SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Complicated asymptotic behavior of solutions for a porous medium equation with nonlinear sources

Liangwei Wang1* and Jingxue Yin2

Author Affiliations

1 School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, China

2 School of Mathematical Sciences, South China Normal University, Guangzhou, China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:35  doi:10.1186/1687-2770-2013-35

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/35


Received:12 July 2012
Accepted:4 February 2013
Published:21 February 2013

© 2013 Wang and Yin; licensee Springer

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

Abstract

In this paper, we investigate the complicated asymptotic behavior of the solutions to the Cauchy problem of a porous medium equation with nonlinear sources when the initial value belongs to a weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M1">View MathML</a> space.

AMS Subject Classification: 35K55, 35B40.

Keywords:
complexity; asymptotic behavior; porous medium equation

1 Introduction

In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem of the porous medium equation with nonlinear sources

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M2">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M3">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M5">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M6">View MathML</a>.

It is well known that any positive solutions of problem (1.1)-(1.2) blow up in finite time if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M7">View MathML</a>[1-3], while positive global solutions do exist if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M8">View MathML</a>[4-7]. In 2000, Mukai, Mochizuki and Huang in [6] found that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M11">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M12">View MathML</a>, then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M13">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M14">View MathML</a>, the solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15">View MathML</a> of problem (1.1)-(1.2) with the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M16">View MathML</a> are global and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M17">View MathML</a>

(1.3)

Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M18">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M19">View MathML</a>

uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M21">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M22">View MathML</a> is a semigroup generated by the Cauchy problem of the porous medium equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M23">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M24">View MathML</a>

(1.5)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M25">View MathML</a>.

On the other hand, regarding problem (1.4)-(1.5), in 2002, Vázquez and Zuazua [8] found that for any bounded sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M26">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M27">View MathML</a>, there exists an initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M28">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M29">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M30">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M31">View MathML</a> uniformly on any compact subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. In our previous papers [9], for any bounded sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M33">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M34">View MathML</a>, we have shown that there exists an initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M35">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M36">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M30">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M38">View MathML</a>

uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M41">View MathML</a>. For more details on the study of complicated asymptotic behavior of solutions for the heat equation and other evolution equations, we refer the readers to [10-14].

In this paper, we are quite interested in the above mentioned same topic for the equation with strongly nonlinear sources, namely equation (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M42">View MathML</a>. We will show that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M43">View MathML</a>, there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M44">View MathML</a> and an initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M45">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M47">View MathML</a>, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M48">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M49">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M50">View MathML</a>

uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M52">View MathML</a>. For this purpose, we first show that if the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M53">View MathML</a>, then the solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15">View MathML</a> are global and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M55">View MathML</a>

(1.6)

One can easily see that (1.6) captures (1.3). From this, we can follow the framework by Kamin and Peletier [15] to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M56">View MathML</a>

(1.7)

So, we can get our results by following the framework in [9] and using (1.6)-(1.7).

The rest of this paper is organized as follows. The next section is devoted to giving a sufficient condition for the global existence of solutions for problem (1.1)-(1.2) and the upper bounded estimates on these solutions. In the last section, we investigate the complicated asymptotic behavior of solutions.

2 Preliminaries and estimates

In this section we state the definition of a weak solution of problem (1.1)-(1.2) and give the upper bounded estimates on the global solutions. We begin with the definition of the weak solution of problem (1.1)-(1.2).

Definition 2.1[16,17]

By a weak solution of problem (1.1)-(1.2) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M57">View MathML</a>, we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M57">View MathML</a> such that

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M60">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M62">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M63">View MathML</a>.

2. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M64">View MathML</a> and any nonnegative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M65">View MathML</a> which vanishes for large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M66">View MathML</a>, the following equation holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M67">View MathML</a>

(2.1)

A supersolution [or subsolution] is similarly defined with equality of (2.1) replaced by ≥ [or ≤]. The weak solutions for problem (1.4)-(1.5) can be defined in a similar way as above. It is well known that problem (1.1)-(1.2) has a unique, nonnegative and bounded weak solution, at least locally in time [16,17]. Now we state the comparison principle for problem (1.1)-(1.2).

