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Asymptotic profile of solutions to the semilinear beam equation

Yunpeng Zhang1* and Yanling Li2

Author Affiliations

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

2 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 2014, 2014:84  doi:10.1186/1687-2770-2014-84

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/84


Received:26 January 2014
Accepted:1 April 2014
Published:17 April 2014

© 2014 Zhang and Li; 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 credited.

Abstract

In this paper, we investigate the initial value problem for the semilinear beam equation. Under a small condition on the initial value, we prove the global existence and optimal decay estimate of solutions. Moreover, we show that as time tends to infinity, the solution is asymptotic to a diffusion wave, which is given explicitly in terms of the solution of parabolic equation.

MSC: 35L30, 35L75.

Keywords:
beam equation; decay estimate; asymptotic profile; diffusion wave

1 Introduction

We investigate the initial value problem for the following semilinear beam equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M1">View MathML</a>

(1.1)

with the initial value

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M2">View MathML</a>

(1.2)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M3">View MathML</a> is the unknown function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M6">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M7">View MathML</a> are constants. The nonlinear term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M8">View MathML</a> is a given smooth function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M9">View MathML</a>. More precisely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M10">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M12">View MathML</a>.

This initial value problem was studied by [1,2] when f satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M13">View MathML</a>

(1.3)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M14">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M15">View MathML</a>. Here C is independent of v, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M16">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M17">View MathML</a>. [1] proved that there exists a global solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M18">View MathML</a> to the problem (1.1), (1.2) under smallness condition on the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M20">View MathML</a>. In particular, they showed the decay estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M21">View MathML</a>

(1.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M22">View MathML</a>

(1.5)

In addition to the above assumptions, suppose that the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M23">View MathML</a>, Takeda and Yoshikawa [2] established the following asymptotic profile of global solution:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M24">View MathML</a>

(1.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M25">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M26">View MathML</a>

The main purpose of our present paper is two-fold: first, we try to recover all the results about global existence and decay estimate of solution in Takeda and Yoshikawa [1] under some assumptions on the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M27">View MathML</a> and the nonlinear function f, which is much weaker than those needed in Takeda and Yoshikawa’s arguments. More precisely, the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M29">View MathML</a> and (1.3) have been relaxed to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M31">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M32">View MathML</a>) and the nonlinear function f satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M33">View MathML</a> in this paper. Second, we show that the solution is asymptotic to a diffusion wave, given explicitly in terms of the solution of parabolic equation that is different from the one in [2]. For the details, we refer to Theorem 4.1. Moreover, under some additional assumptions on the initial data, we also prove that the convergence rates of our new asymptotic profile are better than that obtained by [2]. For details, we refer to Theorem 4.2.

The study of the global existence and asymptotic behavior of solutions to hyperbolic-type equations has a long history. We refer to [3,4] for hyperbolic equations, [5-7] for the damped wave equation and [8-16] for various aspects of dissipation of the plate equation.

The paper is organized as follows. In Section 2, we study the decay property of the solution to the linear problem. Then, in Section 3, we prove the global existence and decay estimate of the solutions. Finally, we prove that the solution is asymptotic to a diffusion wave, which is given explicitly in terms of the solution of the parabolic equation in Section 4.

Notations We give some notations which are used in this paper. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M34">View MathML</a> denote the Fourier transform of u defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M35">View MathML</a>

and we denote its inverse transform by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M36">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M38">View MathML</a> denotes the usual Lebesgue space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M39">View MathML</a>. The usual Sobolev space of s is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M40">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M41">View MathML</a>.

Finally, in this paper, we denote every positive constant by the same symbol C or c which will not lead to any confusion. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M42">View MathML</a> is the Gauss symbol.

2 Linear problem

2.1 Solution formula

The aim of this subsection is to derive the solution formula for the problem (1.1), (1.2). We first investigate the linearized equation of (1.1):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M43">View MathML</a>

(2.1)

with the initial data in (1.2), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M44">View MathML</a>. We apply the Fourier transform to (2.1). This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M45">View MathML</a>

(2.2)

The corresponding initial values are given as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M46">View MathML</a>

(2.3)

The characteristic equation of (2.2) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M47">View MathML</a>

(2.4)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M48">View MathML</a> be the corresponding eigenvalues of (2.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M49">View MathML</a>

(2.5)

The solution to the problem (2.2), (2.3) in the Fourier space is then given explicitly in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M50">View MathML</a>

(2.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M51">View MathML</a>

(2.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M52">View MathML</a>

(2.8)

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M54">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M55">View MathML</a>

(2.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M56">View MathML</a>

(2.10)

respectively, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M36">View MathML</a> denotes the inverse Fourier transform. Then, applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M36">View MathML</a> to (2.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M59">View MathML</a>

(2.11)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M60">View MathML</a>

(2.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M61">View MathML</a> is a smooth function; it satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M11">View MathML</a>.

