Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2009, Article ID 280790, 17 pages
doi:10.1155/2009/280790
Research Article
Boundary Stabilization of Memory Type for the
Porous-Thermo-Elasticity System
Abdelaziz Soufyane,1 Mounir Afilal,2 and Mama Chacha1
1
College of Engineering and Applied Sciences, ALHOSN University, P.O. Box 38772,
Abu Dhabi, United Arab Emirates
2
Département Mathématiques & Informatiques, Faculté Polydisciplinaire de Safi, Université Cadi Ayyad,
Route Sidi Bouzid, BP 4162, Safi 46000, Morocco
Correspondence should be addressed to Abdelaziz Soufyane, asoufyane@hotmail.com
Received 10 January 2009; Accepted 14 March 2009
Recommended by Irena Lasiecka
We consider the one-dimensional viscoelastic Porous-Thermo-Elastic system. We establish a
general decay results. The usual exponential and polynomial decay rates are only special cases.
Copyright q 2009 Abdelaziz Soufyane 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
An increasing interest has been developed in recent years to determine the decay behavior
of the solutions of several elasticity problems. It is known that combining the elasticity
equations with thermal effects provokes stability of solutions in the one-dimensional case
1. Several results concerning the exponential or the polynomial decay of solutions for the
thermoelastic systems were obtained by 2–6.
A sample model describing the one-dimensional porous-thermo-elasticity, which was
developed in 7, 8, is given by the following system:
ρutt μuxx bvx − βθx ,
in 0, L × R ,
Jvtt αvxx − bux − ξv mθ − τvt ,
cθt κθxx − βuxt − mvt ,
in 0, L × R ,
1.1
in 0, L × R ,
where t denotes the time variable, x is the space variable, the functions uis the displacement,
v is the volume fraction of the solid elastic material, and the function θ is the temperature
difference. The coefficients ρ, μ, J, α, ξ, τ, c, and κ are positive constants. b is a constant such
that b2 < μξ.
2
Abstract and Applied Analysis
Casas and Quintanilla 7 considered the above system and used the semigroup theory
and the method developed by Liu and Zheng 4 to establish the exponential decay of the
solution under the boundary conditions of the form
ux, t vx x, t θx x, t 0,
x 0, L, t ∈ 0, ∞.
1.2
Soufyane 9 considered the following system:
utt uxx vx − θx , in 0, L × R ,
t
vtt vxx − ux − v θ − gt − svxx sds, in 0, L × R ,
0
θt θxx − uxt − vt ,
ux, 0 u x,
0
in 0, L × R ,
ut x, 0 u x,
1
vt x, 0 v1 x,
1.3
vx, 0 v x,
0
θx, 0 θ0 x,
u0, t uL, t vx 0, t vx L, t θ0, t θL, t 0,
t ≥ 0.
He proved that the solution of 1.3 decays exponentially if the function g decays exponentially, and the solutions 1.3 decay polynomially if the function g decays polynomially.
Recently Pamplona et al. 10 considered the follwing system:
ρutt μuxx bvx − βθx γuxxt ,
in 0, π × R ,
Jvtt αvxx − bux − ξv mθ,
in 0, π × R ,
cθt κθxx − βuxt − mvt ,
ux, 0 u0 x,
ut x, 0 u1 x,
vt x, 0 v1 x,
in 0, π × R ,
vx, 0 v0 x,
1.4
θx, 0 θ0 x,
u0, t uπ, t v0, t vπ, t θx 0, t θx π, t 0,
t ≥ 0.
They proved that the system is not exponential stable, and they showed that the solution
decays polynomially.
In this paper we are concerned with the following model:
ρutt μuxx bvx − βθx ,
in 0, L × R ,
Jvtt αvxx − bux − ξv mθ,
cθt κθxx − βuxt − mvt ,
in 0, L × R ,
1.5
in 0, L × R ,
with the initial conditions
ux, 0 u0 x,
ut x, 0 u1 x,
vt x, 0 v1 x,
vx, 0 v0 x,
θx, 0 θ0 x,
1.6
Abstract and Applied Analysis
3
and the boundary conditions
u0, t v0, t θ0, t θL, t 0, t ≥ 0,
t
uL, t − g1 t − sμux L, s bvL, sds,
1.7
1.8
0
vL, t −
t
1.9
g2 t − sαvx L, sds.