Lemma 2.1[16,17]

Suppose that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M69">View MathML</a>are supersolution and subsolution of the problem (1.1)-(1.2), respectively. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M70">View MathML</a>

then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M71">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M72">View MathML</a>

To study the asymptotic behavior of solutions for problem (1.1)-(1.2), we adopt the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M74">View MathML</a> as that in [16-18]. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M76">View MathML</a>, the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M74">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M78">View MathML</a>

with the obvious norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M79">View MathML</a> and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M73">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M81">View MathML</a>

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M82">View MathML</a>. Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M83">View MathML</a>

Hence they are both Banach spaces. The existence and uniqueness of a weak solution of problem (1.4)-(1.5) with the initial-value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M84">View MathML</a> is shown in [16,17], and this solution satisfies the following proposition.

Proposition 2.1[17]

Problem (1.4)-(1.5) generates a continuous bounded semigroup in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M73">View MathML</a>given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M86">View MathML</a>

In other words, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M87">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M88">View MathML</a>, then the semigroup<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M22">View MathML</a>is a contraction.

We now introduce the definitions of scalings and the commutative relations between the semigroup operators and the dilation operators. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M91">View MathML</a>, the dilation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M92">View MathML</a> is defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M93">View MathML</a>

From the definitions of the dilation operator and the semigroup operator, we can get that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M95">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M96">View MathML</a>

(2.2)

see details in [19,20].

In the rest of this section, we give a sufficient condition for the existence of global solutions of problem (1.1)-(1.2) and establish the upper bounded estimates of these solutions.

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M43">View MathML</a>. There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M44">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M101">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M102">View MathML</a>, the solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15">View MathML</a>of problem (1.1)-(1.2) with the initial value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M104">View MathML</a>are global. Moreover, the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M105">View MathML</a>

(2.3)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M106">View MathML</a>is a constant dependent only onMandη.

Remark 2.1 Notice that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M108">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M109">View MathML</a>. So, our results capture Theorem 3 in [6]. Here we use some ideas of them.

Proof To prove this theorem, we need the fact that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M110">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M111">View MathML</a>

(2.4)

which has been given in Lemma 2.6 of [20]. We give the proof here for completeness. In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M112">View MathML</a>

This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M113">View MathML</a>. Therefore, from Proposition 2.1, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M114">View MathML</a> is well defined. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M116">View MathML</a> in (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M117">View MathML</a>

(2.5)

Now taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M120">View MathML</a> in (2.5), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M121">View MathML</a>

(2.6)

The fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M122">View MathML</a> clearly means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M123">View MathML</a>

(2.7)

see [21]. This implies that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M124">View MathML</a>, the following limit holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M125">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M126">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M127">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M128">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M129">View MathML</a>. This means that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M130">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M131">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M132">View MathML</a>

(2.8)

From (2.7), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M133">View MathML</a>, there exists a constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M134">View MathML</a>

(2.9)

Combining (2.8) and (2.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M135">View MathML</a>

By (2.6), we thus obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M136">View MathML</a>

So, we complete the proof of (2.4). Now taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M137">View MathML</a>

we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M138">View MathML</a>

Therefore, by the comparison principle and (2.4), for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M139">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M140">View MathML</a>

(2.10)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M141">View MathML</a> (see [17,21]), there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M142">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M144">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M145">View MathML</a>

Combining this with (2.6) and using the comparison principle, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M146">View MathML</a>

In other words,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M147">View MathML</a>

(2.11)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M148">View MathML</a>, (2.3) clearly holds. In the rest of proof, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M149">View MathML</a>. The hypothesis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46">View MathML</a> indicates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M151">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M152">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M153">View MathML</a> is the constant given by (2.11). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M154">View MathML</a>, taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M155">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M156">View MathML</a>

we obtain from (2.11) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M157">View MathML</a> is an increasing function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M158">View MathML</a>

(2.12)

Now letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M159">View MathML</a> to satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M160">View MathML</a>

(2.13)

and then taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M161">View MathML</a>

one can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M162">View MathML</a> is a supersolution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M163">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M164">View MathML</a>

(2.14)

(2.12) and (2.13) clearly mean that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M165">View MathML</a>

From this and (2.14), we can get (2.3). So, we complete the proof of this theorem. □

3 Complicated asymptotic behavior

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M43">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M44">View MathML</a> be as given by Theorem 2.1. We introduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M168">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M169">View MathML</a>

In the rest of this section, we show that the complexity may occur in the asymptotic behavior of solutions of problem (1.1)-(1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M170">View MathML</a>. Our main result is the following theorem.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M42">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46">View MathML</a>. Then there is a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M170">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M47">View MathML</a>, there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M175">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M49">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M177">View MathML</a>

uniformly on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M15">View MathML</a>is the solution of problem (1.1)-(1.2).

