Applied Mathematics, 2011, 2, 1134-1139
doi:10.4236/am.2011.29157 Published Online September 2011 (http://www.SciRP.org/journal/am)
Quenching Rate for the Porous Medium Equation with a
Singular Boundary Condition*
Zhengce Zhang, Yanyan Li
College of Science, Xi’an Jiaotong University, Xi’an, China
Received July 3, 2011; revised July 24, 2011; accepted August 1, 2011
Abstract
We study the porous medium equation


=,0<<,>
m
tx t0
t
xx
. We prove finite time quenching for the solution at the boundary . We also estab-
uu with a singular boundary condition

 
0, =0,
m
x
utu
 =0x
lish the quenching rate and asymptotic behavior on the quenching point.
Keywords: Porous Medium Equation, Quenching Profile, Quenching Rate, Singular Boundary Condition
1. Introduction
The nonlinear diffusion equation


=m
t
uu
with exponent , is usually called the porous
medium equation, written here PME for short. In the
particular case , it is called Boussinesq's equation.
The PME equation is one of the simplest examples of a
nonlinear evolution equation of parabolic type. It appears
in the description of different natural phenomena, and its
theory and properties depart strongly from the heat
equation , its most famous relative.
>1m
=2m
u=
t
u
There are a number of physical applications where this
simple model appears in a natural way, mainly to
describe processes involving fluid flow, heat transfer or
diffusion. Maybe the best known of them is the descrip-
tion of the flow of an isentropic gas through a porous
medium, modelled independently by Leibenzon [1] and
Muskat [2] around 1930, where represents the den-
sity of the gas and is a constant. The most
striking manifestation of this nonlinear degeneracy is that
in porous medium flow there is a finite speed of propa-
gation of disturbances from rest.
u


2,m
Once the theory for the PME began to be known, a
number of applications have been proposed. Some of
them concern the fast diffusion equation, the generalized
PME and the inhomogeneous versions already com-
mented. There are numerous examples with lower order
terms, in the areas of reaction-diffusion, where the PME
is only responsible for one of the various mechanisms of
the equation or system.
In [3], it is devoted to present a detailed account of the
asymptotic behavior as of the solutions
t


,uxt
of the equation

=m
t
uu
with exponent . The study extends the well-known
theory of the classical heat equation (HE, the case )
into a nonlinear situation, which needs a whole set of
new tools. The space dimension can be any integer
. They also present the extension of the results to
exponents (fast-diffusion equation, FDE).
>1m
<1
=1m
1N
m
In this paper we study the problem



 
 
0
=,>0,>0,
0,=0,,> 0,
,0=,> 0,
m
txx
m
x
uux t
ututt
uxu xx
 







(1.1)
where >0

, , is a smooth nonnegative >1m0 u
'
function satisfying , for


00
m
ux

00
''
m
u>0x
and the compatibility conditions at .
=0x
The study of quenching (in general the solution is
defined up to but some term in the problem
ceases to make sense) began with the work of Kawarada
[4] appeared in 1975. In that paper he studied the
semilinear heat equation as a singular reaction at level
. He proved that not only the reaction term, but also
the time derivative blows up wherever reaches this
value, see also [5]. Quenching problems have been
studied by many authors, see [6-10] and the references
=tT
=1u
u
*Supported by the Fundamental Research Funds for the Central Uni-
ve
r
sities of China.

