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Global existence and asymptotic behavior of solutions to a class of fourth-order wave equations

Zhitao Zhuang and Yuanzhang Zhang*

Author Affiliations

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

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Boundary Value Problems 2013, 2013:168  doi:10.1186/1687-2770-2013-168

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/168


Received:9 May 2013
Accepted:27 June 2013
Published:16 July 2013

© 2013 Zhuang and Zhang; 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

This paper is concerned with the Cauchy problem for a class of fourth-order wave equations in an n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, a solution space is defined. We prove the global existence and optimal decay estimate of the solution in the corresponding Sobolev spaces by the contraction mapping principle provided that the initial value is suitably small.

MSC: 35L30, 35L75.

Keywords:
fourth-order wave equation; global existence; decay estimate

1 Introduction

We investigate the Cauchy problem for a class of fourth-order wave equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M1">View MathML</a>

(1.1)

with the initial value

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M2">View MathML</a>

(1.2)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M3">View MathML</a> is the unknown function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M5">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M6">View MathML</a> is a constant. The nonlinear term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M7">View MathML</a> is a smooth function with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M8">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M9">View MathML</a>.

Equation (1.1) is reduced to the classical Cahn-Hilliard equation if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M10">View MathML</a> (see [1]), which has been widely studied by many authors. Galenko et al.[2-5] proposed to add inertial term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M11">View MathML</a> to the classical Cahn-Hilliard equation in order to model non-equilibrium decompositions caused by deep supercooling in certain glasses. For more background, we refer to [4-6] and references therein. It is obvious that (1.1) is a fourth-order wave equation. For global existence and asymptotic behavior of solutions to more higher order wave equations, we refer to [7-14] and references therein.

Very recently, global existence and asymptotic behavior of solutions to the problem (1.1), (1.2) were established by Wang and Wei [7] under smallness condition on the initial data. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M13">View MathML</a>, they obtained the following decay estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M14">View MathML</a>

(1.3)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M16">View MathML</a>. The main purpose of this paper is to refine the result in [7] and prove the following decay estimate for the solution to the problem (1.1), (1.2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M17">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M18">View MathML</a> data,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M19">View MathML</a>

(1.4)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M21">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M22">View MathML</a> is assumed to be small. We also establish the decay estimate for the solution to the problem (1.1), (1.2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M17">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M24">View MathML</a> data,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M25">View MathML</a>

(1.5)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M21">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M28">View MathML</a> is assumed to be small. Compared to the result in [7], we obtain a better decay estimate of solutions for small initial data.

The paper is organized as follows. In Section 2, we study the decay property of the solution operators appearing in the solution formula. We prove global existence and asymptotic behavior of solutions for the Cauchy problem (1.1), (1.2) in Sections 3 and 4, respectively.

Notations We introduce some notations which are used in this paper. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M29">View MathML</a> denote the Fourier transform of u defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M30">View MathML</a>

We denote its inverse transform by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M31">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M33">View MathML</a> denotes the usual Lebesgue space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M34">View MathML</a>. The usual Sobolev space of order s is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M35">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M36">View MathML</a>. The corresponding homogeneous Sobolev space of order s is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M37">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M38">View MathML</a>; when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M39">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M41">View MathML</a>. We note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M42">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M43">View MathML</a>.

For a nonnegative integer k, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M44">View MathML</a> denotes the totality of all the kth order derivatives with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M45">View MathML</a>. Also, for an interval I and a Banach space X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M46">View MathML</a> denotes the space of k-times continuously differential functions on I with values in X.

Throughout the paper, we denote every positive constant by the same symbol C or c without confusion. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M47">View MathML</a> is the Gauss symbol.

