Volume 2007, Article ID 87650,8pages 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 equationut=Δum+λ|u|p−1u−βu, 0< m <1, is studied. Sufficient conditions about the extinction and decay estimates of solutions are obtained by usingLp-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
ut=Δum+λ|u|p−1u−βu, x∈Ω,t >0, (1.1)
u(x,t)=0, x∈∂Ω,t >0, (1.2)
u(x, 0)=u0(x)≥0, x∈Ω, (1.3)
with 0< m <1 andp,λ,β >0, whereΩ⊂RN(N >2) is a bounded domain with smooth boundary.
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 estimacondi-tions of solucondi-tions for problem (1.1)–(1.3). For the proof of our result, we employLp-integral model estimate methods and two crucial lemmas on differential inequality.
Our main results read as follows.
Theorem1.1. Assume that0≤u0(x)∈L∞(Ω)∩W01,2(Ω),0< m=p <1, andλ1is the
first eigenvalue of
−Δψ(x)=λψ(x), ψ|∂Ω=0, (1.4)
andϕ1(x)≥0withϕ1∞=1is the eigenfunction corresponding to the eigenvalueλ1.
(1)Ifλ <4mλ1/(m+ 1)2, then the weak solution of problem (1.1)–(1.3) vanishes in the
sense of · 2ast→ ∞.
(2)If(N−2)/(N+ 2)≤m <1withλ < λ1or0< m <(N−2)/(N+ 2)withλ < λ∗, then
the weak solution of problem (1.1)–(1.3) vanishes in finite time, and
u(·,t) m+1≤
u01m−+1m+ C1
β
e(m−1)βt−C1 β
1/(1−m)
, N−2
N+ 2≤m <1,
u(·,t) r+1≤
u01r−+1m+C2 β
e(m−1)βt−C2 β
1/(1−m)
, 0< m <N−2 N+ 2,
(1.5)
fort∈[0,T∗), where
0< T∗≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
T1, (N−2)
(N+ 2)≤m <1,
T2, 0< m < (N−2)
(N+ 2),
(1.6)
r=N(1−m)
2 −1,
λ∗=(r+m)2λ 4rm < λ1,
(1.7)
Theorem1.2. Let0< m <1,m < p. Then the weak solution of problem (1.1)–(1.3) vanishes in finite time, and
u(·,t)
m+1≤B1e−α1t, t∈
0,T01
,
u(·,t) m+1≤
u(·,T01 1−m m+1+
C3 β
e(m−1)β(t−T01)−C3
β
1/(1−m)
, t∈T01,T3
,
u(·,t)
m+1≡0, t∈
T3, +∞,
(1.8)
for(N−2)/(N+ 2)≤m <1, u(·,t)
r+1≤B2e−α2t, t∈
0,T02
,
u(·,t) r+1≤
u(·,T02 1−m r+1 +
C4 β
e(m−1)β(t−T02)−C4
β
1/(1−m)
, t∈T02,T4
,
u(·,t)
r+1≡0, t∈
T4, +∞
,
(1.9)
for0< m <(N−2)/(N+ 2), whereC3,C4,T3, andT4are given by (2.29), (2.34), (2.31),
and (2.36), respectively.
To obtain the above results, we will use the following lemmas which are of crucial importance in the proofs of decay estimates.
Lemma1.3 [7]. Let y(t)≥0be a solution of the differential inequality
dy
dt +Cyk+βy≤0 (t≥0), y
T0
≥0, (1.10)
whereC >0is a constant andk∈(0, 1). Then one has the decay estimate
y(t)≤
yT0
1−k +C
β
e(k−1)β(t−T0)−C
β
1/(1−k)
, t∈T0,T∗,
y(t)≡0, t∈T∗, +∞,
(1.11)
whereT∗=(1/(1−k)β) ln(1 + (β/C)y(T0)1−k).
