Abstract

This paper is devoted to understand the blow-up properties of reaction-diffusion equations which combine a localized reaction term with nonlinear diffusion. In particular, we study the critical exponent of a -Laplacian equation with a localized reaction. We obtain the Fujita exponent of the equation.

1. Introduction

This short paper is devoted to the Fujita critical exponent of the following -Laplacian equation: where , , and the initial data is nonnegative continuous and compactly supported.

We take the coefficient which is a compactly supported function, which means that the reaction term acts only locally. Thus, the model may be used to describe a chemical reaction-diffusion process in which, due to the effect of the catalyst, the reaction takes place only at some local sites [1].

As a representative example of quasilinear reaction-diffusion equation, classical -Laplacian equation with no source term ( in (1)) and its variants had been extensively studied in the past few decades (see [2–6] and references therein). It is well known that -Laplacian equation in general does not allow for classical solvability due to the nonlinearity and degeneracy. Therefore, the “solution” to a -Laplacian means the weak solution in the usual integral sense (see, e.g., [4]). Among other things, previous studies show that blow-up happens for the problem with homogeneous reaction; that is, : It is shown that if , every solution is global in time, if , all solutions blow up, and if , both global in time solutions and blowing up solutions exist. In this case, the numbers and are called the global existence exponent and the Fujita exponent, respectively. On the other hand, we remark that there is a close connection of problem (1) with the problem of diffusion with flux conditions on the boundary: In fact, if we take a sequence of reaction coefficients converging to a Dirac delta at the origin (i.e., if ), the corresponding solutions of (1) should converge to a solution of problem (3). In [7], authors proved that the global existence exponent of (3) is and the Fujita exponent is . After that, there are plenty of works that extended their study to doubly degenerate equation or fast diffusion case (see, e.g., [8–10] and references therein).

The motivation of this short paper is the wish to understand the blow-up properties of reaction-diffusion equations which combine a nonlinear diffusion and localized reaction term, and this is the main difference with the existing studies of blow-up for similar reaction-diffusion equations. For the porous medium equation case , Ferreira et al. [11] proved that , .

In our previous work, we has obtained the global existence exponent of (1) is . In this note, we will prove that the Fujita exponent of (1) is . Thus, it is the same as that of (3), and this is coincident with our intuitive judgement. We will modify several techniques developed in [11] to prove our main result.

2. Some Preliminary Lemmas

In this section, we will give some preliminary lemmas, whose proofs may be independent and interesting. We first have the following lemma from [3].

Lemma 1. There exists a positive constant such that problem has a unique nonnegative solution .

When , we need to introduce the energy functional of (1) as follows:

Lemma 2. When , if there exists such that , then all solutions to problem (1) blow up in finite time.

Proof. It is easy to verify that which shows that is decreasing. Thus, for any .
Define For any , we have and From , it follows that . Thus, Namely, for any , we have for some positive constant .
Since and , we obtain that , where is a constant. Therefore, as .
Next, we will prove that there exists such that as .
It follows from that Since , we have . Thus, by (9), Therefore, by using Hölder inequality, we could obtain Due to the fact that as and since , there exists a constant such that when is large enough. This is equivalent to the proposition that is a concave function.
Since and is increasing, there exists such that . Therefore, as .
Furthermore, we claim that as . Otherwise, is bounded in . Assume that for , where is a positive constant.
Integrating the inequality twice over the interval , we obtain that The right side is bounded in , and so is , which contradicts the fact that as . Hence, in the limit .
Next, we prove that blows up in finite time. By (1), Therefore, blows up in the sense of norm from (16).

Lemma 3. If and , Cauchy problem has a global supersolution.

Proof. Consider the following Cauchy problem: The existence and uniqueness of the solution to this problem was established in [5, 6, 12]. The solution has the form where is a positive, bounded, and decreasing function and satisfies the following boundary value problem:
Set , where is to be determined later; is a constant between and .
One can see that and show that . From (21), we know that for some . Therefore, is the solution of the following problem:
Next, we introduce the function , where and are solutions of the following ordinary differential system: Then, it is easy to verify that satisfies Therefore, is a supersolution of (18) if . If , it is a global supersolution from the following Lemma 4.

Lemma 4. If and , then there exists a positive constant such that problem (25) has a global solution with bounded in if .

Proof. The local existence and uniqueness of solution of (25) follows from the standard theory of initial value problem on ordinary differential equation. From (25), for , we have , and the solution continues as long as the solution exists and is finite.
By (25), is uniquely defined by . Moreover, for all . From and , we have . So
Let be a positive constant defined by and, given a fixed , by (28), we have as long as exists globally. Meanwhile, it is easy to see that for all ; thus, is global.

3. Main Results and Their Proofs

We first deal with the case of , which is easier than the other cases.

Theorem 5. If , then there exist both global solutions and blowing-up solutions to problem (1).

Proof. Lemma 3 states that when and , equation has a global supersolution.
Note that the reaction coefficient of this equation is bigger than , multiplied by a constant if necessary. So, we conclude that problem (1) has global solutions if and .
On the other hand, it is well known (see, e.g., [13]) that problem admits a blowing-up solution, which is a subsolution to our problem for suitable . We thus complete our proof.

Our remainder objective is to prove that if , then all solutions blow up in finite time. And this result is composed of several theorems.

To simplify the exposition, we may take , a characteristic function; for some in the remainder of this paper. Note that such assumption imposing on is not essential since it can be modified to a more general compactly function.

Theorem 6. If , then all the solutions to problem (1) blow up in finite time.

Proof. We will prove this result by the similar idea of Lemma  10 in [11]. We firstly construct here a blow-up subsolution matching of a self-similar function with a blowing-up parabola.
Fix a point and consider the even function obtained by the reflection of where and are to be determined later and is a self-similar solution of the problem It is known from [7] that problem (33) admits blowing-up self-similar solution if , and they have the form where .
In order to have a function , we need To this end, we take Henceforth, for , the function has the following form: Notice that if , . In order to see that is a subsolution to problem (1), we only have to look at the interval . Direct calculations show Therefore, when is small, is a subsolution to problem (1) if the inequality is valid.
This inequality could be achieved if we take . Since , . So, we only need to choose to get the desired blow-up subsolution.
Now, we claim that all solutions to (1) blow up in finite time. Suppose on the contrary that exist globally. From Lemma 6 of [11], we know that there exists some such that . Henceforth, by comparison theorem. And this implies that blows up. This contradicts the hypothesis that exist globally.

Theorem 7. If , then every solution to problem (1) blows up in finite time.

Proof. We first prove that problem (1) admits, for any length , a unique symmetric blow-up solution in the self-similar form Substituting into the first equation of (1), we have that the profile satisfies the equation plus the boundary condition .
If , that is, , is the desired blow-up solution, where is defined in Lemma 1.
If , we construct by putting together two pieces as in [11]. For , since , it is easy to verify that satisfies the above equation for any , where . If , we consider the problem Taking in Lemma 3.1 of [14], we know that, for every length , there exists a positive solution to this problem. Therefore, is the desired self-similar blow-up solution.
We now check that the above self-similar solution constructed can be put below any solution if we let pass enough time. In fact, the self-similar solution has small initial value if is large, but its support is not small, since the length is not small. We then use the penetration property of the solutions to the -Laplacian equation to guarantee that there exists such that the support of contains the interval . Therefore, we could get by taking which is large enough.
By comparison, must blow up in finite time.
Now, we turn to the case of .