- Research Article
- Open Access
A Quasilinear Parabolic System with Nonlocal Boundary Condition
© Botao Chen et al. 2011
- Received: 8 May 2010
- Accepted: 11 August 2010
- Published: 18 August 2010
We investigate the blow-up properties of the positive solutions to a quasilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate. These extend the resent results of Wang et al. (2009), which considered the special case , and Wang et al. (2007), which studied the single equation.
- Global Existence
- Comparison Principle
- Parabolic System
- Nonlocal Boundary
- Nonlocal Boundary Condition
where , and is a bounded connected domain with smooth boundary. and for the sake of the meaning of nonlocal boundary are nonnegative continuous functions defined for and , while the initial data , are positive continuous functions and satisfy the compatibility conditions and for , respectively.
with , , , and . They obtained that solutions of (1.6) are global if , and may blow up in finite time if . For the critical case of , there should be some additional assumptions on the geometry of .
Recently, the genuine degenerate situation with zero boundary values for (1.7) has been discussed by Lei and Zheng . Clearly, problem (1.6) is just the special case by taking in (1.7) with zero boundary condition.
with nonlocal boundary condition (1.2). The typical characterization of systems (1.10) and (1.11) is the complete couple of the nonlocal sources, which leads to the analysis of simultaneous blowup.
where and are positive parameters. They gave the criteria for finite time blowup or global existence, and established blow-up rate estimate.
To our knowledge, there is no work dealing with the parabolic system (1.1) with nonlocal boundary condition (1.2) except for the single equation case, although this is a very classical model. Therefore, the main purpose of this paper is to understand how the reaction terms, the weight functions and the nonlinear diffusion affect the blow-up properties for the problem (1.1)–(1.3). We will show that the weight functions play substantial roles in determining blowup or not of solutions. Firstly, we establish the global existence and finite time blow-up of the solution. Secondly, we establish the precise blowup rate estimates for all solutions which blow up.
Our main results could be stated as follows.
Suppose that for any . If and hold, then any solution to (1.1)–(1.3) with positive initial data blows up in finite time.
Theorem 1.2 ..
Suppose that for any .
(1)If , and , then every nonnegative solution of (1.1)–(1.3) is global.
(2)If , or , then the nonnegative solution of (1.1)–(1.3) exists globally for sufficiently small initial values and blows up in finite time for sufficiently large initial values.
To establish blow-up rate of the blow-up solution, we need the following assumptions on the initial data
(1) for some ;
where , , and will be given in Section 4.
Theorem 1.3 ..
This paper is organized as follows. In the next section, we give the comparison principle of the solution of problem (1.1)–(1.3) and some important lemmas. In Section 3, we concern the global existence and nonexistence of solution of problem (1.1)–(1.3) and show the proofs of Theorems 1.1 and 1.2. In Section 4, we will give the estimate of the blow-up rate.
In this section, we give some basic preliminaries. For convenience, we denote that for . As it is now well known that degenerate equations need not posses classical solutions, we begin by giving a precise definition of a weak solution for problem (1.1)–(1.3).
Definition 2.1 ..
A vector function defined on , for some , is called a sub (or super) solution of ( 1.1 )–( 1.3 ), if all the following hold:
A weak solution of (1.1) is a vector function which is both a subsolution and a supersolution of (1.1)-(1.3).
Lemma 2.2 (Comparison principle).
Let and be a subsolution and supersolution of (1.1)–(1.3) in , respectively. Then in , if
By Gronwall's inequality, we know that , can be obtained in similar way, then .
Local in time existence of positive classical solutions of the problem (1.1)–(1.3) can be obtained using fixed point theorem (see ), the representation formula and the contraction mapping principle as in . By the above comparison principle, we get the uniqueness of the solution to the problem. The proof is more or less standard, so is omitted here.
From Lemma 2.2, it is easy to see that the solution of (1.1)–(1.3) is unique if .
where are bounded functions and , and and is not identically zero. Then for imply that in . Moreover, if or if , then for imply that in
We give some lemmas that will be used in the following section. Please see  for their proofs.
If , and , then there exist two positive constants , such that . Moreover, for any .
Lemma 2.6 ..
If , or , then there exist two positive constants , such that . Moreover, for any .
Compared with usual homogeneous Dirichlet boundary data, the weight functions and play an important role in the global existence or global nonexistence results for problem (1.1)–(1.3).
Proof of Theorem 1.1..
where , and we use the assumption
It is easy to check that is the unique solution of the ODE problem (3.1), then and imply that blows up in finite time. Under the assumption that for any , is a subsolution of problem (1.1)–(1.3). Therefore, by Lemma 2.2, we see that the solution of problem (1.1)–(1.3) satisfies and then blows up in finite time.
where are positive constant such that . We remark that and ensure the existence of such .
here, we used , and .
Now, it follows from (3.8)–(3.15) that defined by (3.7) is a positive supersolution of (1.1)–(1.3).
So we can choose . Furthermore, assume that are small enough to satisfy (3.15). It follows that defined by (3.7) is a positive supersolution of (1.1)–(1.3). Hence, exists globally.
Due to the requirement of the comparison principle we will construct blow-up subsolutions in some subdomain of in which . We use an idea from Souplet  and apply it to degenerate equations. Let be a nontrivial nonnegative continuous function and vanished on . Without loss of generality, we may assume that and . We will construct a blow-up positive subsolution to complete the proof.
notice that is sufficiently small.
Since and is continuous, there exist two positive constants and such that , for all . Choose small enough to insure , hence on . Under the assumption that and for any , we have and Furthermore, choose so large that . By comparison principle, we have . It shows that solution to (1.1)–(1.3) blows up in finite time.
where , ; , ; , , , ; , ; , . By the conditions (4.1), we have and satisfy that . Under this transformation, assumptions - become
() , for some ;
where will be given later.
By the standard method [16, 42], we can show that system (4.2) has a smooth nonnegative solution , provided that satisfy the hypotheses - . We thus assume that the solution of problem (4.2) blows up in the finite time . Denote . We can obtain the blow-up rate from the following lemmas.
by virtue of Young's inequality. Integrating (4.6) from to , we can obtain (4.4).
where are positive constants independent of . It follows from Lemma 4.1 and (4.19), we have the following lemma.
According the transform and Lemma 4.3, we can obtain Theorem 1.3.
The authors would like to thank the anonymous referees for their suggestions and comments on the original manuscript. This work was partially supported by NSF of China (10771226) and partially supported by the Educational Science Foundation of Chongqing (KJ101303), China.
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