Open Access Research

Blow-up phenomena and global existence for the weakly dissipative generalized periodic Degasperis-Procesi equation

Caochuan Ma1*, Ahmed Alsaedi2, Tasawar Hayat23 and Yong Zhou24

Author Affiliations

1 Department of Mathematics, Tianshui Normal University, Tianshui, Gansu, 741001, P.R. China

2 Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

3 Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad, 44000, Pakistan

4 Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, P.R. China

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Boundary Value Problems 2014, 2014:123  doi:10.1186/1687-2770-2014-123


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/123


Received:19 February 2014
Accepted:6 May 2014
Published:20 May 2014

© 2014 Ma et al.; 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

In this paper, we investigate the Cauchy problem of a weakly dissipative generalized periodic Degasperis-Procesi equation. The precise blow-up scenarios of strong solutions to the equation are derived by a direct method. Several new criteria guaranteeing the blow-up of strong solutions are presented. The exact blow-up rates of strong solutions are also determined. Finally, we give a new global existence results to the equation.

MSC: 35G25, 35Q35, 58D05.

Keywords:
weakly dissipative; generalized periodic Degasperis-Procesi equation; blow-up; global existence; blow-up rate

1 Introduction

Recently, the following generalized periodic Degasperis-Procesi equation (μDP) was introduced and studied in [1-3]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2">View MathML</a> is a time-dependent function on the unite circle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M4">View MathML</a> denotes its mean. The μDP equation can be formally described as an evolution equation on the space of tensor densities over the Lie algebra of smooth vector fields on the circle . In [2], the authors verified that the periodic μDP equation describes the geodesic flows of a right-invariant affine connection on the Fréchet Lie group <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M6">View MathML</a> of all smooth and orientation-preserving diffeomorphisms of the circle .

Analogous to the generalized periodic Camassa-Holm (μCH) equation [4-6], μDP equation possesses bi-Hamiltonian form and infinitely many conservation laws. Here we list some of the simplest conserved quantities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M8">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M10">View MathML</a> is an isomorphism between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M12">View MathML</a>. Moreover, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M13">View MathML</a> is also a conserved quantity for the μDP equation.

Obviously, under the constraint of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M14">View MathML</a>, the μDP equation is reduced to the μBurgers equation [7].

It is clear that the closest relatives of the μDP equation are the DP equation [8-11]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M15">View MathML</a>

which was derived by Degasperis and Procesi in [8] as a model for the motion of shallow water waves, and its asymptotic accuracy is the same as for the Camassa-Holm equation.

Generally speaking, energy dissipation is a very common phenomenon in the real world. It is interesting for us to study this kind of equation. Recently, Wu and Yin [12] considered the weakly dissipative Degasperis-Procesi equation. For related studies, we refer to [13] and [14]. Liu and Yin [15] discussed the blow-up, global existence for the weakly dissipative μ-Hunter-Saxton equation.

In this paper, we investigate the Cauchy problem of the following weakly dissipative periodic Degasperis-Procesi equation [16]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M16">View MathML</a>

(1.1)

the constant λ is a nonnegative dissipative parameter and the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M17">View MathML</a> models energy dissipation. Obviously, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M18">View MathML</a> then the equation reduces to the μDP equation. we can rewrite the system (1.1) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M19">View MathML</a>

(1.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M21">View MathML</a> be the associated Green’s function of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M22">View MathML</a>, then the operator can be expressed by its associated Green’s function,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M23">View MathML</a>

where ∗ denotes the spatial convolution. Then equation (1.1) takes the equivalent form of a quasi-linear evolution equation of hyperbolic type:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M24">View MathML</a>

(1.3)

It is easy to check that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M10">View MathML</a> has the inverse

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M26">View MathML</a>

(1.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M28">View MathML</a> commute, the following identities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M29">View MathML</a>

(1.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M30">View MathML</a>

(1.6)

The paper is organized as follows. In Section 2, we briefly give some needed results, including the local well-posedness of equation (1.1), and some useful lemmas and results which will be used in subsequent sections. In Section 3, we establish the precise blow-up scenarios and blow-up criteria of strong solutions. In Section 4, we give the blow-up rate of strong solutions. In Section 5, we give two global existence results of strong solutions.

Remark 1.1 Although blow-up criteria and global existence results of strong solutions to equation (1.1) are presented in [16], our blow-up results improve considerably earlier results.