0
Our main interest concerns the asymptotic behavior of the solution of the system above. That
is, whether the dissipation given by the boundary memory effect is strong enough to stabilize
the whole system. And what type of rate of decay may we expect exponential decay or
polynomial decay?. We obtain an exponential decay or polynomial decay result under some
conditions on gi i 1, 2. Our proof is based on the multiplier techniques.
This work is divided into four sections. In Section 2 we introduce some notations and
some material needed for our work. In Section 3 we state and prove the exponential decay of
the solutions of our studied system. Section 4 is devoted to the polynomial decay.
2. Preliminaries
In this section we introduce some notations and we study the existence of regular and weak
solutions to system 1.5–1.9. First, we will use 1.8 and 1.9 to estimate the boundary
terms ux L, t and vx L, t.
Defining the convolution product operator by
g ∗ ϕt t
gt − sϕsds,
2.1
0
and differentiating equation 1.8 we obtain
μux L, t bvL, t 1
1
g1 ∗ uL, t −
ut L, t
g1 0
g1 0
∀t ≥ 0.
2.2
Applying Volterra’s inverse operator, we get
μux L, t bvL, t −
1
ut L, t k1 ∗ uL, t,
g1 0
∀t ≥ 0,
2.3
where the resolvent kernel k1 satisfies
k1 t 1
1
g ∗ k1 t −
g t.
g1 0 1
g1 0 1
2.4
4
Abstract and Applied Analysis
Denoting by η1 1/g1 0, we arrive at
μux L, t bvL, t −η1 ut L, t k1 0uL, t − k1 tuL, 0 k1 ∗ uL, t,
∀t ≥ 0. 2.5
A similar procedure leads to
αvx L, t −η2 vt L, t k2 0vL, t − k2 tvL, 0 k2 ∗ vL, t,
∀t ≥ 0,
2.6
where η2 1/g2 0.
Reciprocally, taking, in a natural way, the initial data u0 L v0 L 0, the identities
2.5 and 2.6 imply 1.8 and 1.9.
Since we are interested in relaxation functions of exponential or polynomial type and
identities 2.5-2.6 involve the resolvent kernels ki i 1, 2, we want to know if ki has the
same properties. The following lemma answers this question. Let h be a relaxation function
and k its resolvent kernel, that is,
kt − k ∗ ht ht.
2.7
Lemma 2.1 see 11. If h is a positive continuous function, then k is also positive and continuous.
Moreover,
1 If there exist positive constants c0 and γ with c0 < γ such that
ht ≤ c0 e−γt ,
2.8
then the function k satisfies
kt ≤
c0 γ − −t
e ,
γ − − c0 2.9
for all 0 < < γ − c0 .
2 If
t
cp : sup
t∈R
1 tp 1 t − s−p 1 s−p ds < ∞,
2.10
0
for a given p > 1 and if there exists a positive constant c0 with c0 cp < 1, for which
ht ≤ c0 1 t−p ,
2.11
then the function k satisfies
kt ≤
c0
1 t−p .
1 − c0 cp
2.12
Abstract and Applied Analysis
5
Based on Lemma 2.1, we will use 2.5-2.6 instead of 1.8-1.9. We then define
g o ϕt :
t
gt − s|ϕt − ϕs|2 ds,
0
g ϕt :
t
2.13
gt − sϕt − ϕsds.
0
By using Hölder’s inequality for 0 ≤ μ ≤ 1, we have
|g ϕt|2 ≤
t
|gs|21−μ ds |g|2μ oϕt.
2.14
0
Lemma 2.2 see 12. If g, ϕ ∈ C1 R , then
t
1
1
1 d
g oϕ −
gsds |ϕt|2 .
g ∗ ϕϕt − gt|ϕt|2 g o ϕ −
2
2
2 dt
0
2.15
3. Exponential Decay
In this section we study the asymptotic behavior of the solutions of system 1.5–1.9, when
the resolvent kernels ki i 1, 2 satisfy, for γi > 0, the following conditions:
ki 0 > 0,
ki t ≥ 0,
ki t ≤ 0,
ki t ≥ −γi ki t.
3.1
These assumptions imply that ki converges exponentially to 0, that is,
0 ≤ −ki t ≤ Ce−γi t .
3.2
We define the first-order energy of system 1.5–1.9 by
Et :
1
2
L
ρ|ut x, t|2 J|vt x, t|2 c|θx, t|2 μ|ux x, t|2 dx
0
1
2
L
α|vx x, t|2 ξ|vx, t|2 2bux x, tvx, t dx
3.3
0
η1
η2
k1 tu2 L, t − k1 ouL, t k2 tv2 L, t − k2 ovL, t.
2
2
In the sequel we define by V1 : {u ∈ H 1 0, L : u0 0}. We are now ready to state our first
result.
6
Abstract and Applied Analysis
2
Theorem 3.1. Given u0 , u1 , v0 , v1 , θ0 ∈ V1 × L2 0, L × L2 0, L, assume that 3.1 holds
with
sup ki t small enough.
3.4
t∈t0 ,∞
Assume further that b is a small number, then the energy E satisfies the following decay estimates:
Et ≤ c1 E0e−ωt ,
if u0 L v0 L 0.
3.5
Otherwise,
t
t
Et ≤ c1 E0e−ωt 1 k12 seωs ds k22 seωs ds ,
0
3.6
0
where c1 and ω are positive constants independent of the initial data.
Proof. The main idea is to construct a Lyapunov functional Lt equivalent to Et. To do this
we use the multiplier techniques. The proof of Theorem 3.1 will be achieved with the help of
a sequence of lemmas.
Lemma 3.2. Under the assumptions of Theorem 3.1, the energy of the solution of 1.5–1.9 satisfies
L
η1
η1
dE
2
≤ −κ |θx |2 dx − u2t L, t k12 t|u0 L|
dt
2
2
0
η1 η1
k1 t|uL, t|2 − k1 ouL, t
2
2
3.7
η2
η2
2
− vt2 L, t k22 t|v0 L|
2
2
η2 η2
k2 t|vL, t|2 − k2 ovL, t.
2
2
Proof. Multiplying first equation of 1.5 by ut , multiplying second equation of 1.5 by vt and
third equation of 1.5 by θ, and integrating by parts over 0, L, we obtain
d
2dt
L
2
2
ρ|ut | μ|ux | dx −b
0
d
2dt
L
L
utx vdx β
0
J|vt |2 α|vx |2 ξ|v|2 dx −b
0
d
2dt
L
0
L
L
uxt θdx μux L, t bvL, tut L, t,
0
vt ux dx m
0
L
θvt dx αvx Lvt L,
0
L
L
L
2
c|θ| dx −κ |θx | dx − m θvt dx − β θuxt dx.
2
0
0
0
3.8
Abstract and Applied Analysis
7
By a summation of these three identities, we get
d
2dt
Ω
ρ|ut |2 μ|ux |2 J|vt |2 α|vx |2 ξ|v|2 c|θ|2 2bux vdx
L
≤ −κ |θx |2 dx μux L, t bvL, tut L, t αvx L, tvt L, t,
3.9
0
using 2.5, 2.6, and Lemma 2.2, we obtain
L
η1
η1
dE
2
≤ −κ |θx |2 dx − u2t L, t k12 t|u0 L|
dt
2
2
0
η1 η1
k t|uL, t|2 − k1 ouL, t
2 1
2
η2 2
η2 2
2
− vt L, t k2 t|v0 L|
2
2
η2
η2
k2 t|vL, t|2 − k2 ovL, t,
2
2
3.10
which ends the proof of Lemma 3.2.
Lemma 3.3. Under the assumptions of Theorem 3.1, the energy of the solution of 1.5–1.9 satisfies
d
dt
L
d
2xux 1 − ε0 uρut dx dt
0
≤ −ε0
L
0
− ε0
ρu2t dx
2 − ε0 μ
−
2
L
0
L
2xvx 1 − ε0 vJvt dx
0
u2x dx
− 2b1 − ε0 L
ux vdx
0
L
2 − ε0 L 2
2
α vx dx C θx2 dx
Jvt dx −
2
0
0
0
L
3.11
1 − ε0 μuLux L ρLu2t L μLu2x L − 1 − ε0 αv0vx 0,
where ε0 is a small positive number.
Proof. We multiply first equation of 1.5 by 2xux 1 − ε0 u to obtain
d
dt
L
2xux 1 − ε0 uρut dx
0
L
0
2xutx 1 − ε0 ut ρut dx L
0
3.12
2xux 1 − ε0 uμuxx bvx − βθx dx.
8
Abstract and Applied Analysis
Integrating by parts, we get
d
dt
L
2xux 1 − ε0 uρut dx −ε0
L
0
0
ρu2t dx − 2 − ε0 μ
− b1 − ε0 L
0
L
0
u2x dx b
L
2xux vx dx
0
ux vdx 1 − ε0 μuL, tux L, t ρLu2t L, t
μLu2x L, t − β
L
2xux 1 − ε0 uθx dx.
0
3.13
Similarly, we multiply second equation of 1.5 by 2xvx 1 − ε0 v and integrate over 0, L,
using integration by parts, to arrive at
d
dt
L
2xvx 1 − ε0 vJvt dx −ε0
L
0
0
Jvt2 dx
L
L
2
− 2 − ε0 α vx dx ε0 ξv2 dx αLvt2 L, t
0
0
− ξLv L, t − 1 − ε0 αvL, tvx L, t
L
L
− b2xvx 1 − ε0 vux dx m 2xvx 1 − ε0 vθdx.
μLvx2 L, t
2
0
0
3.14
Summing the above two identities and using Poincare’s and Young’s inequalities and taking
ε0 small, we deduce that
d
dt
L
2xux 1 − ε0 uρut dx 0
≤ −ε0
L
0
− ε0
ρu2t dx
L
0
2 − ε0 μ
−
2
Jvt2 dx
d
dt
L
0
L
2xvx 1 − ε0 vJvt dx
0
u2x dx
− 2b1 − ε0 L
ux vdx
0
L
2 − ε0 L 2
α vx dx C θx2 dx
−
2
0
0
3.15
1 − ε0 μuL, tux L, t ρLu2t L, t μLu2x L, t
− αLvt2 L, t μLvx2 L, t − ξLv2 L, t 1 − ε0 αvL, tvx L, t.
The proof of Lemma 3.3 is completed.
Now, we introduce the Lyapunov functional. So, for N > 0 large enough, let
Lt NEt L
0
2xux 1 − ε0 uρut dx L
0
2xvx 1 − ε0 vJvt dx.
3.16
Abstract and Applied Analysis
9
Applying Young’s inequality and Poincaré’s inequality to the boundary terms, we have, for
ε > 0,
1 − ε0 μuL, tux L, t 1 − ε0 αvL, tvx L, t
≤ εμu2 L, t αv2 L, t Cε μu2x L, t αvx2 L, t
L
L
|ux |2 dx |vx |2 dx Cε μu2x L, t αvx2 L, t.
≤ εC
0
3.17
0
By rewriting the boundary conditions 2.5-2.6 as
μux L, t −bvL, t − η1 ut k1 tu − k1 tu0 − k1 uL, t,
αvx L, t −η2 vt k2 tv − k2 tv0 − k2 vL, t,
3.18
and combining all above relations, using the fact that b is a small number, the condition 17,
taking N large enough, ε0 very small, and ε ε0 , we obtain
dL
κN
t ≤ −
dt
2
L
0
|θx |2 dx − N
η1 2
2
u L, t Nη1 k12 t|u0 L|
8 t
Nη1 Nη2 2
2
k1 ouL, t −
vt L, t Nη2 k22 t|v0 L|
2
8
L
Nη2 1 − ε0 L 2
2
k ovL, t − ε0 ρut dx −
μ ux dx
−
2 2
8
0
0
L
L
1 − ε0 α vx2 dx
− ε0 Jvt2 dx −
8
0
0
−
3.19
Cε −k1 u2 −k2 v L, t.
2
Applying inequality 2.14 with μ 1/2, using the trace formula we have, for some positive
constant c0 , the following estimate:
d
2
2
Lt ≤ −c0 Et C k12 t|u0 L| k22 t|v0 L| .
dt
3.20
Also, by direct computations, it is easy to check that, for N large, we have
N
Et ≤ Lt ≤ 2NEt.
2
3.21
10
Abstract and Applied Analysis
Therefore 3.20 becomes
d
2
2
Lt ≤ −ωLt C k12 t|u0 L| k22 t|v0 L| ,
dt
3.22
for some positive constants ω. At this point we distinguish two cases.
