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.
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.
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.
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.
Theorem 1.2 ..
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 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).
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.
We give some lemmas that will be used in the following section. Please see  for their proofs.
Lemma 2.6 ..
3. Global Existence and Blowup in Finite Time
Proof of Theorem 1.1..
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.
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.
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.
4. Blow-Up Rate Estimates
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.
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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