Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 173, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
COMPACTNESS OF THE DIFFERENCE BETWEEN THE
POROUS THERMOELASTIC SEMIGROUP AND ITS
DECOUPLED SEMIGROUP
EL MUSTAPHA AIT BENHASSI, JAMAL EDDINE BENYAICH,
HAMMADI BOUSLOUS, LAHCEN MANIAR
Abstract. Under suitable assumptions, we prove the compactness of the difference between the porous thermoelastic semigroup and its decoupled one.
This will be achieved by proving the norm continuity of this difference and
the compactness of the difference between the resolvents of their generators.
Applications to porous thermoelastic systems are given.
1. Introduction
An increasing interest to determine the decay behavior of solutions of several
porous elastic and thermoelastic problems has been discovered recently. The theory
of porous elastic material was established first by Cowin and Nunziato [5, 6, 7].
In a recent paper the authors of [25] proved a slow decay of solution of porous
elastic system with boundary Dirichlet conditions in one dimensional case. After,
Casas and Quintanilla [8], proved the exponential decay of a porous thermoelastic
system. This problem has recently been the focus of interest of Glowinsky and Lada
[13, 14, 15]. In this work, we consider the abstract porous thermoelastic model
ẅ1 (t) + A1 w1 (t) + C1 w2 (t) + C2 θ(t) = 0,
ẅ2 (t) + A2 w2 (t) −
C1∗ w1 (t)
θ̇(t) + A3 θ(t) −
w1 (0) =
w10 ,
ẇ1 (0) =
w11 ,
t ≥ 0,
∗
− C3 θ(t) + DD ẇ2 (t) = 0,
(1.1)
t ≥ 0,
C2∗ ẇ1 (t)
w2 (0)
+ C3∗ ẇ2 (t) = 0, t ≥ 0,
= w20 , ẇ2 (0) = w21 , θ(0)
(1.2)
(1.3)
0
=θ ,
(1.4)
with its decoupled system
−1 ∗
∗
ẅ1 (t) + A1 w1 (t) + C1 w2 (t) + C2 A−1
3 C2 ẇ1 (t) − C2 A3 C3 ẇ2 (t)
= 0,
t ≥ 0,
−1 ∗
∗
∗
ẅ2 (t) + A2 w2 (t) − C1∗ w1 (t) − C3 A−1
3 C2 ẇ1 (t) + (C3 A3 C3 + DD )ẇ2 (t)
= 0,
t ≥ 0,
θ̇(t) = −A3 θ(t) + C2∗ ẇ1 (t) − C3∗ ẇ2 (t),
t ≥ 0,
2010 Mathematics Subject Classification. 34G10, 47D06.
Key words and phrases. Porous thermoelastic system; semigroup; compactness;
norm continuity; fractional powers; essential spectrum.
c
2015
Texas State University - San Marcos.
Submitted October 14, 2014. Published June 22, 2015.
1
(1.5)
(1.6)
(1.7)
2
E. M. AIT BENHASSI, J. E. BENYAICH, H. BOUSLOUS, L. MANIAR
w1 (0) = w10 ,
ẇ1 (0) = w11 ,
w2 (0) = w20 ,
ẇ2 (0) = w21 ,
EJDE-2015/173
θ(0) = θ0 .
(1.8)
The corresponding porous elastic system is given by the first and second equations
in the decoupled system (1.5)-(1.8),
−1 ∗
∗
ẅ1 (t) + A1 w1 (t) + C1 w2 (t) + C2 A−1
3 C2 ẇ1 (t) − C2 A3 C3 ẇ2 (t)
= 0,
t ≥ 0,
−1 ∗
∗
∗
ẅ2 (t) + A2 w2 (t) − C1∗ w1 (t) − C3 A−1
3 C2 ẇ1 (t) + (C3 A3 C3 + DD )ẇ2 (t)
= 0,
t ≥ 0,
w1 (0) = w10 ,
ẇ1 (0) = w11 ,
w2 (0) = w20 ,
ẇ2 (0) = w21 .
(1.9)
(1.10)
(1.11)
In this article, we first show the existence of solution of problems determined by
systems (1.1)-(1.4), (1.5)-(1.8) and (1.9)-(1.11) using the Lumer-Phillips theorem
from the theory of semigroups [9, Corollary 3.20]]. Second we address the problem of compactness of difference between the porous-thermoelasticity C0 -semigroup
(T (t))t≥0 generated by the system (1.1)-(1.4) and the C0 -semigroup (Td (t))t≥0 generated by its decoupled system (1.5)-(1.8). As in [1], we prove the norm continuity
of t 7−→ T (t) − Td (t) for t > 0, and we show the compactness of the difference
R(λ, A) − R(λ, Ad ) for every λ in ρ(A) ∩ ρ(Ad ), where A and Ad are the generators of (T (t))t≥0 and (Td (t))t≥0 , respectively. These two results together with [20,
Theorem 2.3] lead to the compactness of the difference T (t) − Td (t). This yields
that the essential spectrums σe (T (t)), and σe (Td (t)) coincide. In the case where
−1/2
the operators A−1
C1 A−1
3 and A1
2 are compact, following a similar argument as
in [11], we prove that σe (S(t)) = σe (Td (t)), where (S(t))t≥0 is the C0 -semigroup
generated by the system (1.9)-(1.11).
Consequently one can derive stability results on the first semigroup from the
ones of the third semigroup. Finally two applications to a porous thermoelastic
system are given. In the first application where A−1
i , i = 1, 2 are compact but
A−1
is
not
compact,
we
show
that
only
the
two
essential
spectrums σe (T (t)), and
3
σe (Td (t)) coincide. The second application is similar to the one given by Glowinsky
and Lada in [15], where the exponential stability of porous thermoelastic system
is derived from the corresponding decoupled system. In this application, following
a different approach and using the compactness of A−1
i , i = 1, 2, 3, we obtain the
same stability result first for the simpler porous elastic system, then the property
is derived for the original porous thermoelastic system.
2. Main results
In what follows, Ai : D(Ai ) ⊂ Hi → Hi , i = 1, 2, 3, be self-adjoint positive
operators with bounded inverses, and Hi be Hilbert spaces equipped with the norm
k · kHi , i = 1, 2, 3. The operator Ai can be extended (or restricted) to each Hi,α ,
such that it becomes a bounded operator
Ai : Hi,α → Hi,α−1 ,
D(Aα
i ),
∀α ∈ R,
(2.1)
kAα
i zkHi
where for α ≥ 0, Hi,α =
with the norm kzki,α =
and for α ≤ 0,
∗
Hi,α = Hi,−α
, the dual of Hi,−α with respect to the pivot space Hi . The operator
D ∈ L(H2 ) and D∗ its adjoint. The coupled operators Ci , i=1,2,3, satisfy
1/2
(C1) D(C1 ) ⊂ H2 → H1 , with adjoint C1∗ such that D(A2 ) ,→ D(C1 ) and
1/2
D(A1 ) ,→ D(C1∗ ).
EJDE-2015/173
COMPACTNESS OF THE DIFFERENCE . . .
3
1/2
(C2) D(C2 ) ⊂ H3 → H1 with adjoint C2∗ such that D(A3 ) ,→ D(C2 ) and
1/2
D(A1 ) ,→ D(C2∗ ).
(C3) D(C3 ) ⊂ H3 → H2 with adjoint C3∗ such that
1/2
D(A3 ) ,→ D(C3 )
1/2
and D(A2 ) ,→ D(C3∗ ).
(2.2)
Set
H := H1,1/2 × H2,1/2 × H1 × H2 × H3 ,
in this Hilbert space we introduce the new inner product
!  we1 
1
E
D w
w
f2
w2
v1
e1 iH1,1/2 + hw2 , w
f2 iH2,1/2 + hv1 , ve1 iH1 + hv2 , ve2 iH2
,  ve1  = hw1 , w
v
2
θ
v
e2
θe
e H + <(hC ∗ w1 , w
e2 iH2 − hw2 , C1∗ w
e1 iH2 ).
+ hθ, θi
1
3
The associated norm of this inner product coincides with the canonical norm of H.
We can rewrite (1.1)-(1.4) and (1.5)-(1.8) as the first order evolution equations
in H,
dη
= Aη, η ∈ H,
dt
η(0) = (w10 , w20 , w11 , w21 , θ0 ),
and
dη
= Ad η, η ∈ H,
dt
η(0) = (w10 , w20 , w11 , w21 , θ0 ),
respectively, where A is the unbounded linear operator defined by

