A Quasilinear Parabolic System with Nonlocal Boundary Condition

Abstract

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.

1. Introduction

In this paper, we deal with the following degenerate parabolic system:

(1.1)

with nonlocal boundary condition

(1.2)

and initial data

(1.3)

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.

Problem (1.1)–(1.3) models a variety of physical phenomena such as the absorption and "downward infiltration" of a fluid (e.g., water) by the porous medium with an internal localized source or in the study of population dynamics (see [1]). The solution of the problem (1.1)–(1.3) is said to blow up in finite time if there exists called the blow-up time such that

(1.4)

while we say that exists globally if

(1.5)

Over the past few years, a considerable effort has been devoted to the study of the blow-up properties of solutions to parabolic equations with local boundary conditions, say Dirichlet, Neumann, or Robin boundary condition, which can be used to describe heat propagation on the boundary of container (see the survey papers [2, 3] and references therein). The semilinear case of (1.1)–(1.3) has been deeply investigated by many authors (see, e.g., [2–11]). The system turns out to be degenerate if ; for example, in [12, 13], Galaktionov et al. studied the following degenerate parabolic equations:

(1.6)

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 .

Song et al. [14] considered the following nonlinear diffusion system with coupled via more general sources:

(1.7)

Recently, the genuine degenerate situation with zero boundary values for (1.7) has been discussed by Lei and Zheng [15]. Clearly, problem (1.6) is just the special case by taking in (1.7) with zero boundary condition.

For the more parabolic problems related to the local boundary, we refer to the recent works [16–20] and references therein.

On the other hand, there are a number of important phenomena modeled by parabolic equations coupled with nonlocal boundary condition of form (1.2). In this case, the solution could be used to describe the entropy per volume of the material (see [21–23]). Over the past decades, some basic results such as the global existence and decay property have been obtained for the nonlocal boundary problem (1.1)–(1.3) in the case of scalar equation (see [24–28]). In particular, in [28], Wang et al. studied the following problem:

(1.8)

with . They obtained the blow-up condition and its blow-up rate estimate. For the special case in the system (1.8), under the assumption that , Seo [26] established the following blow-up rate estimate:

(1.9)

for any For the more nonlocal boundary problems, we also mention the recent works [29–34]. In particular, Kong and Wang in [29], by using some ideas of Souplet [35], obtained the blow-up conditions and blow-up profile of the following system:

(1.10)

subject to nonlocal boundary (1.2), and Zheng and Kong in [34] gave the condition for global existence or nonexistence of solutions to the following similar system:

(1.11)

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.

Recently, Wang and Xiang [30] studied the following semilinear parabolic system with nonlocal boundary condition:

(1.12)

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.

Theorem 1.1.

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 ;

(2) There exists a constant , such tha

(1.13)

where , , and will be given in Section 4.

Theorem 1.3 ..

Suppose that for any ; and satisfy and ; assumptions (H1)-(H2) hold. If the solution of (1.1)–(1.3) with positive initial data blows up in finite time , then there exist constants such that

(1.14)

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.

2. Preliminaries

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:

(1);

(2) for , and for almost all ;

• (3)
  

  (2.1)

 where is the unit outward normal to the lateral boundary of . For every and any ϕ belong to the class of test functions,

(2.2)

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

Proof.

Let , the subsolution satisfies

(2.3)

On the other hand, the supersolution satisfies the reversed inequality

(2.4)

Set , we have

(2.5)

where

(2.6)

Since and are bounded in , it follows from , , that  are bounded nonnegative functions. is a function between and . Noticing that and are nonnegative bounded function and on , we choose appropriate function as in [36] to obtain that

(2.7)

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 [37]), the representation formula and the contraction mapping principle as in [38]. 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.

Remark 2.3.

From Lemma 2.2, it is easy to see that the solution of (1.1)–(1.3) is unique if .

