Open Access

A Quasilinear Parabolic System with Nonlocal Boundary Condition

Boundary Value Problems20102011:750769

DOI: 10.1155/2011/750769

Received: 8 May 2010

Accepted: 11 August 2010

Published: 18 August 2010

Abstract

We investigate the blow-up properties of the positive solutions to a quasilinear 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 blow-up rate estimate. These extend the resent results of Wang et al. (2009), which considered the special case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq1_HTML.gif , and Wang et al. (2007), which studied the single equation.

1. Introduction

In this paper, we deal with the following degenerate parabolic system:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ1_HTML.gif
(1.1)
with nonlocal boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ2_HTML.gif
(1.2)
and initial data
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq2_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq3_HTML.gif is a bounded connected domain with smooth boundary. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq5_HTML.gif for the sake of the meaning of nonlocal boundary are nonnegative continuous functions defined for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq7_HTML.gif , while the initial data https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq9_HTML.gif are positive continuous functions and satisfy the compatibility conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq11_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq12_HTML.gif , respectively.

Problem (1.1)–(1.3) models a variety of physical phenomena such as the absorption and "downward infiltration" of a fluid (e.g., water) by the porous medium with an internal localized source or in the study of population dynamics (see [1]). The solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq13_HTML.gif of the problem (1.1)–(1.3) is said to blow up in finite time if there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq14_HTML.gif called the blow-up time such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ4_HTML.gif
(1.4)
while we say that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq15_HTML.gif exists globally if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ5_HTML.gif
(1.5)
Over the past few years, a considerable effort has been devoted to the study of the blow-up properties of solutions to parabolic equations with local 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 [2, 3] and references therein). The semilinear case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq16_HTML.gif of (1.1)–(1.3) has been deeply investigated by many authors (see, e.g., [211]). The system turns out to be degenerate if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq17_HTML.gif ; for example, in [12, 13], Galaktionov et al. studied the following degenerate parabolic equations:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ6_HTML.gif
(1.6)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq20_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq21_HTML.gif . They obtained that solutions of (1.6) are global if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq22_HTML.gif , and may blow up in finite time if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq23_HTML.gif . For the critical case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq24_HTML.gif , there should be some additional assumptions on the geometry of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq25_HTML.gif .

Song et al. [14] considered the following nonlinear diffusion system with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq26_HTML.gif coupled via more general sources:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ7_HTML.gif
(1.7)

Recently, the genuine degenerate situation with zero boundary values for (1.7) has been discussed by Lei and Zheng [15]. Clearly, problem (1.6) is just the special case by taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq27_HTML.gif in (1.7) with zero boundary condition.

For the more parabolic problems related to the local boundary, we refer to the recent works [1620] 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 (see [2123]). 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 [2428]). In particular, in [28], Wang et al. studied the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ8_HTML.gif
(1.8)
with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq28_HTML.gif . They obtained the blow-up condition and its blow-up rate estimate. For the special case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq29_HTML.gif in the system (1.8), under the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq30_HTML.gif , Seo [26] established the following blow-up rate estimate:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ9_HTML.gif
(1.9)
for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq31_HTML.gif For the more nonlocal boundary problems, we also mention the recent works [2934]. In particular, Kong and Wang in [29], by using some ideas of Souplet [35], obtained the blow-up conditions and blow-up profile of the following system:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ10_HTML.gif
(1.10)
subject to nonlocal boundary (1.2), and Zheng and Kong in [34] gave the condition for global existence or nonexistence of solutions to the following similar system:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ11_HTML.gif
(1.11)

with nonlocal boundary condition (1.2). The typical characterization of systems (1.10) and (1.11) is the complete couple of the nonlocal sources, which leads to the analysis of simultaneous blowup.

Recently, Wang and Xiang [30] studied the following semilinear parabolic system with nonlocal boundary condition:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ12_HTML.gif
(1.12)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq32_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq33_HTML.gif are positive parameters. They gave the criteria for finite time blowup or global existence, and established blow-up rate estimate.

To our knowledge, there is no work dealing with 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 main purpose of this paper is to understand how the reaction terms, the weight functions and the nonlinear diffusion affect the blow-up properties for the problem (1.1)–(1.3). We will show that the weight functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq34_HTML.gif play substantial roles in determining blowup or not of solutions. Firstly, we establish the global existence and finite time blow-up of the solution. Secondly, we establish the precise blowup rate estimates for all solutions which blow up.

Our main results could be stated as follows.

Theorem 1.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq35_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq36_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq38_HTML.gif hold, then any solution to (1.1)–(1.3) with positive initial data blows up in finite time.