 0 0 I
0
0
0
A : D(A) ⊂ H → H,
1
A =  −A
C∗
0
−C1
−A2
0
0
I
0
0
0
−C2
∗
0 −DD C3
C2∗ −C3∗ −A3
,
(2.3)
D(A) = D(A1 ) × D(A2 ) × D(A1 ) × D(A2 ) × D(A3 ),
(2.4)
1
0
with
1/2
1/2
and the operator Ad associated to the decoupled system
Ad : D(Ad ) = D(A) ⊂ H → H, Ad

=
0
0
I
0
0
0
0
0
I
0
−1 ∗
∗
 −A1 −C1 −C2 A−1
0
C2 A3 C3
3 C2
 ∗
−1 ∗
∗
∗
C1 −A2 C3 A−1
0
3 C2 −C3 A3 C3 −DD
0
0
C2∗
−C3∗
−A3


.
We rewrite the coupled second order system (1.9)-(1.11) on the Hilbert space
Hc := H1,1/2 × H2,1/2 × H1 × H2 ,
as the first order evolution equation
de
η
= Me
η , ηe ∈ Hc ,
dt
ηe0 = (w10 , w20 , w11 , w21 ),
(2.5)
4
E. M. AIT BENHASSI, J. E. BENYAICH, H. BOUSLOUS, L. MANIAR
EJDE-2015/173
and M : D(M) ⊂ Hc → Hc , is the unbounded linear operator defined by
!
0
0
I
0
0
0
0
I
∗
∗
−A1 −C1 −C2 A−1
C2 A−1
3 C2
3 C3
−1 ∗
−1 ∗
∗
C1 −A2 C3 A3 C2 −C3 A3 C3 −DD ∗
M=
,
(2.6)
1/2
(2.7)
with
1/2
D(M) = D(A1 ) × D(A2 ) × D(A1 ) × D(A2 ).
Now we formulate the main results of this paper.
Theorem 2.1. The operators A, Ad and M generate strongly continuous contraction semigroups (T (t))t≥0 , (Td (t))t≥0 on H and (S(t))t≥0 on Hc .
Theorem 2.2. Assume that
−1/2
A1
−1/2
C2 A−1
3 ,
A1
C1 A−1
2 ,
−1/2
A2
C3 A−1
3 ,
(2.8)
are compact operators from H3 to H1 , from H2 to H1 and from H3 to H2 respectively. Then T (t) − Td (t) is compact for every t ≥ 0.
As a consequence of Theorem 2.2, we have the following particular results.
Corollary 2.3. Assume that the operators A−1
i , i = 1, 2, are compact. Then
T (t) − Td (t) is compact for every t ≥ 0.
−1/2
Corollary 2.4. Assume that the operators A−1
and A1
3
Then σe (T (t)) = σe (S(t)) for t ≥ 0.
C1 A−1
are compact.
2
3. Well-posedness results
In this section we use Lumer-Phillips theorem (see [9, Corollary 3.20]) for the
proof of Theorem 2.1.
3.1. Porous thermoelastic system. To show that the operator (A, D(A)) defined by (2.3)-(2.4) generates a contraction semigroup on the Hilbert H, we need
the following technical lemma.
Lemma 3.1. The operator A is invertible in H and A−1 is bounded on H.
 f1 
!
w1
f2
Proof. Given a vector  f3  ∈ H, we need
f4
f5
A
w1
w2
v1
v2
w3
!
w2
v1
v2
w3
∈ D(A), such that
 f1 
f2
=  f3  .
f4
f5
We have
A


v1 = f1 ,



 f1 

!

w1
v2 = f2 ,
f2
w2


v1
f
=
⇔ A1 w1 + C1 w2 + C2 w3 = −f3 ,
3
v2

f4

w3
−C1∗ w1 + A2 w2 + DD∗ v2 − C3 w3 = −f4 ,
f5



−C ∗ v + C ∗ v + A w = −f .
3 3
5
2 1
3 2
EJDE-2015/173
COMPACTNESS OF THE DIFFERENCE . . .
5
Hence


v1 = f1 ,



 f1 

!

w1
v2 = f2 ,
f2
w2
A vv1 =  f3  ⇔ A1 w1 + C1 w2 + C2 w3 = −f3 ,
2

f4

w3

f5
−C1∗ w1 + A2 w2 − C3 w3 = −f4 − DD∗ f2 ,



w = −A−1 (f − C ∗ f + C ∗ f ),
3
5
2 1
3 2
3


v1 = f 1 ,





v2 = f2 ,
∗
∗
⇔ A1 w1 + C1 w2 = C2 A−1
3 (f5 − C2 f1 + C3 f2 ) − f3 = K1 ,


∗
∗
∗

−C1∗ w1 + A2 w2 = −C3 A−1

3 (f5 − C2 f1 + C3 f2 ) − f4 − DD f2 = K2 ,


w = −A−1 (f − C ∗ f + C ∗ f ),
3
5
2 1
3 2
3


v1 = f 1 ,





v2 = f2 ,
−1
⇔ w1 = −A−1
1 C1 w2 + A1 K1 ,


∗ −1

(C1∗ A−1

1 C1 + A2 )w2 = K2 + C1 A1 K1 ,


w = −A−1 (f − C ∗ f + C ∗ f ).
3
5
2 1
3 2
3
We have
v1 = f1 ∈ H1,1/2 ,
v2 = f2 ∈ H2,1/2 ,
∗
∗
w3 = −A−1
3 (f5 − C2 f1 + C3 f2 ) ∈ D(A3 ).
Suppose that we have found w2 with the appropriate regularity. Then,
−1
w1 = −A−1
1 C1 w2 + A1 K1 ∈ D(A1 ).
We now solve the equation
∗ −1
(C1∗ A−1
1 C1 + A2 )w2 = K2 + C1 A1 K1 .
(3.1)
1/2
To find w2 we introduce a bilinear form Λ on D(A2 ), defined by
−1/2
Λ(η, ζ) = hA1
−1/2
C1 η, A1
1
1
C1 ζi + hA22 η, A22 ζi.
1/2
Since Λ is a bilinear continuous and coercive form on D(A2 ), the Lax-Milgram
1/2
Lemma leads to the existence and uniqueness of w2 ∈ D(A2 ) solution to the
equation (3.1).
∗ −1
−1
Moreover K2 + C1∗ A−1
H2 = D(A2 ),
1 K1 − C1 A1 C1 w2 ∈ H2 and [(A2 )−1 ]
(where (A2 )−1 is an extension of A2 ), then w2 ∈ D(A2 ), (see [3, Proposision 5]).
−1
Set B1 = (C1∗ A−1
, then we have
1 C 1 + A2 )


v1 = f1 ,



 f1 

!