To get this theorem, we need to prove the following lemma first.

Lemma 3.1Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M42">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M43">View MathML</a>. Letube a solution of problem (1.1)-(1.2). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M182">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M46">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M184">View MathML</a>

Proof

We first define the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M185">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M186">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M188">View MathML</a>. Using the comparison principle, we know that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M189">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M190">View MathML</a>

and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M191">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M192">View MathML</a>

The results of Theorem 2.1 imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M193">View MathML</a>

(3.1)

Here we have used the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M194">View MathML</a>. So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M195">View MathML</a>

Now we estimate the integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M196">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M197">View MathML</a> in several steps. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M198">View MathML</a>, we take λ large enough to satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M199">View MathML</a> and assume, without loss of generality, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M200">View MathML</a> in the rest of this proof. Then using the same method as above, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M201">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M202">View MathML</a>. Similarly, we can get the integral estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M203">View MathML</a>, which have been given in [22]. By using the same methods as in [15], we can get that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M204">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M205">View MathML</a>

(3.3)

uniformly on any compact subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M207">View MathML</a>, we can obtain from (3.1) that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M208">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M209">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M210">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M212">View MathML</a>. Taking R as given by (3.4), from (3.3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M213">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M214">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M215">View MathML</a>

(3.6)

Therefore, from (3.4)-(3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M216">View MathML</a>

(3.7)

Now letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M217">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M119">View MathML</a> in (3.7), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M219">View MathML</a>

So, we complete the proof of this lemma. □

Now we can prove our main result.

Proof of Theorem 3.1

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M220">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M221">View MathML</a>

From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M222">View MathML</a>, we obtain that there exists a countable set F such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M223">View MathML</a>

and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M47">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M226">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M227">View MathML</a>

(3.8)

Therefore, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M228">View MathML</a> such that

I. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M229">View MathML</a>, there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M230">View MathML</a> of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M231">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M232">View MathML</a>

II. There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M233">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M234">View MathML</a>

Now we can follow the methods given in [9] to construct an initial value as follows. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M235">View MathML</a>

(3.9)

Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M236">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M237">View MathML</a> is the cut-off function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M238">View MathML</a> relatively to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M239">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M240">View MathML</a> is selected large enough to satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M241">View MathML</a>

Notice first that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M47">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M243">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M244">View MathML</a>

By (3.9) and (3.10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M245">View MathML</a>

So, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M246">View MathML</a>

Using the same method as that in [9], we can get that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M247">View MathML</a>, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M248">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M49">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M250">View MathML</a>

(3.11)

uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M252">View MathML</a>, from (1.2), we know that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M253">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M254">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M255">View MathML</a>

(3.12)

uniformly on any compact subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. This uses the fact that the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M257">View MathML</a> is regularizing since the images of bounded sets are relatively compact subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M258">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M259">View MathML</a> in compact sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>[21]. And notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M261">View MathML</a>. We thus obtain from Theorem 2.1 that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M262">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M208">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M264">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M265">View MathML</a>

(3.13)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M266">View MathML</a>

(3.14)

Combining (3.12), (3.13) with (3.14), we thus have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M267">View MathML</a>

(3.15)

uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. By Lemma 3.1, (3.11) and (3.15), we can get that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M252">View MathML</a>, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M175">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M271">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M272">View MathML</a>

uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. So, we complete the proof of Theorem 3.1. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The paper is the result of joint work of both authors who contributed equally to the final version of the paper. Both authors read and approved the final manuscript.

Acknowledgements

This work is supported by NSFC, the Research Fund for the Doctoral Program of Higher Education of China, the Natural Science Foundation Project of ‘CQ CSTC’ (cstc2012jjA00013), the Scientific and Technological Projects of Chongqing Municipal Commission of Education (KJ121105).

