- Open Access
Non-Newtonian polytropic filtration systems with nonlinear boundary conditions
© Du and Li; licensee Springer. 2011
- Received: 9 November 2010
- Accepted: 21 June 2011
- Published: 21 June 2011
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
- Polytropic filtration systems
- Nonlinear boundary conditions
- Global existence
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 [1–4] and the references therein). For the Neuman problem (1.1), the local existence of solutions in time have been established; see the monograph .
We note that most previous works deal with special cases of (1.1) (see [5–13]). For example, Sun and Wang  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 m11 ≤ k1 when k1 ≤ m1; and exist globally if and only if when k1 > m1. In , Wang studied the case n = 2 of (1.1) in one dimension. Recently, Li et al.  extended the results of  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 , and the case k i ≤ 1(i = 1,..., n) (porous medium equations) is discussed in . 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. , 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.
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 [5–13]. 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 .
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.
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
(ii). (u1(x, 0),..., u n (x, 0)) ≥ (≤)(u10(x),..., un 0(x));
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 functions and (u1,..., u n ) is any weak solution of (1.1). Also assume that (u1,..., u n ) ≥ (δ,..., δ) > 0 and are the lower and upper solutions of (1.1) in Q T , respectively, with nonlinear boundary flux and , where . Then we have in Q T .
When n = 2, the proof of Lemma 2.1 is given in . 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 2.7-2.9 in .
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).
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 - m1 ≤ k2 - m2 ≤ · · · k n - m n .
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.
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.
then . Therefore, we have shown that is an upper solution of (1.1) and exists globally. Therefore, , and hence the solution (u1,..., u n ) of (1.1) exists globally.
Then, as in the calculations of cases 1 and 2, we have . We prove that is an upper solution of (1.1), so (u1,..., u n ) exists globally.
Proof of the necessity.
Note that . 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.
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.
We have by (3.16), (3.18), and (3.19) that .
It follows from (3.16), (3.17), and (3.20) that .
(3.15), (3.21), and (3.22) imply that . Therefore, (u1,..., u1) is a lower solution of (1.1).
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 (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.
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.
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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