w1
v2 = f2 ,
f2
w2
−1


v
f
A v1 =
⇔ w1 = −A−1
3
1 C1 w2 + A1 K1 ,
2

f4

w3
∗
w2 = B1 K2 + B1 C1 A−1
f5

1 K1 ,


w = −A−1 (f − C ∗ f + C ∗ f ),
3
5
2 1
3 2
3
6
E. M. AIT BENHASSI, J. E. BENYAICH, H. BOUSLOUS, L. MANIAR
EJDE-2015/173

v1 = f1 ,





v2 = f2 ,




−1 ∗
−1
−1 ∗
∗ −1

w1 = (−A−1

1 C1 B1 C3 A3 C2 + A1 C1 B1 C1 A1 C2 A3 C2


−1 ∗
−1
−1 ∗
−1

∗

−A−1

1 C2 A3 C2 )f1 + (A1 C1 B1 C3 A3 C3 + A1 C1 B1 DD


−1
−1
−1
−1
−1
∗
∗
∗


−A1 C1 B1 C1 A1 C2 A3 C3 + A1 C2 A3 C3 )f2

−1
−1
∗ −1
⇔
+(A−1
1 C1 B1 C1 A1 − A1 )f3 + A1 C1 B1 f4


−1
−1
−1
−1
−1
−1

+(A1 C1 B1 C3 A3 − A1 C1 B1 C1∗ A−1

1 C2 A3 + A1 C2 A3 )f5 ,



−1 ∗
∗
∗ −1
w2 = (B1 C3 A−1

3 C2 − B1 C1 A1 C2 A3 C2 )f1


−1 ∗

∗
∗
∗ −1

+(−B1 C3 A−1

3 C3 − B1 DD + B1 C1 A1 C2 A3 C3 )f2


−1
−1
−1
∗
∗


−B1 C1 A1 f3 − B1 f4 + (−B1 C3 A3 + B1 C1 A−1

1 C2 A3 )f5 ,


∗
∗
w3 = −A−1
3 (f5 − C2 f1 + C3 f2 ).
Thus,
a15
a11
a12
a13
A−1
1 C1 B1
a22
−B1 C1∗ A−1
−B
a
1
25
 a21
1
 I
0
0
0
0
0
I
0
0
0
−1 ∗
∗
A−1
0
0
−A−1
3 C2 −A3 C3
3

A−1 =


,
(3.2)
where
−1 ∗
−1
−1 ∗
−1
−1 ∗
∗ −1
a11 = −A−1
1 C1 B1 C3 A3 C2 + A1 C1 B1 C1 A1 C2 A3 C2 − A1 C2 A3 C2 ,
−1 ∗
−1
−1
−1 ∗
∗
∗ −1
a12 = A−1
1 C1 B1 C3 A3 C3 + A1 C1 B1 DD − A1 C1 B1 C1 A1 C2 A3 C3
−1 ∗
+ A−1
1 C 2 A3 C 3 ,
−1
∗ −1
a13 = A−1
1 C 1 B 1 C 1 A1 − A1 ,
−1
−1
−1
−1
−1
∗ −1
a15 = A−1
1 C1 B1 C3 A3 − A1 C1 B1 C1 A1 C2 A3 + A1 C2 A3 ,
−1 ∗
∗
∗ −1
a21 = B1 C3 A−1
3 C2 − B1 C1 A1 C2 A3 C2 ,
−1 ∗
∗
∗
∗ −1
a22 = −B1 C3 A−1
3 C3 − B1 DD + B1 C1 A1 C2 A3 C3 ,
−1
∗ −1
a25 = −B1 C3 A−1
3 + B1 C1 A1 C2 A3 .
The boundedness of the operator A−1 follows by the assumptions (2.2).
Now, to prove that the operator A generates a strongly continuous contraction
semigroup (T (t))t≥0 on H, we have only to show that (A, D(A)) is a dissipative
operator on H and! λI − A is surjective for some λ > 0.
For every
<
A
w1
w2
v1
v2
w3
w1
w2
v1
v2
w3
∈ D(A), by the Cauchy-Schwartz inequality, we have
!
,
w1
w2
v1
v2
w3
! =<
v1
v2
2 w3
 ∗ −A1 w1 −C1 w2 −C
C1 w1 −A2 w2 −DD ∗ v2 +C3 w3
∗
∗
C2 v1 −C3 v2 −A3 w3


,
w1
w2
v1
v2
w3
= < hv1 , w1 iH1,1/2 + hv2 , w2 iH2,1/2 − hA1 w1 , v1 iH1 − hC1 w2 , v1 iH1
− hC2 w3 , v1 iH1 + hC1∗ w1 , v2 iH2 − hA2 w2 , v2 iH2 − hDD∗ v2 , v2 iH2
+ hC3 w3 , v2 iH2 + hC2∗ v1 , w3 iH3 − hC3∗ v2 , w3 iH3 − hA3 w3 , w3 iH3
+ hC1∗ v1 , w2 iH2 − < v2 , C1∗ w1 iH2
1/2
= −kD∗ v2 k2H2 − kA3 w3 k2H3 ≤ 0.
! EJDE-2015/173
COMPACTNESS OF THE DIFFERENCE . . .
7
Finally, A is dissipative. By a standard argument, one shows that (λI − A) is
surjective for λ ∈ (0, kA1−1 k ). Thus, [9, Corollary 3.20] leads to the claim.
3.2. Decoupled system. We show that the operator (Ad , D(Ad )), associated with
the decoupled system (1.5)-(1.8), generates a contraction semigroup on the Hilbert
space H. For this, we first show the following lemma.
Lemma 3.2. The operator Ad is boundedly invertible in H.
Proof. Following the argument of the proof of Lemma 3.1, we show that the operator Ad is invertible and


b11
b12
b13
A−1
0
1 C1 B1
 b21
b22
−B1 C1∗ A−1
−B1
0 
1


−1

(3.3)
Ad =  I
0
0
0
0 
,
 0
I
0
0
0 
∗
∗
A−1
−A−1
0
0
−A−1
3 C2
3 C3
3
where
−1 ∗
−1
−1 ∗
−1
−1 ∗
∗ −1
b11 = −A−1
1 C1 B1 C3 A3 C2 + A1 C1 B1 C1 A1 C2 A3 C2 − A1 C2 A3 C2 ,
−1 ∗
−1
−1
−1 ∗
∗
∗ −1
b12 = A−1
1 C1 B1 C3 A3 C3 + A1 C1 B1 DD − A1 C1 B1 C1 A1 C2 A3 C3
−1 ∗
+ A−1
1 C2 A3 C3 ,
−1
∗ −1
b13 = A−1
1 C1 B1 C1 A1 − A1 ,
−1 ∗
∗
∗ −1
b21 = B1 C3 A−1
3 C 2 − B 1 C 1 A1 C 2 A3 C 2 ,
−1 ∗
∗
∗
∗ −1
b22 = −B1 C3 A−1
3 C3 − B1 DD + B1 C1 A1 C2 A3 C3 .
Now we show the dissipativity of the operator (Ad , D(Ad )) on H. Take
w1
w2
v1
v2
w3
D(Ad ), by the Cauchy-Schwarz inequality, we have
! !
w1
w1
w2
w2
< Ad vv1 , vv1
2
w3
2
w3
= < hv1 , w1 iH1,1/2 + hv2 , w2 iH2,1/2 − hA1 w1 , v1 iH1
−1 ∗
∗
− hC1 w2 , v1 iH1 − hC2 A−1
3 C2 v1 , v1 iH1 + hC2 A3 C3 v2 , v1 iH1
∗
+ hC1∗ w1 , v2 iH2 − hA2 w2 , v2 iH2 + hC3 A−1
3 C2 v1 , v2 iH2
∗
∗
∗
∗
− h(C3 A−1
3 C3 + DD )v2 , v2 iH2 + hC2 v1 , w3 iH3 − < C3 v2 , w3 iH3
− hA3 w3 , w3 iH3 + hC1∗ v1 , w2 iH2 − < C1∗ w1 , v2 iH2
1/2
−1/2
−1/2
= < − kD∗ v2 k2H2 − kA3 C2∗ v1 k2H3 + 2hA3 C3∗ v2 , A3 C2∗ v1 i
−1/2
− kA3 C3∗ v2 k2
≤ −kD∗ v2 k2H2 ≤ 0.
The proof of λI − A is surjective for some λ > 0, follows as in Theorem 2.1.
!
∈
8
E. M. AIT BENHASSI, J. E. BENYAICH, H. BOUSLOUS, L. MANIAR
EJDE-2015/173
3.3. Porous elastic system. As above, we can compute the operator


b11 b12
b13
A−1
1 C 1 B1
b21 b22 −B1 C ∗ A−1
−B1 
1 1
,
M−1 = 
I

0
0
0
0
I
0
0
(3.4)
−1
where B1 = (C1∗ A−1
, and
1 C 1 + A2 )
−1 ∗
−1
−1 ∗
−1
−1 ∗
∗ −1
b11 = −A−1
1 C1 B1 C3 A3 C2 + A1 C1 B1 C1 A1 C2 A3 C2 − A1 C2 A3 C2 ,
−1 ∗
−1
−1
−1 ∗
∗
∗ −1
b12 = A−1
1 C1 B1 C3 A3 C3 + A1 C1 B1 DD − A1 C1 B1 C1 A1 C2 A3 C3
−1 ∗
+ A−1
1 C2 A3 C3 ,
−1
∗ −1
b13 = A−1
1 C 1 B 1 C 1 A1 − A1 ,
−1 ∗
∗
∗ −1
b21 = B1 C3 A−1
3 C2 − B1 C1 A1 C2 A3 C2 ,
−1 ∗
∗
∗
∗ −1
b22 = −B1 C3 A−1
3 C3 − B1 DD + B1 C1 A1 C2 A3 C3 ,
and show that the operator M generates a strongly continuous contraction semigroup (S(t))t≥0 on Hc .
4. Compactness result
In this section we prove the compactness of the difference T (t)−Td (t), we use [20,
Theorem 2.3], where it is sufficient to prove the norm continuity of the difference
between the two semigroups, and the compactness of the difference between the
resolvents of their generators. To show the first assertion, we need the following
technical lemma, see [21, Theorem 1.4.3] .
−A3 t
Lemma 4.1. The map t 7→ Aα
is norm continuous on (0, ∞) for all α ≥ 0.
3e
Now we can show the following norm continuity result.
Theorem 4.2. The map t 7→ T (t) − Td (t) is norm continuous on (0, ∞).
 0
w1
 w20 
1 
Proof. Let t > 0 and x0 = 
 w11  ∈ D(A) such that kx0 k ≤ 1. Let us write
w2
w30

T (t)x0 − Td (t)x0 =

w1 (t)−w1 (t)
(t)−w2 (t)

 wv2 (t)−v
1 (t) 
 1
v2 (t)−v 2 (t)
w3 (t)−w3 (t)
Z

t
T (t −
=
0

0
0


f
(s)
s)
g(s)
0
ds,
where
−1 ∗
∗
f (s) = C2 A−1
3 C2 v 1 (s) − C2 A3 C3 v 2 (s) − C2 w 3 (s),
−1 ∗
∗
g(s) = −C3 A−1
3 C2 v 1 (s) + C3 A3 C3 v 2 (s) + C3 w 3 (s).
Let 0 < h < 1, we begin by checking that kf (s + h) − f (s)k → 0 as h → 0.
Rt
Rt
We have w3 (t) = e−A3 t w30 + 0 e−A3 (t−σ) C2∗ v 1 (σ)dσ − 0 e−A3 (s−σ) C3∗ v 2 (σ)dσ.
Then
−1 ∗
∗
−A3 s 0
f (s) = C2 A−1
w3
3 C2 v 1 (s) − C2 A3 C3 v 2 (s) − C2 e
Z s
−1/2
1/2
− C 2 A3
A3 e−A3 (s−σ) C2∗ v 1 (σ)dσ
0
EJDE-2015/173
COMPACTNESS OF THE DIFFERENCE . . .
−1/2
Z
+ C 2 A3
s
9
1/2
A3 e−A3 (s−σ) C3∗ v 2 (σ)dσ
0
−1/2
−1/2
−1/2 ∗
−1/2
)(C3 A3 )∗ v 2 (s)
Z s
−1/2
1/2
−1/2
−1/2
− (C2 A3 )A3 e−A3 s w30 − (C2 A3 )
A3 e−A3 (s−σ) (C2 A3 )∗ v 1 (σ)dσ
0
Z s
−1/2
−1/2 ∗
−A3 (s−σ)
+ (C2 A3 )
A3 e
(C3 A3 ) v 2 (σ)dσ.
= (C2 A3
) v 1 (s) − (C2 A3
)(C2 A3
0
−1/2
C 2 A3
−1/2
Since
and C3 A3
are bounded operators from H3 to H1 and from H3
1/2
to H2 respectively, and s 7→ e−A3 s , s 7→ A3 e−A3 s are norm continuous on (0, ∞),
−1/2
1/2
the map s 7→ (C2 A3 )A3 e−A3 (s) is norm continuous on (0, ∞), and there exists
−1/2
a positive constant α(s) and β(s) such that k(C2 A3 )∗ v 1 (σ)k ≤ α(s)kv 1 (σ)k and
−1/2
k(C3 A3 )∗ v 2 (σ)k ≤ β(s)kv 2 (σ)k, for every σ ∈ [0, s). By the inequality
k(w1 , w2 , v 1 , v 2 , w3 )kH ≤ kx0 kH ,
for all t ≥ 0,
we deduce
−1/2 ∗
k(C2 A3
) v 1 (σ)k ≤ α(s)kx0 k,
−1/2
k(C3 A3 )∗ v 2 (σ)k
≤ β(s)kx0 k,
for every σ ∈ [0, s). Thus
s
Z
−1/2 ∗
A3 e−A3 (s−σ) (C2 A3
s 7→
) v 1 (σ)dσ,
0
Z
s
s 7→
−1/2 ∗
A3 e−A3 (s−σ) (C3 A3
) v 2 (σ)dσ
0
are continuous on (0, ∞) uniformly with respect to kx0 k ≤ 1.
Finally kf (s + h) − f (s)k → 0, as h → 0, uniformly in x0 . Using the same
argument, we have kg(s + h) − g(s)k → 0, as h → 0, uniformly in x0 .
Let us write