Lemma1.4 [8]. Let0< k < p, and lety(t)≥0be a solution of the differential inequality
dy dt +Cy
k+βy≤γyp (t≥0), y(0)≥0, (1.12)
whereC,γ >0andk∈(0, 1). Then there existα > β,B >0, such that
2. Proofs of theorems
In this section, we will give detailed proofs for our result. Let · p and · 1,pdenote
Lp(Ω) andW1,p(Ω) norms, respectively, 1≤p≤ ∞.
2.1. Proof ofTheorem 1.1. (1) First of all, we show that u(·,t)
∞≤u0(·)∞:=M. (2.1)
Multiplying (1.1) by (u−M)+and integrating overΩ, we obtain
1 2
d dt
Ω(u−M) 2 +dx+
AM(t)∇u
m· ∇udx
≤λ
Ωu
m(u−M)
+dx−β
Ωu(u−M)+dx≤λ
AM(t)u
m+1dx,
(2.2)
whereAM(t)= {x∈Ω|u(x,t)> M}. Sinceλ1is the first eigenvalue, then we have
Ω∇u
m· ∇udx≥ 4m (m+ 1)2λ1
Ωu
m+1
dx, (2.3)
for anyu∈W01,2(Ω). We further have
AM(t)∇u
m· ∇udx≥ 4m (m+ 1)2λ1
AM(t)u
m+1dx. (2.4)
Therefore, we have
d dt
Ω(u−M) 2
+dx≤0. (2.5)
SinceΩ(u0−M)2+dx=0, it follows that
Ω(u−M) 2
+dx≡0, ∀t≥0, (2.6)
which implies thatu(·,t)∞≤ u0(·)∞.
Multiplying (1.1) byuand integrating overΩ, we conclude that
1 2 d dt Ωu 2 dx+
Ω∇u
m· ∇udx≤λ
Ωu
m+1 dx−β
Ωu 2
dx. (2.7)
We further have
1 2 d dt Ωu 2
dx+mλ1−λ
Ωu
m+1 dx+β
Ωu 2
dx≤0. (2.8)
Letv=u/M. Then, we have
d dt
Ωv
2dx+ 2Mm−1mλ 1−λ
Ωv
m+1dx+ 2βMm−1
Ωv
Since 0< m <1, we have
d dt
Ωv
2dx+ 2Mm−1mλ 1−λ+β
Ωv
2dx≤0, (2.10)
which implies that
Ωv
2dx≤e−2Mm−1mλ1−λ+βt
Ωv 2
0dx, (2.11)
that is,
Ωu 2
dx≤e−2u0m−∞1
mλ1−λ+βt
Ωu 2
0dx. (2.12)
Therefore, we conclude thatu(·,t)2→0 ast→ ∞.
(2) We consider first the case (N−2)/(N+ 2)≤m <1. Multiplying (1.1) byumand integrating overΩ, we have [9]
1
m+ 1
d dtum
+1
m+1+um 2
1,2=λu22mm−βumm+1+1. (2.13)
Noticing thatλ1=infv∈W1,2 0 (Ω),v=0
Ω|∇v|2dx/Ωv2dx, we obtain
1
m+ 1
d dtum
+1
m+1+
1− λ
λ1
um2
1,2+βumm+1+1≤0. (2.14)
By the H¨older inequality, we have
um+1
m+1=
Ω1·u
m·udx≤ |Ω|m/(m+1)−(N−2)/2Num
2N/(N−2)um+1. (2.15)
The embedding theorem gives that
um
m+1≤ |Ω|m/(m+1)−(N−2)/2Num2N/(N−2)≤C0|Ω|m/(m+1)−(N−2)/2Num1,2, (2.16)
whereC0is the embedding constant. By (2.14)–(2.16), we obtain the differential
inequal-ity
d
dtum+1+C1umm+1+βum+1≤0, (2.17)
where
C1=C0−2|Ω|(N−2)/N−2m/(m+1)
1− λ
λ1
. (2.18)
Settingy(t)= u(·,t)m+1,y(0)= u0(·)m+1, byLemma 1.3, we obtain
um+1≤
u01m−+1m+C1 β
e(m−1)βt−C1 β
1/(1−m)
, t∈0,T1,
um+1≡0, t∈
T1, +∞
,
where
T1= 1
(1−m)βln
1 + β
C1
u01−m m+1
. (2.20)
We now turn to the case 0< m <(N−2)/(N+ 2) withλ < λ∗=(r+m)2λ/4rm < λ 1.
