Open Access

Blowup Analysis for a Semilinear Parabolic System with Nonlocal Boundary Condition

Boundary Value Problems20092009:516390

DOI: 10.1155/2009/516390

Received: 23 July 2009

Accepted: 26 October 2009

Published: 22 November 2009

Abstract

This paper deals with the properties of positive solutions to a semilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blowup rate estimate for small weighted nonlocal boundary.

1. Introduction

In this paper, we devote our attention to the singularity analysis of the following semilinear parabolic system:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ1_HTML.gif
(1.1)

with nonlocal boundary condition

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ2_HTML.gif
(1.2)

and initial data

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq1_HTML.gif is a bounded connected domain with smooth boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq3_HTML.gif are positive parameters. Most physical settings lead to the default assumption that the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq4_HTML.gif defined for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq5_HTML.gif are nonnegative and continuous, and that the initial data https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq7_HTML.gif are nonnegative, which are mathematically convenient and currently followed throughout this paper. We also assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq8_HTML.gif satisfies the compatibility condition on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq9_HTML.gif , and that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq11_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq12_HTML.gif for the sake of the meaning ofnonlocal boundary.

Over the past few years, a considerable effort has been devoted to studying the blowup properties of solutions to parabolic equations withlocal boundary conditions, say Dirichlet, Neumann, or Robin boundary condition, which can be used to describe heat propagation on the boundary of container (see the survey papers [1, 2]). For example, the system (1.1) and (1.3) with homogeneous Dirichlet boundary condition

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ4_HTML.gif
(1.4)

has been studied extensively (see [35] and references therein), and the following proposition was proved.

Proposition 1.1.
  1. (i)
    All solutions are global if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq13_HTML.gif , while there exist both global solutions and finite time blowup solutions depending on the size of initial data when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq14_HTML.gif (See [4]). (ii) The asymptotic behavior near the blowup time is characterized by
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ5_HTML.gif
    (1.5)
     

for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq15_HTML.gif (See [3, 5]).

For the more parabolic problems related to the local boundary, we refer to the recent works [69] and references therein.

On the other hand, there are a number of important phenomena modeled by parabolic equations coupled with nonlocal boundary condition of form (1.2). In this case, the solution could be used to describe the entropy per volume of the material [1012]. Over the past decades, some basic results such as the global existence and decay property have been obtained for the nonlocal boundary problem (1.1)–(1.3) in the case of scalar equation (see [1316]). In particular, for the blowup solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq16_HTML.gif of the single equation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ6_HTML.gif
(1.6)

under the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq17_HTML.gif , Seo [15] established the following blowup rate estimate

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ7_HTML.gif
(1.7)

for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq18_HTML.gif . For the more nonlocal boundary problems, we also mention the recent works [1722]. In particular, Kong and Wang in [17], by using some ideas of Souplet [23], obtained the blowup conditions and blowup profile of the following system:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ8_HTML.gif
(1.8)

subject to nonlocal boundary (1.2), and Zheng and Kong in [22] gave the condition for global existence or nonexistence of solutions to the following similar system:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ9_HTML.gif
(1.9)

with nonlocal boundary condition (1.2). The typical characterization of systems (1.8) and (1.9) is the complete couple of the nonlocal sources, which leads to the analysis of simultaneous blowup.

To our surprise, however, it seems that there is no work dealing with singularity analysis of the parabolic system (1.1) with nonlocal boundary condition (1.2) except for the single equation case, although this is a very classical model. Therefore, the basic motivation for the work under consideration was our desire to understand the role of weight function in the blowup properties of that nonlinear system. We first remark by the standard theory [4, 13] that there exist local nonnegative classical solutions to this system.

Our main results read as follows.

Theorem 1.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq19_HTML.gif . All solutions to (1.1)–(1.3) exist globally.

It follows from Theorem 1.2 and Proposition 1.1(i) that any weight perturbation on the boundary has no influence on the global existence when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq20_HTML.gif , while the following theorem shows that it plays an important role when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq21_HTML.gif . In particular, Theorem 1.3(ii) is completely different from the case of the local boundary (1.4) (by comparing with Proposition 1.1(i)).

Theorem 1.3.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq22_HTML.gif .
  1. (i)

    For any nonnegative https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq24_HTML.gif , solutions to (1.1)–(1.3) blow up in finite time provided that the initial data are large enough.

