Abstract and Applied Analysis

Volume 2013 (2013), Article ID 898546, 6 pages

http://dx.doi.org/10.1155/2013/898546

Research Article

Asymptotic Behavior of Solutions to Fast Diffusive Non-Newtonian Filtration Equations Coupled by Nonlinear Boundary Sources

Wang Zejia,¹ Wang Shunti,² and Zhang Chengbin²

¹College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China

²Institute of Mathematics, Jilin University, Changchun 130012, China

Received 6 February 2013; Accepted 27 March 2013

Academic Editor: Sining Zheng

Copyright © 2013 Wang Zejia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerning the asymptotic behavior of solutions to the fast diffusive non-Newtonian filtration equations coupled by the nonlinear boundary sources. We are interested in the critical global existence curve and the critical Fujita curve, which are used to describe the large-time behavior of solutions. It is shown that the above two critical curves are both the same for the multidimensional problem we considered.

1. Introduction

In this paper, we study the non-Newtonian filtration equations coupled by the nonlinear boundary sources

where , , , is the unit ball in with boundary , is the inward normal vector on , and are nonnegative, suitably smooth, and bounded functions satisfying the appropriate compatibility conditions.

The system (1)–(3) can be used to describe the models in population dynamics, chemical reactions, heat propagation, and so on. It is well known that the classical solutions do not exist because the equations in (1) are degenerate in , while the local existence and the comparison principle of the weak solutions can be obtained; see [1, 2]. In this paper, we investigate the asymptotic behavior of solutions to the system (1)–(3), including blowup in a finite time and global existence in time.

Since the beginning work on critical exponent done by Fujita in [3], there are a lot of Fujita type results established for various equations; see the survey papers [4, 5] and the references therein and also the papers [6–9]. We recall some results on the fast diffusion case. In [10], the authors obtained the critical exponents for the single one-dimensional fast diffusive equation on , that is,

They showed that the global critical exponent is and the critical Fujita exponent is . After that, the corresponding results on the coupled -laplace equations can be found in the paper [11], in which the authors consider multiple equations coupled by boundary sources. For the system,

the results in [11] are that the critical global existence curve is , and when , the critical Fujita curve is , where , and are constants depending on , and . It can be seen by some computations that the critical Fujita curve is on the strictly right of the critical global existence curve.

As for the multidimensional problem, the single equation case of (1)–(3) was discussed in [12]; that is,

with , , . It was shown that the critical global exponent and the critical Fujita exponent both are .

Motivated by the papers mentioned above, the aim of this paper is to study the asymptotic behavior of solutions to the system (1)–(3). We show that the phenomenon that the two critical exponents for multi-dimensional equation coincide also occurs in the coupled equations.

Furthermore, by virtue of the radial symmetry of the exterior domain of the unit ball, we note that the above result can be extended to the following more general problems:

with ,  , .

We will state our main results and prove them in the next section.

2. Main Results and Their Proofs

In this section, we first state our main results and then prove them. Our main results are as follows.

Theorem 1. The critical global existence curve and the critical Fujita curve for the system (1)–(3) with are the same one, that is, the curve given by

Namely, if , then all nonnegative solutions of the system (1)–(3) exist globally in time, while if , then there exist both blow-up solutions for large initial data and global existent solution for small initial data.

Theorem 2. Assume that , and . Then both the critical global existence curve and the critical Fujita curve for the system (7)–(9) are the curve given by

Since the proofs of Theorems 1 and 2 are similar, we shall give the proof of Theorem 2 only. We note that the equations in (1) and (7) are degenerate at the points where , and classical solutions may not exist generally. It is therefore necessary to consider weak solutions in the distribution sense.

To prove Theorem 2, we first establish some preliminary results. Let and consider the problem:

where , ,  , , are nonnegative, suitably smooth, and bounded functions. It is not difficult to check that the solution of the system (12)–(14) is also the solution of the system (7)–(9) if and are radially symmetrical. We now study the asymptotic behavior of solutions to the system (12)–(14).

Proposition 3. If , then all nonnegative solutions of the system (12)–(14) exist globally in time.

Proof of Proposition 3. We prove this proposition by constructing a kind of global upper solutions. Let

where , , , , ,  and and are large constants satisfying that

Obviously, we have  and for , and a direct computation yields that

Noticing that for , we have

Due to the choice of and ,, then we get

On the other hand,

Similarly, we have

Since that , then

This indicates that

Noticing the global existence in time of , we get that the solution of the problem (12)–(14) exists globally by the comparison principle. The proof is complete.

Proposition 4. If , then the nonnegative nontrivial solutions of the system (12)–(14) blow up in finite time for large initial data.

Proof of Proposition 4. The proposition is proved by constructing a kind of lower blow-up solutions. Set

where , and

Due to , we see that and

We claim that is a lower solution to the problem (12)–(14) with , if the following inequalities hold:

Note that

So if we assume that satisfy

then (27)– (30) hold provided that

Take

where , are the constants to be determined. It is clear that the above satisfy (32). Now, we verify that satisfy (33)–(36) in the distribution sense. We claim that (33) is valid for . In fact, we only need to verify that

that is,

In the first place, we compute each term in (40) as follows:

Then, the inequalities in (40) are valid provided that, for ,

So, we choose , and is small enough that

Similarly, choose , satisfying that

to get the validity of (35).

Next, we verify that and given by (37) and (38) also satisfy the boundary conditions (34) and (36):

From (34), we need

Set , where is to be determined. Then rewriting the last inequality, we get

which can be obtained by choosing small enough, if

In a similar discussion on the boundary condition (36), for enough small , the following inequality is needed:

Note that and ; therefore, (48)-(49) are equal to

Recall the assumption that , so there exists a constant , such that (50) holds. Furthermore, we can choose as small as needed.

Therefore, the solution of the problem (12)–(14) blows up in a finite time if is large enough such that

The proof is completed.

Proposition 5. If , then every nonnegative nontrivial solution of the system (12)–(14) with small initial data exists globally.

Proof of Proposition 5. We seek the steady-state solution of the system (12)–(14):

A direct calculation shows that should satisfy

which implies that

with , . Integrating the above equalities yields

In particular, we let

Define , , and then , . Noticing that , we get

Therefore, the bounded positive functions:

are just a couple of steady-state solutions of the problem (12)–(14) with the initial data . By the comparison principle, for any initial data which is small enough to satisfy

the solutions of the problem (12)–(14) exist globally in time.

Remark 6. Due to the fact that for , , it is seen from Propositions 3 and 5 that all nonnegative solutions to the system (12)–(14) with enough small initial data exist globally in time.

Now, we prove the main result for the system (7)–(9), that is, Theorem 2.

Proof of Theorem 2. Noticing that the functions are bounded, we can choose two bounded, radially symmetrical functions denoted by satisfying that , respectively. By using Proposition 3 and the comparison principle, we can obtain the global existence of solutions to the system (7)–(9).

For the initial data is large enough such that , , here are defined in the proof of Proposition 4 if , then the solutions of the system (7)–(9) with such blow up in a finite time by the comparison principle and Proposition 4.

On the other hand, using the comparison principle again and combining with Proposition 5, we see that the solution of (7)–(9) exists globally if

where

The proof is complete.

Acknowledgments

The authors would like to express their many thanks to the Editor and Reviewers for their constructive suggestions to improve the previous version of this paper. This work was supported by the NNSF.

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