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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

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/269


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

© 2013 Zeng et al.; licensee Springer.

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 u t u m = a ( x ) u p ( 0 , t ) + b ( x ) u q ( x , t ) . 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:

u t u m = a ( x ) u p ( 0 , t ) + b ( x ) u q ( x , t ) , x Ω , 0 < t < T , (1.1)

u ( x , τ ) = 0 , x Ω , t > 0 , (1.2)

u ( x , 0 ) = u 0 ( x ) , x Ω , (1.3)

where Ω is an open ball of R N , N 2 with radius R, and p q > m > 0 .

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 m = 1 (see [4-7]). Let us mention, for instance, when a ( x ) = b ( x ) = 1 , blow-up properties have been investigated by Okada and Fukuda [7]. Moreover, they proved that if p q > 1 and u 0 ( x ) is sufficiently large, every radial symmetric solution (maximal solution) has a global blow-up and the solution satisfies

C 1 ( T t ) 1 / ( p 1 ) u ( x , t ) C 2 ( T t ) 1 / ( p 1 ) , (1.4)

in all compact subsets of Ω as t is near the blow-up time T , where C 1 and C 2 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

u t u m = λ 1 u p ( 0 , t ) + λ 2 u q ( x , t ) , x Ω , 0 < t < T , (1.5)

with Dirichlet boundary condition (1.2) and initial condition (1.3). In the case of m = 1 and m > 1 , 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 λ 1 and λ 2 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 a ( x ) , b ( x ) and u 0 ( x ) satisfy the following two conditions:

(A1) a ( x ) , b ( x ) and u 0 ( x ) C 0 2 ( Ω ) ; a ( x ) , b ( x ) and u 0 ( x ) are positive in Ω.

(A2) a ( x ) , b ( x ) and u 0 ( x ) are radially symmetric; a ( r ) , b ( r ) and u 0 ( r ) are non-increasing for r [ 0 , R ] .

Theorem 2.1Suppose that u 0 ( t ) satisfies (A1) and (A2). If max { p , q } > m , 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 ( p > q ), problem (1.1)-(1.3) admits some uniform blow-up profile.

Theorem 2.2Assume (A1) and (A2). Let u ( x , t ) be the blow-up solution of (1.1)-(1.3) and u ( x , t ) is non-decreasing in time. If p > max { q , m } , then we have

lim t T u ( x , t ) ( T t ) 1 / ( p 1 ) = a ( x ) ( ( p 1 ) a p ( 0 ) ) 1 / ( 1 p ) , (2.1)

uniformly in all compact subsets of Ω.

Throughout this paper, we denote

g ( t ) = u p ( 0 , t ) and G ( t ) = 0 t g ( s ) d s .

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

{ φ = λ φ , in  Ω , φ ( x ) = 0 , on  Ω .

Denote by λ the first eigenvalue and by φ the corresponding eigenfunction with φ > 0 in Ω, normalized by Ω a ( x ) φ ( x ) d x = 1 . In the following, C is different from line to line. Also, we will sometimes use the notation u v for lim t T u ( t ) / v ( t ) = 1 with T 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:

{ u t u m = a ( x ) g ( t ) , x Ω , 0 < t < T , u ( x , t ) = 0 , x Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , x Ω . (2.2)

Lemma 2.1Assume (A1), (A2) and p > q . Let u ( x , t ) be the blow-up solution of (2.2) and assume that u ( x , t ) is non-decreasing in time, we then have

lim t T u ( x , t ) G ( t ) = a ( x ) (2.3)

uniformly in all compact subsets of Ω.

Proof Assumption (A2) implies u ( 0 , t ) = max x Ω ¯ u ( x , t ) and u m ( 0 , t ) 0 on ( 0 , T ) . From (2.2), we have

u t ( 0 , t ) a ( 0 ) u p ( 0 , t ) + b ( 0 ) u q ( 0 , t ) , 0 < t < T ,

which implies

lim t T sup u ( 0 , t ) G ( t ) a ( 0 ) . (2.4)

Thus lim t T G ( t ) = and lim t T g ( t ) = .

