On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
The relation between the thermoelastic Cattaneo model and the thermoelastic Gurtin-Pipkin model is established. The existence of the compact global attractor of the Cattaneo-Mindlin plate model is proved and its properties are studied. Установлена зависимость между моделями термоупругости Каттанео и...
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| Zitieren: | On the Long-Time Behavior of the Thermoelastic Plates with Second Sound / T.B. Fastovska // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 191-206. — Бібліогр.: 14 назв. — англ. |
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| citation_txt | On the Long-Time Behavior of the Thermoelastic Plates with Second Sound / T.B. Fastovska // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 191-206. — Бібліогр.: 14 назв. — англ. |
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| description | The relation between the thermoelastic Cattaneo model and the thermoelastic Gurtin-Pipkin model is established. The existence of the compact global attractor of the Cattaneo-Mindlin plate model is proved and its properties are studied.
Установлена зависимость между моделями термоупругости Каттанео и Гертина-Пипкина. Доказано существование компактного глобального аттрактора модели термоупругости Каттанео-Миндлина и изучены его свойства.
|
| first_indexed | 2025-12-07T17:17:57Z |
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Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 2, pp. 191–206
On the Long-Time Behavior of the Thermoelastic Plates
with Second Sound
T.B. Fastovska
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University,
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail: Tamara.B.Fastovska@univer.kharkov.ua
Received December 27, 2011, revised March 27, 2012
The relation between the thermoelastic Cattaneo model and the ther-
moelastic Gurtin–Pipkin model is established. The existence of the com-
pact global attractor of the Cattaneo–Mindlin plate model is proved and its
properties are studied.
Key words: second sound, asymptotic behavior, global attractor.
Mathematics Subject Classification 2010: 35B40, 45K05, 74D05.
1. Introduction and Functional Settings
In this paper we consider the nonlinear Mindlin–Timoshenko model of a ther-
moelastic plate with heat conduction of Cattaneo type (for review see [1]). The
Mindlin–Timoshenko model describes dynamics of a plate in view of transverse
shear effects (see, e.g., [2, 3] and references therein). Unlike the classical Fourier
constitutive law, the Cattaneo model describes heat conduction processes under
the assumption of finite speed propagation of disturbances.
We assume that the plate has a uniform thickness h and, when in equilib-
rium, its middle surface lies in the bounded domain Ω ⊂ (x1, x2, 0) with the
sufficiently smooth boundary ∂Ω. The integro-differential equations for the vec-
tor of angles of deflection of the filament v(x, t) = (v1(x, t), v2(x, t)) ∈ R2, the
transverse displacement of the middle surface w(x, t) ∈ R, the temperature varia-
tion θ(x, t) and the heat flax q(x, t) averaged with respect to the thickness, where
x = (x1, x2) ∈ Ω and t ≥ 0, are the following:
α0vtt + β0vt −Av + µ(v +∇w) + β∇θ +∇vΦ(v) = 0,
α1wtt + β1wt − µdiv(v +∇w) + g(w) = 0,
γθt + κdivq + βdivvt = 0,
ωqt + q +∇θ = 0.
(1)
c© T.B. Fastovska, 2013
T.B. Fastovska
Here the vector function ∇vΦ(v) = (∂v1Φ(v1, v2), ∂v2Φ(v1, v2)) and the scalar
function g(w) are the feedback forcing terms. The parameters α0, α1, β0, β1, β,
γ, κ, µ, ω are positive constants. The operator A has the structure
A =
∂2
x1
+ 1−ν
2 ∂2
x2
1+ν
2 ∂x1x2
1+ν
2 ∂x1x2
1−ν
2 ∂2
x1
+ ∂2
x2
= ∇div− 1− ν
2
rotrot,
where 0 < ν < 1 is the viscoelastic Poisson’s ratio.
The Timoshenko systems have been treated by many authors. There are
several works investigating the presence or the lack of the exponential stability
of linear and nonlinear problems with various types of damping and boundary
conditions for the Timoshenko ( see, e.g., [4, 5]) and Timoshenko–Cattaneo prob-
lems [6, 7]. The existence and the properties of attractors for the related systems
were established in [8–10]. In [8], the existence of a compact global attractor
for the Mindlin–Timoshenko elasticity and its upper semicontinuity, as the shear
modulus tends to infinity, are shown. Paper [10] is devoted to the existence of
a compact global attractor and its properties of the Mindlin–Timoshenko vis-
coelastic system of memory type coupled with Gurtin–Pipkin heat conduction
equations (see [11] for the model description). The long-time behavior of the
Mindlin–Timoshenko problem
α0vtt + β0vt −Av + µ(v +∇w) + β∇θ +∇vΦ(v) = 0,
α1wtt + β1wt − µdiv(v +∇w) + g(w) = 0,
γθt − 1
ω2
∞∫
0
η( s
ω )∆τ(s)ds + βdivvt = 0,
θ = τt + τs, s ≥ 0
(2)
for the model with Gurtin–Pipkin heat conduction with Dirichlet boundary con-
ditions was studied in [9].
