In this article, we study the global existence and the blow-up of non-Newtonian 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 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 2 3 4 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 5 6 7 8 9 10 11 12 13 ). 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 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 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

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 6 7 8 9 10 11 12 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 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.

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

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 2.7-2.9 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 = (α ₁,..., 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

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 = (α ₁,..., α_n ) ^T , with α_i > 0 (i = 1,..., n) such that

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 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 14 15 16 ).

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 lower-order principal minor determinants of A are non-negative, and |A| < 0, for, if not, there exists some lth-order (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 sth-order (1 ≤ s ≤ l - 1) principal minor determinants detA _{s × s} of A _{l × l} are non-negative. 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

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

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

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

The authors declare that they have no competing interests.

DW carried out all studies in the paper. LZ participated in the design of the study in the paper.