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Blow up problems for a degenerate parabolic equation with nonlocal source and nonlocal nonlinear boundary condition

Guangsheng Zhong12* and Lixin Tian1

Author Affiliations

1 Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China

2 School of Science, Nantong University, Nantong 226007, China

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Boundary Value Problems 2012, 2012:45  doi:10.1186/1687-2770-2012-45

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/45


Received:5 September 2011
Accepted:18 April 2012
Published:18 April 2012

© 2012 Zhong and Tian; licensee Springer.

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

Abstract

This article deals with the blow-up problems of the positive solutions to a nonlinear parabolic equation with nonlocal source and nonlocal boundary condition. The blow-up and global existence conditions are obtained. For some special case, we also give out the blow-up rate estimate.

Keywords:
parabolic equation; nonlocal source; nonlocal nonlinear boundary condition; existence; blow-up

1. Introduction

In this article, we consider the positive solution of the following degenerate parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M1">View MathML</a>

(1.1)

where a, l > 0 and Ω is a bounded domain in RN (N ≥ 1) with smooth boundary ∂Ω.

There have been many articles dealing with properties of solutions to degenerate parabolic equations with homogeneous Dirichlet boundary condition (see [1-4] and references therein). For example, Deng et al. [5] studied the parabolic equation with nonlocal source

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M2">View MathML</a>

(1.2)

which is subjected to homogeneous Dirichlet boundary condition. It was proved that there exists no global positive solution if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M4">View MathML</a>, where φ(x) is the unique positive solution of the linear elliptic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M5">View MathML</a>

(1.3)

However, there are some important phenomena formulated into parabolic equations which are coupled with nonlocal boundary conditions in mathematical modeling such as thermoelasticity theory (see [6,7]). Friedman [8] studied the problem of nonlocal boundary conditions for linear parabolic equations of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M6">View MathML</a>

(1.4)

with uniformly elliptic operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M7">View MathML</a> and c(x)≤ 0. It was proved that the unique solution of (1.4) tends to 0 monotonically and exponentially as t →+∞ provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M8">View MathML</a>

Parabolic equations with both nonlocal sources and nonlocal boundary conditions have been studied as well (see [9-12]). Lin and Liu [13] considered the problem of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M9">View MathML</a>

(1.5)

They established local existence, global existence, and nonexistence of solutions, and discussed the blow-up properties of solutions.

Chen and Liu [14] considered the following nonlinear parabolic equation with a localized reaction source and a weighted nonlocal boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M10">View MathML</a>

(1.6)

Under certain conditions, they obtained blow-up criteria. Furthermore, they derived the uniform blow-up estimate for some special f(u).

In recent few years, reaction-diffusion problems coupled with nonlocal nonlinear boundary conditions have also been studied. Gladkov and Kim [15] considered the following problem for a single semilinear heat equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M11">View MathML</a>

(1.7)

where p, l > 0. They obtained some criteria for the existence of global solution as well as for the solution to blow-up in finite time.

For other works on parabolic equations and systems with nonlocal nonlinear boundary conditions, we refer readers to [16-20] and the references therein.

Motivated by those of works above, we will study the problem (1.1) and want to understand how the function f(u) and the coefficient a, the weight function g(x, y) and the nonlinear term ul (y, t) in the boundary condition play substantial roles in determining blow-up or not of solutions.

In this article, we give the following hypotheses:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M12">View MathML</a> for α∈(0,1),u0(x) > 0 in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M13">View MathML</a> on ∂Ω.

(H2) g(x, y)≢0 is a nonnegative and continuous function defined for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M14">View MathML</a>.

(H3) fC([0,∞))∩C1(0,∞), f > 0, f' ≥ 0 in (0,∞).

The main results of this article are stated as follows.

Theorem 1.1. Assume that 0 < l ≤ 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M15">View MathML</a> for all x∈∂Ω.