Z. C. ZHANG ET AL.1135
therein.
The nonlinear parabolic equation


1
=1
p
mm
tx
xx
m
uumu
m

 p
u (1.2)
with is a mathematical model for many physical
problems corresponding to nonlinear diffusion with con-
vection. The source term on the right-hand side of
is of convective nature. In the theory of un-natured
porous medium equation, the convective part may re-
present the effect of gravity. Moreover, with is
also a Boussinesq equation of hydrology, which is
involved in various fields of petroleum technology and
ground water hydrology. For instance, in [11], Zhang and
Wang studied the following equation:
0m
(1.2)
=2m



 
 
  
22
0
1
=1
1||,
(0,) = 0,1,=1,,
,0=,0,1 ,
p
mmp
tx
xx
m
x
mm
xx
m
uumuu
m
mu u
ut utmu
uxu xx












q
t
(1.3)
for
 
,0,10,
x
tT
(1.3)
>>1qm =qm
, where and
are parameters, and is continuous and
satisfies the compatibility conditions. They proved that,
every solution of will blow up in finite time for
or for and . And they got
>1qm
<2p

<2p

0>1ux
>1
the blow up rate
 
12 1qm
,~uxtT t for
>>1qm and
 

11
,~ m uxtT t

 for =>1qm
and .
<2p
The porous medium equation without convection has
been considered extensively in the past few years. For
instance, in [12], Galaktionov and Levine studied the
following equation:

 


 
0
=,,0,0,
0,=0, ,0,,
,0 =,0,.
m
txx
mq
x
uu xtT
ututtT
uxu xx







,
 (1.4)
They proved that, if

0<1 2qm , then all
nonnegative solutions to were global; while for
(1.4)

>1qm2, the solutions to the equation would blow
up in finite time. Moreover, if

12< 1mqm, all
nonnegative solutions blow up in finite time; if >1qm

,
global nontrivial nonnegative solutions existed.
Pablo, Quiros and Rossi [13] firstly distinguished non-
simultaneous quenching from simultaneous one. They
considered a heat system coupled via inner absorptions,
  
  
00
=,=,
0, =0, =1, =1, =0,
,0=,,0=,0,1 ,
pq
txx txx
xxxx
uuvvv u
utvtutvt
uxu x vxv xx






(1.5)
for






,0,10,
x
tT , where






mi min,=0,
xx
vxtv t
,uv
[0,1]
n,=0, ,uxtut[0,1]
under certain assumptions on the initial data 00 .
For the coupled equations (1.5), the following quenching
rates were proved in [13]:
>0
1) If quenching is non-simultaneous and, for instance,
is the quenching component, then v



0, ~vt Tt

for close to T.
t
2) If quenching is simultaneous, then for t close to T,
a)
  
11
11
0, ~,0, ~,
,>1,<1;
pq
pq pq
ut TtvtTt
pqor pq



b)
 
12
0,,0,~,== 1utvt Ttpq;
c)


1
1
1
0,~ log,
0,~log,>= 1.
q
q
q
ut Tt
vt TtTtqp




For the system

 

 
 
00
=,=, ,0,10,,
0, =0,1, =1,,0,,
0,= 0,1,=1,,0,,
,0 =,,0 =,[0,1],
txxtxx
p
xx
q
xx
uu vvxtT
utut vttT
vtvt uttT
uxu x vxv xx









(1.6)
the finite time quenching results with the coupled sin-
gular nonlinear boundary flux were obtained by Zheng
and Song [14], other than the situation in the model of
(1.5) with coupled nonlinear absorption terms. The quen-
ching in (1.6) may be either simultaneous or non-simul-
taneous. This is determined by particular ranges of non-
linear exponents and the initial data. They showed that


=1x is the only quenching point and there are three
kinds of simultaneous quenching rates can be briefly
described in the following conclusions:
1)
  
22
1, ~, 1, ~,
,>1,<1;
ut Ttvt Tt
pqor pq


2)
  
14 14
1,~,1,~,== 1;u tTtvtTtpq
3)
 

1
1
1
1,~ log,
1,~log, 1=<,
q
q
q
ut Tt
v tTtTtpq




where




=1 1ppq 

 and



=1 1qpq  .
And



11
1, ~p vt Tt

 for non-simultaneous
quenching with quenching only. v
In [15], Fila and Levine studied the quenching pro-

Z. C. ZHANG ET AL. 1136
blem for the scalar case
 
 
  
