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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 87650, 8 pages
doi:10.1155/2007/87650
Research Article
Extinction and Decay Estimates of Solutions for
a Class of Porous Medium Equations
Wenjun Liu, Mingxin Wang, and Bin Wu
Received 3 April 2007; Accepted 6 September 2007
Recommended by Michel Chipot
The extinction phenomenon of solutions for the homogeneous Dirichlet boundary value
problem of the porous medium equation u
t
= Δu
m
+ λ|u|
p−1
u −βu,0 <m<1, is studied.
Sufficient conditions about the extinction and decay estimates of solutions are obtained
by using L
p
-integral model estimate methods and two crucial lemmas on differential in-
equality.
Copyright © 2007 Wenjun Liu et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction and main results
This paper is devoted to the extinction and decay estimates for the porous medium equa-
tion
u
t
= Δu
m
+ λ|u|
p−1
u −βu, x ∈ Ω, t>0, (1.1)
u(x,t)
= 0, x ∈ ∂Ω, t>0, (1.2)
u(x,0)
= u
0
(x) ≥ 0, x ∈ Ω, (1.3)
with 0 <m<1andp,λ,β>0, where Ω
⊂ R
N
(N>2) is a bounded domain with smooth
boundary.
The phenomenon of extinction is an important property of solutions for many evo-
lutionary equations which have been studied extensively by many researchers. Especially,
there are also some papers concerning the extinction for the porous medium equation.
For instance, in [1–3], the authors studied the extinction and large-time behavior of solu-
tion of (1.1)forthecaseβ
= 0andλ<0; and in [4], the authors obtained conditions for
the extinction of solutions of (1.1) without absorption by using sub- and supersolution
2 Journal of Inequalities and Applications
methods and an eigenfunction argument. But as far as we know, few works are concerned
with the decay estimates of solutions for the porous medium equation.
The existence and uniqueness of nonnegative solution for problem (1.1)–(1.3)have
been studied in [ 5, 6]. The purpose of the present paper is to establish sufficient condi-
tions about the extinction and decay estimations of solutions for problem (1.1)–(1.3). For
the proof of our result, we employ L
p
-integral model estimate methods and two crucial
lemmas on differential inequality.
Our main results read as follows.
Theorem 1.1. Assume that 0
≤ u
0
(x) ∈ L
∞
(Ω) ∩ W
1,2
0
(Ω), 0 <m= p<1,andλ
1
is the
first eigenvalue of
−Δψ(x) = λψ(x), ψ|
∂Ω
= 0, (1.4)
and ϕ
1
(x) ≥ 0 with ϕ
1
∞
= 1 is the eigenfunction corresponding to the eigenvalue λ
1
.
(1) If λ<4mλ
1
/(m +1)
2
, then the weak solution of problem (1.1)–(1.3) vanishes in the
sense of
·
2
as t →∞.
(2) If (N
− 2)/(N +2)≤ m<1 with λ<λ
1
or 0 <m<(N − 2)/(N +2)with λ<λ
∗
, then
the weak solution of problem (1.1)–(1.3) vanishes in finite time, and
u(·,t)
m+1
≤
u
0
1−m
m+1
+
C
1
β
e
(m−1)βt
−
C
1
β
1/(1−m)
,
N
− 2
N +2
≤ m<1,
u(·,t)
r+1
≤
u
0
1−m
r+1
+
C
2
β
e
(m−1)βt
−
C
2
β
1/(1−m)
,0<m<
N
− 2
N +2
,
(1.5)
for t
∈ [0,T
∗
),where
0 <T
∗
≤
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
T
1
,
(N
− 2)
(N +2)
≤ m<1,
T
2
,0<m<
(N
− 2)
(N +2)
,
(1.6)
r
=
N(1 − m)
2
− 1,
λ
∗
=
(r + m)
2
λ
4rm
<λ
1
,
(1.7)
and C
1
, C
2
, T
1
,andT
2
are given by ( 2.18 ), (2.24), (2.20), and (2.26),respectively.