(1) If a is sufficiently small, then the solution of (1.1) exists globally;

(2) If a is sufficiently large, then the solution of (1.1) also exists globally provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M16">View MathML</a> for some δ > 0.

Theorem 1.2. Assume that l > 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M17">View MathML</a> for all x∈∂Ω. Then the solution of (1.1) exists globally provided that a and u0(x) are sufficiently small. While the solution blows up in finite time if a,u0(x) are sufficiently large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M19">View MathML</a> for some δ > 0.

Theorem 1.3. Assume that l > 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M18">View MathML</a> for all x∈∂Ω. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M19">View MathML</a> for some δ > 0, then the solution of (1.1) blows up in finite time provided that u0(x) is large enough.

Theorem 1.4. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M19">View MathML</a> for some δ > 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M20">View MathML</a>, where φ(x) is the solution of (1.3), then there exists no global positive solution of (1.1).

To describe conditions for blow-up of solutions, we need an additional assumption on the initial data u0.

(H4) There exists a constant ε > ε1 > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M21">View MathML</a>, where ε1 will be given later.

Theorem 1.5. Assume u0(x) satisfies (H1), (H2), and (H4), Δu0 ≤ 0 in Ω holds, and let f(u) = up,0 < p ≤ 1, l = 1, then the following limits converge uniformly on any compact subset of Ω:

(1) If 0 < p < 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M22">View MathML</a>.

(2) If p = 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M23">View MathML</a>.

This article is organized as follows. In Section 2, we establish the comparison principle and the local existence. Some criteria regarding to global existence and finite time blow-up for problem (1.1) are given in Section 3. In Section 4, the global blow-up result and the blow-up rate estimate of blow-up solutions for the special case of f (u) = up, 0 < p ≤ 1 and l = 1 are obtained.

2. Comparison principle and local existence

First, we start with the definition of subsolution and supersolution of (1.1) and comparison principle. Let QT = Ω × (0, T), ST = ∂Ω × (0, T), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M24">View MathML</a>.

Definition 2.1. A function

    u
(x, t) is called a subsolution of (1.1) on QT, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M25">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M26">View MathML</a>

(2.1)

Similarly, a supersolution ū(x, t) of (1.1) is defined by the opposite inequalities.

A solution of problem (1.1) is a function which is both a subsolution and a supersolution of problem (1.1).

The following comparison principle plays a crucial role in our proofs which can be obtained by similar arguments as [10] and its proof is therefore omitted here.

Lemma 2.2. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M27">View MathML</a> and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M28">View MathML</a>

(2.2)

where d(x, t), ci (x, t)(i = 1,2,3,4) are bounded functions and d(x, t)≥ 0, ci (x, t)≥ 0 (i = 2,3,4) in QT, c5 (x, y)≥ 0 for x∈∂Ω, y∈Ω and is not identically zero. Then, w(x, 0) > 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29">View MathML</a> implies w(x, t) > 0 in QT. Moreover, c5 (x, y) ≡ 0 or if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M30">View MathML</a> on ST, then w(x, 0) ≥ 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29">View MathML</a> implies w(x, t) ≥ 0 in QT.

On the basis of the above lemmas, we obtain the following comparison principle of (1.1).

Lemma 2.3. Let u and v be nonnegative subsolution and supersolution of (1.1), respectively, with u(x, 0) ≤ v(x, 0) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29">View MathML</a>. Then, u v in QT if u η or v η for some small positive constant η holds.

Local in time existence of positive classical solutions of (1.1) can be obtained by using fixed point theorem [21], the representation formula and the contraction mapping principle as in [13]. By the above comparison principle, we get the uniqueness of solution to the problem. The proof is more or less standard, so is omitted here.

3. Global existence and blow-up in finite time

In this section, we will use super- and subsolution techniques to derive some conditions on the existence or nonexistence of global solution.

