Non-Newtonian polytropic filtration systems with nonlinear boundary conditions

  • Wanjuan Du1Email author and

    Affiliated with

    • Zhongping Li1

      Affiliated with

      Boundary Value Problems20112011:2

      DOI: 10.1186/1687-2770-2011-2

      Received: 9 November 2010

      Accepted: 21 June 2011

      Published: 21 June 2011

      Abstract

      This article deals with the global existence and the blow-up of non-Newtonian polytropic filtration systems with nonlinear boundary conditions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained by constructing various upper and lower solutions.

      Mathematics Subject Classification (2000)

      35K50, 35K55, 35K65

      Keywords

      Polytropic filtration systems Nonlinear boundary conditions Global existence Blow-up

      Introduction

      In this article, we study the global existence and the blow-up of non-Newtonian polytropic filtration systems with nonlinear boundary conditions
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ1_HTML.gif
      (1.1)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equa_HTML.gif

      Ω ⊂ ℝ N is a bounded domain with smooth boundary ∂Ω, ν is the outward normal vector on the boundary ∂Ω, and the constants k i , m i > 0, m ij ≥ 0, i, j = 1,..., n; ui 0(x) (i = 1,..., n) are positive C1 functions, satisfying the compatibility conditions.

      The particular feature of the equations in (1.1) is their power- and gradient-dependent diffusibility. Such equations arise in some physical models, such as population dynamics, chemical reactions, heat transfer, and so on. In particular, equations in (1.1) may be used to describe the nonstationary flows in a porous medium of fluids with a power dependence of the tangential stress on the velocity of displacement under polytropic conditions. In this case, the equations in (1.1) are called the non-Newtonian polytropic filtration equations which have been intensively studied (see [14] and the references therein). For the Neuman problem (1.1), the local existence of solutions in time have been established; see the monograph [4].

      We note that most previous works deal with special cases of (1.1) (see [513]). For example, Sun and Wang [7] studied system (1.1) with n = 1 (the single-equation case) and showed that all positive (weak) solutions of (1.1) exist globally if and only if m11k1 when k1m1; and exist globally if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq1_HTML.gif when k1 > m1. In [13], Wang studied the case n = 2 of (1.1) in one dimension. Recently, Li et al. [5] extended the results of [13] into more general N-dimensional domain.

      On the other hand, for systems involving more than two equations when m i = 1(i = 1,..., n), the special case k i = 1(i = 1,..., n) (heat equations) is concerned by Wang and Wang [9], and the case k i ≤ 1(i = 1,..., n) (porous medium equations) is discussed in [12]. In both studies, they obtained the necessary and sufficient conditions to the global existence of solutions. The fast-slow diffusion equations (there exists i(i = 1,..., n) such that k i > 1) is studied by Qi et al. [6], and they obtained the necessary and sufficient blow up conditions for the special case Ω = B R (0) (the ball centered at the origin in ℝ N with radius R). However, for the general domain Ω, they only gave some sufficient conditions to the global existence and the blow-up of solutions.

      The aim of this article is to study the long-time behavior of solutions to systems (1.1) and provide a simple criterion of the classification of global existence and nonexistence of solutions for general powers k i m i , indices m ij , and number n.

      Define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equb_HTML.gif

      Our main result is

      Theorem. All positive solutions of (1.1) exist globally if and only if all of the principal minor determinants of A are non-negative.

      Remark. The conclusion of Theorem covers the results of [513]. Moreover, this article provides the necessary and sufficient conditions to the global existence and the blow-up of solutions in the general domain Ω. Therefore, this article improves the results of [6].

      The rest of this article is organized as follows. Some preliminaries will be given in next section. The above theorem will be proved in Section 3.

      Preliminaries

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

      Definition. Let T > 0 and Q T = Ω × (0, T]. A vector function (u1(x, t),.., u n (x, t)) is called a weak upper (or lower) solution to (1.1) in Q T if

      (i). http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq2_HTML.gif ;

      (ii). (u1(x, 0),..., u n (x, 0)) ≥ (≤)(u10(x),..., un 0(x));

      (iii). for any positive functions ψ i (i = 1,..., n) ∈ L1(0, T; W1,2(Ω)) ∩ L2(Q T ), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equc_HTML.gif

      In particular, (u1(x, t),..., u n (x, t)) is called a weak solution of (1.1) if it is both a weak upper and a lower solution. For every T < ∞, if (u1(x, t),..., u n (x, t)) is a solution of (1.1) in Q T , then we say that (u1(x, t),..., u n (x, t)) is global.

      Lemma 2.1 (Comparison Principle.) Assume that ui 0(i = 1,..., n) are positive http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq3_HTML.gif functions and (u1,..., u n ) is any weak solution of (1.1). Also assume that (u1,..., u n ) ≥ (δ,..., δ) > 0 and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq4_HTML.gif are the lower and upper solutions of (1.1) in Q T , respectively, with nonlinear boundary flux http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq6_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq7_HTML.gif . Then we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq8_HTML.gif in Q T .