Wenjun Liu et al. 3
Theorem 1.2. Let 0 <m<1, m<p. Then the weak solution of problem (1.1)–(1.3) vanishes
in finite time, and
u(·,t)
m+1
≤ B
1
e
−α
1
t
, t ∈
0,T
01
,
u(·,t)
m+1
≤
u(·,T
01
1−m
m+1
+
C
3
β
e
(m−1)β(t−T
01
)
−
C
3
β
1/(1−m)
, t ∈
T
01
,T
3
,
u(·,t)
m+1
≡ 0, t ∈
T
3
,+∞
,
(1.8)
for (N
− 2)/(N +2)≤ m<1,
u(·,t)
r+1
≤ B
2
e
−α
2
t
, t ∈
0,T
02
,
u(·,t)
r+1
≤
u(·,T
02
1−m
r+1
+
C
4
β
e
(m−1)β(t−T
02
)
−
C
4
β
1/(1−m)
, t ∈
T
02
,T
4
,
u(·,t)
r+1
≡ 0, t ∈
T
4
,+∞
,
(1.9)
for 0 <m<(N
− 2)/(N +2),whereC
3
, C
4
, T
3
,andT
4
are given by (2.29), (2.34), (2.31),
and (2.36), respectively.
To obtain the above results, we will use the follow ing lemmas which are of crucial
importance in the proofs of decay estimates.
Lemma 1.3 [7]. Let y(t)
≥ 0 be a solution of the differential inequality
dy
dt
+ Cy
k
+ βy ≤ 0(t ≥ 0), y
T
0
≥
0, (1.10)
where C>0 is a constant and k
∈ (0,1). Then one has the decay estimate
y(t)
≤
y
T
0
1−k
+
C
β
e
(k−1)β(t−T
0
)
−
C
β
1/(1−k)
, t ∈
T
0
,T
∗
,
y(t)
≡ 0, t ∈
T
∗
,+∞
,
(1.11)
where T
∗
= (1/(1 − k)β)ln(1+(β/C)y(T
0
)
1−k
).
Lemma 1.4 [8]. Let 0 <k<p,andlety(t)
≥ 0 be a solution of the differential inequality
dy
dt
+ Cy
k
+ βy ≤ γy
p
(t ≥ 0), y(0) ≥ 0, (1.12)
where C,γ>0 and k
∈ (0,1). Then there ex ist α>β, B>0, such that
0
≤ y(t) ≤ Be
−αt
, t ≥ 0. (1.13)
4 Journal of Inequalities and Applications
2. Proofs of theorems
In this section, we will give detailed proofs for our result. Let
·
p
and ·
1,p
denote
L
p
(Ω)andW
1,p
(Ω) norms, respectively, 1 ≤ p ≤∞.
2.1. Proof of Theorem 1.1. (1) First of all, we show that
u(·,t)
∞
≤
u
0
(·)
∞
:= M. (2.1)
Multiplying (1.1)by(u − M)
+
and integrating over Ω,weobtain
1
2
d
dt
Ω
(u − M)
2
+
dx +
A
M
(t)
∇u
m
·∇udx
≤ λ
Ω
u
m
(u − M)
+
dx − β
Ω
u(u − M)
+
dx ≤ λ
A
M
(t)
u
m+1
dx,
(2.2)
where A
M
(t) ={x ∈ Ω | u(x,t) >M}.Sinceλ
1
is the first eigenvalue, then we have
Ω
∇u
m
·∇udx ≥
4m
(m +1)
2
λ
1
Ω
u
m+1
dx, (2.3)
for any u
∈ W
1,2
0
(Ω). We fur ther have
A
M
(t)
∇u
m
·∇udx ≥
4m
(m +1)
2
λ
1
A
M
(t)
u
m+1
dx. (2.4)
Therefore, we have
d
dt
Ω
(u − M)
2
+
dx ≤ 0. (2.5)
Since
Ω
(u
0
− M)
2
+
dx = 0, it follows that
Ω
(u − M)
2
+
dx ≡ 0, ∀t ≥ 0, (2.6)
which implies that
u(·,t)
∞
≤u
0
(·)
∞
.