     
  2. (ii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq26_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq27_HTML.gif , then any solutions to (1.1)–(1.3) with positive initial data blow up in finite time.

     
  3. (iii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq29_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq30_HTML.gif , then solutions to (1.1)–(1.3) with small initial data exist globally in time.

     

Once we have characterized for which exponents and weights the solution to problem (1.1)–(1.3) can or cannot blow up, we want to study the way the blowing up solutions behave as approaching the blowup time. To this purpose, the first step usually consists in deriving a bound for the blowup rate. For this bound estimate, we will use the classical method initially proposed in Friedman and McLeod [24]. The use of the maximum principle in that process forces us to give the following hypothesis technically.

(H)There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq31_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq32_HTML.gif

However, it seems that such an assumption is necessary to obtain the estimates of type (1.5) or (1.10) unless some additional restrictions on parameters https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq33_HTML.gif are imposed (for the related problem, we refer to the recent work of Matano and Merle [25]).

Here to obtain the precise blowup rates, we shall devote to establishing some relationship between the two components https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq35_HTML.gif as our problem involves a system, but we encounter the typical difficulties arising from the integral boundary condition. The following theorem shows that we have partially succeeded in this precise blowup characterization.

Theorem 1.4.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq38_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq39_HTML.gif and assumption (H) holds. If the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq40_HTML.gif of (1.1)–(1.3) with positive initial data https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq41_HTML.gif blows up in finite time https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq42_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ10_HTML.gif
(1.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq43_HTML.gif are both positive constants.

Remark 1.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq44_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq45_HTML.gif , then Theorem 1.4 implies that for the blowup solution of problem (1.6), we have the following precise blowup rate estimate:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ11_HTML.gif
(1.11)

which improves the estimate (1.7). Moreover, we relax the restriction on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq46_HTML.gif .

Remark 1.6.

By comparing with Proposition 1.1(ii), Theorem 1.4 could be explained as the small perturbation of homogeneous Dirichlet boundary, which leads to the appearance of blowup, does not influence the precise asymptotic behavior of solutions near the blowup time and the blowup rate exponents https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq48_HTML.gif are just determined by the corresponding ODE system https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq49_HTML.gif . Similar phenomena are also noticed in our previous work [18], where the single porous medium equation is studied.

The rest of this paper is organized as follows. Section 2 is devoted to some preliminaries, which include the comparison principle related to system (1.1)–(1.3). In Section 3, we will study the conditions for the solution to blow up and exist globally and hence prove Theorems 1.2 and 1.3. Proof of Theorem 1.4 is given in Section 4.

2. Preliminaries

In this section, we give some basic preliminaries. For convenience, we denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq50_HTML.gif We begin with the definition of the super- and subsolution of system (1.1)–(1.3).

Definition 2.1.

A pair of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq51_HTML.gif is called a subsolution of (1.1)–(1.3) if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ12_HTML.gif
(2.1)

A supersolution is defined with each inequality reversed.

Lemma 2.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq53_HTML.gif are nonnegative functions. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq54_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ13_HTML.gif
(2.2)

then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq55_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq56_HTML.gif .

Proof.

Set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq57_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq58_HTML.gif , by continuity, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq59_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq60_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq61_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq62_HTML.gif .

We claim that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq63_HTML.gif will lead to a contradiction. Indeed, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq64_HTML.gif suggests that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq65_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq66_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq67_HTML.gif . Without loss of generality, we suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq68_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq69_HTML.gif , we first notice that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ14_HTML.gif
(2.3)

In addition, it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq70_HTML.gif on boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq71_HTML.gif and at the initial state https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq72_HTML.gif . Then it follows from the strong maximum principle that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq73_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq74_HTML.gif , which contradicts to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq75_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq76_HTML.gif , we shall have a contradiction:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ15_HTML.gif
(2.4)

In the last inequality, we have used the facts that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq77_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq79_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq80_HTML.gif , which is a direct result of the previous case.

Therefore, the claim is true and thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq81_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq82_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq83_HTML.gif .

Remark 2.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq84_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq85_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq86_HTML.gif in Lemma 2.2, we can obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq87_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq88_HTML.gif under the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq89_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq90_HTML.gif . Indeed, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq91_HTML.gif , we can conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq92_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq93_HTML.gif as the proof of Lemma 2.2. Then the desired result follows from the limit procedure https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq94_HTML.gif .

From the above lemma, we can obtain the following comparison principle by the standard argument.

