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Global existence of timelike minimal surface of general co-dimension in Minkowski space time

Yinxia Wang1* and Jingzi Liu2

Author Affiliations

1 School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China

2 College of Electric Power, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China

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Boundary Value Problems 2014, 2014:32  doi:10.1186/1687-2770-2014-32

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/32


Received:13 November 2013
Accepted:23 January 2014
Published:7 February 2014

© 2014 Wang and Liu; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we prove that the global existence of solutions to timelike minimal surface equations having arbitrary co-dimension with slow decay initial data in two space dimensions and three space dimensions, provided that the initial value is suitably small.

MSC: 35L70.

Keywords:
timelike minimal surface; global existence; slow decay initial value

1 Introduction

The theory of minimal surfaces has a long history, originating with the papers of Lagrange (1760) and the famous Plateau problem; we refer to the classical papers by Calabi [1] and by Cheng and Yau [2]. Timelike minimal submanifolds may be viewed as simple but nontrivial examples of D-branes, which play an important role in string theory, and the system under consideration here thus has natural generalizations motivated by string theory. The case of timelike surfaces has been investigated by several authors (see [3-5] and [6]). Huang and Kong [7] studied the motion of a relativistic torus in the Minkowski space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M2">View MathML</a>). They derived the equations for the motion of relativistic torus in the Minkowski space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M2">View MathML</a>). This kind of equation also describes the three dimensional timelike extremal submanifolds in the Minkowski space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M5">View MathML</a>. They showed that these equations can be reduced to a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M6">View MathML</a> dimensional quasilinear symmetric hyperbolic system and the system possesses some interesting properties, such as nonstrict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, and the strong null condition (see [8] and [9]). They also found and proved the interesting fact that all plane wave solutions to these equations are lightlike extremal submanifolds and vice versa, except for a type of special solution. For small initial data with compact support, the global existence problem for timelike minimal hypersurfaces has been considered by Brendle [10] and Lindblad [11].

Paul et al.[12] investigated timelike minimal submanifolds of dimension <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M8">View MathML</a>, of Minkowski spacetimes of dimension <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M10">View MathML</a>. The authors considered an embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M5">View MathML</a> into Minkowski spacetime <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M12">View MathML</a> given by the graph of a map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M13">View MathML</a>. Let Greek indices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M14">View MathML</a> take values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M15">View MathML</a> and let uppercase Latin indices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M16">View MathML</a> take values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M17">View MathML</a>. Introduce cartesian coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M18">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M20">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M21">View MathML</a>. The induced metric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M5">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M23">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M26">View MathML</a> is the Minkowski metric. By variational principles (see [13]), they derived the Euler-Lagrange equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M27">View MathML</a>

(1.2)

Moreover for a small initial value with compact support, they also proved the global existence of classical solutions for (1.2).

In this paper, we consider (1.2) with the initial data

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M28">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M29">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M30">View MathML</a>) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M31">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M32">View MathML</a> is a constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M33">View MathML</a> is a small parameter. The aim of this paper is to prove that the Cauchy problem (1.2), (1.3) has a global classical solution, provided that the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M29">View MathML</a> is sufficiently small and satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M36">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M38">View MathML</a>). We reduce the restriction on compact support of the initial data to some decay. In other words, we show the global existence of solutions to timelike minimal surface in two space dimensions and three space dimensions, provided that the initial value is suitably small.

To study (1.2), we note that (1.2) can be written in divergence form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M39">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M40">View MathML</a> is the Minkowski wave operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M41">View MathML</a>, as well as in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M42">View MathML</a>

(1.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M43">View MathML</a>

(1.6)

We raise and lower Greek (intrinsic) indices using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M44">View MathML</a> and its inverse, while Latin (extrinsic) indices are raised and lowered using the identity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M45">View MathML</a> and its inverse. From (1.6), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M46">View MathML</a> has the symmetries

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M47">View MathML</a>

(1.7)

Due to the symmetries, an energy estimate and local well posedness holds for the system (1.5).

The plan of this paper is as follows. In Section 2, we cite some estimates and prove some estimates on the solution of linear wave equations. The global existence of solutions to timelike minimal surface equations with slow decay initial value in two space dimensions and three space dimensions will be proved in Section 3 and Section 4, respectively.

2 Preliminaries

Following Klainerman [14], we introduce a set of partial differential operators

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M48">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M49">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M50">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M51">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M52">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M53">View MathML</a> denotes a product of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M54">View MathML</a> of the vector fields (2.2), (2.3), (2.4), and (2.5). <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M55">View MathML</a> is a multi-index, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M56">View MathML</a>, σ is the number of partial differential operators in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M57">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M58">View MathML</a>

(2.6)

It is easy to prove Lemma 2.1 (see [15]).