Multiplying (1.1)byu and integrating over Ω,weconcludethat
1
2
d
dt
Ω
u
2
dx +
Ω
∇u
m
·∇udx ≤ λ
Ω
u
m+1
dx − β
Ω
u
2
dx. (2.7)
We further have
1
2
d
dt
Ω
u
2
dx +
mλ
1
− λ
Ω
u
m+1
dx + β
Ω
u
2
dx ≤ 0. (2.8)
Let v
= u/M. Then, we have
d
dt
Ω
v
2
dx +2M
m−1
mλ
1
− λ
Ω
v
m+1
dx +2βM
m−1
Ω
v
2
dx ≤ 0. (2.9)
Wenjun Liu et al. 5
Since 0 <m<1, we have
d
dt
Ω
v
2
dx +2M
m−1
mλ
1
− λ + β
Ω
v
2
dx ≤ 0, (2.10)
which implies that
Ω
v
2
dx ≤ e
−2M
m−1
mλ
1
−λ+β
t
Ω
v
2
0
dx, (2.11)
that is,
Ω
u
2
dx ≤ e
−2u
0
m−1
∞
mλ
1
−λ+β
t
Ω
u
2
0
dx. (2.12)
Therefore, we conclude that
u(·,t)
2
→ 0ast →∞.
(2) We consider first the case (N
− 2)/(N +2)≤ m<1.
Multiplying (1.1)byu
m
and integrating over Ω,wehave[9]
1
m +1
d
dt
u
m+1
m+1
+
u
m
2
1,2
= λu
2m
2m
− βu
m+1
m+1
. (2.13)
Noticing that λ
1
= inf
v∈W
1,2
0
(Ω),v=0
Ω
|∇v|
2
dx/
Ω
v
2
dx,weobtain
1
m +1
d
dt
u
m+1
m+1
+
1 −
λ
λ
1
u
m
2
1,2
+ βu
m+1
m+1
≤ 0. (2.14)
By the H
¨
older inequality, we have
u
m+1
m+1
=
Ω
1 · u
m
· udx ≤|Ω|
m/(m+1)−(N−2)/2N
u
m
2N/(N−2)
u
m+1
. (2.15)
The embedding theorem g ives that
u
m
m+1
≤|Ω|
m/(m+1)−(N−2)/2N
u
m
2N/(N−2)
≤ C
0
|Ω|
m/(m+1)−(N−2)/2N
u
m
1,2
, (2.16)
where C
0
is the embedding constant. By (2.14)–(2.16), we obtain the differential inequal-
ity
d
dt
u
m+1
+ C
1
u
m
m+1
+ βu
m+1
≤ 0, (2.17)
where
C
1
= C
−2
0
|Ω|
(N−2)/N−2m/(m+1)
1 −
λ
λ
1
. (2.18)
Setting y(t)
=u(·,t)
m+1
, y(0) =u
0
(·)
m+1
,byLemma 1.3,weobtain
u
m+1
≤
u
0
1−m
m+1
+
C
1
β
e
(m−1)βt
−
C
1
β
1/(1−m)
, t ∈
0,T
1
,
u
m+1
≡ 0, t ∈
T
1
,+∞
,
(2.19)
6 Journal of Inequalities and Applications
where
T
1
=
1
(1 − m)β
ln
1+
β
C
1
u
0
1−m
m+1
. (2.20)
We now turn to the case 0 <m<(N
− 2)/(N +2)withλ<λ
∗
= (r + m)
2
λ/4rm < λ
1
.
Multiplying (1.1)byu
r
(r = N(1 − m)/2 − 1) and integr ating over Ω,wehave
1
r +1
d
dt
u
r+1
r+1
+
4rm
(r + m)
2
−
λ
λ
1
u
(r+m)/2
2
1,2
+ βu
r+1
r+1
≤ 0. (2.21)
By the embedding theorem and the specific choice of r,weobtain
u
(r+m)/2
r+1
=
Ω
u
((r+m)/2)·(2N/(N−2))
dx
(N−2)/2N
≤ C
0
u
(r+m)/2
1,2
. (2.22)
Therefore,
d
dt
u
r+1
+ C
2
u
m
r+1
+ βu
r+1
≤ 0, (2.23)
where
C
2
= C
−2
0
4rm
(r + m)
2
−
λ
λ
1
> 0. (2.24)
Setting y(t)
=u(·,t)
r+1
, y(0) =u
0
(·)
r+1
,byLemma 1.3,weobtain
u
r+1
≤
u
0
1−m
r+1
+
C
2
β
e
(m−1)βt
−
C
2
β
1/(1−m)
, t ∈
0,T
2
,
u
r+1
≡ 0, t ∈
T
2
,+∞
,
(2.25)
where
T
2
=
1
(1 − m)β
ln
1+
β
C
2
u
0
1−m
m+1
. (2.26)
2.2. Proof of Theorem 1.2. We consider first the case p
≤ 1. When (N − 2)/(N +2)≤
m<1, multiplying (1.1)byu
m
, and by the embedding theorem and the H
¨
older inequality,
we can easily obtain
d
dt
u
m+1
+ C
−2
0
Ω|
(N−2)/N−2m/(m+1)
u
m
m+1
+ βu
m+1
≤ λ|Ω|
1−(m+p)/(m+1)
u
p
m+1
.