References

  1. Galaktionov, VA, Kurdjumov, SP, Mihaĭov, AP, Samarskiĭ, AA: On unbounded solutions of the Cauchy problem for the parabolic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M274">View MathML</a>. Sov. Math. Dokl.. 252(6), 1362–1364 (1980) (Russian)

  2. Galaktionov, VA: Blow-up for quasilinear heat equations with critical Fujita’s exponents. Proc. R. Soc. Edinb. A. 124(3), 517–525 (1994) (English summary)

    (English summary)

    Publisher Full Text OpenURL

  3. Kawanago, T: Existence and behaviour of solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M276">View MathML</a>. Adv. Math. Sci. Appl.. 7(1), 367–400 (1997) (English summary)

  4. Mochizuki, K, Suzuki, R: Critical exponent and critical blow-up for quasilinear parabolic equations. Isr. J. Math.. 98, 141–156 (1997) (English summary)

    (English summary)

    Publisher Full Text OpenURL

  5. Suzuki, R: Asymptotic behavior of solutions of quasilinear parabolic equations with slowly decaying initial data. Adv. Math. Sci. Appl.. 9(1), 291–317 (1999) (English summary)

  6. Mukai, K, Mochizuki, K, Huang, Q: Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values. Nonlinear Anal.. 39(1), 33–45 (2000). Publisher Full Text OpenURL

  7. Suzuki, R: Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity. J. Differ. Equ.. 190(1), 150–181 (2003) (English summary)

    (English summary)

    Publisher Full Text OpenURL

  8. Vázquez, JL, Zuazua, E: Complexity of large time behaviour of evolution equations with bounded data. Chin. Ann. Math., Ser. B. 23(2), 293–310 (2002). Publisher Full Text OpenURL

  9. Yin, J, Liangwei, W, Huang, R: Complexity of asymptotic behavior of solutions for the porous medium equation with absorption. Acta Math. Sci.. 30(6), 1865–1880 (2010)

  10. Cazenave, T, Dickstein, F, Weissler, FB: Universal solutions of the heat equation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. Discrete Contin. Dyn. Syst.. 9(5), 1105–1132 (2003) (English summary)

  11. Cazenave, T, Dickstein, F, Weissler, FB: Universal solutions of a nonlinear heat equation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. Ann. Sc. Norm. Super. Pisa, Cl. Sci.. 2(1), 77–117 (2003) (English summary)

  12. Cazenave, T, Dickstein, F, Weissler, FB: Chaotic behavior of solutions of the Navier-Stokes system in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. Adv. Differ. Equ.. 10(4), 361–398 (2005)

  13. Cazenave, T, Dickstein, F, Weissler, FB: Nonparabolic asymptotic limits of solutions of the heat equation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. J. Dyn. Differ. Equ.. 19(3), 789–818 (2007). Publisher Full Text OpenURL

  14. Carrillo, JA, Vázquez, JL: Asymptotic complexity in filtration equations. J. Evol. Equ.. 7(3), 471–495 (2007) (English summary)

    (English summary)

    Publisher Full Text OpenURL

  15. Kamin, S, Peletier, LA: Large time behaviour of solutions of the heat equation with absorption. Ann. Sc. Norm. Super. Pisa, Cl. Sci.. 12(3), 393–408 (1985)

  16. Bénilan, P, Crandall, MG, Pierre, M: Solutions of the porous medium equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a> under optimal conditions on initial values. Indiana Univ. Math. J.. 33(1), 51–87 (1984). Publisher Full Text OpenURL

  17. Vázquez, JL: The Porous Medium Equation. Mathematical Theory, Clarendon, Oxford (2007)

  18. DiBenedetto, E, Herrero, MA: On the Cauchy problem and initial traces for a degenerate parabolic equation. Trans. Am. Math. Soc.. 314(1), 187–224 (1989)

  19. Yin, J, Wang, L, Huang, R: Complexity of asymptotic behavior of the porous medium equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/35/mathml/M20">View MathML</a>. J. Evol. Equ.. 11(2), 429–455 (2011). Publisher Full Text OpenURL

  20. Wang, L, Yin, J, Jin, C: ω-Limit sets for porous medium equation with initial data in some weighted spaces. Discrete Contin. Dyn. Syst., Ser. B. 18(1), 223–236 (2013)

  21. DiBenedetto, E: Degenerate Parabolic Equations, Springer, New York (1993)

  22. Kamin, S, Peletier, LA: Large time behaviour of solutions of the porous media equation with absorption. Isr. J. Math.. 55(2), 129–146 (1986). Publisher Full Text OpenURL