2 Preliminaries

In this section we recall some elementary results which we want to use in this paper. We list them and skip their proofs for conciseness. Local well-posedness for equation (1.1) can be obtained by Kato’s theory [17], in [16] the authors gave a detailed description on well-posedness theorem.

Theorem 2.1[16]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M32">View MathML</a>; then there is a maximal timeTand a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M33">View MathML</a>

of the Cauchy problems (1.1) which depends continuously on the initial data, i.e. the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M34">View MathML</a>

is continuous.

Remark 2.1 The maximal time of existence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M35">View MathML</a> in Theorem 2.1 is independent of the Sobolev index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>.

Next we present the Sobolev-type inequalities, which play a key role to obtain blow-up results for the Cauchy problem (1.1) in the sequel.

Lemma 2.2[18]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M37">View MathML</a>is such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M38">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M39">View MathML</a>

Lemma 2.3[19]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M40">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M41">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M42">View MathML</a>

wherecis a constant depending only onr.

Lemma 2.4[20]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M44">View MathML</a>, then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M45">View MathML</a>there exists at least one point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M46">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M47">View MathML</a>

and the functionmis almost everywhere differentiable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M48">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M49">View MathML</a>

We also need to introduce the classical particle trajectory method which is motivated by McKean’s deep observation for the Camassa-Holm equation in [21]. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M50">View MathML</a> is the solution of the Camassa-Holm equation and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M51">View MathML</a> satisfies the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M52">View MathML</a>

(2.1)

where T is the maximal existence time of solution, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M53">View MathML</a> is a diffeomorphism of the line. Taking the derivative with respect to x, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M54">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M55">View MathML</a>

(2.2)

which is always positive before the blow-up time.

In addition, integrating both sides of the first equation in equation (1.1) with respect to x on , we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M57">View MathML</a>

it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M58">View MathML</a>

(2.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M59">View MathML</a>

(2.4)

3 Blow-up solutions

In this section, we are able to derive an import estimate for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M60">View MathML</a>-norm of strong solutions. This enables us to establish precise blow-up scenario and several blow-up results for equation (1.1).

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>be given and assume the T is the maximal existence time of the corresponding solutionuto equation (1.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M64">View MathML</a>

(3.1)

Proof The first equation of the Cauchy problem (1.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M65">View MathML</a>

In view of equation (1.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M66">View MathML</a>

A direct computation implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M67">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M68">View MathML</a>

(3.2)

So we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M69">View MathML</a>

In view of equation (2.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M70">View MathML</a>

Combing the above relations, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M71">View MathML</a>

Integrating the above inequality with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M72">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M73">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M74">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M75">View MathML</a>

In view of the diffeomorphism property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M53">View MathML</a>, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M77">View MathML</a>

This completes the proof of Lemma 3.1. □

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>be given and assume thatTis the maximal existence time of the corresponding solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2">View MathML</a>to the Cauchy problem (1.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63">View MathML</a>. If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M82">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M83">View MathML</a>

then the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M84">View MathML</a>-norm of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M85">View MathML</a>does not blow up on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M86">View MathML</a>.

Proof We assume that c is a generic positive constant depending only on s. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M41">View MathML</a>. Applying the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M88">View MathML</a> to the first one in equation (1.3), multiplying by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M89">View MathML</a>, and integrating over , we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M91">View MathML</a>

(3.3)

Let us estimate the first term of the above equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M92">View MathML</a>

(3.4)

where we used Lemma 2.3 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M93">View MathML</a>. Furthermore, we estimate the second term of the right hand side of equation (3.3) in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M94">View MathML</a>

(3.5)

Combing equations (3.4) and (3.5) with equation (3.3) we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M95">View MathML</a>

An application of Gronwall’s inequality and the assumption of the theorem yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M96">View MathML</a>

This completes the proof of the theorem. □

The following result describes the precise blow-up scenario. Although the result which is proved in [16], our method is new, concise, and direct.

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>be given and assume thatTis the maximal existence time of the corresponding solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2">View MathML</a>to the Cauchy problem (1.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63">View MathML</a>. Then the corresponding solution blows up in finite time if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M101">View MathML</a>

Proof Since the maximal existence time T is independent of the choice of s by Theorem 2.1, applying a simple density argument, we only need to consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M102">View MathML</a>. Multiplying the first one in equation (1.2) by y and integrating over with respect to x yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M104">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M105">View MathML</a> is bounded from below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M106">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M107">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M108">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M109">View MathML</a>

Applying Gronwall’s inequality then yields for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M110">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M111">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M112">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M114">View MathML</a>, Lemma 2.2 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M115">View MathML</a>

Theorem 3.1 ensures that the solution u does not blow up in finite time. On the other hand, by the Sobolev embedding theorem it is clear that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M116">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M117">View MathML</a>. This completes the proof of the theorem. □

We now give first sufficient conditions to guarantee wave breaking.