The main goal of the paper is to study the long-time behavior of the semilin-
ear thermoelastic Mindlin–Timoshenko–Cattaneo system with locally Lipschitz
nonlinearities of any polynomial growth (of odd degrees) and to establish the
closeness of the family of attractors to the attractor of the Fourier thermoelastic
model in a suitable sense in limit case ω → 0. In the present paper, we establish
the relation between the dynamics of problems (1) and (2). It is shown that
(1) can be decomposed into two systems. The energy of the first system decays
exponentially to zero. The dynamics of the second one is connected with the
dynamics of system (2) according to the law given in Lemma 2. Additionally, we
describe the relation between the structures of the attractors of systems (1) and
(2). We also establish the upper-semicontinuity of the family of attractors of (1)
with respect to the relaxation time.
192 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
Rewrite the thermoelastic model with second sound (1) in the following way:
Putt + Mut + Au + Rθ = F (u),
γθt + κdivq + βdivvt = 0,
ωqt + q +∇θ = 0,
(3)
and subject it to the initial and boundary conditions
u(x, 0) = u0(x) ∈ [H1
0 (Ω)]3, ut(x, 0) = u1(x) ∈ [L2(Ω)]3, x ∈ Ω,
θ(x, 0) = θ0(x) ∈ L2(Ω), q(x, 0) = q0(x) ∈ [L2(Ω)]2, x ∈ Ω,
θ(x, t) = 0, u(x, t) = 0, x ∈ ∂Ω, t ≥ 0.
(4)
The operator A with the domain
D(A) = {u = (v1, v2, w) ∈ [(H2 ∩H1
0 )(Ω)]3}
has the structure
A =
( −A + µI µ∇
−µdiv −µ∆
)
.
Obviously, A is a positive self-adjoint operator with the square root possessing
the domain D(A1/2) = [H1
0 (Ω)]3. It is easy to see that the operators
{R : H1
0 (Ω) → [L2(Ω)]3, Rθ = β(∂1θ, ∂2θ, 0)}
and
{Q : [H1
0 (Ω)]3 → L2(Ω), Qu = −β(∂1u1 + ∂2u2), u = (u1, u2, u3)}
possess the property (Rθ, u) = (θ, Qu) for any θ ∈ H1
0 (Ω) and u ∈ [H1
0 (Ω)]3. The
bounded in [L2(Ω)]3 operators P and M are defined by the formulas
P =
(
α0I 0
0 α1I
)
, M =
(
β0I 0
0 β1I
)
.
The nonlinear term has the structure
F (u) =
−∂v1Φ(v1, v2)
−∂v2Φ(v1, v2)
−g(w)
, u = (v1, v2, w). (5)
We assume that the memory kernel η(s) possesses the properties
η ∈ C1(R+) ∩ L1(R+), (6)
moreover,
η(s) ≥ 0, (7)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 193
T.B. Fastovska
and for any s ∈ R+ there exists l > 0 such that for any s ∈ R+
η′(s) + lη(s) ≤ 0. (8)
We will use the notation
ηω(s) =
1
ω2
η(
s
ω
). (9)
Introduce a Hilbert space L2
ω(R+,H l(Ω)) of H l-valued functions on R+ (l ∈ R)
such that
‖τ‖2
L2
ω(R+,Hl(Ω)) ≡ ω
∞∫
0
ηω(s)‖τ(s)‖2
Hl(Ω)ds < ∞,
which denotes a norm in this space. We endow L2
ω(R+,H l(Ω)) with the inner
product
(φ1, φ2)L2
ω(R+,Hl(Ω)) = ω
∞∫
0
ηω(s)(φ1(s), φ2(s))Hl(Ω)ds.
We will also need the space
H1
ω(R+,H1
0 (Ω)) = {φ : φ(s), φs(s) ∈ L2
ω(R+, H1
0 (Ω))}.
Define the operator Tω : D(Tω) → L2(Ω) with the domain
D(Tω) = {τ ∈ L2
ω(R+, H1
0 (Ω)) :
∞∫
0
ηω(s)∆τds ∈ L2(Ω), τ |s=0 = 0}
by the formula
Tωτ = −
∞∫
0
ηω(s)∆τds.