0
=, ,0,10,,
0, =0,1, =1,,0,,
,0=> 0,0,1,
txx
q
xx
uu xtT
utut uttT
uxu xx








(1.7)
and obtained that
 

1
21
1, ~q utTt .
In [7], Deng and Xu studied the quenching problem
 
  
 
0
=,0<<1,>0,
0,= 0,1,=1,,> 0,
,0 =,01.
xx
t
xx
uu xt
ututgutt
uxu xx








(1.8)
For the special case and

=m uu 

=
g
uu


,
here 0< ,<m

, it is well known, 0<
corresponds to the porous medium case, refers to
the fast diffusion case, and when , the equation in
reduces to the heat equation. They obtained that x
= 1 is the only quenching point and the quenching rate is
<1m
>1m
=1m
(2.8)


121
1, ~m ut Tt
 
 for the porous medium case.
Our main purpose in this paper is to examine the
quenching behavior of the solution of the problem ,
that is, the solution reaches zero in finite time and the
quenching rate about
(1.1)
x
and t. We get the same quen-
ching rate as in [7]. Furthermore, we give the asymptotic
profile


1
,~m uxT x

.
The paper is organized as follows: In Section 2, we
prove that quenching occurs only at . In Section 3,
we derive estimates for the quenching rate. In Section 4,
we give the precise asymptotic profile near .
=0x
=0x
2. Quenching on the Boundary
We state a lemma that guarantees that, for certain initial
data, the solution of decreases with and
increases with
(1.1) t
x
.
Lemma 2.1. Suppose that and
for , then and


00
'
m
ux


m
x
ux


00
''
m
ux>0x,>0t

,<0
t
uxt in .

0, 0,T



Proof. Let


,
xxt,= m
vxt u and .

,=
t
wxtu
Then and satisfy

,vxt
 
,

,wxt



 



12
0
=1
>0,>0,
0, =0, =0,,>0,
,0=0,>0,
mm
txx xx
m
x
'
m
vmuv mmuuv
xt
vtut utt
vxu xx












(2.1)
and


 


1
11
0
=,>0,>0,
0,=0,,> 0,
,0=0,> 0,
m
txx
m
x
''
m
wmuwx t
mu wtutt
wx ux











(2.2)
respectively. From the maximum principle, it follows
that , , and hence

and >0v<0w

,>0
m
x
uxt


,<0t
t
ux in




0,T0, .
By the monotone of the on

,uxt

x
, we can find
the quenching point in a finite interval


1
0,
x
, where
1
1< < <xK

. Let u be a solution of (1.1) with
0
0<uM

on


1
0,
x
. Then 0< for all in
the existence interval and
uMt


1
0,
x
x
>0
. We now present
the quenching result for the problem .
(1.1)
Theorem 2.1. Assume

. Then every solution
of (1.1 ) quenches in finite time with the only quenching
point .
u
=0x
Proof. By Lemma 2.1, we know and


,>0
m
x
uxt


,<0
t
uxt. Thus,


  
1
0,
min,=0, ,0,.
xx
uxtu ttT

For
 
1
0
=,
xd
F
tuxt
x
d
.
, we have
 





11
00
1
=,d=
=, 0,

xx
'm
txx
mm
xx
F
tuxtx u
uxtut
x
Since


=<0
m
t
xx
uu, we have






1,< 0,.
mm
xx
uxtut
There exist >0

small enough such that








1,1 0,
mm
xx
uxtut   .
Thus









1
=,
0,
=0,







'm m
xx
m
x
0,
,
F
tu xtut
ut
ut M
and so





11 11
0
0, ,
FtF
M
tM
ut M
xx xx



 t
which means that there exists such that
>0T


lim0,= 0
tT
ut


.
To show that is the unique quenching point, it
suffices to prove that the quenching cannot occur at any
inner point
=0x


00,1 2x. Define



2
,= ,,
2
m
x
hxt uxtx
M




Z. C. ZHANG ET AL.1137
where >0

. Since



,2>0
m
x
uxT for ,
there exists
>0x
0>0

such that



0>0,2
m
x
uxT  
for


0,3 4x. If we take 0 32 9M

 , then

,2 0hxT,


0,3 4x. We have





 