Theorem 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>andTbe the maximal time of the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2">View MathML</a>to equation (1.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63">View MathML</a>. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M122">View MathML</a>

then the corresponding solution to equation (1.1) blow up in finite time in the following sense: there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M123">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M124">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M125">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M126">View MathML</a>

Proof As mentioned early, we only need to consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M102">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M128">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M129">View MathML</a> be a point where this minimum is attained by using Lemma 2.4. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M130">View MathML</a>

Differentiating the first one in equation (1.3) with respect to x, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M131">View MathML</a>

From equation (1.6) we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M132">View MathML</a>

(3.6)

Obviously <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M134">View MathML</a>. Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M135">View MathML</a> into equation (3.6), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M136">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M137">View MathML</a>

Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M138">View MathML</a>

Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M139">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M140">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141">View MathML</a>. From the above inequality we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M142">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M143">View MathML</a>

then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M123">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M145">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M146">View MathML</a>. Theorem 3.3 implies that the solution u blows up in finite time. □

We give another blow-up result for the solutions of equation (1.1).

Theorem 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>andTbe the maximal time of the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2">View MathML</a>to equation (1.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63">View MathML</a>is odd satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M152">View MathML</a>, then the corresponding solution to equation (1.1) blows up in finite time.

Proof By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M153">View MathML</a>, we can check the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M154">View MathML</a>

is also a solution of equation (1.1), therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M50">View MathML</a> is odd for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141">View MathML</a>. By continuity with respect to x of u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M157">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M158">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M159">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141">View MathML</a>. From equation (3.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M161">View MathML</a>

Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M162">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M163">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141">View MathML</a>. From the above inequality we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M165">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M166">View MathML</a>

there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M123">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M168">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M146">View MathML</a>. Theorem 3.3 implies that the solution u blows up in finite time. □

4 Blow-up rate

In this section, we consider the blow-up profile; the blow-up rate of equation (1.1) with respect to time can be shown as follows.

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>andTbe the maximal time of the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2">View MathML</a>to equation (1.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63">View MathML</a>. IfTis finite, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M174">View MathML</a>

Proof It is inferred from Lemma 2.4 that the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M175">View MathML</a>

is locally Lipschitz with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M178">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141">View MathML</a>. Then we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M180">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M181">View MathML</a>

(4.1)

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M182">View MathML</a>

Now fix any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M183">View MathML</a>. In view of Theorem 3.1, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M184">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M185">View MathML</a>. Being locally Lipschitz, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M186">View MathML</a> is absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M86">View MathML</a>. It then follows from the above inequality that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M186">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M189">View MathML</a> and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M190">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M186">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M189">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M193">View MathML</a>

It is found from equation (4.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M194">View MathML</a>

(4.2)

Integrating both sides of equation (4.2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M195">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M196">View MathML</a>

(4.3)

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M197">View MathML</a>

(4.4)

By the arbitrariness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M198">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M199">View MathML</a>

(4.5)

This completes the proof of the theorem. □

5 Global existence

In this section, we will present some global existence results. Let us now prove the following lemma.

Lemma 5.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>be given and assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M35">View MathML</a>is the maximal existence time of the corresponding solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2">View MathML</a>to the Cauchy problem (1.1). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M204">View MathML</a>be the unique solution of equation (2.1). Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M205">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M9">View MathML</a>.

Proof By the first one in equation (1.2) and equation (2.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M207">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M208">View MathML</a>

 □

Lemma 5.1 and equation (2.2) imply that y and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M209">View MathML</a> have the same sign.

Theorem 5.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M31">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M212">View MathML</a>does not change sign, then the corresponding solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M2">View MathML</a>to equation (1.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63">View MathML</a>exists globally in time.

Proof By equation (2.1), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M53">View MathML</a> is diffeomorphism of the line and the periodicity of u with respect to spatial variable x, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M141">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M129">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M218">View MathML</a>.

We first consider the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M219">View MathML</a> on , in which case Lemma 5.1 ensures that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M221">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M222">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M223">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M224">View MathML</a>.