 

w1 (t)−w1 (t)
w1 (t+h)−w1 (t+h)
 w2 (t+h)−w2 (t+h)   w2 (t)−w2 (t)   v1 (t+h)−v1 (t+h)  −  v1 (t)−v1 (t)  v (t+h)−v (t+h)
v (t)−v (t)
2
2
2
w3 (t+h)−w3 (t+h)
Z
=
t+h
Z
=
t+h
T (t + h −
0
0

0
0
s)  f (s)  ds
g(s)
0

T
2
w3 (t)−w3 (t)

−
T (t −
0

0
0
(s)  f (t+h−s)  ds
g(t+h−s)
0
Z
−
T
0
0
g(t+h−s)−g(t−s)
0
h

h
T (t +
0
0

0
0
(s)  f (t−s)  ds
g(t−s)
0
g(t+h−s)−g(t−s)
0


0
Z t
Z
0
 f (t+h−s)−f (t−s)  ds
≤
+

0
0
s)  f (s)  ds
g(s)
0

t


0
Z
Z t
0
=
T (s)  f (t+h−s)−f (t−s)  ds +
0

t
Z


0
0
s)  f (h−s)  ds
g(h−s)
0

0
0
 f (h−s)  ds
.
g(h−s)
0
10
E. M. AIT BENHASSI, J. E. BENYAICH, H. BOUSLOUS, L. MANIAR
EJDE-2015/173
In addition, there exists constants N1 and N2 such that
kf (h − s)k ≤ N1 ,
sup
s∈[0,t+1]
sup
kg(h − s)k ≤ N2
s∈[0,t+1]
uniformly with respect to x0 , and 0 < h < 1.
Since kf (s + h) − f (s)k → 0 as h → 0 and kg(s + h) − g(s)k → 0 as h → 0
Rt
uniformly with respect x0 , we deduce that 0 kf (t + h − s) − f (t − s)kds → 0 and
Rt
kg(t + h − s) − g(t − s)kds → 0, as h → 0 uniformly for x0 ∈ D(A) such that
0
kx0 k ≤ 1. Finally, t 7→ T (t) − Td (t) is norm continuous on (0, ∞).
Proof of Theorem 2.2. Since the map t 7→ T (t)−Td (t) is norm continuous on (0, ∞),
we need only to show the compactness of R(λ, A) − R(λ, Ad ), λ ∈ ρ(A) ∩ ρ(Ad ).
From the following result
R(λ, Ad ) − R(λ, A) = A R(λ, A)[A−1 − A−1
d ]Ad R(λ, Ad ),
it is sufficient to prove that A−1 − A−1
d is