Case 1. If u0 L v0 L 0, then 3.22 reduces
d
Lt ≤ −ωLt,
dt
∀t ≥ 0.
3.23
A simple integration over 0, t yields
Lt ≤ L0e−ωt ,
∀t ≥ 0.
3.24
By using 3.21, estimate 3.5 is proved.
Case 2. If u0 L / 0 or v0 L / 0, then 3.22 gives
d
Lt ≤ −ωLt C1 k12 t C2 k22 t,
dt
3.25
where
2
C1 C|u0 L| ,
2
C2 C|v0 L| .
3.26
In this case we introduce the following functional:
Ft : Lt − C1 e
−ωt
t
0
k12 teωs ds
− C2 e
−ωt
t
0
k22 teωs ds.
3.27
A simple differentiation of F, using 3.25, leads to
dF
t ≤ −ωFt.
dt
3.28
Again a simple integration over 0, t yields
Ft ≤ F0e−ωt ,
∀t ≥ 0.
3.29
A combination of 3.21, 3.27, and 3.29 then yields the estimate 3.6. This completes the
proof of Theorem 3.1.
Abstract and Applied Analysis
11
4. Polynomial Decay
In this section we study the asymptotic behavior of the solutions of system 1.5–1.9 when
the resolvent kernels ki i 1, 2 satisfy
ki 0 > 0,
ki t ≤ 0,
ki t ≥ 0,
ki t ≥ γi −ki t
1qi
,
4.1
for q1 , q2 /
0, 0, 0 ≤ qi < 1/2, and some positive constants γi . These assumptions imply that
ki decays polynomially to 0 if qi > 0. That is,
0 ≤ −ki t ≤ C1 t−1/qi .
4.2
The following lemmas will play an important role in the sequel.
Lemma 4.1. (see [13]) Let p > 1, 0 ≤ r < 1, and t ≥ 0. Then for 0 < r,
|k |oφL, t
11/1−r1p
t
1/1−r1p
11/1p
r
2
≤ 2 φL∞ 0,T,H 1 0,L |k s| ds
oφL, t ,
|k |
4.3
0
and for r 0,
|k |oφL, t
11/1p
t
1/1p
11/1p
≤2
φ·, s2H 1 0,L ds tφ·, t2H 1 0,L
oφL, t ,
|k |
4.4
1
− 1.
q
4.5
0
where
p
3
Theorem 4.2. Given that u0 , v0 , u1 , v1 , θ0 ∈ V 2 × L2 0, L , assume that b small number
and 4.1 hold. Then there exists a positive constant λ > 0, for which the energy E satisfies, for all
t ≥ 0, the following decay estimates:
Et ≤
λ
1/q
1 t
,
if u0 L v0 L 0 on.
4.6
Otherwise,
Et ≤
t
1−rp1
2
2
k
s
k
s1
s
ds
,
1
2
1
1 t1−r/q
0
λ
4.7
12
Abstract and Applied Analysis
where
p
1
<r<
,
p1
p1
q max{q1 , q2 },
p min{p1 , p2 }.
4.8
Moreover, if
∞
0
k12 s k22 s1 s1−rp1 ds < ∞
4.9
for some r, satisfying 4.8, then 4.7 reduces to 4.6.
Proof. By using 4.1 in 3.7, we easily see that
L
η1
η1
dE
2
≤ − κ |θx |2 dx − u2t L, t k12 t|u0 L|
dt
2
2
0
η1 η1 γ1
1q
k1 t|uL, t|2 −
−k1 t 1 ouL, t
2
2
η2
η2
2
− vt2 L k22 t|v0 L|
2
2
η2 η2 γ2
1q
−k2 t 2 ovL, t.
k2 t|vL|2 −
2
2
4.10
By defining the functional Lt as in 3.16, we get
dL
κN
t ≤ −
dt
2
L
|θx |2 dx −
0
Nη1 2
2
ut L, t Nη1 k12 t|u0 L|
8
Nη1 γ1
Nη2 2
2
1q
−k1 t 1 ouL, t −
v L, t Nη2 k22 t|v0 L|
2
8 t
L
Nη2 γ2
1 − ε0 L 2
1q
−k2 t 2 ovL, t − ε0 ρu2t dx −
μ ux dx
−
2
8
0
0
L
L
1 − ε0 α vx2 dx
− ε0 Jvt2 dx −
8
0
0
−
4.11
Cε −k1 u2 −k2 v L, t.
