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Asymptotic analysis for reaction-diffusion equations with absorption

Wanjuan Du* and Zhongping Li

Author Affiliations

College of Mathematic and Information, China West Normal University, Nanchong, 637009, P.R. China

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Boundary Value Problems 2012, 2012:84  doi:10.1186/1687-2770-2012-84

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/84


Received:31 May 2012
Accepted:20 July 2012
Published:2 August 2012

© 2012 Du 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 cited.

Abstract

In this paper, we study the blow-up and nonextinction phenomenon of reaction-diffusion equations with absorption under the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then give the blow-up rate estimates for the nonglobal solutions. In addition, the nonextinction of solutions is also concerned.

MSC: 35B33, 35K55, 35K60.

Keywords:
reaction-diffusion; absorption; blow-up; blow-up rate; non-extinction

1 Introduction

In this paper, we consider the reaction-diffusion equations with absorption

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M6">View MathML</a> is a bounded domain with smooth boundary Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M7">View MathML</a> is a nontrivial, nonnegative, bounded, and appropriately smooth function. Parabolic equations like (1.1) appear in population dynamics, chemical reactions, heat transfer, and so on. We refer to [2,8,9] for details on physical models involving more general reaction-diffusion equations.

The semilinear case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M8">View MathML</a>) of (1.1) has been investigated by Bedjaoui and Souplet [3]. They obtained that the solutions exist globally if either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M9">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M10">View MathML</a>, and the solutions may blow up in finite time for large initial value if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M11">View MathML</a>. Recently, Xiang et al.[11] considered the blow-up rate estimates for nonglobal solutions of (1.1) (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M8">View MathML</a>) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M11">View MathML</a>, and obtained that (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M14">View MathML</a>; (ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M15">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M16">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M17">View MathML</a> are positive constants. Liu et al.[7] studied the extinction phenomenon of solutions of (1.1) for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M18">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M19">View MathML</a> and obtained some sufficient conditions about the extinction in finite time and decay estimates of solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M21">View MathML</a>).

Recently, Zhou et al.[10] investigated positive solutions of the degenerate parabolic equation not in divergence form

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M25">View MathML</a>. They at first gave some conditions about the existence and nonexistence of global solutions to (1.2), and then studied the large time behavior for the global solutions.

Motivated by the above mentioned works, the aim of this paper is threefold. First, we determine optimal conditions for the existence and nonexistence of global solutions to (1.1). Secondly, by using the scaling arguments we establish the exact blow-up rate estimates for solutions which blow up in a finite time. Finally, we prove that every solution to (1.1) is nonextinction.

As it is well known that degenerate equations need not possess classical solutions, we give a precise definition of a weak solution to (1.1).

Definition 1.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M29">View MathML</a>, a nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M30">View MathML</a> is called a weak upper (or lower) solution to (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M31">View MathML</a> if for any nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M32">View MathML</a>, one has

In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34">View MathML</a> is called a weak solution of (1.1) if it is both a weak upper and a weak lower solution. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M35">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34">View MathML</a> is a weak solution of (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M31">View MathML</a>, we say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34">View MathML</a> is global. The local in time existence of nonnegative weak solutions have been established (see the survey [1]), and the weak comparison principle is stated and proved in the Appendix in this paper.

The behavior of the weak solutions is determined by the interactions among the multinonlinear mechanisms in the nonlinear diffusion equations in (1.1). We divide the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M39">View MathML</a>-parameter region into three classes: (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M40">View MathML</a>; (ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M41">View MathML</a>; (iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M42">View MathML</a>.

Theorem 1.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M43">View MathML</a>, then all solutions of (1.1) are bounded.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M44">View MathML</a> be the first eigenfunction of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M45">View MathML</a>

(1.3)

with the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M46">View MathML</a>, normalized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M47">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M49">View MathML</a> in Ω.

Theorem 1.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M50">View MathML</a>. Then all solutions are global if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M51">View MathML</a>, and there exist both global and nonglobal solutions if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M52">View MathML</a>.

Theorem 1.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M42">View MathML</a>, then there exist both global and nonglobal solutions to (1.1).

To obtain the blow-up rate of blow-up solutions to (1.1), we need an extra assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M56">View MathML</a>, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M57">View MathML</a>. By the assumption and comparison principle, we know that u is radially decreasing in r with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M58">View MathML</a>.