Proof of Theorem 1.1. (1) Let ψ(x) be the unique positive solution of the linear elliptic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M31">View MathML</a>

(3.1)

where ε0 is a positive constant such that 0 < ψ(x) < 1 (since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M32">View MathML</a>, there exists such ε0). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M34">View MathML</a>.

We define a function w(x, t) as following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M35">View MathML</a>

(3.2)

where M ≥ 1 is a constant to be determined later. Then, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M36">View MathML</a>

(3.3)

On the other hand, we have for x ∈ Ω, t > 0,,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M37">View MathML</a>

(3.4)

We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M38">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M39">View MathML</a>, then it is easy to verify that w(x, t) is a supersolution of (1.1) provided that a a0. By comparison principle, u(x, t) ≤ w(x, t), then u(x, t) exists globally.

(2) Consider the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M40">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M41">View MathML</a>, b1 is a positive constant to be fixed later. It follows from hypothesis (H3) and the theory of ordinary differential equation (ODE) that there exists a unique solution z (t) to problem (3.5) and z (t) is increasing. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M16">View MathML</a> for some positive δ, we know that z (t) exists globally and z (t) ≥ z0.

Let v(x, t) = z (t) ψ (x), where ψ (x) is given by (3.1), then for x ∈ Ω, t > 0, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M42">View MathML</a>

(3.6)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M43">View MathML</a>, if a is sufficiently large such that a > a1, then we can choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M44">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M45">View MathML</a>

(3.7)

On the other hand, for x ∈ ∂Ω, t > 0, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M46">View MathML</a>

(3.8)

Here, we use the conclusions 0 < ψ (x) < 1 and z(t) > 1.

Also for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M47">View MathML</a>

(3.9)

And the inequalities (3.5)-(3.9) show that v(x, t) is a supersolution of (1.1). Again by using the comparison principle, we obtain the global existence of u(x, t). The proof is complete.

Proof of Theorem 1.2. The proof of global existence part is similar to the first case of Theorem 1.1. For any given positive constant M ≤ 1, w (x) = (x) is a supersolution of problem (1.1) provided that u0 (x) ≤ ψ (x) < 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M48">View MathML</a>, so the solution of (1.1) exists globally by using the comparison principle.

To prove the bow-up result, we introduce the elliptic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M49">View MathML</a>

Under the hypothesis (H2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M50">View MathML</a>, we know that it exists a unique positive solution φ(x). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M52">View MathML</a>, and z(t) be the solution of the following ODE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M53">View MathML</a>

(3.10)

Then, z(t) is increasing and z (t) ≥ z1. Due to the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M19">View MathML</a> for some positive constant δ, we know that z (t) of problem (3.10) blows up in finite time.

If a and u0 (x) are so large that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M54">View MathML</a>u0 (x) ≥ z1 (K*)l, then we set v1(x, t) = z(t)φl(x). For x ∈ Ω, t > 0, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M55">View MathML</a>

(3.11)

For x ∈ ∂Ω, t > 0, by Jensen's inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M56">View MathML</a>

(3.12)

Also for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M29">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M57">View MathML</a>

(3.13)

The inequalities (3.10)-(3.13) show that v1(x, t) is a subsolution of problem (1.1). Since v1(x, t) blows up in finite time, u(x, t) also blows up in finite time by comparison principle.

Proof of Theorem 1.3. Let z(t) be the solution of the following ODE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M58">View MathML</a>

(3.14)

where 0 < b2 < a |Ω|. If u0(x) is large enough, we can set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M59">View MathML</a>. Then, z (t) is increasing and satisfies z (t) ≥ z2 > 1. Moreover, z (t) of problem (3.14) blows up in finite time.

Set s (x, t) = z (t), then we have for x ∈ Ω, t > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M60">View MathML</a>

(3.15)

For x ∈ ∂Ω, t > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M61">View MathML</a>

(3.16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M62">View MathML</a>

(3.17)

From (3.14)-(3.17), we see that s (x, t) is a subsolution of (1.1). Hence, u (x, t) ≥ s (x, t) by comparison principle, which implies u (x, t) blows up in finite time. This completes the proof.