We assume that the nonlinearities of the problem satisfy the conditions
g ∈ C1(R), Φ ∈ C2(R2), (10)
and there exist q > 0 and C̃ > 0 such that
|g′(z)| ≤ C̃(1 + |z|q),
|∂2
1Φ(z)|+ |∂2
2Φ(z)|+ |∂1∂2Φ(z)| ≤ C̃(1 + |z|q). (11)
Moreover, there exist bi ∈ R, i = 1, 2 such that
Φ(z1, z2) ≥ −b1,
G(z) ≡
z∫
0
g(ζ)dζ ≥ −b2.
(12)
194 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
We also assume that there exist ai > 0, i = 1, 4 such that
−a1Φ(z) +∇zΦ(z) z ≥ −a2,
−a3G(z) + g(z) z ≥ −a4.
(13)
In the paper we will establish the relation between problem (3)–(4) and the
Gurtin–Pipkin thermoelastic model considered in [9]:
Putt + Mut + Au + Rθ = F (u), (14)
γθt + Tωτ −Qut = 0, (15)
θ = τt + τs, s ≥ 0 (16)
u|t=0 = u0, ut|t=0 = u1, θ|t=0 = θ0, τ(t, s)|t=0 = τ0(s), s ≥ 0. (17)
Introduce the space
Xω = D(A1/2)× [L2(Ω)]3 × L2(Ω)× [L2(Ω)]2
with the inner product
u1
v1
θ1
q1
,
u2
v2
θ2
q2
Xω
=(A1/2u1, A
1/2u2)+(P 1/2v1, P
1/2v2)+γ(θ1, θ2)+ωκ(q1,q2)
and the space
D̃ = {φ ∈ [L2(Ω)]2 : divφ ∈ L2(Ω)}
with the inner product
(q1,q2)D̃ = (divq1, divq2) + (q1,q2).
We define the operator Bω : Xω ⊃ D(Bω) → Xω with the domain
D(Bω) = {(u, u, θ,q) : u ∈ D(A), u ∈ D(A1/2), θ ∈ H1
0 (Ω),q ∈ D̃}
by the formula
Bω =
0 I 0 0
−P−1A −P−1M −P−1R 0
0 1
γ Q 0 −κ
γ div
0 0 − 1
ω∇ − 1
ω
.
We will use the notations Z(t) = (u(t), u(t), θ(t),q(t)) and Z0 = (u0, u1, θ0,q0)
∈ Xω, where u(t) = ut(t). Then the problem (3)–(4) can be rewritten as follows:
d
dtZ(t)− BωZ(t) = f(Z(t)),
Z(0) = Z0.
(18)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 195
T.B. Fastovska
Introduce the space
Hω = D(A1/2)× [L2(Ω)]3 × L2(Ω)× L2
ω(R+,H1
0 (Ω))
with the inner product
u1
v1
θ1
τ1
,
u2
v2
θ2
τ2
Hω
= (A1/2u1, A
1/2u2) + (v1, v2) + γ(θ1, θ2) + 1
ω 〈∇τ1,∇τ2〉.
Denote Aη
ω : D(Aη
ω) → Hω to be
Aη
ω =
0 I 0 0
−A −D −R 0
0 1
γ Q 0 1
γ Tω
0 0 I −∂s
with the domain
D(Aη
ω) = D(A)×D(A1/2)×H1
0 (Ω)× [D(Tω) ∩H1
ω(R+,H1
0 (Ω))].
For problem (14)–(17), we use the result obtained in [9]:
Proposition 1. The operator Aη
ω is the generator of the C0-semigroup Uη
ω(t)
on the space Hω.
Relying on this result we give the definition of the mild solution to (14)–(17).
Definition 1. The function Z = (u, ut, θ, τ) ∈ C(0, T ; Hω) is a mild solution
to (14)–(17) on the interval [0, T ] subjected to the initial conditions Z(0) = Z0 =
(u0, u1, θ0, τ0) if the relation
Sω(t)Z0 = Z(t) = Uη
ω(t)Z0 +
t∫
0
Uη
ω(t− s)f(Z(s))ds
holds true for any t ∈ [0, T ].
The long-time behavior of the dynamical system generated by problem (14)–
(17), i.e., the existence and the properties of the compact global attractor, was
studied in paper [9]. By definition (see, e.g., [12, 13]), a global attractor is a
bounded closed set Aω ⊂ Hω such that Sω(t)Aω = Aω for all t ≥ 0, and
lim
t→+∞ sup
y∈B
dist(S(t)y, Aω) = 0
for any bounded set B ⊂ Hω .