0
0


A−1 − A−1
d = 0
0
0
compact. We have

0 0 0 c15
0 0 0 c25 

0 0 0 0
,
0 0 0 0
0 0 0 0
(4.1)
where
−1
−1
−1
−1
−1
∗ −1
c15 = A−1
1 C1 B1 C3 A3 − A1 C1 B1 C1 A1 C2 A3 + A1 C2 A3 ,
−1
∗ −1
c25 = −B1 C3 A−1
3 + B 1 C 1 A1 C 2 A3 .
From the assumption (2.8), it is clear that the operators c15 and c25 are compact,
and this achieves the proof.
−1/2
Proof of Corollary 2.4. Since the operators A−1
and A1 C1 A−1
are compact,
3
2
assumption (2.8) is satisfied. In view of Theorem 2.2, it is enough to show that for
each t > 0,
Td (t)(w10 , w20 , w11 , w21 , w30 ) − (S(t)(w10 , w20 , w11 , w21 ); 0) : k(w10 , w20 , w11 , w21 , w30 )k ≤ 1
is a compact set in H, i.e. that
Z t
Z t
n
e−A3 t w30 +
e−A3 (t−σ) C2∗ v 1 (σ)dσ −
e−A3 (s−σ) C3∗ v 2 (σ)dσ :
0
0
o
k(w10 , w20 , w11 , w21 , w30 )k ≤ 1
is a compact set in H3 , where (w1 (σ), w2 (σ), v 1 (σ), v 2 (σ)) = S(σ)(w10 , w20 , w11 , w21 ).
Since
(w10 , w20 , w11 , w21 , w30 )
Z t
1/2
−1/2
A3 e−A3 (t−σ) (C2 A3 )∗ v 1 (σ)dσ
→ A3 e−A3 t w30 +
0
Z t
−1/2
−A3 (s−σ)
−
A3 e
(C3 A3 )∗ v 2 (σ)dσ
0
is bounded with values in H3 (we have used the Lemma 4.1 and Lebesgue’s theorem)
−1/2
and A3
is compact, the result follows.
EJDE-2015/173
COMPACTNESS OF THE DIFFERENCE . . .
11
−γ
−γ
Remark 4.3. (1) If we have the conditions C3 A−γ
3 , C2 A3 and C1 A2 are compact
for some γ < 1 then the assumptions (2.8) are satisfied and we have the compactness
of the difference between T (t) − Td (t) for every t ≥ 0, which is similar to Henry’s
condition in [11].
−1
(2) If we suppose that A−1
1 and A2 are compacts we have σe (T (t)) = σe (Td (t))
for t ≥ 0 but to have σe (Td (t)) = σe (S(t)) for t ≥ 0 we need the condition A−1
3 is
compact.
5. Applications
We give two illustrating examples of Theorem 2.2 and Corollary 2.4.
Application 1. We give one application of Theorem 2.2. Let Ω the bounded open
Jelly Roll set proposed in [26],
1
Ω = {(x, y) ∈ R2 : < r < 1} \Γ,
2
where Γ is the curve in R2 given in polar coordinates by
3π
2
+ arctan(φ)
, −∞ < φ < ∞.
2π
We consider the initial and boundary problem
r(φ) =
ü(t, x) − ∆e u(t, x) − b∇φ(t, x) + c∇θ(t, x) = 0
in (0, +∞) × Ω,
φ̈(t, x) − (a∆ − αI)φ(t, x) + b div u(t, x) − dθ(t, x) + rφ̇(t, x) = 0
in (0, +∞) × Ω,
θ̇(t, x) − (∆ − kI)θ(t, x) + c div u̇(t, x) + dφ̇(t, x) = 0
u = 0,
u(0) = u0 ,
in (0, +∞) × Ω,
∂θ
= 0 on (0, +∞) × ∂Ω,
∂n
φ(0) = φ0 , φ̇(0) = φ1 , θ(0) = θ0 ,
(5.1)
φ = 0,
u̇(0) = u1 ,
in Ω,
where n denotes the outer uniter normal vector to ∂Ω, ∆e := µ∆ + (µ + λ)∇ div,
and µ, λ, a, b, c, d, r, α, k are positive constants.
To fit this system into the abstract setting of (1.1)-(1.4), we take
H1 = L2 (Ω)2 ,
H2 = H3 = L2 (Ω),
H = Hc × L2 (Ω),
A1 = −∆e ,
H1, 21 = (H01 (Ω))2 ,
H2, 12 = H01 (Ω),
Hc = (H01 (Ω))2 × H01 (Ω) × L2 (Ω)2 × L2 (Ω),
D(A1 ) = D(−∆D ) = (H 2 (Ω) ∩ H01 (Ω))2 ,
A2 = −(a∆ − αI),
D(A2 ) = D(−∆D ) = H 2 (Ω) ∩ H01 (Ω),
A3 = −(∆ − kI),
D(A3 ) = D(−∆N ).
We recall that u, φ, θ are the displacement vector, the volume fraction and the
temperature. The Dirichlet and Neumann Laplacian operators ∆D and ∆N are
the unique positive self adjoint operators associated to the closed quadratic form
on H01 (Ω) and H 1 (Ω) respectively
Z
h∆f, gi =
∇f ∇gdx.
Ω
The operator DD∗ = rIH2 , and the coupled operators
C1 = −b∇,
C2 = c∇,
C1∗ = b div,
C2∗ = −c div,
C3 = d IH3 ,
12
E. M. AIT BENHASSI, J. E. BENYAICH, H. BOUSLOUS, L. MANIAR
EJDE-2015/173
−
D(C2∗ ) = D(C1∗ ) = {u ∈ H 1 (Ω)2 : u · →
n = 0 in ∂Ω}.
D(C1 ) = D(C2 ) = H 1 (Ω),
−1
Note that the conditions (2.2) are verified and we have A−1
1 and A2 are compact
from H1 and H2 respectively, then the assumptions (2.8) are satisfied, consequently
the Theorem 2.2 is satisfied. To show σe (Td (t)) = σe (S(t)) for t ≥ 0, we need the
−1
compactness of A−1
3 , but from [26], A3 is not compact.
Application 2. We give an application of Corollary 2.4. Let Ω ⊂ R2 a bounded
open domain with boundary ∂Ω having regularity of class C 2 , and satisfies the
following condition:
(A1) If ϕ ∈ (H01 (Ω))2 such that
−∆ϕ = γ 2 ϕ in Ω,
div ϕ = 0
ϕ=0
in Ω,
(5.2)
in ∂Ω.
for some γ ∈ R, then ϕ = 0.
We consider the initial and boundary problem
ü(t, x) − ∆e u(t, x) − b∇φ(t, x) + c∇θ(t, x) = 0
in (0, +∞) × Ω,
φ̈(t, x) − (a ∆ − αI)φ(t, x) + b div u(t, x) + rφ̇(t, x) = 0
θ̇(t, x) − ∆θ(t, x) + c div u̇(t, x) = 0
u = 0,
0
φ = 0,
1
u(0) = u ,
u̇(0) = u ,
θ=0
0
φ(0) = φ ,
in (0, +∞) × Ω,
in (0, +∞) × Ω,
(5.3)
on (0, +∞) × ∂Ω,
φ̇(0) = φ1 , θ(0) = θ0
in Ω,
where ∆e := µ∆ + (µ + λ)∇ div is Lamé operator, µ, λ, a, b, c, r, α are positive
constants, and the condition (λ + µ)α > b2 is satisfied.
To fit this system into the abstract setting of (1.1)-(1.4), we take
H1 = L2 (Ω)2 ,
H1, 21 = (H01 (Ω))2 ,
H2 = H3 = L2 (Ω),
H = Hc × L2 (Ω),
H2, 12 = H01 (Ω),
where Hc = (H01 (Ω))2 × H01 (Ω) × L2 (Ω)2 × L2 (Ω),
D(A1 ) = D(−∆D ) = (H 2 (Ω) ∩ H01 (Ω))2 ,
A1 = −∆e ,
A2 = −(a∆ − αI),
A3 = −∆,
D(A2 ) = H 2 (Ω) ∩ H01 (Ω),
D(A3 ) = D(A2 ).
The operator DD∗ = rIH2 , and the coupled operators
C1 = −b∇,
C1∗ = b div,
C2∗ = −c div,
C3 = 0,
−
D(C2∗ ) = D(C1∗ ) = {u ∈ H 1 (Ω)2 : u · →
n = 0 on ∂Ω}.
D(C1 ) = D(C2 ) = H 1 (Ω),
C2 = c∇,