2 Decay property

The aim of this section is to derive the solution formula to the Cauchy problem (1.1), (1.2). Without loss of generality, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M48">View MathML</a>. We first study the linearized equation of (1.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M49">View MathML</a>

(2.1)

with the initial data in (1.2). Taking the Fourier transform, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M50">View MathML</a>

(2.2)

The corresponding initial value are given as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M51">View MathML</a>

(2.3)

The characteristic equation of (2.2) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M52">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M53">View MathML</a> be the corresponding eigenvalues, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M54">View MathML</a>

(2.4)

The solution to the problem (2.2), (2.3) is given in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M55">View MathML</a>

(2.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M56">View MathML</a>

(2.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M57">View MathML</a>

(2.7)

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M59">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M61">View MathML</a>, respectively, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M62">View MathML</a> denotes the inverse Fourier transform. Then, applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M31">View MathML</a> to (2.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M64">View MathML</a>

(2.8)

By the Duhamel principle, we obtain the solution formula to (1.1), (1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M65">View MathML</a>

(2.9)

The aim of this section is to establish decay estimates of the solution operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M67">View MathML</a> appearing in the solution formula (2.8).

Lemma 2.1The solution of the problem (2.2), (2.3) verifies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M68">View MathML</a>

(2.10)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M69">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M70">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M71">View MathML</a>.

Proof We apply the energy method in the Fourier space to prove (2.10). Such an energy method was first developed in [15]. We multiply (2.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M72">View MathML</a> and take the real part. This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M73">View MathML</a>

(2.11)

Multiplying (2.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M74">View MathML</a> and taking the real part, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M75">View MathML</a>

(2.12)

Combining (2.11) and (2.12) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M76">View MathML</a>

(2.13)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M77">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M78">View MathML</a>

A simple computation implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M79">View MathML</a>

(2.14)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M80">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M81">View MathML</a>

It follows from (2.14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M82">View MathML</a>

(2.15)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M83">View MathML</a>

Using (2.13) and (2.15), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M84">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M85">View MathML</a>

which together with (2.14) proves the desired estimates (2.10). Then we have completed the proof of the lemma. □

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M86">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M87">View MathML</a>be the fundamental solutions to (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then we have the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M88">View MathML</a>

(2.16)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M89">View MathML</a>

(2.17)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M69">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M70">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M71">View MathML</a>.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M93">View MathML</a>, from (2.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M94">View MathML</a>

Substituting the equalities into (2.10) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M93">View MathML</a>, we get (2.16).

In what follows, we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M96">View MathML</a>, it follows from (2.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M97">View MathML</a>

Substituting the equalities into (2.10) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M96">View MathML</a>, we get (2.17). The lemma is proved. □

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M86">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M87">View MathML</a>be the fundamental solutions to (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then there exists a small positive number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M101">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M102">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M70">View MathML</a>, we have the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M104">View MathML</a>

(2.18)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M105">View MathML</a>

(2.19)

Proof For sufficiently small ξ, using the Taylor formula, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M106">View MathML</a>

(2.20)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M107">View MathML</a>

(2.21)

It follows from (2.6) and (2.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M108">View MathML</a>

(2.22)

Equations (2.18) and (2.19) follow from (2.20)-(2.22). The proof of Lemma 2.3 is completed. □

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M109">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M110">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M111">View MathML</a>

(2.23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M112">View MathML</a>

(2.24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M113">View MathML</a>

(2.25)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M114">View MathML</a>

(2.26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M115">View MathML</a>

(2.27)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M116">View MathML</a>

(2.28)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M117">View MathML</a>in (2.23).

Proof By the property of the Fourier transform and (2.16), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M118">View MathML</a>

(2.29)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M101">View MathML</a> is a positive constant in Lemma 2.3, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M121">View MathML</a>.

By a straight computation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M122">View MathML</a>

(2.30)

It follows from the Hausdorff-Young inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M123">View MathML</a>

(2.31)

Combining (2.29)-(2.31) yields (2.23). Similarly, we can prove (2.24)-(2.28). Thus we have completed the proof of the lemma. □

3 Global existence and decay estimate (I)

The purpose of this section is to prove global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M18">View MathML</a> data. We need the following lemma, which comes from [16] (see also [17]).

Lemma 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M125">View MathML</a>is smooth, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M126">View MathML</a>is a vector function. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M127">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M128">View MathML</a>is an integer) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M129">View MathML</a>. Then, for the integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M130">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M131">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M133">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M134">View MathML</a>. Furthermore, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M135">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M136">View MathML</a>

(3.2)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M138">View MathML</a>.