2
Applying inequality 2.14 for ki with μ p1 2/2p1 1 if i 1 and μ p2 2/2p2 1
if i 2, we get
|k1 u| ≤ C−k1 2
1q1
ou,
|k2 v| ≤ C−k2 2
1q2
ov .
4.12
Abstract and Applied Analysis
13
Using the above inequalities and taking N large, then for some positive constant c1 , we
obtain
d
Lt ≤ − c1
dt
L
L
2
|θ| dx 0
0
ρu2t dx
1q
−k1 t 1 ouL, t
L
0
μu2x dx
L
0
Jvt2 dx
1q
−k2 t 2 ovL, t
2
2
Cε k12 t|u0 L| k22 t|v0 L| −
L
0
αvx2 dx
4.13
Nη1 2
Nη2 2
u L, t −
v L, t.
8 t
8 t
Now, we fix 0 < r < 1 such that 1/p 1 < r < p/p 1 and p min{p1 , p2 }. From 4.1 we
get
t
0
|ki s|r ds ≤ C
∞
0
1 t−r/qi ds < ∞,
i 1, 2.
−k1 touL11/1−rp1 ≤ C−k1 t11/p1 1 ouL
4.14
−k2 tovL11/1−rp1 ≤ C−k2 t11/p2 1 ovL.
Consequently we have
CEt11/1−rp1 ≤
L
ρ|ut |2 J|vt |2 c|θ|2 μ|ux |2 α|vx |2 dx
0
−k1 t1q1 ouL
4.15
−k2 t1q2 ovL.
Inserting 4.15 into 4.13 and using 3.21, we deduce that
d
2
2
Lt ≤ −CLt11/1−rp1 Cε k12 t|u0 L| k22 t|v0 L| .
dt
4.16
Here, we distinguish two cases.
Case 1 u0 L v0 L 0. In this case 4.16 reduces to
d
Lt ≤ −CLt11/1−rp1 .
dt
4.17
A simple integration over 0, t gives
Lt ≤
C
1−rp1
1 t
,
∀t ≥ 0.
4.18
14
Abstract and Applied Analysis
As a consequence of 4.18 and the fact that 1 − rp 1 > 1, we easily verify that
∞
Ltdt sup tLt < ∞.
4.19
t≥0
0
Therefore, by using Lemma 4.1, 3.3, and 3.21, we have
|k1 |ouL11/1p ≤ C |k1 |11/1p1 ouL
|k2 |ovL11/1p ≤ C |k2 |11/1p2 ovL ,
4.20
11/1p1 p1/p2
|k1 |
ouL
≥ C|k1 |ouL
11/1p2 p1/p2
|k2 |
ovL
≥ C|k2 |ovL.
4.21
which implies that
Consequently we have
11/p1
CEt
≤
L
ρ|ut |2 J|vt |2 c|θ|2 μ|ux |2 α|vx |2 dx
0
−k1 t1q1 ouL
4.22
−k2 t1q2 ovL.
Inserting 4.22 into 4.13, with u0 L v0 L 0, we deduce that
d
Lt ≤ −CLt11/p1 .
dt
4.23
A simple integration then leads to
Lt ≤
C
1 t
p1
C
1 t1/q
,
∀t ≥ 0.
4.24
where q max{q1 q2 }. By using 3.21, estimate 4.6 is established.
Case 2 u0 L /
0 or v0 L /
0. In this case 4.16 gives that
d
Lt ≤ −CLt11/1−rp1 C1 k12 t C2 k22 t,
dt
4.25
where
2
C1 Cε |u0 L| ,
2
C2 Cε |v0 L| .
4.26
Abstract and Applied Analysis
15
We then introduce the following functional Ht : Lt − gt, where
gt : C1 1 t−1−rp1
t
0
C2 1 t−1−rp1
k12 s1 s1−rp1 ds
t
0
4.27
k22 s1 s1−rp1 ds.
