Blowup for a Non-Newtonian Polytropic Filtration System Coupled via Nonlinear Boundary Flux

Abstract

We study the global existence and the global nonexistence of a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux. We first establish a weak comparison principle, then discuss the large time behavior of solutions by using modified upper and lower solution methods and constructing various upper and lower solutions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained.

1. Introduction

In this paper, we consider the following Neumann problem:

(1.1)

(1.2)

(1.3)

where , , is a bounded domain with smooth boundary , is the outward normal vector on the boundary , , and are positive and satisfy the compatibility conditions.

Parabolic equations like (1.1) appear in population dynamics, chemical reactions, heat transfer, and so on. In particular, (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, (1.1) are called the non-Newtonian polytropic filtration equations (see [ 1, 2, 3, 4, 5, 6] and the references therein). For the Neuman problem (1.1)–(1.3), the local existence of solutions in time has been established, see survey in [4].

Recall the single quasilinear parabolic equation with nonlinear boundary condition

(1.4)

with . It is known that the solutions of (1.4) exist globally if and only if for ; they exist globally if and only if when (see [7–10]).

In [11, 12], M. Wang and S. Wang studied the following nonlinear diffusion system with nonlinear boundary conditions

(1.5)

with . In [11], they obtained the necessary and sufficient conditions to the global existence of solutions for . In [12], they considered the case of or and obtained the necessary and sufficient blowup conditions for the special case (the ball centered at the origin in with radius ). However, for the general domain , they only gave some sufficient conditions to the global existence and the blowup of solutions.

In [2], Wang considered the following system with nonlinear boundary conditions:

(1.6)

with . They obtained the necessary and sufficient conditions on the global existence of all positive (weak) solutions.

Sun and Wang in [13] studied the nonlinear equation with nonlinear boundary condition

(1.7)

They proved that all positive (weak) solutions of (1.7) exist globally if and only if when ; they exist globally if and only if when .

The main purpose of this paper is to study the influence of nonlinear power exponents on the existence and nonexistence of global solutions of (1.1)–(1.3). By using upper- and lower-solution methods, we obtain the necessary and sufficient conditions on the existence of global (weak) solutions to (1.1)–(1.3). Our main results are stated as follows.

Theorem 1.1.

If , then all positive (weak) solutions of (1.1)–(1.3) exist globally if and only if and .

Theorem 1.2.

If , then all positive (weak) solutions of (1.1)–(1.3) exist globally if and only if and .

Theorem 1.3.

If , then all positive (weak) solutions of (1.1)–(1.3) exist globally if and only if and .

Theorem 1.4.

If , then all positive (weak) solutions of (1.1)–(1.3) exist globally if and only if and .

Remark 1.5.

If , the results in [11] are included in Theorem 1.4, and if or , Theorems 1.1–1.3 improve the results of [12].

Remark 1.6.

If we extend the solution to (1.6) to the interval by symmetry, we get a solution to the same problem (1.6) with the condition at , substituted by a condition at , Conversely, symmetric solutions to this latter problem are solutions to the original problem (1.6). The problem (1.1)–(1.3) is the more general -dimensional version of the problem (1.6). Theorems 1.1–1.4 extend the results of the problem (1.6) into multidimensional case and it seems to be a natural extension of Wang [2].

Remark 1.7.

If , the conclusions of Theorems 1.1 and 1.4 are consistent with those of the single equation (1.7). This paper generalizes the results of the single equation (1.7) to the system (1.1)–(1.3).

The rest of this paper is organized as follows. Some preliminaries will be given in Section 2 . Theorems 1.1–1.4 will be proved in Sections 3–5, respectively.

2. Preliminaries

As it 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)–(1.3).

Definition 2.1.

Let and . A vector function is called a weak upper (or lower) solution to (1.1)–(1.3) in if

(i);

(ii);

(iii)for any positive two functions one has

(2.1)

In particular, is called a weak solution of (1.1)–(1.3) if it is both a weak upper and a lower solution. For every , if is a solution of (1.1)–(1.3) in , we say that is global.

Next we give some preliminary propositions and lemmas.

Proposition 2.2 (comparison principle).

Assume that are positive functions and is any weak solution of (1.1)–(1.3). Also assume that and are a lower and an upper solution of (1.1)–(1.3) in , respectively, with nonlinear boundary flux and , where . Then we have in .

Proof.

For small , letting and setting , according to the definition of upper and lower solutions, we have

(2.2)

Define

(2.3)

As in [14], by letting we get

(2.4)

that is,

(2.5)

where . Similarly, we have

(2.6)

Since and , it follows from the continuity of and that there exists a sufficiently small such that for . It follows from (2.5) and (2.6) that in .

Denote . We claim that . Otherwise, from the continuity of and there exists such that and for all By (2.5) and (2.6) we obtain that in , which contradicts the definition of . Hence for all

Obviously, is a lower solution of (1.1)–(1.3) in . Therefore, in . Using this fact, as in the above proof we can prove that in .

For convenience, we denote , which are fixed constants, and let .

Proposition 2.3.

Assume and that or holds. Then the solutions of (1.1)–(1.3) blow up in finite time.

Proof.

Without loss of generality, assume . Consider the single equation

(2.7)

We know from [13] that blows up in finite time. Since , by the comparison principle, is a lower solution of (1.1)–(1.3) and blows up in finite time if .

The following propositions can be proved in the similar procedure.

Proposition 2.4.

Assume and that or holds. Then the solutions of (1.1)–(1.3) blow up in finite time.