(2.27)
By Lemma 1.4, there exist α
1
>β, B
1
> 0, such that
0
≤u
m+1
≤ B
1
e
−α
1
t
, t ≥ 0. (2.28)
Wenjun Liu et al. 7
Furthermore, there exist T
01
,suchthat
C
−2
0
|Ω|
(N−2)/N−2m/(m+1)
− λ|Ω|
1−(m+p)/(m+1)
u
p−m
m+1
≥ C
−2
0
|Ω|
(N−2)/N−2m/(m+1)
− λ|Ω|
1−(m+p)/(m+1)
B
1
e
−α
1
T
01
p−m
:= C
3
> 0
(2.29)
holds for t
∈ [T
01
,+∞). Therefore, (2.27)turnsto
d
dt
u
m+1
+ C
3
u
m
m+1
+ βu
m+1
≤ 0. (2.30)
By Lemma 1.3, we can obtain the desire decay estimate for
T
3
=
1
(1 − m)β
ln
1+
β
C
3
u
·
,T
01
1−m
m+1
. (2.31)
For the case 0 <m<(N
− 2)/(N + 2), we multiply (1.1)byu
r
and obtain
d
dt
u
r+1
+ C
−2
0
4rm
(r + m)
2
u
m
r+1
+ βu
r+1
≤ λ|Ω|
1−(r+p)/(r+1)
u
p
r+1
. (2.32)
By Lemma 1.4, there exist α
2
>β, B
2
> 0, such that
0
≤u
r+1
≤ B
2
e
−α
2
t
, t ≥ 0. (2.33)
Furthermore, there exist T
02
,suchthat
C
−2
0
4rm
(r + m)
2
− λ|Ω|
1−(r+p)/(r+1)
u
p−m
r+1
≥ C
−2
0
4rm
(r + m)
2
− λ|Ω|
1−(r+p)/(r+1)
B
2
e
−α
2
T
02
p−m
:= C
4
> 0
(2.34)
holds for t
∈ [T
02
,+∞). Therefore, (2.32)turnsto
d
dt
u
r+1
+ C
4
u
m
r+1
+ βu
r+1
≤ 0. (2.35)
By Lemma 1.3, we can obtain the desire decay estimate for
T
4
=
1
(1 − m)β
ln
1+
β
C
4
u
·
,T
02
1−m
r+1
. (2.36)
For the case p>1, we can rewrite (2.27)and(2.32)as(e.g.,(2.27))
d
dt
u
m+1
+ C
−2
0
|Ω|
(N−2)/N−2m/(m+1)
u
m
m+1
+ βu
m+1
≤ λk
p−1
u
m+1
m+1
(2.37)
since kϕ
1/m
1
(x)isasupersolutionofproblem(1.1 )–(1.3), where ϕ
1
(x)isgiveninTheorem
1.1. The above argument can also be applied, and hence we omit it.
8 Journal of Inequalities and Applications
Acknowledgments
This work was supported by the National NSF of China (10471022), the NSF of
Jiangsu Province (BK2006088), the NSF of Jiangsu Province Education Department
(07KJD510133), and the Science Research Foundation of NUIST.
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Wenjun Liu: Department of Mathematics, Southeast University, Nanjing 210096, China;
College of Mathematics and Physics, Nanjing University of Information Science and Technology,
Nanjing 210044, China
Email address:
Mingxin Wang: Department of Mathematics, Southeast University, Nanjing 210096, China
Email address:
Bin Wu: College of Mathematics and Physics, Nanjing University of Information Science and
Technology, Nanjing 210044, China
Email address:
