ISRN Mathematical Analysis

Volume 2013 (2013), Article ID 852902, 9 pages

http://dx.doi.org/10.1155/2013/852902

Research Article

Global Existence and Blowup for a Reaction-Diffusion System with Nonlocal Boundary Condition

Jun Zhou

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 23 April 2013; Accepted 16 May 2013

Academic Editors: G. Buttazzo, B. Djafari Rouhani, and A. I. Hai

Copyright © 2013 Jun Zhou. 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 considers the singularity properties of positive solutions for a reaction-diffusion system with nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we establish the blow-up rate estimate for the blow-up solution.

1. Introduction

This paper studies the singularity properties of the following reaction-diffusion system with nonlocal boundary condition:

where is a bounded domain of , , with smooth boundary and is the closure of . , and are positive numbers which ensure that the equations in (1) are completely coupled with the nonlinear reaction terms. The functions defined for , are nonnegative and continuous. The initial values and are nonnegative, which are mathematically convenient and currently followed throughout the paper. We also assume that satisfies the compatibility condition on , and that and for any for the sake of the meaning of nonlocal boundary.

Denote that , , and , where . A pair of functions is called a classical solution of problem (1) if for some , , and satisfies (1). The local existence of classical solution of (1) is standard (see [1, 2]). If , it is easy to see , and we say that the solution of problem (1) blows up at finite time . If , is called a global solution of problem (1).

Over the past few years, many physical phenomena were formulated as nonlocal mathematical models (see [3, 4]). It has also being suggested that nonlocal growth terms present a more realistic model in physics for compressible reactive gases. Problem (1) arises in the study of the heat transfer with local source (see [5, 6]) and in the study of population dynamics (see [7, 8]).

In recent years many authors have investigated the following initial boundary value problem of reaction-diffusion system:

with Dirichlet, Neumanns or Robin boundary condition, which can be used to describe heat propagation on the boundary of container (see [2, 4, 9–23] and the literatures cited therein). Specially, when have the form

a classical result is (see [9, 10, 12, 20]).

Theorem A. The system (2) ( is of the form (3)) with homogenous Dirichlet boundary condition

admits a unique global solution for any nonnegative initial data if and only if , , and .

However, there are some important phenomena formulated as parabolic equations which are coupled with nonlocal boundary conditions in mathematical modeling such as thermoelasticity theory (see [24–26]). In this case, the solution could be used to describe the entropy per volume of the material. The problem of nonlocal boundary conditions for linear parabolic equation of the type

with uniformly elliptic operator

and was studied by Friedman [26]. It was proved that the unique solution of (5) tends to monotonically and exponentially as provided for any .

As for more general discussions on the dynamic of parabolic problem with nonlocal boundary conditions, one can see Pao [27], where the following problem:

was considered, and recently Pao [28] gave the numerical solutions for diffusion equation with nonlocal conditions.

In particular, the following single equation:

under the assumption the for any was consider, by Seo [29] and the following blow-up rate estimate is established:

where is the blow-up time, and is any constant satisfying .

Recently, Kong and Wang [30] obtained the blow-up conditions and blow-up profiles of the following system by using some ideas of Souplet [4]:

Furthermore, Zheng and Kong [31] gave the condition for global existence or nonexistence of solution to the following system:

Motivated by the above cited works, in this paper, we deal with singularity analysis of the parabolic system (1) with nonlocal boundary condition and it is seems that there is no work dealing with this type of systems except the single equations case, although this is a very classical model. Our main results read as follows.

Theorem 1. If , , and , then every nonnegative solution of (1) is global.

Theorem 2. Suppose or or .

(i)For any nonnegative functions and , the nonnegative solution of (1) blows up in finite time provided the initial values are large enough.(ii)If and for any , the nonnegative solution of (1) blows up in finite time with any positive initial values.(iii)If and for any , the nonnegative solution of (1) is global with small initial values.

To estimate the blow-up rate of the blow-up solution of (1), we need to add some assumptions for initial data as follows.

(H1) for some .(H2) If , then there exists a a sufficient small constant (which will be given in Section 4) such that and on , where

(H3) If , then there exists a sufficient small constant (which will be given in Section 4) such that and on , where

Theorem 3. Suppose , and , for all and assumptions (H1)-(H2) hold. If is the smooth solution of (1) and blows up in finite time , then there exist positive constants such that

for .

Theorem 4. Suppose , and , for all and assumptions (H1) and (H3) hold. If is the smooth solution of (1) and blows up in finite time , then there exist positive constants such that

for .

The rest of this paper is organized as follows. In the next section, we give some preliminaries, which include the comparison principle related to system (1). In Section 3, we will study the conditions for the solution to blowup and exist globally and prove Theorems 1 and 2. In Section 4, we will establish the precise blowup rate estimate for small weighted nonlocal boundary and prove Theorems 3 and 4.

2. Preliminaries

In this section, we give some basic preliminaries. We begin with the definition of upper and lower solutions of (1).

Definition 5. A pair of nonnegative functions is called an upper solution of problem (1) if and satisfies

Similarly, is called a lower solution of (1) if it satisfied all the reversed inequalities in (16).

Lemma 6. Let , and be continuous and nonnegative functions, and let satisfy

and then on .

Proof. Set . Since , by continuous, there exists such that for all . Thus, .