Set R 1 ( 0 , R ) , Ω 1 = { x R N , | x | < R 1 } and b ( x ) = 1 / a ( x ) , x Ω 1 . By a ( r ) 0 , we obtain that b ( r ) 0 , for 0 r R 1 .

Introducing a function

w ( x , t ) = b ( x ) u ( x , t ) , x Ω 1 , 0 < t < T .

In the following, we only consider m > 2 . For the case of 0 < m 2 , the proof is similar.

A series calculation yields

b u m = b 1 m ( b u ) m + ( m 2 ) b u m 2 ) | u | 2 m u m b 1 | b | 2 m u m b 4 m u m 1 u b . (2.5)

In addition, note

u ( x , t ) b ( x ) = u r ( r , t ) b ( r ) 0 . (2.6)

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

w t = b ( x ) u t b ( x ) u m + g ( t ) ( a ( x ) ) m 1 w m ( m a ( x ) ( | b | ) 2 + m b ) u m + g ( t ) (2.7)

for x Ω 1 , 0 < t < T .

Set m 1 = min x Ω 1 ¯ | a ( x ) | m 1 , m 2 = max x Ω 1 ¯ { m a ( x ) | b | 2 + m | b | } , ε ( t ) = m 2 u m ( 0 , t ) / g ( t ) . Since p > m and note that g ( t ) = u p ( 0 , t ) , then there exists τ ( 0 , T ) such that 0 < ε ( t ) 1 / 2 .

Therefore, in view of (2.7), we observe

w t m 1 w m + ( 1 ε ( t ) ) g ( t ) + ε ( t ) g ( t ) m 2 u m ( 0 , t ) = m 1 w m + ( 1 ε ( t ) ) g ( t ) , x Ω 1 , τ < t < T .

Set g 1 ( t ) = ( 1 ε ( t ) ) g ( t ) , G ( t ) = τ t g 1 ( s ) d s . We then obtain

lim t T G 1 ( t ) = and lim t T G 1 ( t ) G ( t ) = 1 .

Clearly, w ( x , t ) is a sup-solution of the following equation

{ v t = m 1 v m + g 1 ( t ) , x Ω 1 , τ < t < T , v ( x , t ) = 0 , x Ω 1 , t > 0 , v ( x , 0 ) = v 0 ( x ) , x Ω 1 , (2.8)

where 0 v 0 w ( x , τ ) in Ω 1 and v 0 C 1 ( Ω ¯ 1 ) with v 0 | Ω 1 = 0 . Here we also assume that v 0 ( x ) is a symmetric and non-increasing function of | x | ( r = | x | ).

By the maximum principle, we have 0 v ( x , t ) w ( x , t ) and v r 0 in Ω 1 for τ t < T .

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

lim t T v ( x , t ) G ( t ) = 1

uniformly in all compact subsets of Ω.

By the arbitrariness of Ω 1 , we obtain that the following limit converges uniformly in all compact subsets of Ω 1

lim t T inf u ( x , t ) G ( t ) a ( x ) . (2.9)

In particular,

lim t T inf u ( 0 , t ) G ( t ) a ( 0 ) .

This inequality and (2.4) infer that

lim t T u ( 0 , t ) G ( t ) = a ( 0 ) .

Multiplying both sides of (2.2) by φ and integrating over Ω × ( 0 , t ) , we have, for 0 < t < T ,

Ω u φ d x Ω u 0 φ d x = λ 0 t Ω u m φ d x d s + G ( t ) . (2.10)

Since 0 t Ω u m φ d x d s Ω φ d x 0 t u m ( 0 , t ) d s and lim t T u m ( 0 , t ) / g ( t ) = 0 , it then follows that

lim t T Ω u φ d x G ( t ) = 1 . (2.11)

Next we prove that

lim t T sup u ( x , t ) G ( t ) a ( x )

uniformly in any compact subsets of Ω.