12
12
00
1
1>0
,0,342,,
0, =0, =0, >0,
2, ,
9
34, =34,0
32
2,.
mm
txx xx
mm
x
m
x
m
x
hmuh mmuuh
mummuux
MM
xtT T
htutut
tTT
htutM
tTT



















 



,
(2.3)
By the maximum principle, in 0h

0,3 42,TT

, which means that


2
,=
2
m
x
uxt xh
M


0,

,0,12 2,.
x
tTT
Integrating with respect to
x
, we obtain
 
3
,0,
6
mm
uxt utx
M


,

,0,12 2,,
x
tTT
and hence for any

00,1 2x,

1
3
0
0
lim inf,> 0.
6
m
tT
x
ux tM







We have shown that quenching cannot occur in the
interior of

1
0,

x
. By the monotone, the proof is com-
plete.
3. Bounds for the Quenching Rate
In this section, we establish bounds on the quenching
rate.
Theorem 3.1. Suppose that and
. Then the solution of (1.1) satisfies


00
'
m
ux


00
''
m
ux

1
21
12
0, ,
m
CutTt C



where and are positive constants.
1 2
Proof. We first present the upper bound. Let
C C



,=, ,
mr
x
x
tu uxtuxt

 , where , <0r>0

and =r



. We have


 
 

 

 
12
11
32 2
32
11
32 1
3
1
, 0,0,
10, 0,
10,
,, 1
10,0,0,
mm
x xx
rr
tt
mr
xxx
x
rm
txx
r
xt
m muu
tu utut
uutmruuu t
u ut
umuux tmmuu
u utu utut
u












 




 
21
0,0, 0,
r
xt
utuutut








 
=0
=0
=( 1)
tx
mr
mr
mr
mr
mu
ruuu
mr ru
mr mu
ru ut
mr ru
mr ru




 




0,
>0x>0t

22m
x

for , , and


 
 
=0,0,0,
=0, 0,=0.

 


mr
x
r
tu tutut
utut
0,
By the maximum principle, we have





,=,0,0.
mr
x
uuxt ut

xt


Then


0,
xt0

, i.e.,










10,0,0,0. mr
x
rututut
 
0,
xx
ut


Hence
 
11 2
0,0,=0, .
rm m
t
rr
ut ut ut
mm
     
 (3.1)
Integrating the equality from t to T, we obtain (3.1)


21
2
0, .
m
utCT
  

t
Thus we prove the desired upper bound.
We then give the lower bound. We use a modification
of an argument from [16]. For


,tT   with some

such that


0,< 1ut , set
 
0
=0, ,d

t
y
tu tuxt



x (3.2)
with


=0,
m
tu t




, (3.3)
where


>m

 . By


<0
xx
um, we have







mm

,< 0,
xx
uttu .t 
There exists a >0

small enough such that





 


,<10,.
mm
xx
uttu  t

A routine calculation shows








  
 
()
1
0
1
1
=0,0,(,)d
0,, 0,
0, ,
0,0,0,.















t
'
t
mm
x
m
t
ytututuxtx
ututtu t
mu tutt
ututItu t

Z. C. ZHANG ET AL. 1138
Here
 


()
0
=,d
0,,.
t
m
Ituxt x
mu tutt







Since and in

0
m
x
u

0
m
xx
u
 


0, ,T   ,
we find
 

 
0,,,20,utuxtuttut   

(3.4)
for any

0,
x
t   

t
and . By ,
and , we have


,tT  (3.2) (3.3)
(3.4)
 
11
0,20, ,
mm
utytu
   

or equivalently,


   
11
3
0,0,for ,.
m
ut ytCuttT



 (3.5)
 




  






   

