Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 956564, 12 pagesdoi:10.1155/2012/956564

Research Article

Blow-Up and Global Existence for a Degenerate Parabolic System with Nonlocal Sources

Ling Zhengqiu¹ and Wang Zejia²

¹Institute of Mathematics and Information Science, Yulin Normal University, Guangxi, Yulin 537000, China²Institute of Mathematics, Jilin University, Jilin, Changchun 130024, China

Received 25 October 2011; Revised 14 March 2012; Accepted 21 March 2012

Academic Editor: Yuriy Rogovchenko

Copyright © 2012 Ling Zhengqiu and Wang Zejia. 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 investigates the blow-up and global existence of nonnegative solutions for a class of nonlocal degenerate parabolic system. By using the super- and subsolution techniques, the critical exponent of the system is determined. That is, if , then every nonnegative solution is global, whereas if , there are solutions that blowup and others that are global according to the size of initial values and . When , we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain large enough that is, if it contains a sufficiently large ball, there is no global solution.

1. Introduction and Description of Results

In this paper, we investigate the blowup and global existence of nonnegative solutions for the following degenerate parabolic system with nonlocal sources: where is a bounded domain in with smooth boundary and are nonnegative bounded functions in , constants ,  ,  , where .

Equation (1.1) constitutes a simple example of a reaction diffusion system exhibiting a nontrivial coupling on the unknowns , such as heat propagations in a two-component combustible mixture [1], chemical processes [2], and interaction of two biological groups without self-limiting [3]. And they are worth to study because of the applications to heat and mass transport processes. In addition, there exist interesting interactions among the multi-nonlinearities described by the eight exponents and in the problem (1.1).

In the past two decades, many physical phenomena were formulated into nonlocal mathematical models (see [4–7] and references therein) and studied by many authors. Degenerate parabolic equations involving a nonlocal source, which arise in a population model that communicates through chemical means, were studied in [8, 9]. At the same time, there are many important results that have appeared on blowup problems for nonlinear parabolic system. We will recall some of those results concerning the first initial-boundary problem. For the other related works on the global existence and blowup of solutions of nonlinear parabolic system, we refer the readers to [10, 11] and references therein.

In [4], Escobedo and Herrero studied the system in a bounded domain with null Dirichlet boundary conditions. The authors show that if , every solution of (1.2) is global, whereas if , there are solutions that blowup and others that are global according to the size of initial values and .

In [12], Galaktionov et al. considered the system with homogeneous Dirichlet boundary conditions, and they proved that is the critical exponent. Zheng [13] and Li et al. [14] studied the following systems: respectively. They obtained some results on the global solutions, the blowup solutions and the blowup profiles. Lately, Deng et al. in [15] considered the following nonlocal degenerate parabolic system: with homogeneous Dirichlet boundary conditions. Several interesting results are established as follows.(i)If , then every nonnegative solution of (1.5) is global.(ii)If , then if the domain is sufficiently small, the nonnegative solution of (1.5) is global, whereas if the domain contains a sufficiently large ball and , the nonnegative solution blows up in finite time.(iii)If , then there are solutions of (1.5) that blowup and others that are global according to the size of initial data and .

Our present work is motivated by [12–15] mentioned above. The main purpose of this paper is to extend and improve the results in [15]. At the same time, we will prove that is also the critical exponent of system (1.1). Our main results are as follows, two theorems concern the global existence and blowup conditions of the solutions.

Theorem 1.1. If one of the following conditions holds, then the nonnegative solution of system (1.1) exists globally. (1) and .(2) and the domain is sufficiently small.(3) and the initial data are sufficiently small.(4) or and the initial data are sufficiently small.

Theorem 1.2. If one of the following conditions holds, then the nonnegative solution of system (1.1) blows up in a finite time. (1), and the initial data are sufficiently large.(2), and the domain contains a sufficiently large ball, moreover, and are large enough.(3) or and initial data are sufficiently large.

This paper is organized as follows. In the next Section, we establish the local existence theorem and give some auxiliary lemmas. In Section 3, which concerns global existence, we prove Theorem 1.1. In Section 4, which deals with the blowup phenomenon, we prove Theorem 1.2.

2. Local Existence and Comparison Principle

Similar to the Propositions 2.1 and 2.2 of [15], we give the maximum principle and the comparison principle for the nonlocal parabolic system. For convenience, we denote , where .

As it is now well known that degenerate equation needs not possess classical solution, we begin by giving a precise definition of a weak solution for system (1.1). To this end, define the class of test functions

Definition 2.1. A pair of vector function defined on is called a super-solution of (1.1), if all the following conditions hold:(1); (2)if , and for all , ;(3)for every and any ,  A subsolution can be defined in a similar way.

Next, we state the maximum principle and comparison principle, and the proofs that are quite standard, we omit them here.

Lemma 2.2 (maximum principle). Suppose that and satisfy where and are the continuous and the bounded functions on , respectively, and Then on .

