Existence, uniqueness, and asymptotic stability for a thermoelastic plate
In this note we are concerned with the linear theory of a thermoelastic plate when a rate-type equation is assumed for the heat flux. We consider an initial boundary-value problem for this plate and show the existence, uniqueness, and asymptotic stability of the solution. Thermodynamic restriction...
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irk-123456789-1769292021-02-10T01:25:30Z Existence, uniqueness, and asymptotic stability for a thermoelastic plate Amendola, G. In this note we are concerned with the linear theory of a thermoelastic plate when a rate-type equation is assumed for the heat flux. We consider an initial boundary-value problem for this plate and show the existence, uniqueness, and asymptotic stability of the solution. Thermodynamic restrictions on the assumed constitutive equations are also derived. Finally, we give the expression of a pseudo free energy. Розглядається лiнiйна теорiя для термоеластичної платiвки за умови, що тепловий потiк задовольняє рiвняння швидкiсного типу. Доведено iснування, єдинiсть та асимптотичну стiйкiсть розв’язку граничної задачi з початковими умовами. Знайдено термодинамiчнi обмеження на рiвняння задачi. Також наведено вираз для псевдовiльної енергiї. 2003 Article Existence, uniqueness, and asymptotic stability for a thermoelastic plate / G. Amendola // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 147-165. — Бібліогр.: 14 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176929 517.95 en Нелінійні коливання Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
In this note we are concerned with the linear theory of a thermoelastic plate when a rate-type equation
is assumed for the heat flux. We consider an initial boundary-value problem for this plate and show the
existence, uniqueness, and asymptotic stability of the solution. Thermodynamic restrictions on the assumed
constitutive equations are also derived. Finally, we give the expression of a pseudo free energy. |
| format |
Article |
| author |
Amendola, G. |
| spellingShingle |
Amendola, G. Existence, uniqueness, and asymptotic stability for a thermoelastic plate Нелінійні коливання |
| author_facet |
Amendola, G. |
| author_sort |
Amendola, G. |
| title |
Existence, uniqueness, and asymptotic stability for a thermoelastic plate |
| title_short |
Existence, uniqueness, and asymptotic stability for a thermoelastic plate |
| title_full |
Existence, uniqueness, and asymptotic stability for a thermoelastic plate |
| title_fullStr |
Existence, uniqueness, and asymptotic stability for a thermoelastic plate |
| title_full_unstemmed |
Existence, uniqueness, and asymptotic stability for a thermoelastic plate |
| title_sort |
existence, uniqueness, and asymptotic stability for a thermoelastic plate |
| publisher |
Інститут математики НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/176929 |
| citation_txt |
Existence, uniqueness, and asymptotic stability for a thermoelastic plate / G. Amendola // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 147-165. — Бібліогр.: 14 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT amendolag existenceuniquenessandasymptoticstabilityforathermoelasticplate |
| first_indexed |
2025-07-15T14:52:50Z |
| last_indexed |
2025-07-15T14:52:50Z |
| _version_ |
1837725040824549376 |
| fulltext |
UDC 517.95
EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY
FOR A THERMOELASTIC PLATE*
IСНУВАННЯ, ЄДИНIСТЬ ТА АСИМПТОТИЧНА СТIЙКIСТЬ РОЗВ’ЯЗКУ
ДЛЯ ТЕРМОЕЛАСТИЧНОЇ ПЛАТIВКИ
G. Amendola
Dipartimento di Matematica Applicata “U. Dini”, Facoltà di Ingegneria
via Diotisalvi 2, 56126-Pisa, Italy
In this note we are concerned with the linear theory of a thermoelastic plate when a rate-type equation
is assumed for the heat flux. We consider an initial boundary-value problem for this plate and show the
existence, uniqueness, and asymptotic stability of the solution. Thermodynamic restrictions on the assumed
constitutive equations are also derived. Finally, we give the expression of a pseudo free energy.
