Blowup Analysis for a Semilinear Parabolic System with Nonlocal Boundary Condition

In this paper, we devote our attention to the singularity analysis of the following semilinear parabolic system:

(1.1)

with nonlocal boundary condition

(1.2)

and initial data

(1.3)

where is a bounded connected domain with smooth boundary and are positive parameters. Most physical settings lead to the default assumption that the functions defined for are nonnegative and continuous, and that the initial data , are nonnegative, which are mathematically convenient and currently followed throughout this paper. We also assume that satisfies the compatibility condition on , and that and for any for the sake of the meaning ofnonlocal boundary.

Over the past few years, a considerable effort has been devoted to studying the blowup properties of solutions to parabolic equations withlocal 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 [1, 2]). For example, the system (1.1) and (1.3) with homogeneous Dirichlet boundary condition

(1.4)

has been studied extensively (see [3–5] and references therein), and the following proposition was proved.

for some (See [3, 5]).

For the more parabolic problems related to the local boundary, we refer to the recent works [6–9] 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 [10–12]. 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 [13–16]). In particular, for the blowup solution of the single equation

(1.6)

under the assumption that , Seo [15] established the following blowup rate estimate

(1.7)

for any . For the more nonlocal boundary problems, we also mention the recent works [17–22]. In particular, Kong and Wang in [17], by using some ideas of Souplet [23], obtained the blowup conditions and blowup profile of the following system:

(1.8)

subject to nonlocal boundary (1.2), and Zheng and Kong in [22] gave the condition for global existence or nonexistence of solutions to the following similar system:

(1.9)

with nonlocal boundary condition (1.2). The typical characterization of systems (1.8) and (1.9) is the complete couple of the nonlocal sources, which leads to the analysis of simultaneous blowup.

To our surprise, however, it seems that there is no work dealing with singularity analysis of 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 basic motivation for the work under consideration was our desire to understand the role of weight function in the blowup properties of that nonlinear system. We first remark by the standard theory [4, 13] that there exist local nonnegative classical solutions to this system.

Our main results read as follows.

Theorem 1.2.

Suppose that . All solutions to (1.1)–(1.3) exist globally.

It follows from Theorem 1.2 and Proposition 1.1(i) that any weight perturbation on the boundary has no influence on the global existence when , while the following theorem shows that it plays an important role when . In particular, Theorem 1.3(ii) is completely different from the case of the local boundary (1.4) (by comparing with Proposition 1.1(i)).

Theorem 1.3.

Suppose that .

1. (i)

   For any nonnegative and , solutions to (1.1)–(1.3) blow up in finite time provided that the initial data are large enough.

    
2. (ii)

   If , for any , then any solutions to (1.1)–(1.3) with positive initial data blow up in finite time.

    
3. (iii)

   If , for any , then solutions to (1.1)–(1.3) with small initial data exist globally in time.

    

Once we have characterized for which exponents and weights the solution to problem (1.1)–(1.3) can or cannot blow up, we want to study the way the blowing up solutions behave as approaching the blowup time. To this purpose, the first step usually consists in deriving a bound for the blowup rate. For this bound estimate, we will use the classical method initially proposed in Friedman and McLeod [24]. The use of the maximum principle in that process forces us to give the following hypothesis technically.

(H)There exists a constant , such that

However, it seems that such an assumption is necessary to obtain the estimates of type (1.5) or (1.10) unless some additional restrictions on parameters are imposed (for the related problem, we refer to the recent work of Matano and Merle [25]).

Here to obtain the precise blowup rates, we shall devote to establishing some relationship between the two components and as our problem involves a system, but we encounter the typical difficulties arising from the integral boundary condition. The following theorem shows that we have partially succeeded in this precise blowup characterization.

Theorem 1.4.

Suppose that , , , and assumption (H) holds. If the solution of (1.1)–(1.3) with positive initial data blows up in finite time , then

(1.10)

where are both positive constants.

Remark 1.5.

If and , then Theorem 1.4 implies that for the blowup solution of problem (1.6), we have the following precise blowup rate estimate:

(1.11)

which improves the estimate (1.7). Moreover, we relax the restriction on .

Remark 1.6.

By comparing with Proposition 1.1(ii), Theorem 1.4 could be explained as the small perturbation of homogeneous Dirichlet boundary, which leads to the appearance of blowup, does not influence the precise asymptotic behavior of solutions near the blowup time and the blowup rate exponents and are just determined by the corresponding ODE system . Similar phenomena are also noticed in our previous work [18], where the single porous medium equation is studied.

The rest of this paper is organized as follows. Section 2 is devoted to some preliminaries, which include the comparison principle related to system (1.1)–(1.3). In Section 3, we will study the conditions for the solution to blow up and exist globally and hence prove Theorems 1.2 and 1.3. Proof of Theorem 1.4 is given in Section 4.

In this section, we give some basic preliminaries. For convenience, we denote We begin with the definition of the super- and subsolution of system (1.1)–(1.3).

Definition 2.1.

A pair of functions is called a subsolution of (1.1)–(1.3) if

(2.1)

A supersolution is defined with each inequality reversed.

