Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 71, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
COMPACT DECOUPLING FOR THERMOVISCOELASTICITY IN
IRREGULAR DOMAINS
EL MUSTAPHA AIT BEN HASSI, HAMMADI BOUSLOUS, LAHCEN MANIAR
Abstract. Our goal is to prove the compactness of the difference between the
thermoviscoelasticity semigroup and its decoupled semigroup. To show this, we
prove the norm continuity of this difference, the compactness of the difference
of their resolvents and use [4, Theorem 2.3 ]. We generalize a result by Liu
[5]. An illustrative example of a thermoviscoelastic system with Neumann
Laplacian on a Jelly Roll domain is given.
1. Introduction
Consider the abstract thermoviscoelastic model
Z 0
g(s)A1 w(t + s)ds + Bu(t) = 0,
ẅ(t) + A1 w(t) −
t ≥ 0,
(1.1)
−∞
u̇(t) + A2 u(t) − B ∗ ẇ(t) = 0,
0
w(0) = w ,
1
ẇ(0) = w ,
0
u(0) = u ,
t ≥ 0,
w(s) = f0 (s),
(1.2)
s ∈ (−∞, 0),
(1.3)
where g is a given function satisfying the following conditions:
g ∈ C 1 (−∞, 0] ∩ L1 (−∞, 0),
g(t) ≥ 0,
Z
0
g (t) ≥ 0
(1.4)
for t < 0,
(1.5)
0
g(s)ds < 1.
(1.6)
−∞
By the decoupling technique, we obtain the system
Z 0
∗ ˙
¨
g(s)A1 w̄(t + s)ds + BA−1
w̄(t) + A1 w̄(t) −
2 B w̄(t) = 0,
t ≥ 0,
(1.7)
−∞
˙ + A2 ū(t) − B ∗ w̄(t)
˙
ū(t)
= 0,
0
w̄(0) = w ,
1
˙
w̄(0)
=w ,
0
ū(0) = u ,
t ≥ 0,
w̄(s) = f0 (s),
(1.8)
s ∈ (−∞, 0).
(1.9)
The operators A1 and A2 are positive self adjoint and invertible on two Hilbert
spaces H1 and H2 , and B is an unbounded operator from H2 to H1 . Liu [5] proved
that these two systems are well posed and generate two semigroups T := (T (t))t≥0
2000 Mathematics Subject Classification. 34G10, 47D06.
Key words and phrases. Thermoviscoelasticity; semigroup compactness;
semigroup norm continuity, essential spectrum; fractional power.
c 2011 Texas State University - San Marcos.
Submitted August 24, 2010. Published May 31, 2011.
1
2
E. M. AIT BEN HASSI, H. BOUSLOUS, L. MANIAR
EJDE-2011/71
and Td := (Td (t))t≥0 . Assuming that BA−γ
is compact for some 0 < γ < 1, he
2
proved that their difference is compact. In this paper, proceeding as in [1], we
show that t 7→ T (t) − Td (t) and t 7→ T (t) − S(t) are norm continuous for t > 0
where S := (S(t))t≥0 is the semigroup generated by the first equation (1.7) in the
decoupled system. Consequently, under no compactness assumption, rcrit (T (t)) =
rcrit (Td (t)) = rcrit (S(t)) for t ≥ 0 and ω0 (T ) = max{ωcrit (S), s(L)}.
−1/2
Assuming that A1 BA−1
is a compact operator (in particular if BA−γ
is
2
2
R0
2
compact) and −∞ g(s)s ds < ∞, we prove the compactness of the difference
R(λ, L) − R(λ, Ld ) for every λ ∈ ρ(L) ∩ ρ(Ld ), where L and Ld are the generators of T and Td , respectively. Thus, [4, Theorem 2.3] leads to the compactness of
T (t) − Td (t).
To illustrate this generalization, we consider the thermoviscoelastic system
ẅ − µ∆w − (λ + µ)∇ div w + µg1 ∗ ∆w + (λ + µ)g1 ∗ ∇ div w + m∇u = 0
in Ω × (0, ∞),
u̇ + βu − ∆u − m div ẇ = 0
in Ω × (0, ∞),
∂u
= 0 on Γ × (0, ∞),
∂n
ẇ(x, 0) = w1 (x), u(x, 0) = u0 (x)
w = 0,
w(x, 0) = w0 (x),
w(x, 0) + w(x, s) = f0 (x, s)
in Ω,
in Ω × (−∞, 0)),
where µ, λ are positive constants. The set Ω is the Jelley Roll, a bounded open set
proposed in [9],
Ω = {(x, y) ∈ R2 : 1/2 < r < 1} \ Γ,
where Γ is the curve, in R2 , given in polar coordinates by
3π
2
+ arctan(φ)
, −∞ < φ < ∞.
2π
−1
For this system, we show that A−1