196 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
We recall here the theorem proven in [9].
Theorem 1. Let the assumptions (6)–(8) and (10)–(13) hold. Then for any
ω > 0 the dynamical system (Sω(t),Hω) generated by (14)–(17) possesses a com-
pact global attractor Aω whose fractal dimension is finite. The family of attractors
{Aω} is upper semicontinuous at zero, i.e.,
sup
y∈Aω
distHω(y, A0) → 0, ω → 0,
where
A0 =
y =
u0
u1
θ0
0
:
u0
u1
θ0
∈ A
,
where A is the compact global attractor of the dynamical system generated by the
problem
Putt + Mut + Au + Rθ = F (u),
γθt − κ∆θ −Qut = 0,
u|t=0 = u0, ut|t=0 = u1, θ|t=0 = θ0
(19)
in the space [H1
0 (Ω)]3 × [L2(Ω)]2 × L2(Ω).
2. Long-Time Behavior
To study the dynamics of Mindlin–Timoshenko–Cattaneo system (3)–(4), we
consider the problems
Putt + Mut + Au + Rθ = F (u),
γθt + κ∆q + βdivvt = 0, x ∈ Ω, t > 0,
ωqt + q + θ = 0,
u(x, 0) = u0(x) ∈ [H1
0 (Ω)]3, ut(x, 0) = u1(x) ∈ [L2(Ω)]3,
θ(x, 0) = θ0(x) ∈ L2(Ω), q(x, 0) = q0(x) ∈ H1
0 (Ω)
(20)
and
ωpt + p = 0,
p(0) = p0 ∈ [L2(Ω)]2.
(21)
Define the space
Vω = D(A1/2)× [L2(Ω)]3 × L2(Ω)×H1
0 (Ω)
with the norm
‖(u, v, θ, q)‖2
Vω
= ‖A1/2u‖2 + ‖P 1/2v‖2 + γ‖θ‖2 + ωκ‖∇q‖2.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 197
T.B. Fastovska
Obviously problem (21) generates an exponentially stable dynamical system
in the space [L2(Ω)]2, moreover, p = p0e
− t
ω tends to zero as ω → 0.
Problem (20) can be rewritten as (14)–(17) with the kernel η(s) = κe−s if we
prolong θ in (20) backward in time, for instance, in the following way:
θ(x,−t) = −q0(x), t > 0.
It follows from the last equation in (20) that
q(x, t) = e−
t−ξ
ω q0(x)− 1
ω
t∫
ξ
e−
(t−s)
ω θ(x, s)ds, ξ ≤ t, t ≥ 0.
Consequently, letting ξ tend to −∞ and changing variables, we get
q(x, t) = − 1
ω2
∞∫
0
e−
s
ω τ(x, t, s)ds. (22)
Substituting (22) into (20) and defining
τ0 = −sq0(x) ∈ L2
ω(R+,H1
0 (Ω)),
we arrive at problem (14)–(17). It is easy to see that the kernel η(s) = κe−s
satisfies conditions (6)–(8), and η0 = η1 = κ. Consequently, Theorem 1 is valid
for the obtained problem of the kind (14)–(17). To establish the results analogous
to those stated in Theorem 1 for problem (3)–(4), we have to study the relation
between problems (14)–(17) and (20). At first we will show that the existence
of the dynamical system (Sω(t),Hω) generated by problem (14)–(17) with the
kernel η(s) = κe−s leads to the existence of the C0-semigroup Σω(t) on the space
Vω generated by the mild solutions of problem (20).
To give the definition of the solutions, we introduce the operatorsK : H1
0 (Ω) →
L2
ω(R+,H1
0 (Ω)) and N : L2
ω(R+,H1
0 (Ω)) → H1
0 (Ω) acting by the formulas
Kq = −sq and N τ = − 1
ω2
∞∫
0
e−
s
ω τds.
The operators K : Vω → Hω and N : Hω → Vω are defined by
K =
I 0 0 0
0 I 0 0
0 0 I 0
0 0 0 K
, N =
I 0 0 0
0 I 0 0
0 0 I 0
0 0 0 N
.