2.2 Decay property

The aim of this subsection is to establish decay estimates of the solution operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M64">View MathML</a> appearing in (2.9) and (2.10), respectively.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M65">View MathML</a>

(2.13)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M66">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M67">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M68">View MathML</a>.

Proof Multiplying (2.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M69">View MathML</a> and taking the real part yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M70">View MathML</a>

(2.14)

Multiplying (2.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M71">View MathML</a> and taking the real part, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M72">View MathML</a>

(2.15)

Multiplying both sides of (2.14) by 2 and summing up the resulting equation and (2.15) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M73">View MathML</a>

(2.16)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M74">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M75">View MathML</a>

A simple computation implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M76">View MathML</a>

(2.17)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M77">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M78">View MathML</a>

It follows from (2.17) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M79">View MathML</a>

(2.18)

Using (2.16) and (2.18), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M80">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M81">View MathML</a>

which together with (2.17) proves the desired estimates (2.13). Then we have completed the proof of the lemma. □

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M83">View MathML</a>be the fundamental solution of (2.1) in the Fourier space, which are given in (2.7) and (2.8), respectively. Then we have the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M84">View MathML</a>

(2.19)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M85">View MathML</a>

(2.20)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M66">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M67">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M68">View MathML</a>.

Proof Firstly, we investigate the problem (2.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M89">View MathML</a>, from (2.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M90">View MathML</a>

Substituting the equalities into (3.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M89">View MathML</a>, we get (2.19).

In what follows, we consider the problem (2.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M92">View MathML</a>; it follows from (2.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M93">View MathML</a>

Substituting the equalities into (2.13) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M94">View MathML</a>, we get the desired estimate (2.20). The lemma is proved. □

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M95">View MathML</a>

be the fundamental solution to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M96">View MathML</a>.

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M83">View MathML</a>be the fundamental solution of (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then there is a small positive number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M99">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M67">View MathML</a>, we have the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M102">View MathML</a>

(2.21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M103">View MathML</a>

(2.22)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M104">View MathML</a>

(2.23)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M105">View MathML</a>

(2.24)

We rewrite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M82">View MathML</a> in (2.6) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M107">View MathML</a>

(2.25)

For sufficiently small ξ, from (2.24) and (2.25), we immediately obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M108">View MathML</a>

For sufficiently small ξ, from (2.7) and (2.24), we immediately get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M109">View MathML</a>

Thus we get (2.21). The other estimates are proved similarly and we omit the details. The proof of Lemma 2.3 is completed. □

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M53">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M54">View MathML</a>be the fundamental solutions of (2.1), which are given in (2.9) and (2.10), respectively. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M112">View MathML</a>, and letk, j, andlbe nonnegative integers. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M113">View MathML</a>

(2.26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M114">View MathML</a>

(2.27)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M115">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M116">View MathML</a>in (2.26). Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M117">View MathML</a>

(2.28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M118">View MathML</a>

(2.29)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M119">View MathML</a>in (2.28) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M120">View MathML</a>in (2.29).

Proof We only prove (2.26). By the Plancherel theorem and (2.19), the Hausdorff-Young inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M121">View MathML</a>

For the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M122">View MathML</a>, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M123">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M124">View MathML</a>

where we used the Hölder inequality with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M125">View MathML</a> and the Hausdorff-Young inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M126">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M127">View MathML</a>. On the other hand, we can estimate the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M128">View MathML</a> simply as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M129">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M116">View MathML</a>.

Combining the above three inequalities yields (2.26). This completes the proof of Lemma 2.4. □

From Lemma 2.4, we immediately have the following corollary.

Corollary 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M53">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M54">View MathML</a>be the fundamental solution of (2.1), which are given in (2.8) and (2.9), respectively. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M112">View MathML</a>, and letk, j, andlbe nonnegative integers. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M134">View MathML</a>

(2.30)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M135">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M136">View MathML</a>. Also we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M137">View MathML</a>

(2.31)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M138">View MathML</a>.