Theorem 1.2 ..

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq39_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq40_HTML.gif .

(1)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq41_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq42_HTML.gif , then every nonnegative solution of (1.1)–(1.3) is global.

(2)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq43_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq44_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq45_HTML.gif , then the nonnegative solution of (1.1)–(1.3) exists globally for sufficiently small initial values and blows up in finite time for sufficiently large initial values.

To establish blow-up rate of the blow-up solution, we need the following assumptions on the initial data https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq46_HTML.gif

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq48_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq49_HTML.gif ;

(2) There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq51_HTML.gif , such tha
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ13_HTML.gif
(1.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq52_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq53_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq54_HTML.gif will be given in Section 4.

Theorem 1.3 ..

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq55_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq56_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq57_HTML.gif and satisfy https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq58_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq59_HTML.gif ; assumptions (H1)-(H2) hold. If the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq60_HTML.gif of (1.1)–(1.3) with positive initial data https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq61_HTML.gif blows up in finite time https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq62_HTML.gif , then there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq63_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ14_HTML.gif
(1.14)

This paper is organized as follows. In the next section, we give the comparison principle of the solution of problem (1.1)–(1.3) and some important lemmas. In Section 3, we concern the global existence and nonexistence of solution of problem (1.1)–(1.3) and show the proofs of Theorems 1.1 and 1.2. In Section 4, we will give the estimate of the blow-up rate.

2. Preliminaries

In this section, we give some basic preliminaries. For convenience, we denote that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq64_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq65_HTML.gif . As it is now well known that degenerate equations need not posses classical solutions, we begin by giving a precise definition of a weak solution for problem (1.1)–(1.3).

Definition 2.1 ..

A vector function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq66_HTML.gif defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq67_HTML.gif , for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq68_HTML.gif , is called a sub (or super) solution of ( 1.1 )–( 1.3 ), if all the following hold:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq69_HTML.gif ;

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq70_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq71_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq72_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq73_HTML.gif ;
  1. (3)
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ15_HTML.gif
    (2.1)
     
 where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq74_HTML.gif is the unit outward normal to the lateral boundary of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq75_HTML.gif . For every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq76_HTML.gif and any ϕ belong to the class of test functions,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ16_HTML.gif
(2.2)

A weak solution of (1.1) is a vector function which is both a subsolution and a supersolution of (1.1)-(1.3).

Lemma 2.2 (Comparison principle).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq77_HTML.gif   and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq78_HTML.gif   be a subsolution and supersolution of (1.1)–(1.3) in  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq79_HTML.gif , respectively. Then  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq80_HTML.gif   in   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq81_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq82_HTML.gif

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq83_HTML.gif , the subsolution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq84_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ17_HTML.gif
(2.3)
On the other hand, the supersolution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq85_HTML.gif satisfies the reversed inequality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ18_HTML.gif
(2.4)
Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq86_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ19_HTML.gif
(2.5)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ20_HTML.gif
(2.6)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq88_HTML.gif are bounded in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq89_HTML.gif , it follows from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq90_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq91_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq92_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq93_HTML.gif  are bounded nonnegative functions. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq94_HTML.gif is a function between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq96_HTML.gif . Noticing that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq98_HTML.gif are nonnegative bounded function and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq99_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq100_HTML.gif , we choose appropriate function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq101_HTML.gif as in [36] to obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ21_HTML.gif
(2.7)

By Gronwall's inequality, we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq103_HTML.gif can be obtained in similar way, then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq104_HTML.gif .

Local in time existence of positive classical solutions of the problem (1.1)–(1.3) can be obtained using fixed point theorem (see [37]), the representation formula and the contraction mapping principle as in [38]. By the above comparison principle, we get the uniqueness of the solution to the problem. The proof is more or less standard, so is omitted here.

Remark 2.3.

From Lemma 2.2, it is easy to see that the solution of (1.1)–(1.3) is unique if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq105_HTML.gif .

The following comparison lemma plays a crucial role in our proof which can be obtained by similar arguments as in [24, 3840]

Lemma 2.4.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq106_HTML.gif and satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ22_HTML.gif
(2.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq107_HTML.gif  are bounded functions and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq108_HTML.gif ,  and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq109_HTML.gif and is not identically zero. Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq110_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq111_HTML.gif imply that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq112_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq113_HTML.gif . Moreover,   if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq114_HTML.gif or if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq115_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq116_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq117_HTML.gif imply that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq118_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq119_HTML.gif

Denote that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ23_HTML.gif
(2.9)

We give some lemmas that will be used in the following section. Please see [41] for their proofs.

