Research

# Blow-up profile for a degenerate parabolic equation with a weighted localized term

Weili Zeng1*, Xiaobo Lu1, Shumin Fei1 and Miaochao Chen2

Author Affiliations

1 School of Automation, Southeast University, Nanjing, 210096, China

2 Department of Mathematics, Southeast University, Nanjing, 210096, China

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Boundary Value Problems 2013, 2013:269  doi:10.1186/1687-2770-2013-269

 Received: 23 June 2013 Accepted: 21 November 2013 Published: 12 December 2013

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we investigate the Dirichlet problem for a degenerate parabolic equation . We prove that under certain conditions the solutions have global blow-up, and the rate of blow-up is uniform in all compact subsets of the domain. Moreover, the blow-up profile is precisely determined.

##### Keywords:
degenerate parabolic equation; localized source; uniform blow-up rate

### 1 Introduction

In this paper, we consider the following parabolic equation with nonlocal and localized reaction:

(1.1)

(1.2)

(1.3)

where Ω is an open ball of , with radius R, and .

Many of localized problems arise in applications and have been widely studied. Equations (1.1)-(1.3), as a kind of porous medium equation, can be used to describe some physical phenomena such as chemical reactions due to catalysis and an ignition model for a reaction gas (see [1-3]).

As for our problem (1.1)-(1.3), to our best knowledge, many works have been devoted to the case (see [4-7]). Let us mention, for instance, when , blow-up properties have been investigated by Okada and Fukuda [7]. Moreover, they proved that if and is sufficiently large, every radial symmetric solution (maximal solution) has a global blow-up and the solution satisfies

(1.4)

in all compact subsets of Ω as t is near the blow-up time , where and are two positive constants. Souplet [4,8] investigated that global blow-up solutions have uniform blow-up estimates in all compact subsets of the domain.

The work of this paper is motivated by the localized semi-linear problem

(1.5)

with Dirichlet boundary condition (1.2) and initial condition (1.3). In the case of and , the uniform blow-up profiles were studied in [5,9] and [10], respectively.

It seems that the result of [5,9,10] can be extended to and are two functions. Motivated by this, in this paper, we extend and improve the results of [5,9,10]. Our approach is different from those previously used in blow-up rate studies.

In the following section, we establish the blow-up rate and profile to (1.1)-(1.3).

### 2 Blow-up rate and profile

Throughout this paper, we assume that the functions , and satisfy the following two conditions:

(A1) , and ; , and are positive in Ω.

(A2) , and are radially symmetric; , and are non-increasing for .

Theorem 2.1Suppose thatsatisfies (A1) and (A2). If, then the solutions of (1.1)-(1.3) blow up in finite time for large initial data.

The proof of this theorem bears much resemblance to the result in [7,11,12] and is, therefore, omitted here.

Next we will show that in the situation of localized source dominating (), problem (1.1)-(1.3) admits some uniform blow-up profile.

Theorem 2.2Assume (A1) and (A2). Letbe the blow-up solution of (1.1)-(1.3) andis non-decreasing in time. If, then we have

(2.1)

uniformly in all compact subsets of Ω.

Throughout this paper, we denote

In our consideration, a crucial role is played by the Dirichlet eigenvalue problem

Denote by λ the first eigenvalue and by φ the corresponding eigenfunction with in Ω, normalized by . In the following, C is different from line to line. Also, we will sometimes use the notation for with the blow-up time for (1.1)-(1.3).

In order to prove the results of Section 2, first we derive a fact of the following problem:

(2.2)

Lemma 2.1Assume (A1), (A2) and. Letbe the blow-up solution of (2.2) and assume thatis non-decreasing in time, we then have

(2.3)

uniformly in all compact subsets of Ω.

Proof Assumption (A2) implies and on . From (2.2), we have

which implies

(2.4)

Thus and .

Set , and , . By , we obtain that , for .

Introducing a function

In the following, we only consider . For the case of , the proof is similar.

A series calculation yields

(2.5)

(2.6)

Now, according to (2.5) and (2.6), it follows that

(2.7)

for , .

Set , , . Since and note that , then there exists such that .

Therefore, in view of (2.7), we observe

Set , . We then obtain

Clearly, is a sup-solution of the following equation

(2.8)

where in and with . Here we also assume that is a symmetric and non-increasing function of ().

By the maximum principle, we have and in for .

Similar to the proof of Theorem 3.1 in [9] that

uniformly in all compact subsets of Ω.

By the arbitrariness of , we obtain that the following limit converges uniformly in all compact subsets of

(2.9)

In particular,

This inequality and (2.4) infer that

Multiplying both sides of (2.2) by φ and integrating over , we have, for ,

(2.10)

Since and , it then follows that

(2.11)

Next we prove that

uniformly in any compact subsets of Ω.

Assume on the contrary that there exists () such that

Then there exists a sequence , such that

Using the continuity of , we see that there exists () such that for . Note that and (2.9), we obtain

This contradicts (2.11) and we then complete the proof of Lemma 2.1. □

Lemma 2.2Under the assumption of Theorem 2.2, letbe the blow-up solution of (1.1)-(1.3), then it holds that

(2.12)

uniformly in all compact subsets in Ω.

Proof Proceeding as in (2.4), we have

(2.13)

which implies and .

Now, according to , it then follows that is a sub-solution of the following equation:

By the maximum principle, and in . Using Lemma 2.1, it holds that

uniformly in all compact subsets of Ω.

Hence

(2.14)

uniformly in any compact subsets of Ω, which implies

(2.15)

Combining (2.13) with (2.15), we deduce that

(2.16)

Multiplying both sides of (1.1) by φ and integrating over , we find, for ,

Since and , it then follows that

Therefore,

By analogy with the argument taken in Lemma 2.1, we complete the proof of this lemma. □

Proof of Theorem 2.2 By Lemma 2.2, we infer that

hence

Integrating this equivalence between t and , we obtain

(2.17)

The result finally follows by returning (2.17) to (2.12). □

Remark 2.1 It seems that in the case of , the blow-up rate remains valid in all compact subsets, but we do not know how to treat it. (It is an open problem in this case.)

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All the authors typed, read and approved the final manuscript.

### Acknowledgements

This work was supported by the China Postdoctoral Science Foundation Founded Project (2013M540405), the National Natural Science Foundation of China (61374194), and the Scientific Innovation Research of College Graduate in Jiangsu Province (CXZZ_0163).

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