Proposition 2.5.

Assume and that or holds. Then the solutions of (1.1)–(1.3) blow up in finite time.

Proposition 2.6.

Assume and that or holds. Then the solutions of (1.1)–(1.3) blow up in finite time.

Let be the first eigenfunction of

(2.8)

with the first eigenvalue , normalized by , then in and and on (see [15–17]).

Thus there exist some positive constants such that

(2.9)

We have also provided with and some positive constant . For the fixed , there exists a positive constant such that if

At the end of this section, we describe two simple lemmas without proofs.

Lemma 2.7.

Suppose that positive constants satisfy , then for any two positive constants , there exist two positive constants such that and .

Lemma 2.8.

For any constant , there exist positive constants which depend only on and such that

(2.10)

where is a positive bounded function.

3. Proof of Theorem 1.1

Lemma 3.1.

Suppose , . Then all positive solutions of (1.1)–(1.3) exist globally.

Proof .

Construct

(3.1)

where if , if , and if , if , are defined in (2.8) and (2.9), are positive constants to be determined, and

(3.2)

We know that since for any . Thus for a simple computation shows

(3.3)

In addition, we have

(3.4)

Similarly, we can get

(3.5)

Noting on , we have on the boundary that

(3.6)

Since , there exist constants large such that

(3.7)

By (3.3)–(3.7), we know that is a global upper solution of (1.1)–(1.3). The global existence of solutions to (1.1)–(1.3) follows from the comparison principle.

Lemma 3.2.

Suppose , . Then all positive solutions of (1.1)–(1.3) blow up in finite time.

Proof .

Case 1.

. Let if , if , and if , if . In light of , we choose such that

(3.8)

For the above , we set where , , and

(3.9)

(3.10)

By a direct computation, for , we obtain that

(3.11)

(3.12)

If , we have and thus

(3.13)

On the other hand, since for any , we have

(3.14)

We have by (3.10), (3.13), and (3.14) that for .

If , then and hence

(3.15)

We follow from (3.10), (3.11), and (3.15) that for .

Similarly, we can get for also.

We have on the boundary that

(3.16)

(3.17)

Moreover, by (3.8) we have that

(3.18)

Equations (3.9), (3.16)–(3.18) imply that Therefore is a lower solution of (1.1)–(1.3).

Case 2.

. Set as above with

Case 3.

. Set as above with

Case 4.

. Set

By similar arguments, we conform that is a lower solution of (1.1)–(1.3), which blows up in finite time. We know by the comparison principle that the solution blows up in finite time.

We get the proof of Theorem 1.1 by combining Proposition 2.3 and Lemmas 3.1 and 3.2.

4. Proof of Theorems 1.2 and 1.3

Lemma 4.1.

Suppose with . Then all positive solutions of (1.1)–(1.3) exist globally.

Proof .

Take

(4.1)

for , where if , if , satisfying , and constants are to be determined. By performing direct calculations, we have, for ,

(4.2)

By setting if , if , we have on the boundary that

(4.3)

Since , by Lemma 2.7 there exist two positive constants such that , , and

(4.4)

Set By arguments in Lemma 3.1, for , we have

(4.5)

On the other hand, since , there exist two positive constants such that

(4.6)

(4.7)

By (4.2)–(4.7), it follows that is an upper solution of (1.1)–(1.3). Thus the solutions of (1.1)–(1.3) are global.

Lemma 4.2.

Suppose with . Then all positive solutions of (1.1)–(1.3) blow up in finite time.

Proof .

We first prove that there exist such that

(4.8)

When , yields Hence there exist such that Set , and

When , take ,

When , take ,

Let if , if , and , , .

Define where , and

(4.9)

(4.10)

By a direct computation, for , we have

(4.11)

By similar arguments in Lemma 3.2, we have for .

Moreover, for , we have

(4.12)

By (4.8), we have

(4.13)

By (4.9), (4.12), and (4.13), we have that is a lower solution of (1.1) and (1.3), which with the comparison principle implies that the solutions of (1.1)–(1.3) blow up in finite time.

It has been shown from Proposition 2.4 and Lemmas 4.1 and 4.2 that Theorem 1.2 is true.

In a similar way to the proof of Theorem 1.2, we have Theorem 1.3.

5. Proof of Theorem 1.4

Lemma 5.1.

Suppose with . Then all positive solutions of (1.1)–(1.3) exist globally.

Proof .

Take where and are the undetermined positive constants.

Calculating directly for , we have by Lemma 2.8 that

(5.1)

Let if , if , and if , if . We have on the boundary that

(5.2)

Since we know by Lemma 2.8 that there exist constants such that

(5.3)

For the above constants , we choose a constant so large that

(5.4)

By (5.1)–(5.4), we know that is an upper solution of (1.1)–(1.3), Thus the solutions of (1.1)–(1.3) are global.

Lemma 5.2.

Suppose with . Then all positive solutions of (1.1)–(1.3) blow up in finite time.

Proof .

We first prove that there exist such that

(5.5)

In fact, when , yields . Hence there exists such that . Set and

When and , take

When and , let

Take , and , where , and

(5.6)

(5.7)

By a direct computation for , we have

(5.8)

(5.9)

For , we have

(5.10)

Moreover, (5.5) implies

(5.11)

It follows from (5.6), (5.8)–(5.11) that is a lower solution of (1.1)–(1.3). Because blows up in finite time, and so does .

By Proposition 2.6 and Lemmas 5.1 and 5.2, we see that Theorem 1.4 holds.