Abstract
This article deals with the global existence and the blowup of nonNewtonian 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; BlowupIntroduction
In this article, we study the global existence and the blowup of nonNewtonian polytropic filtration systems with nonlinear boundary conditions
where
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; u_{i0}(x) (i = 1,..., n) are positive C^{1 }functions, satisfying the compatibility conditions.
The particular feature of the equations in (1.1) is their power and gradientdependent 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 nonNewtonian 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 singleequation case) and showed that all positive (weak) solutions of (1.1) exist globally if and only if m_{11 }≤ k_{1 }when k_{1 }≤ m_{1}; and exist globally if and only if when k_{1 }> m_{1}. 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 Ndimensional 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 fastslow 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 blowup of solutions.
The aim of this article is to study the longtime 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
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 nonnegative.
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 blowup 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 (u_{1}(x, t),.., u_{n}(x, t)) is called a weak upper (or lower) solution to (1.1) in Q_{T }if
(i). ;
(ii). (u_{1}(x, 0),..., u_{n}(x, 0)) ≥ (≤)(u_{10}(x),..., u_{n0}(x));
(iii). for any positive functions ψ_{i}(i = 1,..., n) ∈ L^{1}(0, T; W ^{1,2}(Ω)) ∩ L^{2}(Q_{T}), we have
In particular, (u_{1}(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 (u_{1}(x, t),..., u_{n}(x, t)) is a solution of (1.1) in Q_{T}, then we say that (u_{1}(x, t),..., u_{n}(x, t)) is global.
Lemma 2.1 (Comparison Principle.) Assume that u_{i0}(i = 1,..., n) are positive functions and (u_{1},..., u_{n}) is any weak solution of (1.1). Also assume that (
 u
 u
When n = 2, the proof of Lemma 2.1 is given in [5]. When n > 2, the proof is similar.
For convenience, we denote , which are fixed constants, and let .
In the following, we describe three lemmas, which can be obtained directly from Lemmas 2.72.9 in [6].
Lemma 2.2 Suppose all the principal minor determinants of A are nonnegative. If A is irreducible, then for any positive constant c, there exists α = (α_{1},..., α_{n})^{T }such that A α ≥ 0 and α_{i }> c (i = 1,..., n).
Lemma 2.3 Suppose that all the lowerorder principal minor determinants of A are nonnegative and A is irreducible. For any positive constant C, there exist large positive constants L_{i}(i = 1,..., n) such that
Lemma 2.4 Suppose that all the lowerorder principal minor determinants of A are nonnegative and A < 0. Then, A is irreducible and, for any positive constant C, there exists α = (α_{1},..., α_{n})^{T}, with α_{i }> 0 (i = 1,..., n) such that
Proof of Theorem
First, we note that if A is reducible, then the full system (1.1) can be reduced to several subsystems, independent of each other. Therefore, in the following, we assume that A is irreducible. In addition, we suppose that k_{1 } m_{1 }≤ k_{2 } m_{2 }≤ · · · k_{n } m_{n}.
Let be the first eigenfunction of
with the first eigenvalue , normalized by , then , in Ω and and on ∂Ω (see [1416]).
Thus, there exist some positive constants , , , and such that
We also have provided with and some positive constant . For the fixed , there exists a positive constant such that if .
Proof of the sufficiency. We divide this proof into three different cases.
Case 1. (k_{i }< m_{i }(i = 1,..., n)). Let
where Q_{i }satisfies , and constants P_{i}, α_{i }(i = 1,..., n) remain to be determined. Since , by performing direct calculations, we have
in Ω × ℝ^{+}. By setting if m_{i }≥ 1, if m_{i }< 1, we have one the boundary that
we have
if
and
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
then , . Therefore, we have proved that 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
where if m_{i }≥ 1, if m_{i }< 1, , , , are defined in (3.1) and (3.2), α_{i}(i = 1,..., n) are positive constants that remain to be determined, and
Since ye^{y }≥ e^{1 }for any y > 0, we know that . Thus, for (x, t) ∈ Ω × ℝ^{+}, a simple computation shows that
In addition, we have
Noting on ∂Ω, we have on the boundary that
Then, we have
if
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
then . Therefore, we have shown that is an upper solution of (1.1) and exists globally. Therefore, , and hence the solution (u_{1},..., 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 be as in (3.3) and
where , and A_{i }are as in case 2. By Lemma 2.3, we choose P_{i }≥ (log Q_{i})^{1}u_{i0}_{∞ }(i = 1,..., s) and M_{i }≥ max{1, u_{i0}_{∞}} (i = s + 1,..., n) such that
Set
By similar arguments, in cases 1 and 2, we have on the boundary that
Therefore employing (3.8), we see that
if we knew
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
Then, as in the calculations of cases 1 and 2, we have . We prove that is an upper solution of (1.1), so (u_{1},..., u_{n}) exists globally.
Proof of the necessity.
Without loss of generality, we first assume that all the lowerorder principal minor determinants of A are nonnegative, and A < 0, for, if not, there exists some lthorder (1 ≤ l < n) principal minor determinant detA_{l × l }of A = (a_{ij})_{n×n }which is negative. Without loss of generality, we may consider that
and all of the sthorder (1 ≤ s ≤ l  1) principal minor determinants detA_{s × s }of A_{l × l }are nonnegative. Then, we consider the following problem:
Note that . If we can prove that the solution (w_{1},..., w_{l}) of (3.10) blows up in finite time, then (w_{1},... 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
where , , , , the α_{i }are as given in Lemma 2.4 and satisfy ,
By direct computation for , we have
For , we have
Thus, by (3.12) and Lemma 2.4, we have
We confirm that (
 u
 u
Case 2. (k_{i }≥ m_{i }(i = 1,..., n)). Let if m_{i }< 1, if m_{i }≥ 1. for k_{i }≥ m_{i }(i = 1,..., n), set
where α_{i}(i = 1,..., n) are to determined later and
By a direct computation, for x ∈ Ω, 0 < t < c/b, we obtain that
If , we have , and thus
On the other hand, since ye^{y }≥ e^{1 }for any y > 0, we have
We have by (3.16), (3.18), and (3.19) that .
If , then , and then
It follows from (3.16), (3.17), and (3.20) that .
We have on the boundary that
Moreover, by (3.14) and Lemma 2.4, we have that
(3.15), (3.21), and (3.22) imply that . Therefore, (
 u
 u
For k_{i }= m_{i}(i = 1,..., n), let
For k_{i }= m_{i }(i = 1,..., s) and k_{i }> m_{i }(i = s + 1,..., n), let as in (3.13) and (3.23). Using similar arguments as above, we can prove that (
 u
 u
 u
 u
Case 3. (k_{i }< m_{i }(i = 1,..., s); k_{i }≥ m_{i }(i = s + 1,..., n)). Let be as in (3.11) and
where α_{i}'s are to determined later and
Based on arguments in cases 1 and 2, we have for . Furthermore, for , we have
Thus,
holds if
From Lemma 2.4, we know that inequalities (3.24) hold for suitable choices of α_{i }(i = 1,..., n). We show that (
 u
 u
 u
 u
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
DW carried out all studies in the paper. LZ participated in the design of the study in the paper.
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).
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