Lemma 2.3 (comparison principle). Let and be a nonnegative supersolution and a nonnegative subsolution of system (1.1), respectively. Then on if and either hold.

Theorem 2.4 (local existence and continuation). Assume , there is a such that there exists a nonnegative weak solution of (1.1) for each . Furthermore, either or

Proof. Owing to the degeneracy of equations of (1.1), in order to prove the existence of solution, for , we first consider the following corresponding regularized system where and are smooth approximation of with and , respectively. It is known that the system (2.7) has a unique classical solution for by the classical theory for parabolic equations, where is the maximal existence time. By a direct computation and the classical maximum principle, we have . Hence satisfies with the corresponding initial and boundary conditions. At the same time, if , according to Lemma 2.2, we get and . On the other hand, passing to the limit , it follows that and is a weak solution of with the corresponding initial and boundary conditions on , where is the maximal existence time. Here a weak solution of (2.12) is defined in a manner similar to that for (1.1), only the equalities for and ; (2.2) may be replaced with respectively. Then, passing to the limit , it happens that and if .Therefore, the limit exists, and the pointwise limit exists for any . Furthermore, as the convergence of the sequence is monotone, passing to the limit in (2.13), we get that is a nonnegative weak solution of (1.1). Thus the proof is completed.

Denote

We give Lemmas 2.5 and 2.6 that will be used in the following; please see [16] for their proofs.

Lemma 2.5. If and , then there exist two positive constants such that .

Lemma 2.6. If or or , then there exist two positive constants such that .

3. Proof of Theorem 1.1

According to Lemma 2.3, we only need to construct bounded super-solutions for any . Let be the unique position solution of the following linear elliptic problem: Denote , then . We define the functions as follows: where such that , and will be fixed later. Clearly, for any , is a bounded function and . Then, we have Denote

(1) If and , by Lemma 2.5, there exist two positive constants such that Therefore, we can choose sufficiently large that and Now, it follows from (3.3)–(3.6) that is a positive super-solution of (1.1).

(2) If and , then there exist two positive constants such that Without loss of generality, we may assume that , where is a sufficiently large ball. And we denote is the unique positive solution of the following linear elliptic problem: Let , then . Therefore, as long as is sufficiently small and such that Furthermore, choose large enough to satisfy (3.6).Then, it follows from (3.3) and (3.6)–(3.9) that is a positive super-solution of (1.1).

(3) If and , by Lemma 2.6, there exist two positive constants such that Hence, we can choose sufficiently small that , and provided are also sufficiently small to satisfy (3.6). Then, from (3.3) and (3.6), (3.10), we know that is a positive super-solution of (1.1).

(4) Finally, if or , there exist also positive constants such that (3.10) and . Similar to the proof of (3), we get that is a positive super-solution of (1.1).

Thus the proof of Theorem 1.1 is completed.

4. Proof of Theorem 1.2

Due to the requirement of the comparison principle, we will construct blowup subsolutions in some subdomain of in which . We use an idea from Souplet [17] and apply it to degenerate parabolic equation. By translation, one may assume without loss of generality that . Let be an open ball with radius , and is a nontrivial nonnegative continuous function, vanished on and . Set with where and are to be determined later. Clearly, and is nonincreasing since . Note that, for small enough,

Obviously, becomes unbounded as at the point . Calculating directly, we obtain notice is sufficiently small. Therefore, If , we have ; then Hence, Similarly, if , then

(1) If and , by Lemma 2.6, there exist two positive constants large enough that

Then, we can choose sufficiently small such that Hence, for sufficiently small , (4.6)–(4.8) imply that

Since and is continuous, there exist two positive constants and such that for all . Choose small enough to insure , hence on , and from (4.3) it follows that for sufficiently large . By comparison principle, we have provided that and . It follows that blows up in finite time.

(2) Next, we consider the case and . Clearly, there exist two positive constants such that

Denote by and the first eigenvalue and the corresponding eigenfunction of the following eigenvalue problem: It is well known that can be normalized as in and . By the property (let ) of eigenvalues and eigenfunctions we see that and , where and are the first eigenvalue and the corresponding normalized eigenfunction of the eigenvalue problem in the unit ball . Moreover,

Similar to (4.1), we define the functions in the form In the following, we will prove that blows up in finite time in the ball . Because of so, does blow up in the larger domain . Calculating directly, we have where and are constants independent of . Then, in view of , we may assume that , that is, the ball , is sufficiently large that Hence, for sufficiently small , (4.16) implies that

Therefore, is a positive subsolution of (1.1) in the ball , which blows up in finite time provided the initial data is sufficiently large that in the ball .

(3) Finally, if or , there also exist two positive constants to satisfy (4.9). Similar to the proof of case (1), we can get that is a subsolution of (1.1), which blows up in finite time.

Thus the proof of Theorem 1.2 is completed.

Acknowledgments

The first author is supported by National Natural Science Foundation of China. The second author 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 is supported by the NNSF of China (11071100).

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