The decoupled system corresponding to system (5.3) is given by
ü(t, x) − ∆e u(t, x) − b∇φ(t, x) + c2 P u̇(t, x) = 0
in (0, +∞) × Ω,
φ̈(t, x) − (a∆ − αI)φ(t, x) + b div u(t, x) + rφ̇(t, x) = 0
θ̇(t, x) − ∆θ(t, x) + c div u̇(t, x) = 0
u = 0,
0
u(0) = u ,
φ = 0,
1
u̇(0) = u ,
θ=0
0
φ(0) = φ ,
in (0, +∞) × Ω,
(5.4)
in (0, +∞) × Ω,
on (0, +∞) × ∂Ω,
φ̇(0) = φ1 , θ(0) = θ0
in Ω,
EJDE-2015/173
COMPACTNESS OF THE DIFFERENCE . . .
13
where P := ∇(∆)−1 div the orthogonal projection operator from L2 (Ω)2 into the
subspace {∇ϕ; ϕ ∈ H01 (Ω)}. Now we write the porous elastic system given by the
first and second equation in decoupled system (5.4)
ü(t, x) − ∆e u(t, x) − b∇φ(t, x) + c2 P u̇(t, x) = 0
in (0, +∞) × Ω,
φ̈(t, x) − (a∆ − αI)φ(t, x) + b div u(t, x) + rφ̇(t, x) = 0
u = 0,
u(0) = u0 ,
φ=0
u̇(0) = u1 ,
in (0, +∞) × Ω,
on (0, +∞) × ∂Ω,
φ(0) = φ0 ,
φ̇(0) = φ1
(5.5)
in Ω.
Let (T (t))t≥0 the porous-thermoelastic C0 -semigroup generated by the system
(5.3) and (S(t))t≥0 the porous elastic C0 -semigroup generated by the system (5.5).
−1
−1
Note that the operators A−1
1 , A2 and A3 are compact, consequently the assumptions of Corollary 2.4 are satisfied, then
σe (T (t)) = σe (S(t))
for t ≥ 0.
The second aim of this application is to characterize the exponential energy decay
of solution of system (5.5), and then deduce the one of the coupled systems (5.3).
Now we show that (S(t))t≥0 is exponentially stable in Hc , by using a similar
argument as in the proof of [15, theorem 4.4]. Let ε(t) := S(t)ε0 , t ≥ 0, be the
solution of
dε
= Mε, ε(0) = ε0 ,
(5.6)
dt
l
where ε0 := (u0 , φ0 , u1 , φ1 ). We look for ε(t) having the form ε(t) = Σ∞
l=0 b εl (t),
where εl (t) ≡ ul (t), φl (t), u̇l (t), φ̇l (t) , ε(0) = ε0 and l ∈ {0} ∪ N. After the
formal substitution into the equation (5.6) we derive equations for (ul , φl ), where
l ∈ {0} ∪ N. For (u0 , φ0 ) we obtain
ü0 (t, x) − ∆e u0 (t, x) + c2 P u̇0 (t, x) = 0
in (0, +∞) × Ω,
φ̈0 (t, x) − (a∆ − αI)φ0 (t, x) + rφ̇0 (t, x) = 0
u0 = 0,
u0 (0) = u0 ,
φ0 = 0
u̇0 (0) = u1 ,
in (0, +∞) × Ω,
(5.7)
(5.8)
on (0, +∞) × ∂Ω,
φ0 (0) = φ0 ,
φ̇0 (0) = φ1
in Ω.
For k ∈ {0} ∪ N, (uk+1 , φk+1 ) will be the solution of problem
ük+1 (t, x) − ∆e uk+1 (t, x) − b∇φk (t, x) + c2 P u̇k+1 = 0
in (0, +∞) × Ω,
φ̈k+1 (t, x) − (a∆ − αI)φk+1 (t, x) + b div uk (t, x) + rφ̇k+1 (t, x)
=0
in (0, +∞) × Ω,
uk+1 = 0,
uk+1 (0) = 0,
φk+1 = 0
u̇k+1 (0) = 0,
on (0, +∞) × ∂Ω,
φk+1 (0) = 0,
φ̇k+1 (0) = 0
in Ω.
Let ε = (u, φ, v, ψ) ∈ Hc , and define the norms
Z
2
k(u(t), v(t))k1 := [µ|∇u(x, t)|2 + (λ + µ)| div u(x, t)|2 + |v(x, t)|2 ]dx,
Ω
Z
2
k(φ(t), ψ(t))k2 := [a|∇φ(x, t)|2 + α|φ(x, t)|2 + |ψ(x, t)|2 ]dx,
Ω
kεk2 := k(u(t), v(t))k21 + k(φ(t), ψ(t))k22 .
(5.9)
(5.10)
14
E. M. AIT BENHASSI, J. E. BENYAICH, H. BOUSLOUS, L. MANIAR
EJDE-2015/173
From [19], there exists M1 , γ1 > 0, such that
k(φ0 (t), φ̇0 (t))k22 ≤ M1 e−γ1 t k(φ0 , φ1 )k22 ,
t ≥ 0.
Note that the damped Lamé system (5.7) has been studied by Zuazua and Lebeau
in [17] and they proved the exponential decay of solution of (5.7) if the following
inequality of observability holds true for some T, C > 0, i.e,
Z T
0
1
kϕ k(L2 (Ω))2 + kϕ k(H −1 (Ω))2 ≤ C
k div ϕkH −1 (Ω) dt,
(5.11)
0
where ϕ(t) is solution of the Lamé system
ϕ̈(t, x) − ∆e ϕ(t, x) = 0
in (0, +∞) × Ω,
on (0, +∞) × ∂Ω,
ϕ=0
0
ϕ(0) = ϕ , ϕ̇(0) = ϕ1
(5.12)
in Ω.
Under the condition that (5.11) is satisfied, we have
k(u0 (t), u̇0 (t))k21 ≤ M2 e−γ2 t k(u0 , u1 )k21 ,
t ≥ 0,
for positive constants M2 , γ2 . Let γ = inf (γ1 , γ2 ), we have
γ
k(u0 (t), φ0 (t), u̇0 (t), φ̇0 (t))k ≤ M e− 2 t k(u0 , φ0 , u1 , φ1 )k,
t ≥ 0.
(5.13)
Let (G(t))t≥0 and (K(t))t≥0 be the contraction C0 -semigroups generated by the
equations (5.8) and (5.7) respectively, where (φ0 (t), φ̇0 (t)) = G(t)(φ0 , φ1 ), and
(u0 (t), u̇0 (t)) = K(t)(u0 , u1 ). For the solution of system (5.10) we have
Z t
(φk+1 (t), φ̇k+1 (t)) =
G(t − s)(0, − div uk (s))ds.
0
Then
t
Z
M1 e −
k(φk+1 (t), φ̇k+1 (t))k2 ≤
γ1
2
(t−s)
k(0, − div uk (s))k2 ds.
0
Since k(0, − div uk (s))k2 ≤ C1 k(uk (s), u̇k (s))k1 , we have
Z t
γ1
k(φk+1 (t), φ̇k+1 (t))k2 ≤
C1 M1 e− 2 (t−s) k(uk (s), u̇k (s))k1 ds.
0
For the solution of system (5.9) we have
Z t
(uk+1 (t), u̇k+1 (t)) =
K(t − s)(0, b∇φk (s))ds.
0
Then
Z
k(uk+1 (t), u̇k+1 (t))k1 ≤
t
M2 e −
γ2
2
(t−s)
k(0, b∇φk (s))k1 ds.
0
Since k(0, b∇φk (s))k1 ≤ C2 bk(φk (s), φ̇k (s))k2 , we have
Z t
γ2
k(uk+1 (t), u̇k+1 (t))k1 ≤
bC2 M2 e− 2 (t−s) k(φk (s), φ̇k (s))k2 ds.
0
Then we have
k(uk+1 (t), φk+1 (t), u̇k+1 (t), φ̇k+1 (t))k
Z t
γ
≤
M3 e− 2 (t−s) k(uk (s), φk (s), u̇k (s), φ̇k (s))kds.
0
(5.14)
EJDE-2015/173
COMPACTNESS OF THE DIFFERENCE . . .
15
From (5.13) and (5.14) we deduce that
tl γ
k(ul (t), φl (t), u̇l (t), φ̇l (t))k ≤ M M3l e− 2 t k(u0 , φ0 , u1 , φ1 )k.
l!
γ
∞ l
Let 0 < b < 2M
,
the
sequence
Σ
b
ε
(t)
is
convergent
in C([0, τ ]; Hc ) for every
l
l=0
3
n
n
n
l
τ > 0. Let ε (t) = Σl=0 b εl (t), n ∈ N where ε (t) is the solution of the problem
dεn (t)
= Mεn (t) + βn (t); εn (0) = ε0 ,
dt
T
where βn (t) := 0, bn ∇φn (t), 0, −bn div un (t) . We have
Z t
S(t − s)βn (s)ds,
εn (t) = S(t)ε0 +
0
and
0
kε(t) − S(t)ε k =
Z
l