1 BA2 , on the canonical modified energy Hilbert
space, is a compact operator but the operator BA−γ
is not compact for every
2
0 < γ < 1.
r(φ) =
2. Well-posedness
Let H1 and H2 be two Hilbert spaces. The operators A1 : D(A1 ) ⊂ H1 → H1
and A2 : D(A2 ) ⊂ H2 → H2 are self adjoint and positive (with not necessarily
compact inverses), while B : D(B) ⊂ H2 → H1 is a closed operator with adjoint
operator B ∗ . Throughout this paper, we assume the following:
1/2
D(A2 ) ,→ D(B)
∗ 1/2
A−1
2 B A1
and
1/2
D(A1 ) ,→ D(B ∗ ),
extends to a bounded linear operator from H1 to H2 .
(2.1)
(2.2)
Note that the operator −A2 generates an analytic strongly continuous semigroup
−1/2
(e−A2 t )t≥0 . Under assumption (2.1), BA2
is a bounded operator from H2 to H1
−1/2
−1/2 ∗
and BA2 (BA2 ) is a bounded self adjoint non negative operator in H1 .
Setting z(t, s) = w(t) − w(t + s), s ∈ (−∞, 0), the system (1.1)-(1.3) can be
transformed into the system
Z 0
ẅ(t) + kA1 w(t) +
g(s)A1 z(t, s)ds + Bu(t) = 0, t ≥ 0,
(2.3)
−∞
EJDE-2011/71
COMPACT DECOUPLING
3
zt − ẇ + zs = 0,
(2.4)
t ≥ 0,
z(t, 0) = 0,
(2.5)
∗
u̇(t) + A2 u(t) − B ẇ(t) = 0,
0
1
t ≥ 0,
0
s ∈ (−∞, 0),
w(0) = w , ẇ(0) = w , u(0) = u , z(0, s) = f0 (s),
R0
with k = 1 − −∞ g(s)ds. Set the Hilbert space
1/2
(2.6)
(2.7)
1/2
H = D(A1 ) × H1 × L2 (g, (−∞, 0), D(A1 )) × H2
endowed with the norm
1/2
1/2
.
k(w, v, z, u)k = kkA1 wk2H1 + kvk2H1 + kzk2L2 (g,(−∞,0),D(A1/2 )) + kuk2H2
1
1/2
1/2
Here the space L2 (g, (−∞, 0), D(A1 )) consists of D(A1 )-valued functions z on
(−∞, 0) endowed with the norm
Z 0
1/2
kzk2L2 (g,(−∞,0),D(A1/2 )) =
g(s)kA1 z(s)k2 ds.
1
−∞
System (2.3)-(2.7) can also be written as a first order system
ẇ = v,
Z
(2.8)
0
v̇ = −kA1 w −
g(s)A1 z(t, s)ds − Bu,
(2.9)
−∞
ż = v − zs
(2.10)
∗
u̇ = −A2 u + B v,
0
(2.11)
1
0
(w(0), v(0), u(0), z(0)) = (w , w , f0 , u ).
(2.12)
We associate with the system (2.8)-(2.11) the operator
Z 0
L(w, v, z, u) = v, −kA1 w −
g(s)A1 z(t, s)ds − Bu, v − zs , −A2 u + B ∗ v ,
−∞
D(L) = {(w, v, z, u) ∈ H : v ∈
1/2
D(A1 ), u
Z
0
∈ D(C), kw +
g(s)z(s)ds ∈ D(A),
−∞
1/2
z ∈ H 1 (g, (−∞, 0), D(A1 )), z(0) = 0},
1/2
where H 1 (g, (−∞, 0), D(A1 )) is the set
1/2
1/2
{z ∈ L2 (g, (−∞, 0), D(A1 )) : zs ∈ L2 (g, (−∞, 0), D(A1 ))}.
The decoupled system (1.7)-(1.9) can also be transformed into
w̄˙ = v̄,
v̄˙ = −kA1 w̄ −
Z
(2.13)
0
−∞
∗
g(s)A1 z̄(t, s)ds − BA−1
2 B v̄,
z̄˙ = v̄ − z̄s ,
(2.14)
(2.15)
ū˙ = −A2 ū + B ∗ v̄,
0
(2.16)
1
0
(w̄(0), v̄(0, z̄(0), ū(0)) = (w , w , f0 , u ),
(2.17)
4
E. M. AIT BEN HASSI, H. BOUSLOUS, L. MANIAR
EJDE-2011/71
to which we associate the operator
Z 0
∗
∗
Ld (w̄, v̄, z̄, ū) = (v̄, −kA1 w̄ −
g(s)A1 z̄(t, s)ds − BA−1
2 B v̄, v̄ − z̄s , −A2 ū + B v̄),
−∞
with D(Ld ) = D(L). Also, the decoupled second order equation (1.7) can be written
as a first order system
w̄˙ = v̄,
v̄˙ = −kA1 w̄ −
Z
(2.18)
0
−∞
∗
g(s)A1 z̄(t, s)ds − BA−1
2 B v̄,
z̄˙ = v̄ − z̄s ,
(2.19)
(2.20)
0
1
(w̄(0), v̄(0), z̄(0)) = (w , w , f0 ),
(2.21)
1/2
1/2
with generating operator defined on H := D(A1 ) × H1 × L2 (g, (−∞, 0), D(A1 ))
by
Z 0
∗
M (w̄, v̄, z̄) = (v̄, −kA1 w̄ −
g(s)A1 z̄(t, s)ds − BA−1
2 B v̄, v̄ − z̄s ),
−∞
1/2
D(M ) = (w̄, v̄, z̄) ∈ H : v̄ ∈ D(A1 ), k w̄ +
Z
0
g(s)z̄(s)ds ∈ D(A),
−∞
1/2
z̄ ∈ H 1 (g, (−∞, 0), D(A1 )), z̄(0) = 0 .
Remark 2.1. L, Ld and M are respectively the parts in H and H of the matrix
operators


