Applied Mathematics, 2011, 2, 575-578
doi:10.4236/am.2011.25076 Published Online May 2011 (http://www.SciRP.org/journal/am)
The Role of Space Dimension on the Blow up for a
Reaction-Diffusion Equation
Zhilei Liang
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China
Received March 9, 2011; revised March 22, 2011; accepted March 27, 2011
Abstract
This paper deals with the doubly degenerate reaction-diffusion equation

1
1
div ,
mq
uuuuu
t
 




where ,

0,1 ,xB0,t1m1

, 11qm

 and B(0,1) denotes a unit ball in RN with the center in
origin. We prove that the blow up phenomenon can be restrained if the space dimension N is taken sufficiently large.
Moreover, the critical condition guaranteeing the absence (or occurrence) of the blow up is achieved.
Keywords: Reaction-Diffusion, Source Term, Space Dimension, Blow up
1. Introduction
The blow up phenomenon in parabolic equation has been
object of active research in recent years [1-4]. It has at-
tracted lots of interests and attentions because of its
physical importance and mathematical challenge. Here
we formulate the condition that guarantees the absence
of the blow up in terms of space dimension. In order to
fix ideas, we consider the following problem


 
 
1
1
0
div,0,1 ,0,
,0, 0,1,0,
,00,0,1 ,
mq
utuuu uxBt
uxtx Bt
uxu xxB
 

 



 



(1.1)
where ,
1m1

, 11qm

, and


0,1B is
a unit ball in with the center in origin. The initial
function 0 is assumed to be positive bounded.
Problem (1.1) is the well known non-Newtonian poly-
tropic filtration equation, which co mes up in a wide vari-
ety of physical contexts. The local in time existence and
the comparison principle of nonnegative weak solutions
have been established in [5], see also [6]. Due to the pos-
sible degeneration at the level or
N
R

x

u
0u0u, we
understand the weak solution in distributional sense as
follows
Definition 1.1 Let

1m

  

. A nonnegative
 
 
2
0,10,0, ;0,1uLBtCtLB

 with
 
110,1 0,uLB t 


 

is said to be a weak solu-
tion of (1.1), if the integral identity


 
1
1
0
0
dd
,0 d0
tmq
t
B
B
uuuuu x
ux xx









s 
is true for all test function .
   
00,1 0,CB T



When it comes to blow up, we mean that the solution
of (1.1) exists for


0,tT and become unbounded as
for some TtT

.
In this paper, we study the influencing factor of space
dimension N on the blow up phenomenon for the Dirich-
let problem (1.1). Such question was first investigated by
A. Tersenov for a heat equation with a nonlinear source.
By analyzing the stationary equation and then using the
comparison theorem, the author proved in [3] that if the
space dimension N is chosen large enough, the solutions
exist for all positive time. In other word, the high dimen-
sion plays a preventive role on the occurrence of the
blow up. In this current paper we extend the results ob-
tained in [3] to a doubly degenerate equation. The below
theorem is our first result.
Theorem 1.1 Given an initial function


0
ux
 
,xM
and a
constant , there is a , such
that for every space dimension , the solution
0M**
0
NNu*
NN


0
,;uxtu of (1.1) satisfies


00
,;, 0.uxtuuMt


 (1.2)
In particular,

Z. L. LIANG 576



**
0
0
00
,
1,
1
q
NNuM
uM
m
uM u




















 






where

1m

   . Moreover, as
*
N
0
u.
Just as described in Theorem 1.1, there exists a critical
which depends on the given , such that the
blow up can be avoided if the space dimension N satis-
fies . On the other hand, for a given space di-
mension N, there also exists a critical initial datum
0 such that all the non-trivial solutions occur blow
up so long as the size of their initial datum is larger than
that of 0. To demonstrate this we first present the
following proposition
*
N

*
u

0
ux
*
NN

x

*
u

x
Proposition 1.1 For a given space dimension N, there
exist positive constants A, a, which depend on N, as well
as ,,mq

, such that the function



11
_
,; ,
q
uxtNTt


 (1.3)
with
 
 
1
1
1,Aa


 







x
Tt

  and
 
11qq   0,
solves the inequality


1
1
div,,0, .
mqN
uuuuuxRtT
t
 


 (1.4)
The proof of Proposition 1.1 is available in the final
Section 3.
Because of



11qq   0

, the
support of

_
,;utN falls into the set


:
x
xaTt aT




. So, for a fixed N, we choose
T so small such that , and therefore, 1aT



supp_,0;0,1 .uNB (1.5)
We are ready to state our second result
Theorem 1.2 Suppose that (1.5) holds for a given
space dimension N, then there exists a critical initial
datum 0
 
*
_
,0;uxux N, such that for all initial data
u0 satisfying
 
0
*
0
_
,0; ,uxuxu x N (1.6)
the solution occur blow up before T. More-
over, inequality (1.6), along with (3.7) (see Section 3
below), leads to

0
,;uxtu

 
0
*
_
,0;, .uuNA N

 
Remark 1.1 Theorems 1.1 and 1.2 are also valid for a
general convex domain.
2. Proof of the Theorem 1.1
This section is devoted to the proof of Theorem 1.1. First
of all, let us analyze the radially symmetric solution


Ur of the stationary equation to (1.1), i.e.,

1
11
1
10, 0,
Nm q
NrUUUUrx
r
 




  (2.1)
with initial conditions


00,0UKU


0.