198 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
Let the operator Dω : Vω ⊃ D(Dω) → Vω with the domain
D(Dω) = {(u, u, θ, q) : u ∈ D(A), u ∈ D(A1/2), θ ∈ H1
0 (Ω),q ∈ H2 ∩H1
0 (Ω)}
have the structure
Dω =
0 I 0 0
−P−1A −P−1M −P−1R 0
0 1
γ Q 0 −κ
γ ∆
0 0 − κ
ω − 1
ω
.
The operators N and K possess the following properties.
Proposition 2. For any Z ∈ D(Dω),
NAωKZ = DωZ, (23)
where Aω is the generator of the semigroup generated by problem (14)–(17) with
the kernel η(s) = κe−s. Moreover,
NK = I ∈ L(Vω). (24)
P r o o f. It is obvious that if Z ∈ D(Dω), then KZ ∈ D(Aω). Therefore the
statement of the lemma can be easily proved by the straightforward calculations.
Let Z(t) = (u(t), u(t), θ(t), q(t)) and Z0 = (u0, u1, θ0, q0) ∈ Vω, where u(t) =
ut(t). Then problem (20) can be rewritten as follows:
d
dtZ(t)− DωZ(t) = f(Z(t)),
Z(0) = Z0.
(25)
Lemma 1. The operator Dω is the generator of the C0-semigroup Wω(t) =
NUω(t)K on the space Vω, where Uω(t) is the exponentially stable C0-semigroup
on the space Hω generated by the operator Aω.
P r o o f. Consider the operator Wω(t) = NUω(t)K. To prove that Wω(t) is
a C0-semigroup on the space Vω, we have to check the semigroup properties. It
follows from (24) that
Wω(0) = NUω(0)K = NK = I.
Now we prove that
Wω(r)Wω(t) = NUω(r)KNUω(t)K = NUω(t + r)K = Wω(t + r). (26)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 199
T.B. Fastovska
Define the operators Ui(t), i = 1, 4 as follows:
U1(t)Z0 = u(t), U2(t)Z0 = ut(t), U3(t)Z0 = θ(t), U4(t)Z0 = τ(t),
where (u(t), ut(t), θ(t), τ(t)) is a mild solution to problem (14)–(17) with initial
conditions Z0. Note that Uω(t) = (U1(t), U2(t), U3(t), U4(t)).
Point out that
Wω(r)Wω(t)Z0 = NUω(r)(U1(t)Z0, U2(t)Z0, U3(t)Z0,KNU4(t)KZ0)
= (U1(t + r)Z0, U2(t + r)Z0, U3(t + r)Z0,N
r∫
r−s
θ(t + ξ)dξ).
In order to derive (26), we must consider more precisely the fourth component
N
r∫
r−s
θ(t + ξ)dξ = N
r∫
t+r−s
θ(ξ)dξ
=
N [
r+t∫
0
U3(ξ)KZ0dξ + (t + r − s)Kq0], t + r − s < 0
N
r+t∫
0
U3(ξ)Z0dξ, t + r − s ≥ 0
= NU4(t + r)KZ0.
Consequently, (26) holds true.
It is easy to see that N ∈ L(Hω, Vω) and K ∈ L(Vω,Hω). This entails that
for any Z ∈ Hω,
lim
t→0
‖Wω(t)Z − Z‖Vω = lim
t→0
‖NUω(t)KZ −NKZ‖Vω
≤ lim
t→0
‖N‖L(Hω ,Vω)‖Uω(t)KZ − KZ‖Hω = 0.
Thus, the semigroup Wω(t) is strongly continuous.
Now we will describe the generator of this semigroup. It follows from property
(23) that for any Z ∈ D(Dω),
lim
t→0
Wω(t)Z − Z
t
= N lim
t→0
Uω(t)KZ − KZ
t
= NAωKZ = DωZ.
The lemma is proved.
Now we establish the well-posedness of problem (25).
Lemma 2. Assume that conditions (10)–(12) hold true. Then problem (25)
generates the dynamical system (Σω(t), Vω). Its evolution operator has the form
Σω(t) = NSω(t)K, where Sω(t) is the evolution operator of problem (14)–(17)
with η(s) = κe−s.
200 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
P r o o f. For any Z0 = (u0, u1, θ0, q0) ∈ Vω,
Z(t) = Σω(t)Z0 = NSω(t)KZ0 = NUω(t)KZ0
+
t∫
0
NUω(t− ξ)Kf(Z(ξ))dξ = Wω(t)Z0 +
t∫
0
Wω(t− ξ)f(Z(ξ))dξ,
where t ∈ [0, T ] for any 0 < T < ∞. Thus, the function Z(t) = Σω(t)Z0 ∈
C(0, T ; Vω) is a mild solution to (25). The lemma is proved.