0
I
0
0
−k(A1 )−1 0 −G−1
−B 
,
L−1 = 
d


0
0
I
− ds
0
B∗
0
−(A2 )−1


0
I
0
0
−1/2
−1/2

−k(A1 )
0
−BA2 (BA2 )∗ −G−1
−1
,
(Ld )−1 = 
d


0
0
I
− ds
∗
0
B
0
−(A2 )−1


0
I
0
−1/2
M = −k(A1 )−1 −BA−1/2
(BA2 )∗ −G−1  ,
2
d
0
I
− ds
1/2
1/2 R 0
where G−1 z = (A1 )−1 −∞ g(s)A1 z(s)ds.
Using Lumer Phillips theorem, the following result can be proved analogously as
in [5, Theorem 2.1].
Theorem 2.2. The operators L, Ld and M generate contraction strongly continuous semigroups (T (t))t≥0 , (Td (t))t≥0 and (S(t))t≥0 on H, H and H respectively.
3. Norm continuity of the difference between the semigroups
To show the norm continuity of the difference between the two semigroups, we
recall the following technical lemma.
EJDE-2011/71
COMPACT DECOUPLING
5
−A2 t
Lemma 3.1 ([1]). The map t 7→ Aα
is norm continuous from (0, ∞) to L(H2 )
2e
for every 0 ≤ α < 1.
Then we have the following result.
Theorem 3.2. The map t 7→ T (t)−Td (t) is norm continuous from (0, ∞) to L(H).
Proof. Let t > 0 and x0 = (w0 , v 0 , f0 , u0 ) ∈ D(L) such that kx0 k ≤ 1.
T (t)(w0 , v 0 , f0 , u0 ) − Td (t)(w0 , v 0 , f0 , u0 )