Proof of Theorem 1.4. Consider the following equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M63">View MathML</a>

(3.18)

and let v (x, t) be the solution to problem (3.18). It is obvious that v (x, t) is a subsolution of (1.1). By Theorem 1 in [5], we can obtain the result immediately.

4. Blow-up rate estimate

Now, we consider problem (1.1) with f (u) = up, 0 < p ≤ 1 and l = 1, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M64">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M65">View MathML</a> for all x ∈ ∂Ω, and suppose that the solution of (4.1) blows up in finite time T*.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M66">View MathML</a>, then U(t) is Lipschitz continuous.

Lemma 4.1. Suppose that u0 satisfies (H1), (H2), and (H4), then there exists a positive constant c0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M67">View MathML</a>

(4.2)

Proof. By the first equation in (4.1), we have (see [22])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M68">View MathML</a>

(4.3)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M69">View MathML</a>

(4.4)

Integrating (4.3) over (t, T*), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M70">View MathML</a>

(4.5)

Setting c0 = (ap |Ω| p)-1/p, then we draw the conclusion.

Lemma 4.2. Under the conditions of Lemma 4.1, there exists a constant ε1, which will be given below, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M71">View MathML</a>

(4.6)

Proof. Let J(x, t) = ut-ε1up+1 for (x, t) ∈Ω × (0, T*), a series of computations yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M72">View MathML</a>

(4.7)

By virtue of Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M73">View MathML</a>

Furthermore, by Young's inequality, for any θ > 0, the following inequality holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M74">View MathML</a>

(4.8)

Using (4.8) and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M75">View MathML</a>, ε1 = a|Ω|, then (4.7) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M76">View MathML</a>

(4.9)

Fix (x, t) ∈∂Ω × (0, T*), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M77">View MathML</a>

Since ut (y, t) = J(y, t) + ε1up+1 (y, t), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M78">View MathML</a>

Noticing that p > 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M79">View MathML</a>, we can apply Jensen's inequality to the last integral in the above inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M80">View MathML</a>

Hence, for (x, t) ∈∂Ω × (0, T*), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M81">View MathML</a>

(4.10)

On the other hand, (H4) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M82">View MathML</a>

(4.11)

Owing to u(x, t) is a positive continuous function for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M83">View MathML</a>, it follows from (4.9)-(4.11) and Lemma 2.2 that J(x, t) ≥ 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M83">View MathML</a>, i.e., ut ε1up+1. This completes the proof.

Integrating (4.6) from t to T*, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M84">View MathML</a>

(4.12)

where c2 = (ε1p)-1/p is a positive constant independent of t. Combining (4.2) with (4.12), we obtain the following result.

Theorem 4.3. Under the conditions of Lemma 4.1, if u (x, t) is the solution of (4.1) and blows up in finite time T*, then there exist positive constants c1, c2, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M85">View MathML</a>

Lemma 4.4. Assume that u0(x) satisfies (H1), (H2), and (H4), Δu0 ≤ 0 in Ω. u(x, t) is the solution of problem (4.1). Then, Δu ≤ 0 in any compact subsets of Ω × (0, T*).

The proof is similar to that of Lemma 1.1 in [14].

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M86">View MathML</a>

Lemma 4.5. Under the conditions of Lemma 4.4, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M87">View MathML</a>

(4.13)

Proof. From Lemma 4.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M88">View MathML</a>

(4.14)

Integrating (4.14) over (0, t), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M89">View MathML</a>

(4.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M90">View MathML</a>

(4.16)

In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M92">View MathML</a>. Noting that ut ≥ 0 by the assumption of the initial function, then we see that g(t) is monotone nondecreasing. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M93">View MathML</a>.

Lemma 4. 6. Under the conditions of Lemma 4.4, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M94">View MathML</a>

(4.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M95">View MathML</a>

(4.18)

uniformly on any compact subsets of Ω.