Now we are ready to state the existence result for the attractor of system (20).
Lemma 3. Let conditions (10)–(13) hold true. Then for any ω > 0 the dy-
namical system (Σω(t), Vω), generated by (25), possesses a compact global
attractor Qω.
P r o o f. It is easy to see that
‖Σω(t)(u0, u1, θ0, q0)‖Vω ≤ ‖N‖L(Hω ,Vω)‖Sω(t)(u0, u1, θ0, τ0)‖Hω .
Consequently, the dynamical system (Σω(t),Vω) possesses an absorbing ball.
Consider the set
K = {
⋃
Y0∈Aω
⋃
t∈R
NSω(t)Y0} = N{
⋃
Y0∈Aω
⋃
t∈R
Sω(t)Y0}. (27)
Since the operator N is bounded and the set { ⋃
Y0∈Aω
⋃
t∈R
Sω(t)Y0} is compact in
Hω, set (27) is compact in Vω. Let B ∈ Vω be a positively invariant set, i.e.,
Σω(t)B ⊂ B. Then
lim
t→+∞ sup
z∈B
distVω(Σω(t)z,K) ≤ lim
t→+∞ sup
y∈G
distVω [NSω(t)y, Nỹ],
where
G = {y = Kz : z ∈ B} ⊂ Hω
is a bounded set in Hω, ỹ is the element on which the minimal distance in Hω
from the point Sω(t)y to the set Aω is reached. Then, obviously,
lim
t→+∞ sup
y∈B
distVω(Σω(t)y, K) ≤ C lim
t→+∞ sup
z∈G
distHω(Sω(t)y, Aω) = 0.
The above estimate implies that the dynamical system (Σω(t), Vω) is asympto-
tically compact. This property together with the existence of an absorbing ball
is a necessary condition for the existence of a compact global attractor Qω (see,
e.g., [12, 13]).
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 201
T.B. Fastovska
Now we study the relation between problems (20), (21) and initial statement
(3). Let us introduce the operator F : Vω → Xω
F =
I 0 0 0
0 I 0 0
0 0 I 0
0 0 0 ∇
.
It is well known (see [14]) that
[L2(Ω)]2 = E1 ⊕ E2, (28)
where
E1 = {∇φ : φ ∈ H1
0 (Ω)} and E2 = {ψ ∈ [L2(Ω)]2 : (divψ, φ) = 0, φ ∈ H1
0 (Ω)}.
By virtue of (28), the initial condition, the problem (3) is subjected to, can be
split as q0 = ∇q0 + p0, where q0 ∈ H1
0 (Ω) and p0 ∈ E2. Therefore, we can define
the operator P1 : [L2(Ω)]2 → H1
0 (Ω) by the formula P1q0 = −(−∆)−1divq0 = q0,
where −∆ is the Laplace operator with the Dirichlet boundary conditions and the
projector P2 : [L2(Ω)] → E2 as P2q0 = p0. Introduce the operators P1 : Xω → Vω
and P2 : Xω → Xω defined by the formulas
P1 =
I 0 0 0
0 I 0 0
0 0 I 0
0 0 0 P1
, P2 =
I 0 0 0
0 I 0 0
0 0 I 0
0 0 0 P2
.
The following properties of these operators can be easily checked by the straight-
forward calculations.
Proposition 3. For any Z ∈ D(Bω),
[FNAωKP1 +
1
ω
P2]Z = BωZ. (29)
Moreover,
FP1 + P2 = I ∈ L(Xω), (30)
and for any Z ∈ Vω,
P1FZ = Z. (31)
Define the operator
Sω(t)(u0, u1, θ0,q0) = (0, 0, 0, e−
t
ω p0) + FΣω(t)(u0, u1, θ0, q0)
= [e−
t
ω P2 + FΣω(t)P1](u0, u1, θ0,q0) (32)
in the space Xω. We are in position to show that Sω(t) is the evolution operator
of problem (3)–(4).
202 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
Lemma 4. The operator Bω is the generator of the C0-semigroup Ũω(t) =
FNUω(t)KP1 + e−
t
ω P2 on the phase space Xω.
P r o o f. First, we prove that Ũω(t) is the generator of the C0-semigroup
on the space Vω. To this end, we check the semigroup properties. We obtain by
(30) and (31)
Ũω(0) = FWω(0)P1 + P2 = FP1 + P2 = I
and
Ũω(r)Ũω(t) = FWω(r)P1FWω(t)P1 + e−
r+t
ω P2
= FWω(r + t)P1 + e−
r+t
ω P2 = Ũω(r + t).