Based on the decay estimates of solutions to the linear problem (2.1), (1.2), we define the following solution space:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M139">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M140">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M141">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M142">View MathML</a>

The Gagliardo-Nirenberg inequality gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M143">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M145">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M146">View MathML</a>).

Theorem 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M148">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M149">View MathML</a>). Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M150">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M151">View MathML</a>is suitably small, the Cauchy problem (1.1), (1.2) has a unique global solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M152">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M153">View MathML</a>

Moreover, the solution satisfies the decay estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M154">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M155">View MathML</a>

(3.5)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M156">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M157">View MathML</a>.

Proof Let us define the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M158">View MathML</a>

(3.6)

Using (2.23), (2.24), (2.27), (3.1) and (3.3), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M15">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M160">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M161">View MathML</a>

(3.7)

It follows from (3.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M162">View MathML</a>

(3.8)

By exploiting (3.8), (2.25), (2.26), (2.28), (3.1) and (3.3), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M163">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M164">View MathML</a>

The above inequality implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M165">View MathML</a>

(3.9)

Combining (3.7) and (3.9) and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M151">View MathML</a> and R suitably small, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M167">View MathML</a>

(3.10)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M168">View MathML</a>, (3.6) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M169">View MathML</a>

(3.11)

By (2.27), (3.2) and (3.3), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M170">View MathML</a>, we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M171">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M172">View MathML</a>

(3.12)

Similarly, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M173">View MathML</a>, from (3.11), (2.28) and (3.2), (3.3), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M174">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M175">View MathML</a>

(3.13)

Combining (3.12) and (3.13) and taking R suitably small yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M176">View MathML</a>

(3.14)

From (3.10) and (3.14), we know that Φ is a strictly contracting mapping. Consequently, we conclude that there exists a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M177">View MathML</a> of the mapping Φ, which is a solution to (1.1), (1.2). We have completed the proof of the theorem. □

4 Global existence and decay estimate (II)

In the previous section, we have proved global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M18">View MathML</a> data. The purpose of this section is to establish global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M24">View MathML</a> data. Based on the decay estimates of solutions to the linear problem (2.1), (1.2), we define the following solution space:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M180">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M181">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M141">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M183">View MathML</a>

Thanks to the Gagliardo-Nirenberg inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M184">View MathML</a>

(4.1)

Theorem 4.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M186">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M21">View MathML</a>). Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M188">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M151">View MathML</a>is suitably small, the Cauchy problem (1.1), (1.2) has a unique global solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M152">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M191">View MathML</a>

Moreover, the solution satisfies the decay estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M192">View MathML</a>

(4.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M193">View MathML</a>

(4.3)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M194">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M157">View MathML</a>.

Proof Let the mapping Φ be defined in (3.6).

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M196">View MathML</a>, (2.23), (2.24), (2.27), (3.1) and (4.1) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M197">View MathML</a>

Thus we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M198">View MathML</a>

(4.4)

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M199">View MathML</a> to (3.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M200">View MathML</a>

(4.5)

By using (2.25), (2.26), (2.28), (3.1), (4.1), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M157">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M202">View MathML</a>

This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M203">View MathML</a>

(4.6)

Combining (4.4) and (4.6) and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M204">View MathML</a> and R suitably small, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M205">View MathML</a>

(4.7)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M168">View MathML</a>, by using (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M207">View MathML</a>

(4.8)

It follows from (2.27), (3.2) and (4.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M170">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M209">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M210">View MathML</a>

(4.9)

Similarly, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M173">View MathML</a>, from (4.5), (2.28), (3.2) and (4.1), we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M212">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M213">View MathML</a>

(4.10)

Using (4.9) and (4.10) and taking R suitably small yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M214">View MathML</a>

(4.11)

It follows from (4.7) and (4.11) that Φ is a strictly contracting mapping. Consequently, we infer that there exists a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/168/mathml/M177">View MathML</a> of the mapping Φ, which is a solution to (1.1), (1.2). 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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