Lemma 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M53">View MathML</a>be the fundamental solution of (2.1), given in (2.8) and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M140">View MathML</a>be the fundamental solution of (2.1), given in (2.8). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M112">View MathML</a>, and letk, j, andlbe nonnegative integers. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M142">View MathML</a>

(2.32)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M119">View MathML</a>. Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M144">View MathML</a>

(2.33)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M120">View MathML</a>.

Proof The proof of Lemma 2.5 is similar to the proof of Lemma 2.4. By employing (2.21) and (2.23), we can prove Lemma 2.5. We omit the details. □

3 Global existence and asymptotic behavior of solutions to (1.1), (1.2)

The purpose of this section is to prove global existence and optimal decay estimate of solutions to the initial value problem (1.1), (1.2). We need the following lemma, which comes from [17] (see also [18]).

Lemma 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M146">View MathML</a>is a smooth function. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M147">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M148">View MathML</a>is an integer) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M149">View MathML</a>. Then for integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M150">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M151">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M152">View MathML</a>, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M153">View MathML</a>

(3.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M155">View MathML</a>.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M156">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M8">View MathML</a>is a smooth function and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M158">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M160">View MathML</a>. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M161">View MathML</a>

Then there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M162">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M163">View MathML</a>, and the initial value problem (1.1), (1.2) has a unique global solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M164">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M165">View MathML</a>

Moreover, the solution satisfies the decay estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M166">View MathML</a>

(3.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M167">View MathML</a>

(3.3)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M168">View MathML</a>in (3.2) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M169">View MathML</a>in (3.3).

Proof The existence and uniqueness of small solutions can be proved by the contraction mapping principle. Here we only show the decay estimates (3.2) and (3.3) for the solution u of (2.12) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M170">View MathML</a> with some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M171">View MathML</a>. To this end, we introduce the quantity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M172">View MathML</a>

(3.4)

Here we note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M173">View MathML</a>

(3.5)

provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M32">View MathML</a>. This follows from the Gagliardo-Nirenberg inequality, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M175">View MathML</a> and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M176">View MathML</a> in (3.4).

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M177">View MathML</a> to (2.12) and taking the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M178">View MathML</a> norm, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M179">View MathML</a>

(3.6)

Firstly, we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M180">View MathML</a>. We apply (2.26) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M184">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M186">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M187">View MathML</a>). This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M188">View MathML</a>

(3.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M189">View MathML</a>. Similarly, applying (2.27) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183">View MathML</a> to the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M193">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M194">View MathML</a>

(3.8)

We estimate the nonlinear term J. We divide J into two parts and write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M195">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M197">View MathML</a> are corresponding to the time intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M199">View MathML</a>, respectively. For the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M196">View MathML</a>, we apply (2.30) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183">View MathML</a>. This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M204">View MathML</a>

(3.9)

Here we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M205">View MathML</a> by Lemma 3.1. Thus we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M206">View MathML</a>. Therefore we can estimate the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M207">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M208">View MathML</a>

On the other hand, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M209">View MathML</a> by Lemma 3.1. Therefore, using (3.5), we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M210">View MathML</a>. Consequently, we can estimate the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M211">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M212">View MathML</a>

Finally, we estimate the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M197">View MathML</a> on the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M199">View MathML</a>. Applying (2.30) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M216">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183">View MathML</a>, and using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M218">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M219">View MathML</a>

(3.10)

Thus we have shown that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M220">View MathML</a>

Substituting all these estimates into (3.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M221">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M222">View MathML</a>. Consequently, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M223">View MathML</a>, from which we can deduce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M224">View MathML</a>, provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M225">View MathML</a> is suitably small. This proves the decay estimate (3.2).