Lemma 2.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq120_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq121_HTML.gif , then there exist two positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq122_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq123_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq124_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq125_HTML.gif .

Lemma 2.6 ..

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq127_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq128_HTML.gif , then there exist two positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq129_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq130_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq131_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq132_HTML.gif .

3. Global Existence and Blowup in Finite Time

Compared with usual homogeneous Dirichlet boundary data, the weight functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq134_HTML.gif play an important role in the global existence or global nonexistence results for problem (1.1)–(1.3).

Proof of Theorem 1.1..

We consider the ODE system
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ24_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq135_HTML.gif , and we use the assumption https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq136_HTML.gif

Set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ25_HTML.gif
(3.2)
with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ26_HTML.gif
(3.3)

It is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq137_HTML.gif is the unique solution of the ODE problem (3.1), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq139_HTML.gif imply that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq140_HTML.gif blows up in finite time. Under the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq141_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq142_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq143_HTML.gif is a subsolution of problem (1.1)–(1.3). Therefore, by Lemma 2.2, we see that the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq144_HTML.gif of problem (1.1)–(1.3) satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq145_HTML.gif and then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq146_HTML.gif blows up in finite time.

Proof of Theorem 1.2.
  1. (1)
    Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq147_HTML.gif be the positive solution of the linear elliptic problem
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ27_HTML.gif
    (3.4)
     
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq148_HTML.gif be the positive solution of the linear elliptic problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ28_HTML.gif
(3.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq149_HTML.gif are positive constant such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq150_HTML.gif . We remark that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq151_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq152_HTML.gif ensure the existence of such https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq153_HTML.gif .

Denote that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ29_HTML.gif
(3.6)
We define the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq154_HTML.gif as following:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ30_HTML.gif
(3.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq155_HTML.gif is a constant to be determined later. Then, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ31_HTML.gif
(3.8)
In a similar way, we can obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ32_HTML.gif
(3.9)

here, we used https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq156_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq157_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq158_HTML.gif .

On the other hand, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ33_HTML.gif
(3.10)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ34_HTML.gif
(3.11)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ35_HTML.gif
(3.12)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq159_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq160_HTML.gif , by Lemma 2.5, there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq161_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ36_HTML.gif
(3.13)
Therefore, we can choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq162_HTML.gif sufficiently large, such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ37_HTML.gif
(3.14)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ38_HTML.gif
(3.15)

Now, it follows from (3.8)–(3.15) that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq163_HTML.gif defined by (3.7) is a positive supersolution of (1.1)–(1.3).

By comparison principle, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq164_HTML.gif , which implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq165_HTML.gif exists globally.
  1. (2)
    If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq166_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq167_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq168_HTML.gif , by Lemma 2.6, there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq169_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ39_HTML.gif
    (3.16)
     

So we can choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq170_HTML.gif . Furthermore, assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq171_HTML.gif are small enough to satisfy (3.15). It follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq172_HTML.gif defined by (3.7) is a positive supersolution of (1.1)–(1.3). Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq173_HTML.gif exists globally.

Due to the requirement of the comparison principle we will construct blow-up subsolutions in some subdomain of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq174_HTML.gif in which https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq175_HTML.gif . We use an idea from Souplet [42] and apply it to degenerate equations. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq176_HTML.gif be a nontrivial nonnegative continuous function and vanished on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq177_HTML.gif . Without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq179_HTML.gif . We will construct a blow-up positive subsolution to complete the proof.

Set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ40_HTML.gif
(3.17)
with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ41_HTML.gif
(3.18)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq180_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq181_HTML.gif are to be determined later. Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq183_HTML.gif is nonincreasing since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq184_HTML.gif Note that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ42_HTML.gif
(3.19)
for sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq185_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq186_HTML.gif becomes unbounded as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq187_HTML.gif , at the point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq188_HTML.gif . Calculating directly, we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ43_HTML.gif
(3.20)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ44_HTML.gif
(3.21)

notice that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq189_HTML.gif is sufficiently small.