      When n = 2, the proof of Lemma 2.1 is given in [5]. When n > 2, the proof is similar.

      For convenience, we denote http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq7_HTML.gif , which are fixed constants, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq9_HTML.gif .

      In the following, we describe three lemmas, which can be obtained directly from Lemmas 2.7-2.9 in [6].

      Lemma 2.2 Suppose all the principal minor determinants of A are non-negative. If A is irreducible, then for any positive constant c, there exists α = (α1,..., α n ) T such that A α ≥ 0 and α i > c (i = 1,..., n).

      Lemma 2.3 Suppose that all the lower-order principal minor determinants of A are non-negative and A is irreducible. For any positive constant C, there exist large positive constants L i (i = 1,..., n) such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equd_HTML.gif
      Lemma 2.4 Suppose that all the lower-order principal minor determinants of A are non-negative and |A| < 0. Then, A is irreducible and, for any positive constant C, there exists α = (α1,..., α n ) T , with α i > 0 (i = 1,..., n) such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Eque_HTML.gif

      Proof of Theorem

      First, we note that if A is reducible, then the full system (1.1) can be reduced to several sub-systems, independent of each other. Therefore, in the following, we assume that A is irreducible. In addition, we suppose that k1 - m1k2 - m2 ≤ · · · k n - m n .

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq10_HTML.gif be the first eigenfunction of
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ2_HTML.gif
      (3.1)

      with the first eigenvalue http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq11_HTML.gif , normalized by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq12_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq14_HTML.gif in Ω and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq15_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq16_HTML.gif on ∂Ω (see [1416]).

      Thus, there exist some positive constants http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq19_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq20_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ3_HTML.gif
      (3.2)

      We also have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq21_HTML.gif provided http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq22_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq23_HTML.gif and some positive constant http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq24_HTML.gif . For the fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq24_HTML.gif , there exists a positive constant http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq25_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq26_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq27_HTML.gif .

      Proof of the sufficiency. We divide this proof into three different cases.

      Case 1. (k i < m i (i = 1,..., n)). Let
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ4_HTML.gif
      (3.3)
      where Q i satisfies http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq28_HTML.gif , and constants P i , α i (i = 1,..., n) remain to be determined. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq29_HTML.gif , by performing direct calculations, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equf_HTML.gif
      in Ω × ℝ+. By setting http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq30_HTML.gif if m i ≥ 1, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq31_HTML.gif if m i < 1, we have one the boundary that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equg_HTML.gif
      we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equh_HTML.gif
      if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ5_HTML.gif
      (3.4)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ6_HTML.gif
      (3.5)
      Note that k i < m i (i = 1,..., n). From Lemmas 2.2 and 2.3, we know that inequalities (3.4) and (3.5) hold for suitable choices of P i , α i (i = 1,..., n). Moreover, if we choose P i , α i to be large enough such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equi_HTML.gif

      then http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq33_HTML.gif . Therefore, we have proved that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq34_HTML.gif is a global upper solution of the system (1.1). The global existence of solutions to the problem (1.1) follows from the comparison principle.

      Case 2. (k i m i (i = 1,..., n)). Let
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ7_HTML.gif
      (3.6)
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq35_HTML.gif if m i ≥ 1, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq36_HTML.gif if m i < 1, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq19_HTML.gif are defined in (3.1) and (3.2), α i (i = 1,..., n) are positive constants that remain to be determined, and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equj_HTML.gif
      Since -ye-y≥ -e-1 for any y > 0, we know that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq38_HTML.gif . Thus, for (x, t) ∈ Ω × ℝ+, a simple computation shows that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equk_HTML.gif
      In addition, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equl_HTML.gif
      Noting http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq39_HTML.gif on ∂Ω, we have on the boundary that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equm_HTML.gif
      Then, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equn_HTML.gif
      if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ8_HTML.gif
      (3.7)
      From Lemma 2.2, we know that inequalities (3.7) hold for suitable choices of α i (i = 1,..., n). Moreover, if we choose ∞ i to be large enough such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equo_HTML.gif

      then http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq40_HTML.gif . Therefore, we have shown that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq4_HTML.gif is an upper solution of (1.1) and exists globally. Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq41_HTML.gif , and hence the solution (u1,..., u n ) of (1.1) exists globally.