In what follows, we prove the decay estimate (3.3) for the time derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M226">View MathML</a>. For this purpose we differentiate (2.12) with respect to t to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M227">View MathML</a>

(3.11)

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M177">View MathML</a> to (3.11) and taking the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M178">View MathML</a> norm, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M230">View MathML</a>

(3.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M231">View MathML</a>. For the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M232">View MathML</a>, we apply (2.28) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183">View MathML</a> to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M236">View MathML</a>

Also, for the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M237">View MathML</a>, applying (2.29) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M241">View MathML</a>

To estimate the nonlinear term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M242">View MathML</a>, we rewrite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M243">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M245">View MathML</a> correspond to the time intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M199">View MathML</a>, respectively. For the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M248">View MathML</a>, we apply (2.31) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M182">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183">View MathML</a>. This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M252">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M206">View MathML</a> as before, we can estimate the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M254">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M255">View MathML</a>

Also, the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M256">View MathML</a> is estimated similarly as before and we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M257">View MathML</a>. Finally, we estimate the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M245">View MathML</a> by applying (2.31) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M260">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M183">View MathML</a> and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M262">View MathML</a>

where we used the estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M263">View MathML</a>. Consequently we have shown that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M264">View MathML</a>

Substituting all these estimates together with the previous estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M224">View MathML</a> into (3.12), we arrive at the desired estimate (3.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M231">View MathML</a>. This completes the proof of Theorem 3.1. □

The above proof of Theorem 3.1 shows that the solution u to the integral equation (2.12) is asymptotic to the linear solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267">View MathML</a> given by the formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M268">View MathML</a> in (2.11) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M269">View MathML</a>. This result is stated as follows.

Corollary 3.1Assume the same conditions of Theorem 3.1. Then the solutionuof the problem (1.1), (1.2), which is constructed in Theorem 3.1, can be approximated by the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267">View MathML</a>to the linearized problem (2.1), (1.2) as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M269">View MathML</a>. More precisely, we have the following asymptotic relations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M272">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M222">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M231">View MathML</a>, respectively, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M268">View MathML</a>is the linear solution.

4 Asymptotic profile

In this section, our aim is to establish an asymptotic profile to our global solution that is constructed in Theorem 4.1. In the previous section, we have shown that the solution u to the problem (1.1), (1.2) can be approximated by the linear solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267">View MathML</a>. In what follows, we shall derive a simpler asymptotic profile of the linear solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267">View MathML</a>.

Let v be the solution to the initial data problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M278">View MathML</a>

(4.1)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M279">View MathML</a>

(4.2)

gives a asymptotic profile of the linear solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267">View MathML</a>. In fact we have the following.

Lemma 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M32">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M282">View MathML</a>and put<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M283">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M267">View MathML</a>be the linear solution and letvbe defined by (4.2). Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M285">View MathML</a>

(4.3)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M222">View MathML</a>.

Proof Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M287">View MathML</a>, so for the proof of (4.3), it suffices to show the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M288">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M222">View MathML</a>. These estimates can be obtained by (2.27) and (2.32). Here we omit the details. □

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M290">View MathML</a>, we call

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M291">View MathML</a>

(4.4)

the diffusion wave with the amount <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M171">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M293">View MathML</a> satisfies the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M294">View MathML</a>

(4.5)

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M295">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M296">View MathML</a>

(4.6)

It is not difficult to prove the following lemma.

Lemma 4.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M297">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M298">View MathML</a>

(4.7)

Combining Corollary 3.1, Lemma 4.1, and Lemma 4.2, we immediately have the following.

Theorem 4.1Under the same assumption as Theorem 3.1, and also assuming that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M290">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M300">View MathML</a>, we letube the global solution to the problem (1.1), (1.2), which is constructed in Theorem 3.1 and we let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M301">View MathML</a>be the diffusion wave defined by (4.4). Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M302">View MathML</a>

(4.8)

We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M303">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M304">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M305">View MathML</a>. We consider the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M306">View MathML</a>

(4.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M307">View MathML</a>

(4.10)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M308">View MathML</a>, by applying Lemma 4.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M309">View MathML</a>

(4.11)

We call

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M310">View MathML</a>

(4.12)

the diffusion wave with the amount <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M311">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M312">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M313">View MathML</a>

(4.13)

From (4.11) and (4.13), we immediately obtain the following lemma.

Lemma 4.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M314">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M315">View MathML</a>

(4.14)

Corollary 3.1, Lemma 4.1, and Lemma 4.3 immediately give the following result.

Theorem 4.2Under the same assumption as Theorem 3.1, also assuming that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M316">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M317">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M318">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M305">View MathML</a>, we letube the global solution to the problem (1.1), (1.2), which is constructed in Theorem 3.1 and we let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M320">View MathML</a>be the diffusion wave defined by (4.12). Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/84/mathml/M321">View MathML</a>

(4.15)

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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