Similarly, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ45_HTML.gif
(3.22)

Case 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq190_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq191_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ46_HTML.gif
(3.23)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ47_HTML.gif
(3.24)

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq192_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ48_HTML.gif
(3.25)
By Lemma 2.6, there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq193_HTML.gif large enough to satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ49_HTML.gif
(3.26)
and we can choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq194_HTML.gif be sufficiently small that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ50_HTML.gif
(3.27)
Thus, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ51_HTML.gif
(3.28)
Hence, for sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq195_HTML.gif , (3.24) and (3.25) imply that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ52_HTML.gif
(3.29)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ53_HTML.gif
(3.30)

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq196_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq197_HTML.gif is continuous, there exist two positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq198_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq199_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq200_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq201_HTML.gif . Choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq202_HTML.gif small enough to insure https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq203_HTML.gif , hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq204_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq205_HTML.gif . Under the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq207_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq208_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq209_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq210_HTML.gif Furthermore, choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq211_HTML.gif so large that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq212_HTML.gif . By comparison principle, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq213_HTML.gif . It shows that solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq214_HTML.gif to (1.1)–(1.3) blows up in finite time.

4. Blow-Up Rate Estimates

In this section, we will estimate the blow-up rate of the blow-up solution of (1.1). Throughout this section, we will assume that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ54_HTML.gif
(4.1)
To obtain the estimate, we firstly introduce some transformations. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq215_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq216_HTML.gif then problem (1.1)–(1.3) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ55_HTML.gif
(4.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq217_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq218_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq220_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq221_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq223_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq224_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq225_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq226_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq227_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq228_HTML.gif . By the conditions (4.1), we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq229_HTML.gif and satisfy that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq230_HTML.gif . Under this transformation, assumptions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq231_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq232_HTML.gif become

() https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq234_HTML.gif , for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq235_HTML.gif ;

() there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq237_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ56_HTML.gif
(4.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq238_HTML.gif will be given later.

By the standard method [16, 42], we can show that system (4.2) has a smooth nonnegative solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq239_HTML.gif , provided that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq240_HTML.gif satisfy the hypotheses https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq241_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq242_HTML.gif . We thus assume that the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq243_HTML.gif of problem (4.2) blows up in the finite time https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq244_HTML.gif . Denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq245_HTML.gif . We can obtain the blow-up rate from the following lemmas.

Lemma 4.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq246_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq247_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq248_HTML.gif , then there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq249_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ57_HTML.gif
(4.4)

Proof.

By (4.2), we have (see [43])
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ58_HTML.gif
(4.5)
Noticing that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq250_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq251_HTML.gif , hence we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ59_HTML.gif
(4.6)

by virtue of Young's inequality. Integrating (4.6) from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq252_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq253_HTML.gif , we can obtain (4.4).

Lemma 4.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq254_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq255_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq256_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq257_HTML.gif is a solution of (4.2). Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ60_HTML.gif
(4.7)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ61_HTML.gif
(4.8)

Proof.

Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq258_HTML.gif a straightforward computation yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ62_HTML.gif
(4.9)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq259_HTML.gif , obviously we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ63_HTML.gif
(4.10)
Otherwise, noticing that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq260_HTML.gif , by virtue of Young's inequality,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ64_HTML.gif
(4.11)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq261_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ65_HTML.gif
(4.12)
Similarly, we also have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ66_HTML.gif
(4.13)
Fix https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq262_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ67_HTML.gif
(4.14)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq263_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq264_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ68_HTML.gif
(4.15)
Noticing that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq265_HTML.gif by virtue of Jensen's inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ69_HTML.gif
(4.16)
here, we used https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq266_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq267_HTML.gif in the last inequality. Hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq268_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ70_HTML.gif
(4.17)
Similarly, we also have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ71_HTML.gif
(4.18)
On the other hand, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq269_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq270_HTML.gif imply that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq271_HTML.gif Combined inequalities (4.12)-(4.18) and Lemma 2.4, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq272_HTML.gif that is, (4.7) holds.Integrating (4.7) from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq273_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq274_HTML.gif , we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ72_HTML.gif
(4.19)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq275_HTML.gif are positive constants independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq276_HTML.gif . It follows from Lemma 4.1 and (4.19), we have the following lemma.

Lemma 4.3.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq277_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq278_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq279_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq280_HTML.gif is the solution of system (4.2) and blows up in finite time https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq281_HTML.gif , then there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_IEq282_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F750769/MediaObjects/13661_2010_Article_57_Equ73_HTML.gif
(4.20)

According the transform and Lemma 4.3, we can obtain Theorem 1.3.

Declarations

Acknowledgments

The authors would like to thank the anonymous referees for their suggestions and comments on the original manuscript. This work was partially supported by NSF of China (10771226) and partially supported by the Educational Science Foundation of Chongqing (KJ101303), China.

Authors’ Affiliations

(1)
College of Mathematics and Computer Sciences, Yangtze Normal University
(2)
College of Mathematics and Physics, Chongqing University

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© Botao Chen et al. 2011

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