Розглядається лiнiйна теорiя для термоеластичної платiвки за умови, що тепловий потiк за-
довольняє рiвняння швидкiсного типу. Доведено iснування, єдинiсть та асимптотичну стiй-
кiсть розв’язку граничної задачi з початковими умовами. Знайдено термодинамiчнi обмеження
на рiвняння задачi. Також наведено вираз для псевдовiльної енергiї.
1. Introduction. Many authors have recently considered the thermoelastic model of a thin plate
and have studied, in particular, the possibility that the solutions of the thermoelastic plate
equations with Dirichlet or Neumann boundary conditions decay exponentially to zero as ti-
me goes to infinity [1 – 8].
In [9] analogous problems have been investigated for a thermoelastic plate model characteri-
zed by the presence of memory effects on the heat flux vector. Results about existence, uni-
queness, and asymptotic stability of the solutions for an initial boundary-value problem have
been derived as a consequence of the dissipation properties of the material; moreover, the
exponential decay rate of the energy is proved with suitable multiplicative techniques.
In this work we consider the constitutive relation for the heat flux vector proposed by
Cattaneo [10 – 12] and examine the modified system of equations which describe the linear
theory of a thin thermoelastic plate. Thus, we prove the existence, uniqueness, and asymptotic
stability of the solution to the initial boundary-value problem corresponding to homogeneous
conditions, under suitable hypotheses on the sources we have introduced in the equations. Fi-
nally, in the last section, we derive the restrictions placed by the thermodynamic principles on
the physical constants and give the expression of a pseudo free energy.
2. Basic equations and position of the problem. We consider a homogeneous, isotropic,
thermoelastic plate with a thin uniform thickness. The middle surface between its faces, denoted
by Ω, is a bounded and regular domain of the Euclidean two-dimensional space R2 with smooth
boundary Γ. We are concerned with small deformations and small variations of the temperature
referred to a fixed reference configuration with a uniform absolute temperature Θ0.
Let us denote by ϑ the mean variation of the temperature on the cross section of the plate
and by u the vertical deflection in the place x ∈ Ω at time t ∈ R+; within the linear approxi-
∗ Work performed under the auspices of C.N.R. and M.U.R.S.T.
c© G. Amendola, 2003
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2 147
148 G. AMENDOLA
mation theory, small vibrations of the thin thermoelastic plate are described by the following
equations:
utt(x, t)− γ∆utt(x, t) + ∆2u(x, t) + α∆ϑ(x, t) = f(x, t), (2.1)
ρϑt(x, t)− α∆ut(x, t) +∇ · q(x, t) = g(x, t), (2.2)
to which we must add the relation between the heat flux q and the temperature gradient ∇ϑ.
Our purpose is to consider the effects of a change of Fourier’s law; therefore, we assume the
Cattaneo – Maxwell equation
τqt(x, t) + q(x, t) + k∇ϑ(x, t) = l(x, t). (2.3)
In these equations we have introduced the sources f, g and l, which must be considered as
known functions of (x, t) ∈ Ω×R+; τ, k, ρ, α, γ are physical and constant parameters such that
τ > 0, k > 0, ρ > 0, α 6= 0, γ ≥ 0. (2.4)
To investigate an initial boundary-value problem for the thermoelastic plate, we must consi-
der the initial and boundary conditions, which are assumed homogeneous and are expressed
by
u(x, 0) = 0, ut(x, 0) = 0, ϑ(x, 0) = 0, q(x, 0) = 0 ∀x ∈ Ω, (2.5)
u(x, t) = 0,
∂u(x, t)
∂ν
= 0, ϑ(x, t) = 0 ∀(x, t) ∈ Γ×R+, (2.6)
where ν = (ν1, ν2) is the external unit normal to Γ.
3. Existence, uniqueness, and asymptotic stability. In order to give a compact definition of
solution of the initial boundary-value problem (2.1) – (2.3) with (2.5), (2.6), we introduce the
following functional spaces:
H2
0(Ω) =
{
u ∈ H2(Ω) : u = 0,
∂u
∂ν
= 0 ∀x ∈ Γ
}
,
U(Ω,R+) = H1(R+;H1(Ω)) ∩ L2(R+;H2
0(Ω)),
T (Ω,R+) = L2(R+;H1
0 (Ω)), Q(Ω,R+) = L2(R+;L2(Ω)),
F(Ω,R+) = H1(R+;L2(Ω)) ∩ T (Ω,R+),
V(Ω,R+) = H1(R+;L2(Ω)) ∩ L2(R+;H1(Ω)).
