A Class of Nonlocal Coupled Semilinear Parabolic System with Nonlocal Boundaries

Abstract

We investigate the positive solutions of the semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions. The blow-up and global existence criteria are obtained.

1. Introduction

In this paper, we consider the positive solutions of the semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions: where is a bounded domain in , , with smooth boundary . The exponents , . The weighted functions in the boundary conditions are continuous, nonnegative on and on . The initial data with , , and satisfy the compatibility conditions.

Many physical phenomena were formulated into nonlocal mathematical models and studied by many authors [1–13]. For example, in [1], Bebernes and Bressan studied an ignition model for a compressible reactive gas which is a nonlocal reaction-diffusion equation. Furthermore, Bebernes et al. [14] considered a more general model: where or with . Chadam et al. [15] studied another form of (2) with and and proved that the blow-up set is the whole region (including the homogeneous Neumann boundary conditions). Souplet [16, 17] considered (2) with the general function . Pao [18] discussed a nonlocal reaction-diffusion equation arising from the combustion theory.

The problems with both nonlocal sources and nonlocal boundary conditions have been studied as well. To motivate our study, we give a short review of examples of such parabolic equations or systems studied in the literature. For example, Lin and Liu [19] studied the following problem: they established local existence, global existence, and nonexistence of solutions and discussed the blow-up properties of solutions.

Gladkov and Kim [20] considered the problem of the form with . And some criteria for the existence of global solution as well as for the solution to blow up in finite time were obtained.

In [21], Kong and Wang studied system (1) when : they obtained the following results, and we extend them as follows.(i)Assume that and hold; then the solution of (5) exists globally.(ii)If one of the following conditions holds: then the solution of (5) blows up in a finite time for the sufficiently large initial data.(iii)Assume that and for all and one of (6) holds; then the solution of problem (5) blows up in a finite time for any positive initial data.

Recently, Zheng and Kong [22] also studied the following problem: with the same initial and boundary conditions as (5), and they established similar conditions for global and nonglobal solutions and also blow-up solutions.

The main purpose of this paper is to get the blow-up criterion of problem (1) for any positive integer .

In the following, we set , and   with for convenience.

It is known by the standard theory [16, 23] that there exists a local positive solution to (1). Moreover, by the comparison principle (see Lemma 10 in the next section), the uniqueness of solutions holds if , .

Theorem 1. Problem (1) has a positive classical solution for some . Moreover, if , then

Theorem 2. If exponents , satisfy the solution of (1) exists globally for any nontrivial nonnegative initial data.

Theorem 3. If exponents , satisfy one of the following: and if , , for all , then the solution of (1) exists globally for small nonnegative initial data.

Theorem 4. If exponents satisfy one of the following: then the solution of (1) blows up in finite time for large initial data.

If the initial data satisfies we have another blow-up result.

Theorem 5. Assume that and the condition (H) holds. Then the solution of (1) blows up in finite time for any positive initial data.

This paper is organized as follows. Section 2 is devoted to some comparison principles. In Section 3, we prove two global existence results. The blow-up results are proved in the final section.

2. Comparison Principle

Before proving the main results, we give the maximum and comparison principles related to the problem. First, we give the following definition of the upper and lower solutions.

Definition 6. A pair of functions is called an upper solution of (1), if, for every ,   and satisfies

Similarly, a lower solution of (1) is defined by the opposite inequalities.

Lemma 7. Suppose that and , in , on , on , , . If, for every , and satisfies where , then , on .

Proof. Set , with . Then Since , , there exists such that for . Suppose for a contradiction that . Then on , and at least one of vanishes at for some . Without loss of generality, suppose that . If ; by virtue of the first inequality of (16), we find that This leads to the conclusion that in by the strong maximum principle, a contradiction. If , this results in a contradiction too, that due to on . This proves that and consequently . We complete the proof.

Lemma 8. Suppose that, for every , and satisfies where and are continuous, nonnegative functions in , on such that on , and there exist positive constants such that . Then , on .

Proof. Suppose that the strict inequalities of (19) hold; by Lemma 7, we have . Now we consider the general case. Set where is any fixed positive constant, and . By (19), we get, for , Therefore, we have on . Letting , we get the desired result.

If the boundary condition is not necessarily valid, we have the following result. The argument of its proof can be referred to [22, Lemma 2.2].

Lemma 9. Suppose that , , are nonnegative and bounded in , on , on , . If, for every ,   and satisfies where , then , on .

By Lemma 9, we can easily get the following result.

Lemma 10. Let and be nonnegative upper and lower solution of system (1) on , respectively. If one assumes that, for some ,(i) or when ,(ii) or when ,
then on .

3. Global Existence Results

Before proving Theorem 2, we give a global existence result for a scalar equation.

Lemma 11. Let and be continuous, nonnegative functions on and , respectively, and let the nonnegative constants satisfy . Then the solutions of the nonlocal problem exist globally.

Proof. The augment is similar to the proof of [22, Lemma 3.1] or [21, Lemma 6]. For the reader's convenience, we complete it. It is easy to prove that there exists a positive function such that Let be large enough such that Setting for , one readily checks that Let ; it follows that This implies that is a global upper solution of (23). Clearly, 0 is a lower solution of it. So we complete the proof.

Proof of Theorem 2. By (11), we know that there exists , , such that Define . Let , be a continuous function defined for . Suppose that solves where In view of Lemma 11, we know that is global. Moreover, in by the maximum principle. Set , . By (28) and (29) and using Hlder's inequality, we get This means that is a global upper solution of (1).

Proof of Theorem 3. Define Let be the unique solution of the elliptic problem Then there exists a constant such that in . We choose to be large enough such that Set . When , it follows that Now we investigate . Set for convenience. A simple computation yields
(a) If , no matter or , we can choose to be small enough such that . For fixed , there exist , , satisfying , . It follows that
(b) If , and , we can choose to be small enough such that Consequently, there exist , satisfying , . Hence (37) holds too.
By (35) and (37), in any case (a) or (b), we know that the solution of (1) must be global for small data , for .

4. Blow-Up Results

In this section, we assume that is a positive solution of (1) on , where is the maximal existence time.

Proof of Theorem 4. We denote by the first eigenvalue and the corresponding eigenfunction of the linear elliptic problem: and satisfies Define .
(a) If , we claim that there exist positive constants , , such that the inequality holds. First, when , (41) holds for any . When , if , (41) holds for any ; if we can choose . That is, (41) holds too. When , if , (41) holds for any ; if , we can choose such that (41) holds too.
(b) If , , and , we can choose such that Hence (41) holds too.
Hence, for the case (a) or (b), we all have . Now let be the unique solution of the ODE problem where . Then blows up in finite time with being large enough.
Set We will show that is a lower solution of problem (1). A direct computation yields is a blowing up lower solution of (1) provided the initial data are so large that , for . We complete the proof.

Proof of Theorem 5. Since in , on , and by the compatibility conditions, we have on . Denote by the positive constant such that on . The assumption (H) implies that by the comparison principle, and in turn , on . Furthermore, satisfies Let be the solution of the following Cauchy problem: Clearly, blows up under the condition On the other hand, since , by Lemma 9, we have as long as both and exist, and thus blows up for any positive initial data. The proof now is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by Natural Science Fund for Colleges and Universities in Jiangsu Province, 12KJB110018, and College Students' Innovative Projects, 2013.