Proof. Let λ > 0 be the principal eigenvalue of -Δ in Ω with the null Dirichlet boundary condition, and ϕ(x) be the corresponding eigenfunction satisfying ϕ(x) > 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M96">View MathML</a>.

In case of (1). Define z1(x, t) = G (t) - u1-p/(1-p), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M97">View MathML</a>. A direct computation shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M98">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M99">View MathML</a> and using the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M100">View MathML</a>. From (4.15), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M101">View MathML</a>

(4.19)

which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M102">View MathML</a>. Then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M103">View MathML</a>

(4.20)

Integrate (4.20) from 0 to t,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M104">View MathML</a>

(4.21)

Thus, (4.19) and (4.21) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M105">View MathML</a>

Define Kρ = {y∈ Ω: dist(y, ∂Ω)≥ ρ}. Since -Δz1 ≤ 0 in Ω × (0, T*). Using Lemma 4.5 in [1], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M106">View MathML</a>

(4.22)

It follows from (4.22) and (4.15) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M107">View MathML</a>

(4.23)

for any xKρ and t ∈ (0,T*), where k1 and K1 are positive constants.

We know from Theorem 4.3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M108">View MathML</a>

(4.24)

In view of (4.15) and Theorem 4.3, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M109">View MathML</a>

(4.25)

From (4.23)-(4.25), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M110">View MathML</a>

(4.26)

It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M111">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M112">View MathML</a>

In case of (2). We define z2 (x, t) = G (t) - ln u(x, t), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M113">View MathML</a>. Then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M114">View MathML</a>

From (4.16), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M115">View MathML</a>

(4.27)

Then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M116">View MathML</a>

(4.28)

Integrate (4.28) from 0 to t yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M117">View MathML</a>

(4.29)

Thus, (4.29) and (4.27) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M118">View MathML</a>

Define Kζ = {y∈ Ω: dist(y, ∂Ω)≥ ζ}. Since -Δz2 ≤ 0 in Ω × (0, T*), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M119">View MathML</a>

(4.30)

It follows from (4.30) and (4.27) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M120">View MathML</a>

(4.31)

for any xKζ and t ∈ (0,T*).

By Theorem 4.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M121">View MathML</a>

(4.32)

On the other hand, we know form (4.16) and Theorem 4.3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M122">View MathML</a>

(4.33)

From (4.31)-(4.33), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M123">View MathML</a>

It is easy to derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M124">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M125">View MathML</a>

This completes the proof.

Proof of Theorem 1.5. Case 1: 0 < p < 1. Form (4.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M126">View MathML</a>

where the notation u ~ v means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M127">View MathML</a>.

Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M128">View MathML</a>

(4.34)

Integrating (4.34) over (t, T*) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M129">View MathML</a>

(4.35)

So, we can get our conclusion by using (4.17) and (4.35).

Case 2: p = 1. In this case, for any given σ: 0 < σ ≪ 1. By (4.18), there exists 0 < t0 < T* such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M130">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M131">View MathML</a>

(4.36)

In view of the right-hand side of the (4.36), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M132">View MathML</a>

Integrating the above inequality from t to T* yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M133">View MathML</a>

Namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M134">View MathML</a>

(4.37)

Similar arguments to the left-hand side of (3.36) yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M135">View MathML</a>

(4.38)

Consequently, (4.37) and (4.38) guarantee that for t0 t T*,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M136">View MathML</a>

(4.39)

Letting σ → 0, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M137">View MathML</a>

(4.40)

because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/45/mathml/M138">View MathML</a>. Due to ln u(x, t) ~ G(t) uniformly on any compact subset of Ω, the proof is complete.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The main results in this article were derived by GZ and LT. All authors read and approved the final manuscript.

Acknowledgements

The authors express their thanks to the referee for his or her helpful comments and suggestions on the manuscript of this article.

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