      Case 3. (k i < m i (i = 1,..., s); k i m i (i = s + 1,..., n)). Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq42_HTML.gif be as in (3.3) and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equp_HTML.gif
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq37_HTML.gif , and A i are as in case 2. By Lemma 2.3, we choose P i ≥ (log Q i )-1||ui 0|| (i = 1,..., s) and M i ≥ max{1, ||ui 0||} (i = s + 1,..., n) such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ9_HTML.gif
      (3.8)
      Set
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equq_HTML.gif
      By similar arguments, in cases 1 and 2, we have on the boundary that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equr_HTML.gif
      Therefore employing (3.8), we see that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equs_HTML.gif
      if we knew
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ10_HTML.gif
      (3.9)
      We deduce from Lemma 2.2 that (3.9) holds for suitable choices of α i (i = 1,..., n). Moreover, we can choose α i large enough to assure that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equt_HTML.gif

      Then, as in the calculations of cases 1 and 2, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq43_HTML.gif . We prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq4_HTML.gif is an upper solution of (1.1), so (u1,..., u n ) exists globally.

      Proof of the necessity.

      Without loss of generality, we first assume that all the lower-order principal minor determinants of A are non-negative, and |A| < 0, for, if not, there exists some l th-order (1 ≤ l < n) principal minor determinant detAl × lof A = (a ij )n×nwhich is negative. Without loss of generality, we may consider that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equu_HTML.gif
      and all of the sth-order (1 ≤ sl - 1) principal minor determinants detAs × sof Al × lare non-negative. Then, we consider the following problem:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ11_HTML.gif
      (3.10)

      Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq44_HTML.gif . If we can prove that the solution (w1,..., w l ) of (3.10) blows up in finite time, then (w1,... w l , δ,..., δ) is a lower solution of (1.1) that blows up in finite time. Therefore, the solution of (1.1) blows up in finite time.

      We will complete the proof of the necessity of our theorem in three different cases.

      Case 1. (k i < m i (i = 1,..., n)). Let
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ12_HTML.gif
      (3.11)
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq46_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq48_HTML.gif , the α i are as given in Lemma 2.4 and satisfy http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq49_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ13_HTML.gif
      (3.12)
      By direct computation for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq50_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equv_HTML.gif
      For http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq51_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equw_HTML.gif
      Thus, by (3.12) and Lemma 2.4, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equx_HTML.gif

      We confirm that (u1,..., u n ) is a lower solution of (1.1), which blows up in finite time. We know by the comparison principle that the solution (u1,..., u n ) blows up in finite time.

      Case 2. (k i m i (i = 1,..., n)). Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq52_HTML.gif if m i < 1, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq53_HTML.gif if m i ≥ 1. for k i m i (i = 1,..., n), set
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ14_HTML.gif
      (3.13)
      where α i (i = 1,..., n) are to determined later and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ15_HTML.gif
      (3.14)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ16_HTML.gif
      (3.15)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ17_HTML.gif
      (3.16)
      By a direct computation, for x ∈ Ω, 0 < t < c/b, we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ18_HTML.gif
      (3.17)
      If http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq54_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq55_HTML.gif , and thus
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ19_HTML.gif
      (3.18)
      On the other hand, since -ye-y≥ -e-1 for any y > 0, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ20_HTML.gif
      (3.19)

      We have by (3.16), (3.18), and (3.19) that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq56_HTML.gif .

      If http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq57_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq58_HTML.gif , and then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ21_HTML.gif
      (3.20)

      It follows from (3.16), (3.17), and (3.20) that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq59_HTML.gif .

      We have on the boundary that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ22_HTML.gif
      (3.21)
      Moreover, by (3.14) and Lemma 2.4, we have that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ23_HTML.gif
      (3.22)

      (3.15), (3.21), and (3.22) imply that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq60_HTML.gif . Therefore, (u1,..., u1) is a lower solution of (1.1).

      For k i = m i (i = 1,..., n), let
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ24_HTML.gif
      (3.23)

      For k i = m i (i = 1,..., s) and k i > m i (i = s + 1,..., n), let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq61_HTML.gif as in (3.13) and (3.23). Using similar arguments as above, we can prove that (u1,..., u n ) is a lower solution of (1.1). Therefore, (u1,..., u n ) ≤ (u1,..., u n ). Consequently, (u1,..., u n ) blows up in finite time.

      Case 3. (k i < m i (i = 1,..., s); k i m i (i = s + 1,..., n)). Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq62_HTML.gif be as in (3.11) and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equy_HTML.gif
      where α i 's are to determined later and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equz_HTML.gif
      Based on arguments in cases 1 and 2, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq63_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq50_HTML.gif . Furthermore, for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq51_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equaa_HTML.gif
      Thus,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equab_HTML.gif
      holds if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ25_HTML.gif
      (3.24)

      From Lemma 2.4, we know that inequalities (3.24) hold for suitable choices of α i (i = 1,..., n). We show that (u1,..., u n ) is a lower solution of (1.1). Since (u1,..., u n ) blows up in finite time, it follows that the solution of (1.1) blows up in finite time.

      Declarations

      Acknowledgements

      This study was partially supported by the Projects Supported by Scientific Research Fund of SiChuan Provincial Education Department(09ZC011), and partially supported by the Natural Science Foundation Project of China West Normal University (07B046).

      Authors’ Affiliations

      (1)
      College of Mathematic and Information, China West Normal University

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      © Du and Li; licensee Springer. 2011

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