0
w(t) − w̄(t)
Z t
 v(t) − v̄(t) 
BA−1 B ∗ v̄(s) − B ū(s)
2

 ds.
=
T (t − s) 
 z(t) − z̄(t)  =


0
0
u(t) − ū(t)
0
∗
Let 0 < h < 1. Setting F (s) := B ū(s) − BA−1
2 B v̄(s), we check that kF (s + h) −
0
F (s)k → 0 as h → 0 uniformly in x . To this end, we have
Z s
−A2 s 0
∗
F (s) = Be
u +B
e−A2 (s−σ) B ∗ v̄(σ)dσ − BA−1
2 B v̄(s)
0
Z s
−1
−A2 s 0
∗
= Be
u + BA2
A2 e−A2 (s−σ) B ∗ v̄(σ)dσ − BA−1
2 B v̄(s).
0
Using an integration by parts,
−A2 (s−σ) ∗
F (s) = Be−A2 s u0 + [BA−1
B v̄(σ)]s0 − BA−1
2 e
2
s
Z
e−A2 (s−σ) B ∗ v̄ 0 (σ)dσ
0
∗
− BA−1
2 B v̄(s)
−A2 s ∗ 0
= Be−A2 s u0 − BA−1
B v − BA−1
2 e
2
Z
s
e−A2 (s−σ) B ∗ v̄ 0 (σ)dσ
0
Z s
−1/2 −A2 s −1/2 ∗ 0
−A2 s 0
A2 B v + kBA−1
e−A2 (s−σ) B ∗ A1 w̄(σ)dσ
= Be
u − BA2 e
2
0
+ BA−1
2
−1/2
Z
s
e−A2 (s−σ) B ∗ [−
Z
0
−∞
−1/2
0
1/2
∗
g(τ )Az̄(s, τ )dτ + BA−1
2 B v̄(σ)]dσ
−1/2
A2 e−A2 s u0 − BA2 e−A2 s A2 B ∗ v 0
Z s
−1/2
1/2
∗ 1/2 1/2
+ kBA2
A2 e−A2 (s−σ) A−1
2 B A1 A1 w̄(σ)dσ
0
Z s
−1/2
−1/2
∗
+ BA2
e−A2 (s−σ) A2 B ∗ BA−1
2 B v̄(σ)dσ
= BA2
− BA−1
2
Z
0
s
e−A2 (s−σ) B ∗
Z
0
g(τ )Az̄(s, τ )dτ dσ
0
−∞
−1/2 1/2 −A2 s 0
−1/2
−1/2
= BA2 A2 e
u − BA2 e−A2 s A2 B ∗ v 0
Z s
1/2
−1/2
∗ 1/2 1/2
+ kBA2
A2 e−A2 (s−σ) A−1
2 B A1 A1 w̄(σ)dσ
Z s0
−1/2
−1/2
∗
+ BA2
e−A2 (s−σ) A2 B ∗ BA−1
2 B v̄(σ)dσ
0
Z s
Z 0
−1/2
1/2
1/2
∗ 1/2
− BA2
A2 e−A2 (s−σ) A−1
B
A
g(τ )A1 z̄(s, τ )dτ dσ.
2
1
0
−∞
6
E. M. AIT BEN HASSI, H. BOUSLOUS, L. MANIAR
−1/2
EJDE-2011/71
1/2
Since BA2
is a bounded operator and s 7→ e−A2 s , s 7→ A2 e−A2 s are norm
−1/2 1/2
continuous from (0, ∞) to L(H2 ), then the mappings s 7→ BA2 A2 e−A2 s and
−1/2
−1/2
s 7→ BA2 e−A2 s A2 B ∗ are norm continuous from (0, ∞) to L(H1 ).
∗ 1/2
Under the assumption (2.2) we have A−1
is a bounded operator ; so there
2 B A1
∗ 1/2 1/2
0
exists a constant α(s) such that, kA−1
B
A
A
2
1
1 w̄(σ)k ≤ α(s)kx k, for every
R s 1/2 −A (s−σ)
1/2
1/2
∗
σ ∈ [0, s]. Thus, s 7→ 0 A2 e 2
A−1
2 B A1 A1 w̄(σ)dσ is continuous in
0
(0, ∞) uniformly in kx k ≤ 1.
−1/2
∗
Since A2 B ∗ BA−1
2 B is a bounded operator, using the same argument
Z s
−1/2
∗
s 7→
e−A2 (s−σ) A2 B ∗ BA−1
2 B v̄(σ)dσ
0
is continuous in (0, ∞) uniformly in kx0 k ≤ 1. In the other hand
Z 0
1/2 Z 0
Z 0
1/2
1/2
g(τ )kA1 z̄(s, τ )k2 dτ 1/2.
g(τ )dτ
k
g(τ )A1 z̄(s, τ )dτ dσk ≤
−∞
−∞
1/2
∗
So A−1
2 B A1
R0
1/2
−∞
g(τ )A1 z̄(s, τ )dτ is a bounded operator uniformly in kx0 k ≤ 1.
R0
R s 1/2
1/2
∗ 1/2
As a consequence, s 7→ 0 A2 e−A2 (s−σ) A−1
2 B A1 ( −∞ g(τ )A1 z̄(s, τ )dτ )dσ is
norm continuous in (0, ∞) uniformly in kx0 k ≤ 1. Finally, kF (s + h) − F (s)k → 0
as h → 0 uniformly in x0 . We have


w(t) − w̄(t)
 v(t) − v̄(t) 


 z(t) − z̄(t) 
u(t) − ū(t)


0
Z t
BA−1 B ∗ v̄(s) − B ū(s)
2
 ds
=
T (t − s) 


0
0
0




0
0
Z t+h
Z t
F (s)
F (s)


=
T (t + h − s) 
T (t − s) 
 0  ds −
 0  ds
0
0
0
0




0
0
Z t+h
Z t
F (t − s)
F (t + h − s)
 ds −
 ds
T (s) 
=
T (s) 




0
0
0
0
0
0




0
0
Z t
Z t+h
F (s)
F (t + h − s) − F (t − s)
 ds +

T (t + h − s) 
=
T (s) 


 0  ds
0
0
t
0
0




0
0
Z t
Z h
F (t + h − s) − F (t − s)
F (t + s)
 ds +
 ds.
=
T (s) 
T (s) 




0
0
0
0
0
0
−∞
EJDE-2011/71
COMPACT DECOUPLING
7
Hence,

 

w(t + h) − w̄(t + h)
w(t) − w̄(t)
 v(t + h) − v̄(t + h)   v(t) − v̄(t) 
 

k
 z(t + h) − z̄(t + h)  −  z(t) − z̄(t)  k
u(t + h) − ū(t + h)
u(t) − ū(t)




0
0
Z t
Z h
F (t + h − s) − F (t − s)
F (t + s)
 dsk + k
 dsk
≤k
T (s) 
T (s) 




0
0
0
0
0
0
Z t
Z h
≤ sup kT (τ )k
kF (t + h − s) − F (t − s)kds +
kT (s)kkF (t + s)kds.
τ ∈[0,t]
0
0
In addition, there exists a constant N such that sups∈[0,t+1] kF (s)k ≤ N uniformly
in x0 , and thus