It is easy to see that F ∈ L(Xω, Vω). Consequently, for any Z ∈ Xω,
lim
t→0
‖Ũω(t)Z − Z‖Xω
= lim
t→0
[‖F‖L(Xω ,Vω)‖Wω(t)P1Z −P1Z‖Vω + ‖e− t
ω P2Z −P2Z‖Xω ] = 0.
Therefore the semigroup Ũω(t) is strongly continuous.
Now we describe the generator of the semigroup. It follows from (29) that for
any Z ∈ D(Bω),
lim
t→0
Ũω(t)Z − Z
t
= F lim
t→0
Wω(t)− I
t
P1Z + lim
t→0
e−
t
ω − I
t
P2Z
= [FNAωKP1 +
1
ω
P2]Z = BωZ.
The lemma is proved.
Now we are in position to prove the well-posedness result for (18).
Lemma 5. Assume that conditions (10)–(12) hold true. Then (18) generates
the nonlinear dynamical system (Sω(t), Xω) with the evolution operator defined
in (32).
P r o o f. For any Z0 = (u0, u1, θ0,q0) ∈ Xω,
Z(t) = Sω(t)Z0 = FΣω(t)P1Z0 + e−
t
ω P2Z0 = Ũω(t)Z0 +
t∫
0
Ũω(t− ξ)f(Z(ξ))dξ,
where t ∈ [0, T ] for any 0 < T < ∞. Therefore the function Z(t) = Sω(t)Z0 ∈
C(0, T ; Xω) is a mild solution to problem (18). The lemma is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 203
T.B. Fastovska
It follows from the above arguments that (Sω(t), Xω) is a dynamical system
possessing the compact global attractor Uω = {FQω}.
By the definition, the fractal dimension of a set is the value
dimfA = lim
ε→0
ln n(A, ε)
ln 1/ε
,
where n(A, ε) is the minimal number of closed balls with radius ε covering the
compact set A. Obviously,
dimf Qω = dimf Uω. (33)
Thus, to show the finite dimensionality of the attractor of the dynamical system
(Sω(t), Xω) it remains to prove the lemma bellow.
Lemma 6. Let (10)–(12) hold. Then the attractor Qω of the dynamical system
(Σω(t),Vω) generated by problem (25) has the finite fractal dimension.
P r o o f. Consider the restriction of N onto the attractor Aω, P = N|Aω .
Let us show that PAω ⊂ Qω. If Y ∈ Aω, then there exists a whole trajectory
Y (t) = (u(t), ut(t), θ(t), τ(t)) ∈ Aω passing through this point. Then PY (t) is a
bounded whole trajectory passing through the point X = PY , i.e., X belongs to
the attractor Qω. On the contrary, let Z = (u, v, θ, q) ∈ Qω, then Y = KZ ∈
Aω PZ = Y , i.e., Qω ⊂ PAω. Therefore, PAω = Qω, and the mapping P is
continuous. Consequently,
dimf Qω ≤ dimf Aω < ∞,
and the lemma is proved.
Now we will establish the upper semicontinuity of the family of attractors
{Uω} with respect to the parameter ω and show that problem (3)–(4) is a singular
perturbation of the classical thermoelastic Mindlin–Timoshenko problem (19).
Theorem 2. Let the assumptions (6)–(8), (10)–(13) hold. Then problem (3)–
(4) generates the dynamical system (Sω(t), Xω), where the operator Sω(t) is de-
fined by formula (32). For any ω > 0 the dynamical system (Sω(t), Xω) possesses
a compact global finite dimensional attractor Uω. The family of attractors {Uω}
is upper semicontinuous at zero , i.e.,
sup
y∈Uω
distXω(y, B0) → 0, ω → 0,
204 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
where
B0 =
y =
u0
u1
θ0
−∇θ0
:
u0
u1
θ0
∈ A
.
Here A is a compact global attractor of the dynamical system generated by problem
(19) on the space [H1
0 (Ω)]3 × [L2(Ω)]2 × L2(Ω).
P r o o f. The existence of the dynamical system and the compact global
attractor was proved in Lemmas 1–5. The finite dimensionality of the attractor
follows from Lemma 6 and (33).
Since
‖ 1
ω
∇
∞∫
0
e−
s
ω (τ − θ)ds‖ ≤ ω
∣∣∇τt
∣∣ → 0 ω → 0,
it follows from Theorem 1 that
lim
ω→0
sup
y∈Uω
distXω(y, B0) = lim
ω→0
sup
z∈Aω
distVω(Nz,NKP1B0) = 0.