Proposition 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq95_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq96_HTML.gif be a subsolution and supersolution of (1.1)–(1.3) in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq97_HTML.gif , respectively. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq98_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq99_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq100_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq101_HTML.gif .

3. Global Existence and Blowup in Finite Time

In this section, we will use the super and subsolution technique to get the global existence or finite time blowup of the solution to (1.1)–(1.3).

Proof of Theorem 1.2.

As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq102_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq103_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ16_HTML.gif
(3.1)
Then we let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq104_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq105_HTML.gif ) be a continuous function satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq106_HTML.gif and set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ17_HTML.gif
(3.2)

We consider the following auxiliary problem:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ18_HTML.gif
(3.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq107_HTML.gif is the measure of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq109_HTML.gif . It follows from [13, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq110_HTML.gif ] that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq111_HTML.gif exists globally, and indeed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq112_HTML.gif (see [13, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq113_HTML.gif ]).

Our intention is to show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq114_HTML.gif is a global supersolution of (1.1)–(1.3). Indeed, a direct computation yields

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ19_HTML.gif
(3.4)
and thus
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ20_HTML.gif
(3.5)
Here we have used the conclusion https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq115_HTML.gif and inequality (3.1). We still have to consider the boundary and initial conditions. When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq116_HTML.gif , in view of Hölder's inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ21_HTML.gif
(3.6)
Similarly, we have also for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq117_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ22_HTML.gif
(3.7)

It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq118_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq119_HTML.gif . Therefore, we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq120_HTML.gif is a global supersolution of (1.1)–(1.3) and hence the solution to (1.1)–(1.3) exists globally by Proposition 2.4.

Proof of Theorem 1.3.
  1. (i)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq121_HTML.gif be the solution to the homogeneous Dirichlet boundary problem (1.1), (1.4), and (1.3). Then it is well known that for sufficiently large initial data the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq122_HTML.gif blows up in finite time when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq123_HTML.gif (see [4]). On the other hand, it is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq124_HTML.gif is a subsolution of problem (1.1)–(1.3). Henceforth, the solution of (1.1)–(1.3) with large initial data blows up in finite time provided that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq125_HTML.gif .

     
  2. (ii)

    We consider the ODE system:

     
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ23_HTML.gif
(3.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq126_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq127_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq128_HTML.gif blows up in finite time https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq129_HTML.gif (see [26]). Under the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq131_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq132_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq133_HTML.gif is a subsolution of problem (1.1)–(1.3). Therefore, by Proposition 2.4, we see that the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq134_HTML.gif of problem (1.1)–(1.3) satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq135_HTML.gif and then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq136_HTML.gif blows up in finite time.
  1. (iii)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq137_HTML.gif be the positive solution of the linear elliptic problem:

     
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ24_HTML.gif
(3.9)
and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq138_HTML.gif be the positive solution of the linear elliptic problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ25_HTML.gif
(3.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq139_HTML.gif is a positive constant such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq140_HTML.gif . We remark that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq142_HTML.gif ensure the existence of such https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq143_HTML.gif .

Let

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ26_HTML.gif
(3.11)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq144_HTML.gif . We now show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq145_HTML.gif is a supsolution of problem (1.1)–(1.3) for small initial data https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq146_HTML.gif . Indeed, it follows from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq147_HTML.gif that, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq148_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ27_HTML.gif
(3.12)
When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq149_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ28_HTML.gif
(3.13)

Here we used https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq150_HTML.gif . The above inequalities show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq151_HTML.gif is a supsolution of problem (1.1)–(1.3) whenever https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq152_HTML.gif . Therefore, system (1.1)–(1.3) has global solutions if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq153_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq154_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq155_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq156_HTML.gif .

4. Blowup Rate Estimate

In this section, we derive the precise blowup rate estimate. To this end, we first establish a partial relationship between the solution components https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq158_HTML.gif , which will be very useful in the subsequent analysis. For definiteness, we may assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq159_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq160_HTML.gif , we can proceed in the same way by changing the role of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq162_HTML.gif and then obtain the corresponding conclusion.

Lemma 4.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq163_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq164_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq165_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq166_HTML.gif , there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq167_HTML.gif such that the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq168_HTML.gif of problem (1.1)–(1.3) with positive initial data https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq169_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ29_HTML.gif
(4.1)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq170_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq171_HTML.gif is a positive constant to be chosen. For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq172_HTML.gif , a series of calculations show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ30_HTML.gif
(4.2)
If we choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq173_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq174_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ31_HTML.gif
(4.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq175_HTML.gif is a function of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq176_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq177_HTML.gif and lies between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq179_HTML.gif .