 

w(t + h) − w̄(t + h)
w(t) − w̄(t)
 v(t + h) − v̄(t + h)   v(t) − v̄(t) 
 

k
 z(t + h) − z̄(t + h)  −  z(t) − z̄(t)  k
u(t + h) − ū(t + h)
u(t) − ū(t)
Z t
≤ sup kT (τ )k
kF (t + h − s) − F (t − s)kds + c(t)N h.
τ ∈[0,t+1]
0
As kF (s + h) − F (s)k → 0 as h → 0 uniformly in x0 , we conclude that
Z t
kF (t + h − s) − F (t − s)kds → 0, h → 0
0
uniformly for x0 ∈ D(L) verifying kx0 k ≤ 1. This achieves the proof.
Theorem 3.3. The maps t 7→ Td (t)−S(t) and t 7→ T (t)−S(t) are norm continuous
from (0, ∞) to L(H).
Proof. Let x0 = (u0 , v 0 , f0 , w0 ) ∈ H such that kx0 k ≤ 1 and t > 0.
Td (t)x0 − (S(t)((u0 , v 0 , f0 ), 0)
Z t
−A2 t 0
= (0, 0, 0, e
w +
e−A2 (t−s) B ∗ π2 S(s)((u0 , v 0 , f0 )ds)
0
1/2
D(A1 )
1/2
2
where π2 :
× H1 × L (g, (−∞, 0), D(A1 )) → H1 , (u, v, z) 7→ v. Set
∆(t) = Td (t)x0 − (S(t)(u0 , v 0 , f0 ), 0). For h > 0, one has
∆(t + h) − ∆(t) = 0, 0, e−A2 (t+h) w0 − e−A2 t w0
Z t+h
+
e−A2 (t+h−s) B ∗ π2 S(s)(u0 , v 0 , f0 )ds
0
Z t
−
e−A2 (t−s) B ∗ π2 S(s)(u0 , v 0 , f0 )ds .
0
Then
k∆(t + h) − ∆(t)k
−A2 (t+h)
= ke
0
−A2 t
w −e
0
Z
w +
0
t+h
e−A2 (t+h−s) B ∗ π2 S(s)(u0 , v 0 , f0 )ds
8
E. M. AIT BEN HASSI, H. BOUSLOUS, L. MANIAR
Z
t
Z
t+h
EJDE-2011/71
e−A2 (t−s) B ∗ π2 S(s)(u0 , v 0 , f0 )ds)k
0
Z t
−A2 (t+h) 0
= ke
w − e−A2 t w0 +
[e−A2 (t+h−s) − e−A2 (t−s) ]B ∗ π2 S(s)(u0 , v 0 , f0 )ds
−
0
e−A2 (t+h−s) B ∗ π2 S(s)(u0 , v 0 , f0 )ds)k
+
t
−A2 (t+h)
≤ ke
− e−A2 t kkw0 k
t
Z
[e−A2 (t+h−s) − e−A2 (t−s) ]B ∗ π2 S(s)(u0 , v 0 , f0 )dsk
+k
0
t+h
Z
e−A2 (t+h−s) B ∗ π2 S(s)(u0 , v 0 , f0 )ds)k.
+k
t
−1/2
Hence, since A2
B ∗ is a bounded operator, we have
k∆(t + h) − ∆(t)k
Z t
−1/2
1/2
1/2
≤
k[A2 e−A2 (t+h−s) − A2 e−A2 (t−s) ]A2 B ∗ π2 S(s)(u0 , v 0 , f0 )kds
0
t+h
Z
+
Z
−1/2
1/2
kA2 e−A2 (t+h−s) A2
B ∗ π2 S(s)(u0 , v 0 , f0 ))kds + ke−A2 (t+h) − e−A2 t k
t
t
1/2
−1/2
1/2
k[A2 e−A2 (t+h−s) − A2 e−A2 (t−s) ]A2
≤
B ∗ π2 S(s)(u0 , v 0 , f0 )kds
0
Z
t+h
−1/2
1/2
kA2 e−A2 (t+h−s) A2
+
B ∗ π2 S(s)(u0 , v 0 , f0 ))kds + ke−A2 (t+h) − e−A2 t k.
t
Since the semigroup (e−A2 t )t≥0 is analytic, it is immediately norm continuous. Thus
−1/2
−1/2
ke−A2 (t+h) − e−A2 t k → 0 as h → 0. In the other hand A2 B ∗ = (BA2 )∗ is a