The following comparison lemma plays a crucial role in our proof which can be obtained by similar arguments as in [24, 38–40]

Lemma 2.4.

Suppose that and satisfy

(2.8)

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

Denote that

(2.9)

We give some lemmas that will be used in the following section. Please see [41] for their proofs.

Lemma 2.5.

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 .

3. Global Existence and Blowup in Finite Time

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..

We consider the ODE system

(3.1)

where , and we use the assumption

Set

(3.2)

with

(3.3)

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.

Proof of Theorem 1.2.

• (1) Let be the positive solution of the linear elliptic problem

  

  (3.4)

and be the positive solution of the linear elliptic problem

(3.5)

where are positive constant such that . We remark that and ensure the existence of such .

Denote that

(3.6)

We define the functions as following:

(3.7)

where is a constant to be determined later. Then, we have

(3.8)

In a similar way, we can obtain that

(3.9)

here, we used , and .

On the other hand, we have

(3.10)

(3.11)

Let

(3.12)

If , and , by Lemma 2.5, there exist positive constants such that

(3.13)

Therefore, we can choose sufficiently large, such that

(3.14)

(3.15)

Now, it follows from (3.8)–(3.15) that defined by (3.7) is a positive supersolution of (1.1)–(1.3).

By comparison principle, we conclude that , which implies exists globally.

• (2) If , or , by Lemma 2.6, there exist positive constants such that

  

  (3.16)

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 [42] 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.

Set

(3.17)

with

(3.18)

where and are to be determined later. Clearly, and is nonincreasing since Note that

(3.19)

for sufficiently small . Obviously, becomes unbounded as , at the point . Calculating directly, we obtain that

(3.20)

(3.21)

notice that is sufficiently small.

Similarly, we have

(3.22)

Case 1.

If , we have , then

(3.23)

Hence,

(3.24)

Case 2.

If , then

(3.25)

By Lemma 2.6, there exist positive constants large enough to satisfy

(3.26)

and we can choose be sufficiently small that

(3.27)

Thus, we have

(3.28)

Hence, for sufficiently small , (3.24) and (3.25) imply that

(3.29)

(3.30)

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

In this section, we will estimate the blow-up rate of the blow-up solution of (1.1). Throughout this section, we will assume that

(4.1)

To obtain the estimate, we firstly introduce some transformations. Let then problem (1.1)–(1.3) becomes

(4.2)

where , ; , ; , , , ; , ; , . By the conditions (4.1), we have and satisfy that . Under this transformation, assumptions - become

(), for some ;

() there exists a constant , such that

(4.3)

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.

Lemma 4.1.

Suppose that satisfy -, then there exists a positive constant such that

(4.4)

Proof.

By (4.2), we have (see [43])

(4.5)

Noticing that and , hence we have

(4.6)

by virtue of Young's inequality. Integrating (4.6) from to , we can obtain (4.4).

Lemma 4.2.

Suppose that satisfy -, is a solution of (4.2). Then

(4.7)

where

(4.8)

Proof.

Set a straightforward computation yields

(4.9)

If , obviously we have

(4.10)

Otherwise, noticing that , by virtue of Young's inequality,

(4.11)

where we have

(4.12)

Similarly, we also have

(4.13)

Fix , we have

(4.14)

where . Since , we have

(4.15)

Noticing that by virtue of Jensen's inequality, we have

(4.16)

here, we used and in the last inequality. Hence

(4.17)

Similarly, we also have

(4.18)

On the other hand, - imply that Combined inequalities (4.12)-(4.18) and Lemma 2.4, we obtain that is, (4.7) holds.Integrating (4.7) from to , we conclude that

(4.19)

where are positive constants independent of . It follows from Lemma 4.1 and (4.19), we have the following lemma.

Lemma 4.3.

Suppose that satisfy -. If is the solution of system (4.2) and blows up in finite time , then there exist positive constants such that

(4.20)

According the transform and Lemma 4.3, we can obtain Theorem 1.3.