Advances in Pure Mathematics, 2011, 1, 280-283
doi:10.4236/apm.2011.15050 Published Online September 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
Solutions with Dead Cores for a Parabolic P-Laplacian
Equation
Zhengce Zhang, Yanyan Li
College of Science , Xian Jiaotong University, Xian, China
E-mail:zhangzc@mail.xjtu.edu.cn, liyan86911@126.com
Received June 10, 2011; revised July 15, 2011; acc epted July 25, 2011
Abstract
We study the solutions with dead cores and the decay estimates for a parabolic p-Laplacian equation with
absorption by sub- and supersolution method. Special attention is given to the case where the solution of the
steady-state problem vanishes in an interior region.
Keywords: Parabolic P-Laplacian Equation, Steady State, Dead Core, Decay Estimates
1. Introduction
In this paper, we study the following initial-boundary
value problem for :
,uxt

 
 
2
00
=in=
,= 0on=,
,0 =,01in,
pq
t
udiv uuuQ
uxt x
ux uxux
,



(1.1)
with and . Here
,>1p
0< <1qp
x
is
continuous on and 0
u can be extended to a func-
tion on

x
, satisfying the compatibility condition
 
0=,uxx x
.
The domain is smooth and bounded.
>1
NN
Our purpose is to describe how the solution
,uxt
1< <2p
of (1.1) tends to its steady state and the existence of dead
cores. A dead core , i.e. a region
where the solution van ishes identically may ap pear. Such
a region is a waste from the engineering point of view.
We concern its existence. Our method is the weak solu-
tion which is similar to that in [4] where the porous me-
dium equation was considered. For the case of
and of the problem (1.1), Chen, Qi and Wang [3]
proved the existence of the singular solution. In [2], they
also studied the long time behavior of solutions to the
Cauchy pro bl e m of


0:=:= 0xux
>1q


2
=in 0,
pqN
t
udivu uu
 
with nonnegative initial value in
 
,0 =ux x
N
,
where

21<NN p<2 and . For initial data
of various decay rates, especially the critical decay
>1q
=Ox
with
=1pq p
 , they showed that
the solution converges as to a self-similar solu-
tion. t
rong ab
eak abso
We have known the following behavior of the absorp-
tion near :
=0u
<1 stsorption,q
1 wrption.q
When
0x
in (1.1), the steady state vanishes
and
,tux tends to zero as . Strong absorption
yields extinction in finite time, that is, there is a time
such that
t T
,uxt
x
0
for all and t. For small
, the absorption is still relatively large in the case of
strong absorption and will tend to drive the solution more
quickly to zero than in the case of weak absorption. If
T
u
x
is not identically zero in (1.1), the corresponding
steady state
x
does not vanish identically, but it is
still true that
,uxt tends to
x
as .
t
The steady-state problem
n
q
 
2= in ,
= o
p
div
xx


 
 (1.2)
is a special case of the problem


xibx

2
=1 = in ,
=1 on
Np
xi
i
ax uufuD
uD
(1.3)
in [1], where
N
D
be an arbitrary domain,
,:abD
be two continuous functions, and
1p
f
s is a nondecreasing function with
0=f0
.
Z. C. ZHANG ET AL.281
From [1], we know that Problem (1.2) has a unique
solution, a dead core exists if
is large enough and

11/
0,
p
Fs
d<s (1.4)
where
 
0
=d
s
Fs f

.
Proofs of existence and uniqueness for Problem (1.1)
are based on a suitable notion of weak solution, which
we include here for the sake of completeness.
Definition 1.1 Let
=0,
T
Q
T
and let denote
the outward unit normal to . A function n
 
1,
,
pT
uxtCWL QT

;0, is called a weak
solution of Problem (1.1) if it satisfies


2
2
0
0
,,d
d
QT
x
s


dd
=,
0ddd
p
t
QT
Tp
q
uxT xT
uu uxt
n
ux xuxt








 (1.5)
for all

1T
CQ
with
 
1,
0
,, ;0,
pT
x
txtCW LQT

 .
Equation (1.5) is obtained easily by multiplying (1.1) by
, integrating over , and using the divergence theo-
rem. T
Q
A weak supsolution of (1.1) is defined by replacing
the equal sign in (1.5) by and restricting
to be
nonnegative. Similarly, we can define a weak subsolu-
tion. For our purposes, it suffices to consider the usual
super- and subsolu tions defined as follows.
We say that 0u is a supersolution of (1.1) if

 
2
0
0,
,0 ,.
pq
t
udiv uu
uxu xx
 
|
u
u

(1.6)
Similarly, 0u is a subsolution if all the inequa-
lities in (1.6) are reversed. If uu
, then the unique weak
solution of (1.1) satisfies
uuuu, almost every-
where in .
T
It follows from the maximum principle that the
solutions of (1.1) and of (1.2) satisfy in ,
in .
Q