When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq180_HTML.gif , on the other hand, we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ32_HTML.gif
(4.4)
Denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq181_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq182_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq183_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq184_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq185_HTML.gif . It follows from Jensen's inequality, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq186_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq187_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ33_HTML.gif
(4.5)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ34_HTML.gif
(4.6)

For the initial condition, we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ35_HTML.gif
(4.7)

provided that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq188_HTML.gif .

Summarily, if we take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq189_HTML.gif , then it follows from Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq190_HTML.gif in [13] that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq191_HTML.gif , that is,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ36_HTML.gif
(4.8)

which is desired.

Using this lemma, we could establish our blowup rate estimate. To derive our conclusion, we shall use some ideas of [3].

Proof of Theorem 1.4.

For simplicity, we introduce https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq192_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq193_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq194_HTML.gif . A direct computation yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ37_HTML.gif
(4.9)

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq195_HTML.gif , we have from the boundary conditions that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ38_HTML.gif
(4.10)
It follows from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq196_HTML.gif and Jensen's inequality that the difference in the last brace is nonnegative and thus
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ39_HTML.gif
(4.11)
By similar arguments, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ40_HTML.gif
(4.12)

On the other hand, the hypothesis (H) implies that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ41_HTML.gif
(4.13)

Hence, from (4.9)–(4.13) and the comparison principle (see Remark 2.3), we get

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ42_HTML.gif
(4.14)
That is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ43_HTML.gif
(4.15)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq197_HTML.gif Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq198_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq199_HTML.gif are Lipschitz continuous and thus are differential almost everywhere (see e.g., [24]). Moreover, we have from equations (1.1) that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ44_HTML.gif
(4.16)

We claim that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ45_HTML.gif
(4.17)
for some positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq200_HTML.gif . Indeed, if we let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq201_HTML.gif be the points at which https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq202_HTML.gif attains its maximum, then relation (4.1) means that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ46_HTML.gif
(4.18)
At any point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq203_HTML.gif of differentiability of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq204_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq205_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ47_HTML.gif
(4.19)

From (4.15), (4.18), and (4.19), we can confirm our claim (4.17).

Integrating (4.17) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq206_HTML.gif yields

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ48_HTML.gif
(4.20)
which gives the upper estimate for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq207_HTML.gif . Namely, there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq208_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ49_HTML.gif
(4.21)
Then by (4.16) and (4.21), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ50_HTML.gif
(4.22)
Integrating this equality from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq209_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq210_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ51_HTML.gif
(4.23)

for some positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq211_HTML.gif . Thus we have established the upper estimates for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq212_HTML.gif .

To obtain the lower estimate for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq213_HTML.gif , we notice that (4.16) and (4.18) lead to

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ52_HTML.gif
(4.24)
for a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq214_HTML.gif . Integrating above equality on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq215_HTML.gif , we see there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq216_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ53_HTML.gif
(4.25)

Finally, we give the lower estimate for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq217_HTML.gif . Indeed, using the relationship (4.16), (4.23) and (4.25), we could prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq218_HTML.gif is bounded from below; that is, there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq219_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ54_HTML.gif
(4.26)
To see this, our approach is based on the contradiction arguments. Assume that there would exist two sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq220_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq221_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq222_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq223_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq224_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ55_HTML.gif
(4.27)
Then we could choose a corresponding sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq225_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq226_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq227_HTML.gif is a positive constant to be determined later. As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq228_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ56_HTML.gif
(4.28)
From (4.23) and (4.27), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ57_HTML.gif
(4.29)
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq229_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq230_HTML.gif , one can get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ58_HTML.gif
(4.30)

which would contradict to (4.25) as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq231_HTML.gif is large enough since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq232_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq233_HTML.gif .

Declarations

Acknowledgments

The authors are very grateful to the anonymous referees for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by Natural Science Foundation Project of CQ CSTC (2007BB2450), China Postdoctoral Science Foundation, the Key Scientific Research Foundation of Xihua University, and Youth Foundation of Science and Technology of UESTC.

Authors’ Affiliations

(1)
School of Mathematics and Computer Engineering, Xihua University
(2)
School of Applied Mathematics, University of Electronic Science and Technology of China

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Copyright

© Y. Wang and Z. Xiang. 2009

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