bounded operator, thus there exists a constant δ(t) such that
−1/2
kA2
−1/2
B ∗ π2 S(s)(u0 , v 0 , f0 )k ≤ δ(t)kA2
−1/2
kA2 B ∗ π2 S(s)(u0 , v 0 , f0 )k
≤
−1/2
kA2 B ∗ π2 S(s)(u0 , v 0 , f0 )k
B ∗ kk(u0 , v 0 , f0 )k
−1/2
δ(t)kA2 B ∗ kkx0 k
≤
−1/2
δ(t)kA2 B ∗ k
for every s ∈ [0, t]
for every s ∈ [0, t]
for every s ∈ [0, t].
Moreover
Z t
1/2
1/2
−1/2
k
[A2 e−A2 (t+h−s) − A2 e−A2 (t−s) ]A2 B ∗ π2 S(s)(u0 , v 0 , f0 )dsk
0
Z t
−1/2
1/2
1/2
≤ δ(t)kA2 B ∗ k
kA2 e−A2 (t+h−s) − A2 e−A2 (t−s) kds.
0
It follows from Lemma 3.1 and Lebegue theorem that
1/2
1/2
kA2 e−A2 (t+h−s) − A2 e−A2 (t−s) k → 0
as h → 0 and t > s, and
Z t
1/2
1/2
−1/2
k
[A2 e−A2 (t+h−s) − A2 e−A2 (t−s) ]A2 B ∗ π2 S(s)(u0 , v 0 , f0 )dsk → 0
0
EJDE-2011/71
COMPACT DECOUPLING
9
as h → 0 uniformly in x0 . For the third term, we have
Z t+h
1/2
−1/2
k
A2 e−A2 (t+h−s) A2 B ∗ π2 S(s)(u0 , v 0 , f0 )ds)k
t
Z
=k
h
−1/2
1/2
A2 e−A2 s A2
B ∗ π2 S(s)(u0 , v 0 , f0 )dsk.
0
Using the similar argument as in the second term, there exists β(t) such that
Z h
Z h
1/2
−1/2
1/2
k
A2 e−A2 s A2 B ∗ π2 S(s)(u0 , v 0 , f0 )dsk ≤ β(t)
kA2 e−A2 s kdskx0 k
0
0
Z h
≤ kx0 kβ(t)
s−1/2 ds
0
√
≤ 2β(t) h
1/2
(here we used kA2 e−A2 t k = O(t−1/2 ) as t > 0; see for example [6, Theorem 1.4.3]).
R t+h 1/2
−1/2
Consequently, k t A2 e−A2 (t+h−s) A2 B ∗ π2 S(s)(u0 , v 0 , f0 )ds)k → 0 as h → 0
uniformly in x0 . Finally, k∆(t + h) − ∆(t)k → 0 as h → 0 uniformly in x0 . Since,
by Theorem 3.2, t 7→ T (t) − Td (t) is norm continuous on (0, ∞), t 7→ T (t) − S(t) is
norm continuous.
Theorem 3.3 leads to the following result.
Corollary 3.4. rcrit (T (t)) = rcrit (Td (t)) = rcrit (S(t)) for t ≥ 0 and ω0 (T ) =
max{ωcrit (Sw ), s(L)}.
4. Compactness of the difference between the two semigroups
We have also this main result.
R0
−1/2
2
Theorem 4.1. Assume A1 BA−1
2 is compact in L(H2 , H1 ) and −∞ g(s)s ds <
∞. Then R(λ, L) − R(λ, Ld ) is compact on H for every λ ∈ ρ(L) ∩ ρ(Ld ).
Proof. We have
R(λ, Ld ) − R(λ, L) = LR(λ, L)[L−1 − L−1
d ]Ld R(λ, Ld ).
1/2
L2 (g, (−∞, 0), D(A1 ))
1/2
D(A1 )
(4.1)
× H2 . We look for
× H1 ×
