Blowup Analysis for a Semilinear Parabolic System with Nonlocal Boundary Condition

  • Yulan Wang1 and

    Affiliated with

    • Zhaoyin Xiang2Email author

      Affiliated with

      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:

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

      with nonlocal boundary condition

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

      and initial data

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

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq2_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq4_HTML.gif defined for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq6_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq8_HTML.gif satisfies the compatibility condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq9_HTML.gif , and that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq11_HTML.gif for any http://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

      http://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 http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ5_HTML.gif
        (1.5)
         

      for some http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq16_HTML.gif of the single equation

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

      under the assumption that http://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

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

      for any http://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:

      http://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:

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

        For any nonnegative http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq23_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq25_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq26_HTML.gif for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq29_HTML.gif for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq31_HTML.gif , such that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq34_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq38_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq41_HTML.gif blows up in finite time http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq42_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ10_HTML.gif
      (1.10)

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq44_HTML.gif and http://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:
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq47_HTML.gif and http://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 http://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 http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq53_HTML.gif are nonnegative functions. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq54_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ13_HTML.gif
      (2.2)

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

      Proof.

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

      We claim that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq63_HTML.gif will lead to a contradiction. Indeed, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq64_HTML.gif suggests that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq65_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq66_HTML.gif for some http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq68_HTML.gif .

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq70_HTML.gif on boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq71_HTML.gif and at the initial state http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq73_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq74_HTML.gif , which contradicts to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq75_HTML.gif .

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq77_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq79_HTML.gif for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq81_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq82_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq83_HTML.gif .

      Remark 2.3.

      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq85_HTML.gif for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq87_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq88_HTML.gif under the assumption that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq89_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq90_HTML.gif . Indeed, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq91_HTML.gif , we can conclude that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq92_HTML.gif in http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq95_HTML.gif ) and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq97_HTML.gif , respectively. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq98_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq99_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq100_HTML.gif in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq102_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq103_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ16_HTML.gif
      (3.1)
      Then we let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq104_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq105_HTML.gif ) be a continuous function satisfying http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq106_HTML.gif and set
      http://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:

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

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

      Our intention is to show that http://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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ19_HTML.gif
      (3.4)
      and thus
      http://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 http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq117_HTML.gif that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ22_HTML.gif
      (3.7)

      It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq118_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq119_HTML.gif . Therefore, we get http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq122_HTML.gif blows up in finite time when http://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 http://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 http://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:

         
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ23_HTML.gif
      (3.8)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq126_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq127_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq128_HTML.gif blows up in finite time http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq129_HTML.gif (see [26]). Under the assumption that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq131_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq132_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq135_HTML.gif and then http://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 http://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:

         
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ24_HTML.gif
      (3.9)
      and let http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ25_HTML.gif
      (3.10)

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

      Let

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ26_HTML.gif
      (3.11)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq144_HTML.gif . We now show that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq146_HTML.gif . Indeed, it follows from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq147_HTML.gif that, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq148_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ27_HTML.gif
      (3.12)
      When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq149_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ28_HTML.gif
      (3.13)

      Here we used http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq150_HTML.gif . The above inequalities show that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq155_HTML.gif for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq157_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq159_HTML.gif . If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq161_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq163_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq164_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq165_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq166_HTML.gif , there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq167_HTML.gif such that the solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq169_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ29_HTML.gif
      (4.1)

      Proof.

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

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

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

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

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

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

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

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

      For http://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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ38_HTML.gif
      (4.10)
      It follows from http://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
      http://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
      http://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

      http://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

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

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq197_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq198_HTML.gif and http://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

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

      We claim that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ45_HTML.gif
      (4.17)
      for some positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq200_HTML.gif . Indeed, if we let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq201_HTML.gif be the points at which http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ46_HTML.gif
      (4.18)
      At any point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq203_HTML.gif of differentiability of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq204_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq205_HTML.gif ,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq206_HTML.gif yields

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq207_HTML.gif . Namely, there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq208_HTML.gif such that
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ50_HTML.gif
      (4.22)
      Integrating this equality from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq209_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq210_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ51_HTML.gif
      (4.23)

      for some positive constant http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq212_HTML.gif .

      To obtain the lower estimate for http://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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ52_HTML.gif
      (4.24)
      for a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq214_HTML.gif . Integrating above equality on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq216_HTML.gif such that
      http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq219_HTML.gif such that

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq220_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq222_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq223_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq224_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq225_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq226_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq228_HTML.gif , we have
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_Equ57_HTML.gif
      (4.29)
      Choosing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq229_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq230_HTML.gif , one can get
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq231_HTML.gif is large enough since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F516390/MediaObjects/13661_2009_Article_852_IEq232_HTML.gif as http://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

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.