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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

This short paper is devoted to the Fujita critical exponent of the following

We take the coefficient

As a representative example of quasilinear reaction-diffusion equation, classical

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

In our previous work, we has obtained the global existence exponent of (

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

There exists a positive constant

When

When

It is easy to verify that

Define

Since

Next, we will prove that there exists

It follows from

Since

Furthermore, we claim that

Integrating the inequality twice over the interval

Next, we prove that

If

Consider the following Cauchy problem:

Set

One can see that

Next, we introduce the function

If

The local existence and uniqueness of solution

By (

Let

We first deal with the case of

If

Lemma

Note that the reaction coefficient of this equation is bigger than

On the other hand, it is well known (see, e.g., [

Our remainder objective is to prove that if

To simplify the exposition, we may take

If

We will prove this result by the similar idea of Lemma 10 in [

Fix a point

In order to have a

This inequality could be achieved if we take

Now, we claim that all solutions to (

If

We first prove that problem (

If

If

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

By comparison,

Now, we turn to the case of

If

From Lemma

We now check that the so-called Barenblatt function

Finally, we may choose

Now, we deal with the case of

If

Here, we argue as in [

Denote by

First, by applying the maximum principle to the linear parabolic equation for the derivative

In fact, this fails for some

To this end we consider a special function

In order to finish the blow-up argument in this case, we observe that the only self-similar solution to this equation is (see [

Theorems

Comparing the proof of the main results in this paper and that of the corresponding part in [

After this paper was submitted, the authors found that, in [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by Scientific Research Found of Sichuan Provincial Education Department (12ZA288), Xihua University Young Scholars Training Program, and Applied Basic Research Project of Sichuan Province (2013JY0178).