Lemma 2.1For any multi-index<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M59">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M60">View MathML</a>

(2.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M61">View MathML</a>

(2.8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M62">View MathML</a>stands for the Poisson bracket, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M63">View MathML</a>are multi-indices, □ is the wave operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M64">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M66">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M67">View MathML</a>are constants.

We need the following lemma that is basically established in [16] and [17]. For completeness, the proof will also be sketched here.

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M68">View MathML</a>and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M69">View MathML</a>

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M70">View MathML</a>is a solution to the following Cauchy problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M71">View MathML</a>

(2.9)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M72">View MathML</a>

(2.10)

Remark 2.1 Under the condition that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M73">View MathML</a>

Tsutaya [18] has showed that the solution of the Cauchy problem (2.9) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M74">View MathML</a>

Obviously, Lemma 2.2 improves the result in [18].

Proof The solution of (2.9) is given

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M75">View MathML</a>

(2.11)

First, we make an estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M76">View MathML</a>; switching to polar coordinates, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M77">View MathML</a>

(2.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M79">View MathML</a>, and χ is the characteristic function of positive numbers.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M80">View MathML</a> be a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M79">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M83">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M84">View MathML</a>

where, as before, φ is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M85">View MathML</a>

We will use the following proposition, which is proved in Kovalyov [19].

Proposition 2.1 (I) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M86">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M87">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M88">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M89">View MathML</a>

(2.13)

(II) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M90">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M91">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M92">View MathML</a>

(2.14)

whereχis the characteristic function of positive numbers.

We next continue to make an estimate for (2.12); we make an estimate for the right-hand side of (2.12) by dividing into two cases.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93">View MathML</a>.

By (2.14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M94">View MathML</a>

(2.15)

We subdivide into three cases again.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M95">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M96">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M97">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M98">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M99">View MathML</a>

(2.16)

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M100">View MathML</a>.

From (2.15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M101">View MathML</a>

(2.17)

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M102">View MathML</a>.

It follows from (2.15) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M103">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M104">View MathML</a>; we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M105">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M106">View MathML</a>

(2.18)

In other words, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M108">View MathML</a>, from (2.16)-(2.18), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M109">View MathML</a>

(2.19)

In what follows, we prove that (2.19) also holds if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93">View MathML</a>. In this case, we also subdivide into two cases.

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M112">View MathML</a>.

By changing variables, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M113">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M114">View MathML</a>

(2.20)

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M115">View MathML</a>.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93">View MathML</a>, thus we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M118">View MathML</a>. From Cases (i)-(iii), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M119">View MathML</a>

(2.21)

In other words, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M93">View MathML</a>, from (2.19)-(2.21), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M121">View MathML</a>

(2.22)

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M122">View MathML</a>.

From (2.12), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M123">View MathML</a>

(2.23)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M124">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M125">View MathML</a>

In what follows, we make an estimate for I and II, respectively, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M126">View MathML</a>.

It follows from (2.14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M127">View MathML</a>

(2.24)

By changing variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M128">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M129">View MathML</a>

(2.25)

By (2.13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M130">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M131">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M132">View MathML</a>

In what follows, we make estimate II by dividing into three cases.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M133">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M134">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M135">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M136">View MathML</a>

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M137">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M138">View MathML</a>

In other words, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M126">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M140">View MathML</a>

(2.26)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M141">View MathML</a>, by changing variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M113">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M143">View MathML</a>

(2.27)

Combining (2.25)-(2.27) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M144">View MathML</a>

(2.28)

Thus (2.22) and (2.28) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M145">View MathML</a>

(2.29)

By Tsutaya [18], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M146">View MathML</a>

(2.30)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M147">View MathML</a>

(2.31)

Equation (2.10) follows from (2.29)-(2.31), and (2.11) immediately. Then we have completed the proof of lemma.  □

The following lemma plays a key role in our main results. It is basically established in [20] and [21].

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M148">View MathML</a>and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M149">View MathML</a>

Assume thatuis a solution to the following Cauchy problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M150">View MathML</a>

(2.32)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M151">View MathML</a>

(2.33)

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M152">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M153">View MathML</a>

and assume thatϕdecays to 0 at infinity. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M154">View MathML</a>

It follows for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M155">View MathML</a>that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M156">View MathML</a>

(2.34)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M157">View MathML</a>.

For the proof of Lemma 2.4, see Klainerman [22].

Using Lemmas 2.2, 2.3 and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M158">View MathML</a> estimate of the linear wave equation with zero initial data, it is not difficulty to prove the following.

Lemma 2.5Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M159">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M160">View MathML</a>be the solution to the Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M161">View MathML</a>

(2.35)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M162">View MathML</a>

(2.36)

By Lemmas 2.2 and 2.3, we can prove the following lemma.