By using 4.1, it is easy to show that, for some t0 > 0, we have
gt11/1−rp1 1 t−1−1−rp1
×
t
0
11/1−rp1
C1 k12 s C2 k22 s1 s1−rp1 ds
≥ γ0 1 t−1−1−rp1
t
×
C1 k12 s C2 k22 s1 s1−rp1 ds ,
4.28
∀t ≥ t0 ,
0
where
γ0 t0
0
1/1−rp1
C1 k12 s C2 k22 s1 s1−rp1 ds
.
4.29
Hence, simple calculations give
g t Cgt11/1−rp1 ≥ C1 k12 t C2 k22 t,
∀t ≥ t0 .
4.30
Thanks to 4.25–4.30, we have
H t ≤ −CLt11/1−rp1 − gt11/1−rp1 .
4.31
By using the fact that
Lt11/1−rp1 Ht gt11/1−rp1 ≥ Ht11/1−rp1 gt11/1−rp1 4.32
estimate 4.31 gives
H t ≤ −CHt11/1−rp1 ,
∀t ≥ t0 .
4.33
A simple integration of 4.33 over t0 , t then leads to
Ht ≤
C
1 t
1−rp1
,
∀t ≥ t0 .
4.34
16
Abstract and Applied Analysis
Therefore, using 4.27 we get, for all, t ≥ t0 ,
Lt ≤
t
1−rp1
2
2
k
s
k
s1
s
ds
.
1
2
1
1−rp1
C
1 t
4.35
0
Again, recalling 3.21 and using the continuity and the boundedness of E, estimate 4.7 is
established.
If, in addition,
∞
0
k12 s k22 s1 s1−rp1 ds < ∞
4.36
for some r, satisfying 4.8 then 4.35 takes the form
Lt ≤
C
1−rp1
1 t
.
4.37
By repeating 4.18–4.24, the desired estimate is established.
Acknowledgment
The authors are deeply grateful to the referees for their valuable comments.
References
1 M. Slemrod, “Global existence, uniqueness, and asymptotic stability of classical smooth solutions in
one-dimensional nonlinear thermoelasticity,” Archive for Rational Mechanics and Analysis, vol. 76, no.
2, pp. 97–133, 1981.
2 S. Jiang, “Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 19, no. 2, pp. 107–121, 1992.
3 G. Lebeau and E. Zuazua, “Decay rates for the three-dimensional linear system of thermoelasticity,”
Archive for Rational Mechanics and Analysis, vol. 148, no. 3, pp. 179–231, 1999.
4 Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, vol. 398 of Chapman &Hall/CRC
Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 1999.
5 R. Racke, “Thermoelasticity with second sound—exponential stability in linear and non-linear 1-d,”
Mathematical Methods in the Applied Sciences, vol. 25, no. 5, pp. 409–441, 2002.
6 X. Zhang and E. Zuazua, “Decay of solutions of the system of thermoelasticity of type III,”
Communications in Contemporary Mathematics, vol. 5, no. 1, pp. 25–83, 2003.
7 P. S. Casas and R. Quintanilla, “Exponential decay in one-dimensional porous-thermo-elasticity,”
Mechanics Research Communications, vol. 32, no. 6, pp. 652–658, 2005.
8 R. Quintanilla, “Slow decay for one-dimensional porous dissipation elasticity,” Applied Mathematics
Letters, vol. 16, no. 4, pp. 487–491, 2003.
9 A. Soufyane, “Energy decay for porous-thermo-elasticity systems of memory type,” Applicable
Analysis, vol. 87, no. 4, pp. 451–464, 2008.
10 P. X. Pamplona, J. E. Muñoz Rivera, and R. Quintanilla, “Stabilization in elastic solids with voids,”
Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 37–49, 2009.
11 M. M. Cavalcanti, V. N. Domingos Cavalcanti, and M. L. Santos, “Existence and uniform decay rates
of solutions to a degenerate system with memory conditions at the boundary,” Applied Mathematics
and Computation, vol. 150, no. 2, pp. 439–465, 2004.
Abstract and Applied Analysis
17
12 M. M. Cavalcanti and H. P. Oquendo, “Frictional versus viscoelastic damping in a semilinear wave
equation,” SIAM Journal on Control and Optimization, vol. 42, no. 4, pp. 1310–1324, 2003.
13 M. de Lima Santos, “Asymptotic behavior of solutions to wave equations with a memory condition
at the boundary,” Electronic Journal of Differential Equations, vol. 2001, no. 73, pp. 1–11, 2001.