; licensee Springer.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M2">View MathML</a>. The local well-posedness of the associated Cauchy problem is established in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M3">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M4">View MathML</a> 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M5">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M6">View MathML</a>

(1.2)

Kenig et al.[1] established that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M7">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M8">View MathML</a>

(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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M9">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M2">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M11">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M12">View MathML</a> in the sense that (1.5) cannot have its flow map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M13">View MathML</a> at the origin in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M14">View MathML</a>. Very recently, Guo et al.[3] studied the Cauchy problem for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M15">View MathML</a>

(1.6)

and he proved that it is locally well-posed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M3">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M17">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M18">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M4">View MathML</a>.

Now we give some notations and definitions. Throughout this paper, we always assume that ψ is a smooth function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M20">View MathML</a>, satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M22">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M26">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M27">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M28">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M29">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M30">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M31">View MathML</a> denotes the Fourier transformation of u with respect to its all variables. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M32">View MathML</a> denotes the Fourier inverse transformation of u with respect to its all variables. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M33">View MathML</a> denotes the Fourier transformation of u with respect to its space variable. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M34">View MathML</a> denotes the Fourier inverse transformation of u with respect to its space variable. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M35">View MathML</a> is the Schwarz space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M36">View MathML</a> is its dual space. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M3">View MathML</a> is the Sobolev space with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M38">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M40">View MathML</a> is the Bourgain space with phase function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M41">View MathML</a>. That is, a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M42">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M43">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M44">View MathML</a> iff

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M45">View MathML</a>

For any given interval L, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M46">View MathML</a> is the space of the restriction of all functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M40">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M48">View MathML</a>, and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M49">View MathML</a> its norm is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M50">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M52">View MathML</a> is abbreviated as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M53">View MathML</a>.

The main result of this paper is as follows.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M54">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M4">View MathML</a>. 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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M56">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M57">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M58">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M59">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M60">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M61">View MathML</a>

(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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M62">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M63">View MathML</a>

(2.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M64">View MathML</a>

Lemma 2.2 is the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M65">View MathML</a> of Lemma 3.1 of [5].

Lemma 2.3For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M66">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M67">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M68">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M69">View MathML</a>

(2.7)

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M70">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M71">View MathML</a>

(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.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M72">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M73">View MathML</a>

(3.1)

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M62">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M77">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M78">View MathML</a>

(3.2)

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M79">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M80">View MathML</a>

(3.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M81">View MathML</a>

(3.4)

Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M83">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M27">View MathML</a>). To derive (3.3), it suffices to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M85">View MathML</a>

(3.5)

By using the symmetry between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M87">View MathML</a>, without loss of generality, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M88">View MathML</a>. Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M89">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M90">View MathML</a>

We will denote the integrals in (3.5) corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M91">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M92">View MathML</a>) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M92">View MathML</a>), respectively. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M27">View MathML</a>.

(1) Subregion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M98">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M99">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M100">View MathML</a>, which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M101">View MathML</a>

Then, by the Plancherel identity, the Hölder inequality, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M102">View MathML</a>, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M103">View MathML</a>

(2) Subregion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M104">View MathML</a>. In this subregion, obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M105">View MathML</a>.

It is easily checked that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M106">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M107">View MathML</a>

(3) Subregion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M108">View MathML</a>. In this subregion, we derive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M109">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M110">View MathML</a>

(i) Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M111">View MathML</a>. By (3.1), we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M112">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M113">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M114">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M115">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M116">View MathML</a>

This case can be proved similarly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M104">View MathML</a>.

(ii) Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M118">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M119">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M120">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M113">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M122">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M123">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M115">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M125">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M126">View MathML</a>

This case can be proved similarly to the above case.

(iii) Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M127">View MathML</a>. This case is similar to (ii) case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M128">View MathML</a>.

(4) Subregion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M129">View MathML</a>. In this subregion, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M130">View MathML</a>, and it is easy to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M131">View MathML</a>

(i) Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M111">View MathML</a>. By using, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M130">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M134">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M135">View MathML</a>

This case can be proved similarly to Subregion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M104">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M137">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M138">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M111">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M140">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M141">View MathML</a>

By using the Cauchy-Schwarz inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M142">View MathML</a>

(ii) Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M143">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M119">View MathML</a>, by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M140">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M146">View MathML</a>

This case can be proved similarly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M147">View MathML</a>.

(iii) Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M148">View MathML</a>.

This case can be proved similarly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M143">View MathML</a>.

(5) Subregion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M150">View MathML</a>. In this region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M151">View MathML</a>, thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M152">View MathML</a>

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M111">View MathML</a>, by using (3.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M154">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M155">View MathML</a>

By using the Plancherel identity, the Hölder inequality, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M156">View MathML</a> as well as (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M157">View MathML</a>

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M118">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M159">View MathML</a>. By using (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M160">View MathML</a>

By using the Plancherel identity, the Hölder inequality, (2.5) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M156">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M162">View MathML</a>

(iii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M127">View MathML</a>.

This case can be proved similarly to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M118">View MathML</a>.

(6) Subregion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M165">View MathML</a>. In this region, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M166">View MathML</a>.

This case can be proved similarly to the Subregion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M150">View MathML</a>.

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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M168">View MathML</a>

(4.1)

We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/122/mathml/M169">View MathML</a>

(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.

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  5. Li, Y, Yan, W, Yang, X: Well-posedness of a higher order modified Camassa-Holm equation in spaces of low regularity. J. Evol. Equ.. 10, 465–486 (2010)

  6. Li, Y, Li, S, Yan, W: Sharp well-posedness and ill-posedness of a higher-order modified Camassa-Holm equation. Differ. Integral Equ.. 25, 1053–1074 (2012)