Lemma 2.6Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M160">View MathML</a>is the solution to the Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M165">View MathML</a>

(2.37)

where the coefficients<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M166">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M167">View MathML</a>) are constants. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M168">View MathML</a>

(2.38)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M169">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M170">View MathML</a>depends on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M172">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M173">View MathML</a>.

3 Global existence in three space dimensions

Theorem 3.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M174">View MathML</a>and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M175">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M176">View MathML</a>is a constant. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M178">View MathML</a>the Cauchy problem (1.2), (1.3) has a global classical solution for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M179">View MathML</a>.

Proof The local existence argument follows from the method of Picard iteration [23] (see also [24] and [12]). In what follows, we will prove the global existence of the classical solutions by a continuous induction, or a bootstrap argument. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M180">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M181">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M182">View MathML</a>

To set up the bootstrap argument, we assume that there is a positive constant K so that on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M183">View MathML</a> we have the following estimates for the norms defined in (3.1):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M184">View MathML</a>

(3.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M185">View MathML</a>

To close the bootstrap, we can prove that we can in fact choose K sufficiently large and ε suitably small so that the above inequalities hold independent of T with K replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M186">View MathML</a>.

From Lemma 2.1 and (1.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M187">View MathML</a>

(3.3)

It follows from Lemma 2.4 and (3.2), (3.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M188">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M189">View MathML</a>

(3.4)

if K is sufficiently large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177">View MathML</a> is suitably small.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M191">View MathML</a>; from (1.4) and Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M192">View MathML</a>

(3.5)

where again at most one of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M193">View MathML</a> can satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M194">View MathML</a>.

Applying Lemma 2.6 to (3.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M195">View MathML</a>

(3.6)

if K is sufficiently large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177">View MathML</a> is suitably small.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M197">View MathML</a>, (3.3) may also be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M198">View MathML</a>

(3.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M199">View MathML</a>.

Using Lemma 2.5, (3.2), and (3.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M200">View MathML</a>

if K is sufficiently large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177">View MathML</a> is suitably small.

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M202">View MathML</a>

(3.8)

if K is sufficiently large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177">View MathML</a> is suitably small.

From (3.1), we know that the estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M204">View MathML</a> implies the desired estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M205">View MathML</a>. We have completed the proof of Theorem 3.1. □

4 Global existence in two space dimensions

Theorem 4.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M206">View MathML</a>and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M207">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M176">View MathML</a>is a constant. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M178">View MathML</a>the Cauchy problem (1.2), (1.3) has a global classical solution for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M179">View MathML</a>.

Proof The local existence argument follows from the method of Picard iteration [23] (see also [24] and [12]). In what follows, we will prove global existence of classical solutions by a continuous induction, or bootstrap argument. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M212">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M213">View MathML</a>

(4.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M214">View MathML</a>

To set up the bootstrap argument, we assume that there is a positive constant K so that on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M183">View MathML</a> we have following estimates for the norms defined in (4.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M216">View MathML</a>

(4.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M217">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M218">View MathML</a> is a fixed, arbitrary constant.

To close the bootstrap, we can prove that we can in fact choose K sufficiently large and ε suitably small so that the above inequalities hold independent of T with K replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M186">View MathML</a>.

It follows from Lemma 2.4 and (3.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M188">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M221">View MathML</a>

(4.3)

if K is sufficiently large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177">View MathML</a> is suitably small.

Applying Lemma 2.6 to (3.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M223">View MathML</a>

(4.4)

if K is sufficiently large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177">View MathML</a> is suitably small.

In what follows, we make an estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M204">View MathML</a>. In order to make this estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M204">View MathML</a>, define the following null forms:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M227">View MathML</a>

(4.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M228">View MathML</a>

(4.6)

Let Q symbolically stand for any of the full forms (4.5) and (4.6). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M229">View MathML</a>

(4.7)

for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M230">View MathML</a>.

Let Q be one of null form in (4.5)-(4.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M231">View MathML</a>

(4.8)

Note that the Lagrangian associated to the volume element of the induced metric is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M232">View MathML</a>. For small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M233">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M234">View MathML</a>

and thus the Euler-Lagrange equations take the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M235">View MathML</a>

For small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M233">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M237">View MathML</a>

So we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M238">View MathML</a>

(4.9)

By Lemma 2.1, (4.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M239">View MathML</a>

(4.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M240">View MathML</a>.

From Lemma 2.5 and (4.1), (4.2), (4.8), (4.10) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M241">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M212">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M243">View MathML</a>

(4.11)

if K is sufficiently large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M177">View MathML</a> is suitably small and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M245">View MathML</a>.

From (4.1), we know that the estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M204">View MathML</a> implies the desired estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/32/mathml/M205">View MathML</a>. We have completed the proof of the theorem. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to each part of this work equally and read and approved the final manuscript.

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