Let (ϕ, ψ, η, ξ) ∈ H =
(w, v, z, u) ∈ D(L) such that L(w, v, z, u) = (ϕ, ψ, η, ξ). Note that the equation
L(w, v, z, u) = (ϕ, ψ, η, ξ) is safistied is equivalent to the system
v=ϕ
Z
0
−kA1 w −
g(s)A1 z(t, s)ds − Bu = ψv − zs = η
−∞
−A2 u + B ∗ v = ξ
which is equivalent to the system
v=ϕ
Z
0
−kA1 w −
g(s)A1 z(t, s)ds − Bu = ψ
−∞
zs = ϕ − η
−A2 u + B ∗ v = ξ
10
E. M. AIT BEN HASSI, H. BOUSLOUS, L. MANIAR
EJDE-2011/71
which is equivalent to the system
Z 0
−kA1 w −
g(s)A1 z(t, s)ds − Bu = ψ
−∞
v=ϕ
Z s
z = sϕ −
η(τ )dτ
0
−1
∗
u = A−1
2 B ϕ − A2 ξ.
By assumption,
R0
1/2
−∞
1/2
L2 (g, (−∞, 0), D(A1 )).
∀s ∈ (−∞, 0),
g(s)s2 ds < ∞, and since ϕ ∈ D(A1 ), we have sϕ ∈
Using Hölder theorem,
Z
Z 0
2
kA1 1/2η(τ )kdτ ≤ −s
s
0
kA1 1/2η(τ )k2 dτ.
s
We can assume that η has compact support in (0, ∞), and then
Z 0
Z 0
−
g(s)s
kA1 1/2η(τ )k2 dτ ds < ∞.
−∞
2
s
1/2
(g, (−∞, 0), D(A1 )).
Thus z ∈ L
lent to the system
−1/2
w = −k −1 A1
−
Z
[(
Note that L(w, v, z, u) = (ϕ, ψ, η, ξ) is equiva-
0
Z
1/2
0
g(s)sds)A1 ϕ +
−∞
−1 ∗
−1
(kA1 ) BA2 B ϕ
+
1/2
0
Z
g(s)A1
−∞
−1
−1
(kA1 ) BA2 ξ −
η(τ )dτ ds]
s
(kA1 )−1 ψ
v=ϕ
Z s
z = sϕ −
η(τ )dτ
u=
0
−1 ∗
A2 B ϕ
− A−1
2 ξ
which is equivalent to the system
Z 0
Z
−1
−1/2
w = −(k) (
g(s)sds)ϕ − (kA1 )
−
−∞
∗
(kA1 )−1 BA−1
2 B ϕ
+
0
−∞
(kA1 )−1 BA−1
2 ξ−
1/2
g(s)A1
Z
0
η(τ )dτ ds
s
(kA1 )−1 ψ
v=ϕ
Z s
z = sϕ −
η(τ )dτ
0
−1
∗
u = A−1
2 B ϕ − A2 ξ.
∗
Replacing Bu by BA−1
2 B v̄ and repeating the above procedure for L, we can prove
that the equation Ld (w̄, v̄, z̄, ū) = (ϕ, ψ, η, ξ) is equivalent to the system
v̄ = ϕ
Z
0
−kA1 w̄ −
−∞
∗
g(s)A1 z̄(t, s)ds − BA−1
2 B v̄ = ψ
z̄s = ϕ − η
−A2 ū + B ∗ v̄ = ξ
EJDE-2011/71
COMPACT DECOUPLING
which is equivalent to the system
h Z
−1 −1/2
w̄ = (−k) A1
0
Z
−
−∞
1/2
Z
0
−∞
s
g(s)A1
11
1/2
g(s)sds A1 ϕ
∗
η(τ )dτ ds + BA−1
2 B ϕ+ψ
i
0
v̄ = ϕ
Z s
z̄ = sϕ −
η(τ )dτ
0
−1
∗
ū = A−1
2 B ϕ − A2 ξ
which is equivalent to the system
Z
Z 0
g(s)sds)ϕ + (kA1 )−1/2
w̄ = −k −1 (
−∞
−∞
−1
− (kA1 )
0
1/2
Z
g(s)A1
s
η(τ )dτ ds
0
∗
−1
BA−1
ψ
2 B ϕ − (kA1 )
v̄ = ϕ
Z s
z̄ = sϕ −
η(τ )dτ
ū =
0
−1 ∗
A2 B ϕ
− A−1
2 ξ.
Therefore, by an easy computation one obtains