We claim that will lead to a contradiction. Indeed, suggests that or for some . Without loss of generality, we suppose that .

If , we first notice that

In addition, it is clear that on the boundary and at initial state . Then it follows from the strong maximum principle that in , which contradicts .

If , we will have a contradiction

In the last inequality, we have used the facts that for any and for any , which is a direct result of the previous case.

Therefore, the claim is true and thus , which implies that on . The proof is complete.

Remark 7. If and for any in Lemma 6, we can obtain on under the assumption that for any . Indeed, for any , we can conclude that on as the proof of Lemma 6. Then the desired result follows from the limit procedure .

Lemma 8. Let and be a upper and lower solution of (1) in , respectively. If for , then on .

Proof. Let and , and then

where

So, the functions and satisfy

Lemma 6 ensures that on , that is, on . The proof is complete.

3. Global Existence and Blowup

In this section, we will use the upper and lower solutions and their corresponding comparison principle developed in Section 2 to get the global existence or finite blowup of the solution to (1). Let us first give the proof of Theorem 1.

Proof of Theorem 1. Using the condition , and , we have . Thus, we can choose two positive constant and such that

Then, let be a continuous function such that and set

We consider the following auxiliary problem:

where is the measure of and . It follows from [32, Theorem  4.2] that exists globally and indeed on [32, Theorem  2.1].

Our aim is to show that is a global upper solution of (1). Indeed, a direct computation yields

Here, we have used that conclusion and the inequality (23). We still have to consider the boundary and initial conditions. When , we have

Similarly, we have

It is clear that and . Therefore, we get that is a global upper solution of (1) and hence the solution of (1) exists globally by Lemma 8. The proof is complete.

Proof of Theorem 2. (i) Let be the solution of (2) and (3) with homogeneous Dirichlet boundary. Then it is well known for sufficiently large initial data that the solution blows up in finite time when or or (Theorem A). On the other hand, it is obvious that is a lower solution of (1). Hence, the solution of (1) with large initial data blows up in finite time.

(ii) We consider the following ODE system:

If or , it is clear that the solution of (29) blows up in finite time. For the case and , it follows that

Thus, we get

Then implies that blows up in finite time, and so does . From the above analysis, we see that or or implies that blows up in finite time. Under the assumption and for any , is a lower solution of problem (1). Therefore, by Lemma 8, we see that the solution of (1) satisfies and then blows up in finite time.

(iii) Let be the positive solution of the linear elliptic problem

and be the positive solution of the linear elliptic problem

Since and for any , we can choose such that , .

Let and , where are positive constants which satisfy . We remark that under the assumption or or , we can choose such easily. We now show that is an upper solution of (1) for small initial data . Indeed, for any , we have

When ,

Here, we have used , . The above inequalities show that is an upper solution of (1) whenever and . The proof is complete.

4. Blowup Rate

In this section, we will estimate the blow-up rate of (1). By the standard methods (see [1, 2, 6]), we can show that system (1) has a smooth solution provided that satisfy the hypotheses (H1). We thus assume that the smooth solution of (1) blows up at finite time and set , . We can obtain the blow-up rate from the following lemmas.

Lemma 9. Suppose that satisfy (H1), and then there exists a positive constant such that

Proof. By the equations in (1), we have [33, Theorem  4.5]

Noticing that and , we have

by virtue of Young's inequality. Integrating (38) from to , we can get (36). The proof is complete.

Lemma 10. If , , and for any , then there exists a positive constant such that the solution of (1) with positive initial value satisfies

Proof. Let , where is a positive constant to be chosen. For , a series of calculations shows that

If we choose , then . So, we have

where is a function of and , which lines between and .

When , we have

Denote that for any . Since for any , . It follows from Jensen's inequality and that

Combining the above inequality with (42), we obtain

For the initial condition, we have

on provided that .

Summarily, if we take small enough such that

it follows from (41), (44), (45), and [32, Theorem  2.1] that , which implies (39). The proof is complete.

Combining (39) with (1), we know that the solution () of (1) satisfies

if , where ,,  , and . It is easy to see that and if .

Let be the solution of the following system:

It is easy to see that by Remark 7 if for any .

Lemma 11. Suppose that and satisfy (H1)-(H2) and that the assumptions in Lemma 10 hold; then the solution of (48) satisfies

if is small enough such that and .

Proof. Let and . For , a series of calculations shows that

For , using the boundary conditions we have

It follows from for any and Jensen's inequality that

Thus,

Similarly, we have

For the initial condition, under assumption (H2), we have

if is small enough such that and . Then (49) follows from (50)–(55). The proof is complete.

Proof of Theorem 3. Integrating the inequality for in (49) on yields

where . Since , we obtain

Combining (36) and (57), we get

where and are two positive constants.

Since , it follows from (57) that

Integrating this inequality from to , we get

where is a positive constant. Combining (36) and (60), we get

where and are two positive constants. We completed the proof of Theorem 3.

Proof of Theorem 4. The proof is similar to the proof of Theorem 3, and so we omit it.

Acknowledgments

This study is partially supported by the NSFC Grant 11201380, the Fundamental Research Funds for the Central Universities Grant XDJK2012B007, Doctor Fund of Southwest University Grant SWU111021, and Educational Fund of Southwest University Grant 2010JY053.

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