Theorem 1.4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M59">View MathML</a>. If the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34">View MathML</a>of (1.1) blows up in finite timeT, then there exists a positive constantcsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M61">View MathML</a>

Furthermore, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M62">View MathML</a>, then we have also the upper estimate, that is, there exists a positive constantCsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M63">View MathML</a>

We remark that in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M64">View MathML</a>, Liang [6] studied the blow up rate of blow-up solutions to the following Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M65">View MathML</a>

(1.4)

with the bounded initial function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M66">View MathML</a>, and obtained that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M67">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M68">View MathML</a>. By using the same scaling arguments in this paper, we can find that Theorem 1.4 is correct for (1.4) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M69">View MathML</a>.

Now, we pay attention to the nonextinction property of solutions and have the following result.

Theorem 1.5Any solution of (1.1) does not go extinct in finite time for any nontrivial and nonnegative initial value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M7">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M71">View MathML</a>.

The rest of this paper is organized as follows. In the next section, we discuss the global existence and nonexistence of solutions, and prove Theorems 1.1-1.3. Subsequently, in Sects. 3 and 4, we consider the estimate of the blow-up rate and study the nonextinction phenomenon for the problem (1.1). The weak comparison principle is stated and proved in the Appendix.

2 Global existence and nonexistence

Proof of Theorem 1.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M72">View MathML</a>, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M73">View MathML</a>, then by the comparison principle, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M74">View MathML</a>, where w satisfies

(2.1)

We know from [4,5] that w is bounded.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M76">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M77">View MathML</a>. It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M78">View MathML</a> is a time-independent upper solution to (1.1). □

Proof of Theorem 1.2 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M50">View MathML</a> imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M81">View MathML</a>. Due to the fact that the solution of (2.1) is an upper solution of (1.1), the conclusions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M51">View MathML</a> is obvious true; see [4,5].

Now consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M52">View MathML</a> with small initial data. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M84">View MathML</a> be the unique solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M85">View MathML</a>

(2.2)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86">View MathML</a> solves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M87">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M88">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M89">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M90">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M91">View MathML</a>

provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M92">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M93">View MathML</a> is an upper solution of (1.1), and consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M94">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M52">View MathML</a> with large initial data, we first introduce some transformations. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M97">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M98">View MathML</a>, then (1.1) becomes the following equations not in divergence form:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M102">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M103">View MathML</a>, where ϕ is given in (1.3). Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M104">View MathML</a>

(2.3)

By using Hölder’s inequality, we discover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M105">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M106">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M107">View MathML</a>

(2.5)

Inserting (2.4) into (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M108">View MathML</a>

(2.6)

According to (2.5), (2.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M109">View MathML</a>

(2.7)

as long as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M110">View MathML</a>

Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M111">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M112">View MathML</a>

we then follow from (2.7) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M113">View MathML</a>, and consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M34">View MathML</a>, blows up in finite time since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M113">View MathML</a> is increasing and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M116">View MathML</a>. □

Proof of Theorem 1.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86">View MathML</a> solves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M118">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M88">View MathML</a>, and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M90">View MathML</a>, where ψ is defined in (2.2). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M121">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M42">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M123">View MathML</a> small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M124">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M93">View MathML</a> is an upper solution of (1.1) provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M126">View MathML</a>, and consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M127">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95">View MathML</a>.

Now deal with the nonexistence of global solutions, we seek a blow-up self-similar lower solution of the problem (1.1). Without loss of generality, we may assume that Ω contains the origin. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M42">View MathML</a>, we can choose constant α such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M130">View MathML</a>

and consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M131">View MathML</a>

(2.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M132">View MathML</a>. Note that the support of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M133">View MathML</a> is contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M134">View MathML</a>, which is included in Ω if T is sufficiently small.

After some computations, we have

It will be obtained from the above equalities that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M136">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M137">View MathML</a> satisfies

(2.9)

It is easy to see that

To satisfy (2.9), we distinguish the two zones <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M140">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M141">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M142">View MathML</a>

(2.10)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M141">View MathML</a>, we have

then

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M140">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M147">View MathML</a>. It follow from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M148">View MathML</a> that (2.9) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M141">View MathML</a> if T is sufficiently small. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M151">View MathML</a> given by (2.8) is a blow-up lower solution of the problem (1.1) with appropriately large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M111">View MathML</a>. And consequently, there exist nonglobal solutions to (1.1). □

3 Blow-up rate

In this section, we study the speeds at which the solutions to (1.1) blow up. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M56">View MathML</a>, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M57">View MathML</a>. Then we know from the assumption and comparison principle that u is radially decreasing in r with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M58">View MathML</a>. In this section, denote by T the blow-up time for the nonglobal solutions to (1.1).