On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M225">View MathML</a> on , then Lemma 5.1 ensures that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M227">View MathML</a>. Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M222">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M229">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M224">View MathML</a>. By using Theorem 3.2, we immediately conclude that the solution is global. This completes the proof of the theorem. □

Corollary 5.3If the initial value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M231">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M232">View MathML</a>

then the corresponding solutionuof the initial value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M63">View MathML</a>exists globally in time.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M234">View MathML</a>, by Lemma 2.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M235">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M236">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M237">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M238">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/123/mathml/M239">View MathML</a>

 □

Thus the theorem is proved by using Theorem 5.2.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

This work is partially supported by the NSFC (Grant No. 11101376) the HiCi Project (Grant No. 27-130-35-HiCi).

References

  1. Lenells, J, Misiolek, G, Tiğlay, F: Integrable evolution equations on spaces of tensor densities and their peakon solutions. Commun. Math. Phys.. 299, 129–161 (2010). Publisher Full Text OpenURL

  2. Escher, J, Kohlmann, M, Kolev, B: Geometric aspects of the periodic μDP equation. Prog. Nonlinear Differ. Equ. Appl.. 80, 193–209 (2011)

  3. Fu, Y, Liu, Y, Qu, C: On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations. J. Funct. Anal.. 262, 3125–3158 (2012). Publisher Full Text OpenURL

  4. Khesin, B, Lenells, J, Misiołek, G: Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms. Math. Ann.. 324, 617–656 (2008)

  5. Gui, G, Liu, Y, Zhu, M: On the wave-breaking phenomena and global existence for the generalized periodic Camassa-Holm equation. Int. Math. Res. Not.. 21, 4858–4903 (2012)

  6. Chen, R, Lenells, J, Liu, Y: Stability of the μ-Camassa-Holm peakons. J. Nonlinear Sci.. 23, 97–112 (2013). Publisher Full Text OpenURL

  7. Christov, O: On the nonlocal symmetries of the μ-Camassa-Holm equation. J. Nonlinear Math. Phys.. 19, 411–427 (2012). Publisher Full Text OpenURL

  8. Degasperis, A, Procesi, M: Asymptotic integrability. Symmetry and Perturbation Theory, pp. 23–37. World Scientific, River Edge In: SPT 98, Rome, 1998. (1999)

  9. Zhou, Y: Blow up phenomena for the integrable Degasperis-Procesi equation. Phys. Lett. A. 328, 157–162 (2004). Publisher Full Text OpenURL

  10. Chen, W: On solutions to the Degasperis-Procesi equation. J. Math. Anal. Appl.. 379, 351–359 (2011). Publisher Full Text OpenURL

  11. Liu, Y, Yin, Z: Global existence and blow-up phenomena for the Degasperis-Procesi equation. Commun. Math. Phys.. 267, 801–820 (2006). Publisher Full Text OpenURL

  12. Wu, S, Yin, Z: Blow up, blow-up and decay of the solution of the weakly dissipative Degasperis-Procesi equation. SIAM J. Math. Anal.. 40, 475–490 (2008). Publisher Full Text OpenURL

  13. Guo, Z: Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa-Holm equation. J. Math. Phys.. 49, Article ID 033516 (2008)

  14. Guo, Z: Some properties of solutions to the weakly dissipative Degasperis-Procesi equation. J. Differ. Equ.. 246, 4332–4344 (2009). Publisher Full Text OpenURL

  15. Liu, J, Yin, Z: On the Cauchy problem of a weakly dissipative μ HS equation. Preprint. arXiv:1108.4550 (2011)

  16. Kohlmann, M: Global existence and blow-up for a weakly dissipative μDP equation. Nonlinear Anal.. 74, 4746–4753 (2011). Publisher Full Text OpenURL

  17. Kato, T: Quasi-linear equations of evolution, with applications to partial differential equations. Spectral Theory and Differential Equations, pp. 25–70. Springer, Berlin (1975)

  18. Constantin, A: On the blow-up of solutions of a periodic shallow water equation. J. Nonlinear Sci.. 10, 391–399 (2000). Publisher Full Text OpenURL

  19. Kato, T, Ponce, G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math.. 41, 891–907 (1988). Publisher Full Text OpenURL

  20. Constantin, A, Escher, J: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math.. 181, 229–243 (1998). Publisher Full Text OpenURL

  21. McKean, HP: Breakdown of a shallow water equation. Asian J. Math.. 2, 767–774 (1998)