Lemma 2.2.

Suppose that and are nonnegative functions. If satisfy

(2.2)

then on .

Proof.

Set . Since , by continuity, 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

(2.3)

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

If , we shall have a contradiction:

(2.4)

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 .

Remark 2.3.

If and for any in Lemma 2.2, we can obtain in under the assumption that for . Indeed, for any , we can conclude that in as the proof of Lemma 2.2. Then the desired result follows from the limit procedure .

From the above lemma, we can obtain the following comparison principle by the standard argument.

Proposition 2.4.

Let ) and be a subsolution and supersolution of (1.1)–(1.3) in , respectively. If for , then in .

In this section, we will use the super and subsolution technique to get the global existence or finite time blowup of the solution to (1.1)–(1.3).

Proof of Theorem 1.2.

As , there exist such that

(3.1)

Then we let ( ) be a continuous function satisfying and set

(3.2)

We consider the following auxiliary problem:

(3.3)

where is the measure of and . It follows from [13, Theorem ] that exists globally, and indeed (see [13, Theorem ]).

Our intention is to show that is a global supersolution of (1.1)–(1.3). Indeed, a direct computation yields

(3.4)

and thus

(3.5)

Here we have used the conclusion and inequality (3.1). We still have to consider the boundary and initial conditions. When , in view of Hölder's inequality, we have

(3.6)

Similarly, we have also for that

(3.7)

It is clear that and . Therefore, we get is a global supersolution of (1.1)–(1.3) and hence the solution to (1.1)–(1.3) exists globally by Proposition 2.4.

(3.8)

where . Then implies that blows up in finite time (see [26]). Under the assumption that and for any , is a subsolution of problem (1.1)–(1.3). Therefore, by Proposition 2.4, we see that the solution of problem (1.1)–(1.3) satisfies and then blows up in finite time.

1. (iii)

   Let be the positive solution of the linear elliptic problem:

    

(3.9)

and let be the positive solution of the linear elliptic problem:

(3.10)

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

Let

(3.11)

where . We now show that is a supsolution of problem (1.1)–(1.3) for small initial data . Indeed, it follows from that, for ,

(3.12)

When ,

(3.13)

Here we used . The above inequalities show that is a supsolution of problem (1.1)–(1.3) whenever . Therefore, system (1.1)–(1.3) has global solutions if and , for any .

In this section, we derive the precise blowup rate estimate. To this end, we first establish a partial relationship between the solution components and , which will be very useful in the subsequent analysis. For definiteness, we may assume . If , we can proceed in the same way by changing the role of and and then obtain the corresponding conclusion.

Lemma 4.1.

If , and for any , there exists a positive constant such that the solution of problem (1.1)–(1.3) with positive initial data satisfies

(4.1)

Proof.

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

(4.2)

If we choose such that , we have

(4.3)

where is a function of and and lies between and .

When , on the other hand, we have

(4.4)

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

(4.5)

which implies that

(4.6)

For the initial condition, we have

(4.7)

provided that .

Summarily, if we take , then it follows from Theorem in [13] that , that is,

(4.8)

which is desired.

Using this lemma, we could establish our blowup rate estimate. To derive our conclusion, we shall use some ideas of [3].

Proof of Theorem 1.4.

For simplicity, we introduce . Let and . A direct computation yields

(4.9)

For , we have from the boundary conditions that

(4.10)

It follows from and Jensen's inequality that the difference in the last brace is nonnegative and thus

(4.11)

By similar arguments, we have

(4.12)

On the other hand, the hypothesis (H) implies that

(4.13)

Hence, from (4.9)–(4.13) and the comparison principle (see Remark 2.3), we get

(4.14)

That is,

(4.15)

Let Then and are Lipschitz continuous and thus are differential almost everywhere (see e.g., [24]). Moreover, we have from equations (1.1) that

(4.16)

We claim that

(4.17)

for some positive constant . Indeed, if we let be the points at which attains its maximum, then relation (4.1) means that

(4.18)

At any point of differentiability of , if ,

(4.19)

From (4.15), (4.18), and (4.19), we can confirm our claim (4.17).

Integrating (4.17) on yields

(4.20)

which gives the upper estimate for . Namely, there exists a constant such that

(4.21)

Then by (4.16) and (4.21), we get

(4.22)

Integrating this equality from to , we obtain

(4.23)

for some positive constant . Thus we have established the upper estimates for .

To obtain the lower estimate for , we notice that (4.16) and (4.18) lead to

(4.24)

for a constant . Integrating above equality on , we see there exists a positive constant such that

(4.25)

Finally, we give the lower estimate for . Indeed, using the relationship (4.16), (4.23) and (4.25), we could prove that is bounded from below; that is, there exists a positive constant such that

(4.26)

To see this, our approach is based on the contradiction arguments. Assume that there would exist two sequences with and with as such that

(4.27)

Then we could choose a corresponding sequence such that , where is a positive constant to be determined later. As , we have

(4.28)

From (4.23) and (4.27), we obtain

(4.29)

Choosing such that , one can get

(4.30)

which would contradict to (4.25) as is large enough since as .