Proof of Theorem 1.4 Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M68">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M159">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M160">View MathML</a>

and define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M161">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M162">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M163">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M164">View MathML</a> blows up at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M165">View MathML</a>, moreover, it is a solution of the following problem:

(3.1)

We now construct an upper solution for this problem. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M167">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M169">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M170">View MathML</a>

After a direct computation, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M171">View MathML</a>, we have

Then

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M174">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M175">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M176">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M178">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M179">View MathML</a>. We have an upper solution independent of M, for all M large enough. Therefore, the blow-up time of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M164">View MathML</a> is greater than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M181">View MathML</a>, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M182">View MathML</a>. This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M183">View MathML</a>

and the lower estimate is obtained.

In order to obtained the upper estimates for the blow-up rate, we look for a lower solution to (3.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M184">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M185">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M189">View MathML</a> and θ is given in (2.10). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M190">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M191">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M192">View MathML</a>, μ are to be determined later. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M193">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M194">View MathML</a>. As the same arguments in the proof of Theorem 1.3, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M195">View MathML</a> that

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M140">View MathML</a>, we have that

Now, in order to deal with the initial data, consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M199">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M201">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M202">View MathML</a>.

After a direct computation, we have

Then

Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M205">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M206">View MathML</a>. In addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M207">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M179">View MathML</a>. Therefore, by the comparison principle, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M209">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M210">View MathML</a>. By the virtue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M211">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M212">View MathML</a>.

We have a lower solution independent of M, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M213">View MathML</a>. Therefore, the blow-up time of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M164">View MathML</a> is less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M215">View MathML</a>, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M216">View MathML</a>. This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M217">View MathML</a>

and the upper estimate is obtained. □

4 Nonextinction

We discuss the nonextinction of the solution to the problem (1.1) in this section. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M218">View MathML</a>, the uniqueness of the weak solution to (1.1) may not hold. In this case, we only consider the maximal solution, which can be obtained by standard regularized approximation methods. Clearly, the comparison principle is valid for the maximal solution.

Proof of Theorem 1.5 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M71">View MathML</a>, there exists a region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M220">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M221">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M222">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M224">View MathML</a> is the first Dirichlet eigenvalue of −Δ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223">View MathML</a> with corresponding eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M226">View MathML</a>, normalized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M227">View MathML</a>, and prolong solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M228">View MathML</a> by 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M229">View MathML</a>. We treat the five subcases for the proof.

(a) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M230">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231">View MathML</a>, where

Then

By the comparison principle, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223">View MathML</a>.

(b) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M237">View MathML</a>, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M239">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M240">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M241">View MathML</a>

Then we know by the comparison principle that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223">View MathML</a>.

(c) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M245">View MathML</a>, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231">View MathML</a>, and

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M248">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86">View MathML</a> is nonincreasing and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M250">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M252">View MathML</a>

And consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223">View MathML</a>.

(d) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M245">View MathML</a>, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231">View MathML</a>, where

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86">View MathML</a> is nonincreasing and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M260">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M262">View MathML</a>

Thus, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223">View MathML</a>.

(e) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M245">View MathML</a>, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M231">View MathML</a>, and

where c satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M269">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M86">View MathML</a> is nonincreasing and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M260">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M95">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M273">View MathML</a>

By the comparison principle, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M234">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M223">View MathML</a>.

 □

Appendix

Theorem A.1 (Comparison principle)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M151">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M93">View MathML</a>are a weak lower and a weak upper solutions of (1.1) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M31">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M23">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M93">View MathML</a>has a positive lower bound, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M281">View MathML</a>a.e. in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M31">View MathML</a>.

Proof From the definition of weak upper and lower solutions, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M283">View MathML</a>, we obtain

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M285">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M68">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M287">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M288">View MathML</a> is the characteristic function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M289">View MathML</a>, Then we arrive at

By a simple calculation, we have

(A.1)

Noticing

we get

(A.2)

where L is a positive constant. By (A.1), (A.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M294">View MathML</a>

It follows immediately by using the Gronwall’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M295">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M68">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M281">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/84/mathml/M298">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

DW carried out all studies in the paper. LZ participated in the design of the study in the paper.

Acknowledgements

This work was partially supported by Projects Supported by Scientific Research Fund of Sichuan Provincial Education Department (09ZA119).

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