Multiplying (1.1) byur(r=N(1−m)/2−1) and integrating overΩ, we have
1
r+ 1
d dtur
+1
r+1+
4rm
(r+m)2− λ λ1
u(r+m)/22
1,2+βurr+1+1≤0. (2.21)
By the embedding theorem and the specific choice ofr, we obtain
u(r+m)/2
r+1 =
Ωu
((r+m)/2)·(2N/(N−2))dx
(N−2)/2N
≤C0u(r+m)/21,2. (2.22)
Therefore,
d
dtur+1+C2umr+1+βur+1≤0, (2.23)
where
C2=C0−2
4rm
(r+m)2− λ λ1
>0. (2.24)
Settingy(t)= u(·,t)r+1,y(0)= u0(·)r+1, byLemma 1.3, we obtain
ur+1≤
u01r−+1m+ C2
β
e(m−1)βt−C2
β
1/(1−m)
, t∈0,T2
,
ur+1≡0, t∈
T2, +∞
,
(2.25)
where
T2= 1
(1−m)βln
1 + β
C2
u01−m m+1
. (2.26)
2.2. Proof ofTheorem 1.2. We consider first the case p≤1. When (N−2)/(N+ 2)≤
m <1, multiplying (1.1) byum, and by the embedding theorem and the H¨older inequality, we can easily obtain
d
dtum+1+C
−2
0 Ω|(N−2)/N−2m/(m+1)umm+1+βum+1≤λ|Ω|1−(m+p)/(m+1)ump+1.
(2.27)
ByLemma 1.4, there existα1> β,B1>0, such that
Furthermore, there existT01, such that
C−2
0 |Ω|(N−2)/N−2m/(m+1)−λ|Ω|1−(m+p)/(m+1)ump−+1m
≥C−2
0 |Ω|(N−2)/N−2m/(m+1)−λ|Ω|1−(m+p)/(m+1)
B1e−α1T01
p−m:=
C3>0
(2.29)
holds fort∈[T01, +∞). Therefore, (2.27) turns to
d
dtum+1+C3umm+1+βum+1≤0. (2.30)
ByLemma 1.3, we can obtain the desire decay estimate for
T3= 1
(1−m)βln
1 + β
C3
u ·,T01
1−m m+1
. (2.31)
For the case 0< m <(N−2)/(N+ 2), we multiply (1.1) byurand obtain
d
dtur+1+C
−2 0
4rm
(r+m)2urm+1+βur+1≤λ|Ω|1−(r+p)/(r+1)urp+1. (2.32)
ByLemma 1.4, there existα2> β,B2>0, such that
0≤ ur+1≤B2e−α2t, t≥0. (2.33)
Furthermore, there existT02, such that
C−2 0
4rm
(r+m)2−λ|Ω|
1−(r+p)/(r+1)up−m
r+1
≥C0−2
4rm
(r+m)2−λ|Ω|
1−(r+p)/(r+1)
B2e−α2T02
p−m
:=C4>0
(2.34)
holds fort∈[T02, +∞). Therefore, (2.32) turns to
d
dtur+1+C4urm+1+βur+1≤0. (2.35)
ByLemma 1.3, we can obtain the desire decay estimate for
T4= 1
(1−m)βln
1 + β
C4
u
·,T021r+1−m
. (2.36)
For the casep >1, we can rewrite (2.27) and (2.32) as (e.g., (2.27))
d
dtum+1+C− 2
0 |Ω|(N−2)/N−2m/(m+1)umm+1+βum+1≤λkp−1umm+1+1 (2.37)
sincekϕ11/m(x) is a supersolution of problem (1.1)–(1.3), whereϕ1(x) is given inTheorem
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:[email protected]
Mingxin Wang: Department of Mathematics, Southeast University, Nanjing 210096, China Email address:[email protected]
Bin Wu: College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China