The theorem is proved.
References
[1] D.S. Chandrasekharaiah, Hyperbolic Thermoelasticity: A Review of Recent Litera-
ture. — Appl. Mech. Rev. 51 (1998), 705–729.
[2] J. Lagnese, Boundary Stabilization of Thin Plates. Philadelphia: SIAM, 1989.
[3] P. Schiavone and R.J. Tait, Thermal Effects in Mindlin-Type Plates. — Q. Jl. Mech.
Appl. Math. 46 (1993), 27–39.
[4] J.E. Muñoz Rivera and R. Racke, Global Stability for Damped Timoshenko Systems.
— Disc. Cont. Dyn. Sys. 9 (2003), 1625–1639.
[5] J.E. Muñoz Rivera and R. Racke, Mildly Dissipative Nonlinear Timoshenko Systems
— Global Existence and Exponential Stability. — J. Math. Anal. Appl. 276 (2002),
No. 1, 248–278.
[6] H.D. Fernández Sare and R. Racke, On the Stability of Damped Timoshenko Sys-
tems: Cattaneo Versus Fourier Law. — Arch. Rational Mech. Anal. 194 (2009),
221–251.
[7] S.A. Messaoudi, M. Pokojovy, and B. Said-Houari, Nonlinear Damped Timoshenko
Systems with Second Sound — Global Existence and Exponential Stability. — Math.
Meth. Appl. Sci. 32 (2009), No. 5, 505–534.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 205
T.B. Fastovska
[8] I. Chueshov and I. Lasiecka, Global Attractors for Mindlin–Timoshenko Plates and
for Their Kirchhoff Limits. — Milan J. Math. 74 (2006), 117–138.
[9] T. Fastovska, Upper Semicontinuous Attractor for a 2D Mindlin–Timoshenko Ther-
moelastic Model with Memory. — Commun. Pure Appl. Anal. 6 (2007), No. 1,
83–101.
[10] T. Fastovska, Upper Semicontinuous Attractor for a 2D Mindlin–Timoshenko Ther-
mo-Viscoelastic Model with Memory. — Nonlinear Analysis TMA 71 (2009), No. 10,
4833–4851.
[11] M.E. Gurtin and A.C. Pipkin, A General Theory of Heat Conduction with Finite
Wave Speeds. — Arch. Rational Mech. Anal. 31 (1968), 113–126.
[12] I.D. Chueshov, Introduction to the theory of infinite-dimensional dissipative sys-
tems. Acta, Kharkov, 1999. (Russian). (Engl. transl.: Acta, Kharkov, 2002).
[13] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics.
Springer, New York (1988).
[14] S. Jiang and R. Racke, Evolution Equations in Thermoelasticity. π Monographs
Surveys Pure Appl. Math. 112, Chapman&Hall/CRC, Boca Raton, 2000.
206 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
|
| id | nasplib_isofts_kiev_ua-123456789-106745 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T17:17:57Z |
| publishDate | 2013 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Fastovska, T.B. 2016-10-04T17:16:44Z 2016-10-04T17:16:44Z 2013 On the Long-Time Behavior of the Thermoelastic Plates with Second Sound / T.B. Fastovska // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 191-206. — Бібліогр.: 14 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106745 The relation between the thermoelastic Cattaneo model and the thermoelastic Gurtin-Pipkin model is established. The existence of the compact global attractor of the Cattaneo-Mindlin plate model is proved and its properties are studied. Установлена зависимость между моделями термоупругости Каттанео и Гертина-Пипкина. Доказано существование компактного глобального аттрактора модели термоупругости Каттанео-Миндлина и изучены его свойства. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On the Long-Time Behavior of the Thermoelastic Plates with Second Sound Article published earlier |
| spellingShingle | On the Long-Time Behavior of the Thermoelastic Plates with Second Sound Fastovska, T.B. |
| title | On the Long-Time Behavior of the Thermoelastic Plates with Second Sound |
| title_full | On the Long-Time Behavior of the Thermoelastic Plates with Second Sound |
| title_fullStr | On the Long-Time Behavior of the Thermoelastic Plates with Second Sound |
| title_full_unstemmed | On the Long-Time Behavior of the Thermoelastic Plates with Second Sound |
| title_short | On the Long-Time Behavior of the Thermoelastic Plates with Second Sound |
| title_sort | on the long-time behavior of the thermoelastic plates with second sound |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106745 |
| work_keys_str_mv | AT fastovskatb onthelongtimebehaviorofthethermoelasticplateswithsecondsound |