,uxt1Q

1x
To show monotonicity of
,uxt in time, we need to
impose a natural condition on the initial value
0
ux:
 
2
000
<0, .
pq
divuxuxuax
 
(1.7)
This condition holds automatically if is a posi-
tive constant. If satisfies (1.7), it is an upper
solution to (1.1) so that

0
ux

0
ux
,uxt

0
u x for any .
Now let t

,=vxt
,uxt
; then satisfies

,vxt
  
2=,
,0 =,,,=.
pq
t
vdivvvv
vxuxt vtx


Since
0
,uxu x
, is subsolution of (1.1), and
hence v
t,uxt u
 ,x. Hence, is monotoni-
cally decreasing. Standard theorems can be used to show
that
(, )ut
,uxt tends to the steady-state

x
as t. 
2. Monotonicity and Other Comparison
Theorems
Consider problem (1.1) when only one part of the data is
changed. We then have the following monotonicity pro-
perties:
1) Let 1 and 2
u be the solutions corresponding to u
1
, 2
, respectively, with 12
; then in Q.
21
2) If the initial or the boundary value is decreased so is
the solution.
uu
3) Let
01ux
, and consider two
domains

1x
12
 . Then on .
21 1
These all are easy to prove using super- and subsolu-
tion techniques. Here we omit the proofs.
uuQ
Next we look at the “lumped-parameter” problem and
the steady-state problem with a view to using them as
comparison problems for (1.1). The lumped parameter
problem has no diffusion term. It can be obtained as a
special case of (1.1) with initial value

01ux
and
boundary condition of vanishing normal derivative. We
can then seek a solution independent of
()zt
x
:

=,>0;0=
q
t
zzt z
1.
(2.1)
The solution is given explicitly by
 


 

1/ 1
1/ 1
=11 >1
= =1
=11 0<<1
q
t
q
ztq tq
zt eq
zt qtq






(2.2)
Therefore, extinction occurs in finite time if and only
if . Then
0< <1q
=1 1t
q and for

>0zt
<1 1tq
.
Comparison with (2.1) leads immediately to two re-
sults for (1.1).
Theorem 2.1 (a) If
=0x
ux
, then is a super-
solution of (1.1), so that and, if the
absorption is strong, there is extinction in finite time for

zt

zt

,t
,uxt.
(b) If
0
min =ux > 0
, then the solution
,zt
of (2.1) with initial value
is a subsolution of (1.1) so
that
,,uxt zt
. If the absorption is weak, then
,>0
zt for all t, and hence in Q.

,>0uxt
The following theorem shows that (1.4) is necessary
and sufficient for the existence of a dead core for
Copyright © 2011 SciRes. APM
Z. C. ZHANG ET AL.
282
sufficiently large
.
Theorem 2.2 For , , (1.4) is
satisfied. Then a dead core exists for sufficiently large
0< <1qp>1p
.
Proof. We shall construct, for
sufficiently large,
an upper solution to (1.2) for a ball

vr
B, with
,
0v0< <2rR. We begin by observing that on the
positive real line, the function by
wx



1
1
1
()= 1
11
pp q
pp q
p
p
q
p
wx pq
px
qp















satisfies

2
1=
pq
pwww

, ; >0x
0=ww0=0
. Now we choose
so that
=1
2
R

w
 and consider the function
=2
R
vwr



in the ball in
rR
N
(having extend v to be zero
for <2rR). We then obtain

 

 





22
1
1
1
1
11
1
=1
1
= in 2,
1
=11
:= ,
p1
p
p
p
q
p
p
qp
q
qp
qp
qq
N
div vvpvvv
r
N
vvRR
r
Np
vv
rqp
vCv
Cv v



 
 











where we use and
0v1

11qp q
p

. Since

2=0vR ,

2=0vR
, the last inequality can be ex-
tended to . Furthermore, so that
0,R

=1vR
vr
is an upper solution of the elliptic problem (1.2) for
. Since vanishes for v<2rR, so does
.
Now consider (1.2) on an arbitrary domain
with
. Then contains a ball

 1
B on whose
boundary 1
. Therefore, v
and contains a
dead core for
.
For any 0, we can take 0 the distance
from 0
x ==Rr
x
to the boundary. Theorem 2.2 shows that for
large enough, 0
x
belongs to the dead core. This
suggests making the following definition.
Definition 2. 1 Let . Define
0
x