kΣ∞
l=n+1 b εl (t) +
t
S(t − s)βn (s)dsk → 0,
as n → ∞, ∀n ∈ N.
0
This means that S(t)ε0 = ε(t) and
l
l
∞ l
l t − γt
2 kε0 k ≤ M e−%t kε0 k,
kS(t)ε0 k ≤ Σ∞
l=0 b kεl (t)k ≤ M Σl=0 b M3 e
l!
where % := γ2 − M3 b. Consequently (S(t))t≥0 is exponentially stable and we (M) <
0. Since σe (T (t)) = σe (S(t)) for t ≥ 0, then
we (A) < 0.
Now we prove that (T (t))t≥0 is exponentially stable in H, i.e, kT (t)k ≤ M e−δt ,
t ≥ 0, where M, δ > 0. From [14, Theorem 2.9] the semigroup (T (t))t≥0 is asymptotically stable in H i.e, limt→∞ kT (t)xk = 0, for every x ∈ H. Then s1 (A) ≤ 0,
where
s1 (A) = sup{<λ/λ ∈ σ(A)\σe (A)}.
To show that w0 (A) < 0, it suffices to prove that s1 (A) < 0. Suppose that s1 (A) =
λn t0
0, then there exists {λn }∞
1 ⊂ σ(A)\σe (A), such that <λn → 0, as n → ∞. e
λ n t0
λ n t0
is an eigenvalue of T (t0 ), we have |e
| ≤ 1 and |e
| → 1 as n → ∞. Let y be
in
C.
Then
y
∈
σ
(T
(t0 )) and |y| = 1. Thus,
the accumulation point of {eλn t0 }∞
e
1
re (T (t0 )) ≥ 1,
furthermore
re (T (t0 )) = ewe (A)t0 < 1.
This contradiction implies that s1 (A) < 0, using we (A) < 0, we obtain w0 (A) < 0.
Finally we have proved the uniform stabilization of the energy of solution of system
(5.3).
References
[1] E. Ait Ben Hassi, H. Bouslous, L. Maniar; Compact decoupling for thermoelasticity in irregular domains, Asymptotic Analysis., 58 (2008), 47-56.
[2] K. Ammari, E. M. Ait Ben Hassi, S. Boulite, L. Maniar; Stabilization of coupled second order
systems with delay, Semigroup Forum, 86, (2013), 362-382.
[3] B. Amir, L. Maniar; Application de la théorie d’extrapolation pour la résolution des équations
différentielles à retard homogènes, Extracta Mathematicae., 13, (1998), 95-105.
[4] F. Ammar-Khodja, A. Bader, A. Benabdallah; Dynamic stabilization of systems via decoupling techniques, ESAIM Control Optim.Calc.Var., 4 (1999), 577-593.
16
E. M. AIT BENHASSI, J. E. BENYAICH, H. BOUSLOUS, L. MANIAR
EJDE-2015/173
[5] S. C. Cowin, W. Nunziato; A nonlinear theory of elastic materials with voids, Arch. Rational
Mech. Anal., 72 (1979), 175-201.
[6] S. C. Cowin, W. Nunziato; Linear elastic materials with voids, J. Elasticity., 13 (1983),
125-147.
[7] S. C. Cowin; The viscoelastic behavior of linear elastic materials with voids, J. Elasticity.,
15 (1985), 185-191.
[8] P. S. Casas, R. Quintanilla; Exponential decay in one-dimensional porous-thermo- elasticity,
Mech. Res. Comm., 32 (2005), 652-658.
[9] K. J. Engel, R. Nagel; One-parameter semigroups for linear evolution equations, Graduate
Texts in Mathematics, Vol. 194 , Springer-Verlag, 2000.
[10] D. E. Edmunds, W. D. Evans; Spectral theory and differential operators, Clarendon Press,
Oxford, 1987.
[11] D. Henry, O. Lopes, A. Perissinotto; On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. T.M.A., 21 (1993), 65-75.
[12] D. Henry; Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics,
vol. 840, Springer-Verlag, Berlin, 1981.
[13] P. Glowinski, A. Lada; The compact decoupling for system of thermoelasticity in viscoporous
media and exponential decay, challenges of Technology., 2 (2011), 3-6.
[14] P. Glowinski, A. Lada; Stabilization of elasticity-viscoporosity system by linear boundary
feedback, Math. Methods Appl. Sci., 32 (2009), 702-722.
[15] P. Glowinski, A. Lada; On exponential decay for linear porous-thermo-elasticity system,
Demonstratio Mathematica., 45 (2012), 847-868.
[16] B. Z. Guo; On the exponential stability of c0 -semigroups on Banach spaces with compact
perturbations, Semigroup Forum., 59 (1999), 190-196.
[17] G. Lebeau, E. Zuazua; Decay rates for the three-dimensional linear system of thermoelasticity,
Arch. Ration. Mech. Anal., 148 (1999), 179-231.
[18] W. J. Liu; Compactness of the difference between the thermoviscoelastic semigroup and its
decoupled semigroup, Rocky Mount. J. Math., 30 (2000), 1039-1056.
[19] W. J. Liu, E. Zuazua; Decay rate for dissipative wave equations, Ricerche di Matematica.,
48 (1999), 61-75.
[20] M. Li, G. Xiaohui, F. Huang; Unbounded perturbations of semigroups, compactness and
norm continuity, Semigroup Forum., 65 (2002), 58-70.
[21] A. Lunardi; Analytic Semigroups and optimal regularity in parabolic problems, Birkhauser,
Basel, 1995.
[22] J. E. M. Rivera, R. Racke; Mildly dissipative nonlinear Timoshenko systems, Math. Anal.
Appl., 276 (2002), 248-278.
[23] J. E. M. Rivera, R. Racke; Global stability for damped Timoshenko systems, Discrete.
Contin. Dyn. Syst., 9 (2003), 1625-1639.
[24] A. Pazy; Semigroups of linear operators and application to partial differential equations,
App. Math. Sci. 44, Springer-Verlag, 1983.
[25] R. Quintanilla; Slow decay for one-dimensional porous dissipation elasticity, Appl. Mathematics Letters., 16 (2003), 487-491.
[26] B. Simon; The Neumann Laplacian of a jelly roll, Proc. Amer. Math. Soc., 114 (1992),
783-785.
Université Cadi Ayyad, Faculté des Sciences Semlalia, LMDP, UMMISCO (IRD- UPMC),
Marrakech 40000, B.P. 2390, Maroc. Fax. 00212524437409
E-mail address: m.benhassi@uca.ma
E-mail address: jbenyaich@gmail.com
E-mail address: bouslous@uca.ma
E-mail address: maniar@uca.ma