Open Access

Non-Newtonian polytropic filtration systems with nonlinear boundary conditions

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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ1_HTML.gif
(1.1)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equa_HTML.gif

Ω 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 https://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 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
https://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). https://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
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq6_HTML.gif , where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq7_HTML.gif . Then we have https://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 https://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 https://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
https://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
https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq11_HTML.gif , normalized by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq12_HTML.gif , then https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq14_HTML.gif in Ω and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq15_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq19_HTML.gif , and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq20_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq21_HTML.gif provided https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq22_HTML.gif with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq23_HTML.gif and some positive constant https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq24_HTML.gif . For the fixed https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq25_HTML.gif such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq26_HTML.gif if https://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
https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equf_HTML.gif
in Ω × +. By setting https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq30_HTML.gif if m i ≥ 1, https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equg_HTML.gif
we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equh_HTML.gif
if
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ5_HTML.gif
(3.4)
and
https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equi_HTML.gif

then https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq32_HTML.gif , https://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 https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ7_HTML.gif
(3.6)
where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq35_HTML.gif if m i ≥ 1, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq36_HTML.gif if m i < 1, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq18_HTML.gif , https://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
https://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 https://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 × +, a simple computation shows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equk_HTML.gif
In addition, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equl_HTML.gif
Noting https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equm_HTML.gif
Then, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equn_HTML.gif
if
https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equo_HTML.gif

then https://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 https://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, https://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 https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equp_HTML.gif
where https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ9_HTML.gif
(3.8)
Set
https://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
https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equs_HTML.gif
if we knew
https://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
https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq43_HTML.gif . We prove that https://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
https://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:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ11_HTML.gif
(3.10)

Note that https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ12_HTML.gif
(3.11)
where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq46_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq47_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq49_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq50_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equv_HTML.gif
For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq51_HTML.gif , we have
https://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
https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq52_HTML.gif if m i < 1, https://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
https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ15_HTML.gif
(3.14)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ16_HTML.gif
(3.15)
https://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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equ18_HTML.gif
(3.17)
If https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq54_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq55_HTML.gif , and thus
https://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
https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq56_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq57_HTML.gif , then https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq58_HTML.gif , and then
https://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 https://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
https://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
https://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 https://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
https://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 https://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 https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq63_HTML.gif for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq50_HTML.gif . Furthermore, for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_IEq51_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equaa_HTML.gif
Thus,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-2/MediaObjects/13661_2010_Article_2_Equab_HTML.gif
holds if
https://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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