  
0
0
0
0
1
22
=,d
,,
=,d
,d
0,
0, ,
2
0,
t
t
t
t
x
m
m
x
Itmuxt x
muxtutt
muxtx
muttxt
mut
mututt









 





 

 








dx
x
where

0< <tt

 , and >0
x
u

  
2
0d= 2
t
x
tx t

 
. Then it follows that

1
4
0, .
'm
ytut Cyt

 



 
Integrating the above equality from t to T, we obtain
 
21
1
5.
m
m
y
tCTt



 
That is,

1
21
1
1,
m
m
yCTt




which in conjunction with yields the desired
lower bound.
(3.5)
4. Asymptotic Profile
In this section, we shall derive the following precise
asymptotic profile near x = 0.
Theorem 4.1. Suppose that is the solution of (1.1)
and assume that the quenching occurs at a finite time
, then there exist , such that
u
>0=tT 1 c2 c



11
12
,,0,0 <1.
mm
cxuxtcxxTt
 
 
We first prove a lemma as follows.
Lemma 4.1. Assume that is the solution of (1.1)
and assume that the quenching occurs at a finite time
, then there exists a such that
u
>0=tT 3 c




3
,,,0,0<
x
xuxtcu xtxTt1.
Proof. Let




2
4
=,
mm
x

,
J
xuxtcxu xt, where
4
2
>1
m
cm

. Then
J
satisfies




 

 


 


12
11
44
2
4
22 1
44
12
4
2
4
22
4
1
=4
12
122
=4 1
12
1
0,
mm
txx xx
mmm
txx
mm
x
x
mm
2
()
m
x
x
mm m
x
xx
mm
x
x
m
x
JmuJ mm uuJ
cmxuumc xuu
mmc xuuu
cm muumcuu
mxuuc mmuu
mmc xuuu
cm muu










 




and


0,= 0Jt . Then






2
4
,,
mm
x,
x
uxtcxuxt
and the lemma is proved, where 3
2
>1 cm.
Proof of Theorem 4.1. We first present the lower
bound. Let






:=,, .
m
x
J
uxtuxt
 

Then
J
satisfies



12
2
3
1
=1 >0,
mm
txx
m
x
xx
J
muJmmuu J
muu






and








0, =0,0, =0.
m
x
Jt utut
 

By the maximum principle, we obtain




=, ,
m
x
Ju xtuxt
 
0.
Then
11. m
x
uum
  
Integrating the above equality from 0 to x, we obtain
that


1
1
,.
m
uxt cx



We then give the upper bound. Let


5
=,
m
x
x
ucu




Z. C. ZHANG ET AL. 1139
where

21
0< =<1
2m



, 1
=2  and .
5>0c



 


 



 
121
121
5
2
32
55
21
2
3
5
22 121
5
21 3
5
12
=1
1
1
1
22
=11
mm
txx xx
mm
tx
mm
xxx
mm
xx
m
x
mm m
x
x
mm m
x
x
mum muumxu
cu umxuu
cmuucm uu
mmuxu u
cm muu
mxu ucmuxu
mxuucmuu








 



 


 


 
 
 

 



 

 




 



2
21221
21
5
21
21
2
32
55
22
322 1
55
21
12
2
=1
1
12
=11
12
mm m
xx
m
x
mm
x
mm
xx
mm
x
mm
x
x
mm
x
m
mmuxu umxuu
cmux u
mxuu
mmuxu u
cmuucmux u
mx uuumxu
cmuucmux u
mx u












 

 

 







 



 


 




3
322 1
55
11
12
0.
m
x
mm
x
ucm
cmuucmux u


 

  

 

1m


1
m
x
x
x
x
.
x
On the other hand,
 
5
0,=0,< 0.tcut


By the maximum principle, we have

5
=0
m
x
xu cu


Then
1
5.
m
x
uucx

  

Integrating the above equality, we obtain


1
1
22
,=.
m
m
uxtcxcx






Remark 4.1. Let , we can get tT



11
12
,,
mm
cxuxTcxx
 
 0.
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[