0 0 (kA1 )−1 BA−1
2

0 0
0
−1
−1

L − Ld = 
0 0
0
0 0
0

0
0

0
0
which is a compact operator by assumption. The claim follows from the equality
(4.1).
Now by the Theorem 4.1, Theorem 3.2 and [4, Theorem 3.2], we obtain the main
result of this section.
−1/2
Theorem 4.2. Assume that A1 BA−1
2 is compact in L(H2 , H1 ), (1.4)-(1.6) and
(2.2) hold. Then T (t) − Td (t) is compact on H for all t ≥ 0.
As a consequence of this theorem, we have the following result.
Corollary 4.3. ress (T (t)) = ress (Td (t)) for t ≥ 0 and ω0 (T ) = max{ωess (Td ), s(L)}.
Remark 4.4. The result of Theorem 4.2 has been shown in [5] directly, assuming
the compactness of the operator BA−γ
for some 0 < γ < 1. It is clear that this
2
−1/2
−1
last assumption implies that A1 BA2 is compact from H2 to H1 .
5. Application
We consider the following model for a linear viscoelastic body Ω of Boltzmann
type with thermal damping
ẅ − µ∆w − (λ + µ)∇ div w + µg1 ∗ ∆w + (λ + µ)g1 ∗ ∇ div w + m∇u = 0
(5.1)
in Ω × (0, ∞),
u̇ + βu − ∆u − m div ẇ = 0
in Ω × (0, ∞),
(5.2)
12
E. M. AIT BEN HASSI, H. BOUSLOUS, L. MANIAR
∂u
= 0 on Γ × (0, ∞),
∂n
ẇ(x, 0) = w1 (x), u(x, 0) = u0 (x)
EJDE-2011/71
w = 0,
w(x, 0) = w0 (x),
w(x, 0) + w(x, s) = f0 (x, s)
(5.3)
in Ω,
in Ω × (−∞, 0)),
(5.4)
(5.5)
where λ, µ > 0 the Lame’s constants and m > 0 is the thermal strain parameter, β is
a positive constant and g is a given function which satisfies the following conditions
(C1) g1 ∈ C 1 [0, ∞) ∩ L1 (0, ∞).
(C2) gR1 (t) ≥ 0 and g10 (t) ≤ 0 for t > 0,
∞
(C3) 0 g1 (s)ds < 1.
The set Ω is the bounded open Jelly Roll set defined in [9],
1
Ω = {(x, y) ∈ R2 : < r < 1} \ Γ,
2
where Γ is the curve in R2 given in polar coordinates by
r(φ) =
Note that
Z
3π
2
+ arctan(φ)
,
2π
−∞ < φ < ∞.
t
g1 (t − s)∆w(x, s)ds
−∞
Z
0
=
g1 (−s)∆w(x, t + s)ds
−∞
Z 0
Z
0
g1 (−s)∆(w(x, t + s) − w(x, t))ds +
=
−∞
Z 0
g1 (−s)∆w(x, t)ds
−∞
Z
=−
0
g1 (−s)∆z(x, t, s)ds +
g1 (−s)ds∆w(x, t).
−∞
−∞
A similar expression can be establish for g1 ∗ ∇div w. In order to fit the system
(5.1)-(5.5) into the setting abstract system (1.1)-(1.3), we take
H1 = L2 (Ω, R2 ),
H2 = L2 (Ω, R),
z(x, t, s) = w(x, t) − w(x, t + s),
g(s) = g1 (−s),
and define the operators A1 , A2 , B by
A1 w = −µ∆D w − (λ + µ)∇ divw,
A2 u = (βI − ∆N )u,
Bu = m∇u,
D(A1 ) = D(∆D ) = H01 (Ω, R2 ),
D(A2 ) = D(∆N ) = H 1 (Ω, R),
D(B) = H 1 (Ω, R).
Here, as the domain Ω is irregular the Dirichlet Laplacian ∆D and the Neumann
Laplacian ∆N are defined via quadratic forms. More precisely, ∆D is the unique
positive self adjoint operator associated to the closed quadratic form on H01 (Ω)
Z
h∆f, gi =
∇f ∇g dx,
Ω
and ∆N is the unique non negative self adjoint operator associated to the closed
quadratic form on H 1 (Ω)
Z
h∆f, gi =
∇f ∇g dx.
Ω
EJDE-2011/71
COMPACT DECOUPLING
13
It is clear that the adjoint B ∗ of B is
→
B ∗ w = −m div w, D(B ∗ ) = {w ∈ H 1 (Ω, R2 ) : w · −
n = 0 in ∂Ω},
→
−
where n is the outward unit normal vector on the boundary ∂Ω. Note that
1/2
D(A2 ) = D(B)
and
1/2
D(A1 ) ,→ D(B ∗ ).
We have the following facts.
(i) A−1
2 is not compact on H2 , see [9], but A1 has a compact resolvent on H1 .
−1/2
−1/2
Consequently, A1
and A1 BA−1
2 are compact .
(ii) For every γ ∈ (0, 1], BA−γ
is not compact from H2 into H1 . In fact, it is
2
enough to show that BA−1
is not compact from H2 to H1 . For this, we
2
have
−1
−1
−1
−1
−2
∗
2 −1
2 −1
2
A−1
2 B BA2 = m A2 (−∆N )A2 = m A2 (A2 − βI)A2 = m (A2 − βA2 ).
−1
∗
2
−1
Using the spectral mapping theorem, we have σ(A−1
−
2 B BA2 ) = m (σ(A2 )
−1 ∗
−1
−2
βσ(A2 ) ). As in [9], (β, ∞) ⊂ σ(A2 ), so A2 B BA2 is not compact on H2 .
Consequently BA−1
2 is not compact from H2 to H1 .
References
[1] E. M. Ait Ben Hassi, H. Bouslous and L. Maniar; Compact decoupling for thermoelasticity
in irregular domains, Asymptot. Anal. 58 (2008), 47–56.
[2] K. J. Engel and R. Nagel; One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer-Verlag, 2000.
[3] D. B. Henry, A. Perissinitto and O. Lopes; On the essential spectrum of a semigroup of
thermoelasticity, Nonlinear Anal., TMA 21 (1993), 65–75.
[4] M. Li, X. Gu and F. Huang; Unbounded Perturbations of Semigroups: Compactness and
Norm Continuity, Semigroup Forum 65 (2002), 58–70.
[5] W. J. Liu; Compactness of the difference between the thermoviscoelastic semigroup and its
decoupled semigroup, Rocky Mountain J. Math. 30 (2000), 1039–1056.
[6] A. Lunardi; Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser,
Basel-Boston-Berlin, 1995.
[7] A. F. Neves, H. S. Ribeiro and O. Lopes; On the spectrum of evolution operators generated
by hyperbolic systems, J. Funct. Anal. 67 (1986), 320–344.
[8] Lopes, Orlando; On the structure of the spectrum of a linear time periodic wave equation, J.
Analyse Math. 47 (1986), 55–68.
[9] B. Simon; The Neumann Laplacian of a Jelly Roll, Proc. Amer. Math. Soc. 114 (1992),
783–785.
Département de Mathématiques, Faculté des Sciences Semlalia,
Université Cadi Ayyad, Marrakech 40000, B.P. 2390, Maroc
E-mail address, E. M. Ait Ben Hassi: m.benhassi@ucam.ac.ma
E-mail address, H. Bouslous: bouslous@ucam.ac.ma
E-mail address, L. Maniar: maniar@ucam.ac.ma