Assume on the contrary that there exists x 0 Ω ( x 0 0 ) such that

lim t T sup u ( x 0 , t ) G ( t ) = c > a ( x 0 ) .

Then there exists a sequence { t n } , t n T such that

lim t n T u ( x 0 , t n ) / G ( t n ) = c .

Using the continuity of a ( x ) , we see that there exists x 1 Ω ( | x 1 | < | x 0 | ) such that c > a ( x ) for | x 1 | | x | | x 0 | . Note that u r 0 and (2.9), we obtain

lim t T Ω u φ d x G ( t ) = lim t n T ( | x | < | x 1 | u φ d x G ( t n ) + | x 1 | < | x | < | x 0 | u φ d x G ( t n ) + | x 0 | < | x | < R u φ d x G ( t n ) ) | x | < | x 1 | a ( x ) φ ( x ) d x + c | x 1 | < | x | < | x 0 | φ d x + | x 0 | < | x | < R a ( x ) φ ( x ) d x > | x | < | x 1 | a ( x ) φ ( x ) d x + | x 1 | < | x | < | x 0 | a ( x ) φ d x + | x 0 | < | x | < R a ( x ) φ ( x ) d x = 1 .

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

Lemma 2.2Under the assumption of Theorem 2.2, let u ( x , t ) be the blow-up solution of (1.1)-(1.3), then it holds that

lim t T u ( x , t ) G ( t ) = a ( x ) (2.12)

uniformly in all compact subsets in Ω.

Proof Proceeding as in (2.4), we have

lim t T sup u ( 0 , t ) G ( t ) a ( 0 ) , (2.13)

which implies lim t T G ( t ) = and lim t T g ( t ) = .

Now, according to u t u + a ( x ) g ( t ) , it then follows that u ( x , t ) is a sub-solution of the following equation:

{ v t v m = a ( x ) g ( t ) , x Ω 1 , 0 < t < T , v ( x , t ) = 0 , x Ω 1 , t > 0 , v ( x , 0 ) = u 0 ( x ) , x Ω 1 .

By the maximum principle, u ( x , t ) v ( x ) and v r 0 in Ω × ( 0 , T ) . Using Lemma 2.1, it holds that

lim t T v ( x , t ) G ( t ) = a ( x )

uniformly in all compact subsets of Ω.

Hence

lim t T sup u ( x , t ) G ( t ) a ( x ) (2.14)

uniformly in any compact subsets of Ω, which implies

lim t T sup u ( 0 , t ) G ( t ) = a ( 0 ) . (2.15)

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

lim t T sup u ( 0 , t ) G ( t ) = a ( 0 ) . (2.16)

Multiplying both sides of (1.1) by φ and integrating over Ω × ( 0 , t ) , we find, for 0 < t < T ,

Ω u φ d x Ω u 0 φ d x = λ 0 t Ω u m φ d x d s + G ( t ) 0 t Ω b ( x ) u q φ d x d s .

Since 0 t Ω u m φ d x d s 0 t u ( 0 , s ) d s Ω φ d x and p > q , it then follows that

lim t T 0 t Ω v m φ d x d s G ( t ) = 0 and lim t T 0 t Ω v q φ d x d s G ( t ) = 0 .

Therefore,

lim t T Ω u φ d x G ( t ) = 1 .

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

u ( 0 , t ) a ( 0 ) G ( t ) as  t T ,

hence

G ( t ) = g ( t ) a p ( 0 ) G p ( t ) or ( G 1 p ) ) ( p 1 ) a p ( 0 ) .

Integrating this equivalence between t and T , we obtain

G ( t ) [ ( p 1 ) a p ( 0 ) ( T t ) ] 1 1 p . (2.17)

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

Remark 2.1 It seems that in the case of p = q > m , 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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