*
00 0
=,=0;=
inf inf
x
x
We claim that
,, ,,
*
02
0
,,
N
pqN pq
p
PP
r

 (2.4)
where

1
,, =1 2
11
p
Npq pp
Pp N
pq pq





,
00
=,rdistx
; =
inradius of . The proof con-
sists in noting that the function
1
0
0
=
p
pq
xx
wr



satisfies

2,,
0
=0
pNpq q
p
P
div www
r
  in
and
1w
 . Thus, is a supersolution of (1.2) for any w
,,
0
N
pq
p
P
r
w. Since vanishes at 0
x
, so does
. This
proves the first part of (2.4) and the second part follows
at once.
3. The Corresponding Evolution Problem
Since (1.4) is satisfied with
large enough, the steady
state has a dead core. Does the corresponding evolu- tion
problem have a dead core and, if so, how does it behave
for large t? The answer is given by the fol- lowing
theorem.
Theorem 3.1 For fixed 0, choose 0
x >
, wh ere
0
is defined in (2.3). Then
(a) if 0<<11qp
, for

uxt
0,=0


00
1
:= ;
1
tt q

(b) if 1<qp1
and

0
min>0ux
, then
0
uxt,>0 for all t.
Proof. Part (b) is equivalent to Theorem 2.1(b). To
prove part (a), it suffices to exhibit a supersolution
,wxt such that
0,=0wx t for . We try a func-
tion 0
tt
=wz
,
where
,zt
is the solution of the lumped--parameter
problem with
to be choosen satisfies the differential
inequality for a supersolution and
is the solution of
the steady-state problem with 0
=
. Note that vani-
shes at 0
w
x
for
11tq

. Since and wzw
,
it is clear that
,0wx u

x
0 and

x
,wt .
From the definition of , we have
w




2
2
00
=
.
pq
t
pq
t
q
qq
wdivwww
zdiv z
zz

 

 
 q
w
0
.
 
(2.3)
Copyright © 2011 SciRes. APM
Z. C. ZHANG ET AL.
Copyright © 2011 SciRes. APM
283





2
2
1
1
=
=
=0
pq
t
pq
t
qqt
q
q
vdiv vvv
div v
q
q
By choosing 0
=

, we obtain the desired result.
.

 
 
 





 

4. Decay Estimates
We consider (1.1) with , where
 
0
ux x
x
is
the solution of the corresponding steady-state problem
(1.2). It then follows, since
is a lower solution, that
. If we assume in addition that
, then is bounded a.e. in .

,uxt x

0
ux L
uQ
Our estimates hold for almost all . If the data
are smooth, the solution is continuous and the estimates
hold pointwise . We see k deca y estim a tes for
x
 
,= ,
x
tuxt x
Because we also have v
and v
, we see that
is a supersolution of (1.1). Then v
==uv .


1
. Our principal results are con-
tained in the following theorem. Theorem 4.1 is proved.
Theorem 4.1 (a) if , then
1<qp5. Acknowledgments
0, ,
x
tzt

where
zt
0< is the solution of (2.1). The authors are grateful to the Fundamental Research
Funds for the Central Universities of China.
(b) if , then
<1 1qp
 
0,,
x
tt

 6. References
where is the solution of

t
[1] C. Bandle and S. Vernier-Piro, “Estimates for Solutions
of Quasilinear Problems with Dead Cores,” Zeitschrift für
Angewandte Mathematik und Physik, Vol. 54, No. 5,
2003, pp. 815-821.

1
=,0 =1,=2.
q
tq

(4.1)
Proof. (a) We choose

,=wxt ztx
, where
satisfies (2.1) and

zt
x
satisfies (1.2). Then, we
have [2] X. F. Chen, Y. W. Qi and M. X. Wang, “Long Time Be-
havior of Solutions to P-Laplacian Equation with Ab-
sorption,” SIAM Journal on Mathematical Analysis, Vol.
35, No. 1, 2003, pp. 123-134.
doi:10.1137/S0036141002407727



2
2
=
=
=0,
pq
t
pq
t
qq
t
qq qq
wdivwww
zdiv w
zw
zuz

 




 
[3] X. F. Chen, Y. W. Qi and M. X. Wang, “Singular Solu-
tions of Parabolic P-Lapacian with Absorption,” Trans-
actions of the American Mathemarical Society, Vol. 359
2007, pp. 5653-5668.
[4] C. Bandle, T.Nanbu, I. Stakgold, “Porous Medium Equa-
tion with Absorption,” SIAM Journal on Mathematical
Analysis, Vol. 29, 1998, pp. 1268-1278.
doi:10.1137/S0036141096311423
where we use . Because we also have and
1qwz
w
, we see that is a supersolution of (1.1). Then w
==uw .z

 
(b) We choose


,=vxtt x

, where
t
satisfies (4.1) and
satisfies (1.2). Then, we have