 (2.2)
The local and unique positive solution of (2.1)-(2.2)
follows from the analysis of the equivalent integral equa-
tion, by using Banach contraction mapping theorem.
Moreover, the solution is decreasing and can be extended
whenever it is positive. (See [2] and [7].) Let




*sup :0,0rrUrr

. Integrating (2.1) over


0,r,


*
0,rr, yields
1  
11 1
0d,
r
Nm qN
rU UUUss

 
 (2.3)
which implies

11 1
0
1
d.
r
N
mqNq
rUUUssKr
N

 
  
N
i.e.,


1
qr
Ur K
N



  



(2.4)
Integrating (2.4) once more to get


11
0
11
d
11 ,
1
qr
q
K
UrK ss
N
m
KK
N

 













 


r
i.e.,

1
11*
11 ,0 .
1
q
m
UrKKrr r
N











 




(2.5)
Clearly,
11
1
*11
1.
1
mq
rNK
m











Since the initial function , from
(2.5) we have
 

00, 1ux LB






 
0,0ux UxxB,1,
(2.6)
provided

1
1
0
11
1,
1
q
m
UKK u
N










 





Z. L. LIANG577
which is reduced to

*
0
1.
1
q
mK
NN
Ku










 




(2.7)
To guarantee the validity of (2.7), it suffices to take
0
K
u

M with a constant . That is, 0M


0*
00
1.
1
q
uM
m
NN
uM u









 
 




Apparently, (2.7) also ensures

 
0,,forall 0,1,0Uxuxtx Bt 
0
. (2.8)
In terms of (2.7) and (2.8), the comparison theorem
conclude
 
,,0,,uxtUxxBR t
This deduces that
  
0
,0,0,1,uxtUxUuMx Bt

 0.
(2.9)
This completes the proof of Theorem 1.1.
Remark 2.1 In case of 1qm



Ur, all the solutions of the (1.1) can be bounded by for arbitrary N,
only if the is chosen large enoug h (see (2.5)).
We also refer to [4].

0UK
3. Proof of Theorem 1.2
The proof of Theorem 1.2 is a direct consequence of
Proposition 1.1. So the remain task is to prove the valid-
ity of the Proposition 1.1. In fact, to verify the function

1
1
_,; q uxtNTt



 satisfies (1.4), we turn to
prove the following equivalent inequality obtained after
relatively computation

1
11
1
1
10, ,
1
Nm
N
qa
q


 

 









(3.1)
in which

'dd

. Due to the explicit form
 

1
1a

 




 
, the derivative of

is non-
increasing, i.e.,
1
11
11
110.
1
A
aaa




 






 

 
 

 

By this we compute

 
  
1
11
1
1
1
1
11
12
1
11
11
.
Nm
N
N
N
N
II




 
 

 











































(3.2)
A direct but tedious computation shows


1
1
1
1
1
11
11
1
11
1,
11
A
Iaa
A
aa

















 



 


 

 


 

 

and
 
1
2
1
1
1,
1
A
INaa







 

 
 


where
1
1a


















. Hence, (3.2) is equals to



1
11
1
1
1
1
1
1
11
11
1
11
.
11
Nm
N
A
aa
AN
aa

 






 

 

 
 








 
 

 


   



 




(3.3)
Similarly,
1
11
11
,
11
AA

 


 




 

(3.4)
1
11 .
11
q
qq
AA
qq 1


 


 


(3.5)
Summing up (3.3)-(3.5), we receive

1
110,
q
q
kl A






 (3.6)
where

1
1
1
11
,
11
A
ka



 




   


 1
1 


 
and

Z. L. LIANG 578

1
1
1
1
11
11
.
11
A
lN
a
q



 





 
 



 







1
0 Firstly, because , we have

0k


0  
for all


0,kl   with


0, 1kl. As to


,1kl  ,
inequality (3.6) holds so long as

1
1
1
10
q
q
q
q
kl A
k
klAl











 


which is valid if
 


11 1
1
1
1
1
1
1
1
11
11
1
111
111
1
111
1111
q
q
q
kA
AlkN
lq
a
A
a
ANq
a
 












 


 

 












 
 


 

 




 


 

 

  

 

 






1
1

 (3.7)
Clearly, For any fixed

, m, q and N, (3.7) is true if
we choose A large enough, and at the same vary a such
that the ratio

1
Aa
 1
keeps stable. Moreover, A
is increasing with respect to N. This completes the proof.
4. References
[1] Z. L. Liang, “Blow up Rate for a Porous Medium Equa-
tion with Power Nonlinearity,” Nonlinear Analysis: The-
ory, Methods & Applications, Vol. 73, No. 11, 2010, pp.
3507-3512. doi:10.1016/j.na.2010.06.078
[2] A. Samarskii, V. Galaktionov and V. Kurdyumov, “Blow
up in Quasilinear Parabolic Equations,” Walter de Gruyter,
Berlin, 1995.
[3] A. Tersenov, “Space Dimension Can Prevent the Blowup
of Solution for Parabolic Problems,” Electronic Journal
of Differential Equations Vol. 165, No. 165, 2007, pp. 1-
6.
[4] J. N. Zhao, “Existence and Nonexistence of Solutions for



,,,uuxt,” Journal of Mathe-
matical Analysis and Applications, Vol. 172, No. 1, 1993,
pp. 130-146.
2
div p
t
uuuf


doi:10.1006/jmaa.1993.1012
[5] D. Andreucci and A. F. Tedeev, “Universal Bounds at the
Blow-Up Time for Nonlinear Parabolic Equation,” Ad-
vances in Differential Equations, Vol. 10, No. 1, 2005, pp.
89-120.
[6] Z. Q. Wu, J. N. Zhao and J. X. Yin, et al., “Nonlinear
Diffusion Equations,” World Scientific Publishing Co.,
Inc., River Edge, 2001.
[7] Z. L. Liang and J. N. Zhao, “Localization for the Evolu-
tion p-Laplacian Equation with Strongly Nonlinear
Source Term,” Journal of Differential Equations, Vol.
246, No. 1, 2009, pp. 391-407.
doi:10.1016/j.jde.2008.07.038