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EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY FOR A THERMOELASTIC PLATE 149
Definition 3.1. A triplet (u, ϑ,q) ∈ U(Ω,R+) × T (Ω,R+) × Q(Ω,R+) is said to be a weak
solution to the problem (2.1) – (2.3) with (2.5) – (2.6) and sources (f, g, l) ∈ [Q(Ω,R+)]3 if the
following identity
+∞∫
0
∫
Ω
[−ut(x, t)vt(x, t)− γ∇ut(x, t) · ∇vt(x, t) + ∆u(x, t)∆v(x, t)−
− α∇ϑ(x, t) · ∇v(x, t)− ρϑ(x, t)φt(x, t) + α∇ut(x, t) · ∇φ(x, t)− q(x, t) ·
· ∇φ(x, t)− τq(x, t) · pt(x, t) + q(x, t) · p(x, t)− kϑ(x, t)∇ · p(x, t)]dxdt =
=
+∞∫
0
∫
Ω
[f(x, t)v(x, t) + g(x, t)φ(x, t) + l(x, t) · p(x, t)]dxdt (3.1)
is satisfied for every triplet (v, φ,p) ∈ U(Ω,R+)×F(Ω,R+)× V(Ω,R+).
To study the existence and uniqueness of the solution we identify any function w : R+ →
→ Rn with its causal extension on R and introduce the time-Fourier transform ŵ. We remember
that if w and ŵ belong to L2(R) then also the Fourier transforms ŵ and ŵ′ are L2-functions.
Thus, we have
ŵ(ω) =
+∞∫
−∞
w(s)e−iωsds, ŵ′(ω) = iωŵ(ω)− w(0), w(0) =
1
π
+∞∫
−∞
ŵ(ω)dω. (3.2)
Because of the isomorphisms which exist between each functional space, we have introduced,
and the corresponding space of the Fourier transforms of its functions, denoted with a
circumflex ^ , our problem can be transformed as follows.
Plancherel’s theorem applied to (3.1), taking account of (3.2)2 where the initial data are
zero both for the solutions and for the text functions, yields
1
2π
+∞∫
−∞
∫
Ω
{−iωû(x, ω)[iωv̂(x, ω)]∗ − iωγ∇û(x, ω) · [iω∇v̂(x, ω)]∗ +
+ ∆û(x, ω)[∆v̂(x, ω)]∗ − α∇ϑ̂(x, ω) · [∇v̂(x, ω)]∗ − ρϑ̂(x, ω)[iωφ̂(x, ω)]∗ +
+ iωα∇û(x, ω) · [∇φ̂(x, ω)]∗ − q̂(x, ω) · [∇φ̂(x, ω)]∗ − τ q̂(x, ω) · [iωp̂(x, ω)]∗ +
+q̂(x, ω) · p̂∗(x, ω)− kϑ̂(x, ω)[∇ · p̂(x, ω)]∗
}
dxdω =
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
150 G. AMENDOLA
=
1
2π
+∞∫
−∞
∫
Ω
[f̂(x, ω)v̂∗(x, ω) + ĝ(x, ω)φ̂∗(x, ω) + l̂(x, ω) · p̂∗(x, ω)]dxdω, (3.3)
where ∗ denotes the complex conjugate.
We can now choose v̂(x, ω) = v1(x)v2(ω), φ̂(x, ω) = φ1(x)φ2(ω), p̂(x, ω) = p1(x)p2(ω).
The arbitrariness of (v2, φ2, p2) allows us to derive from (3.3) the following identity:∫
Ω
{
−ω2û(x, ω)v∗1(x)− ω2γ∇û(x, ω) · ∇v∗1(x) + ∆û(x, ω)∆v∗1(x)− α∇ϑ̂(x, ω) ·
· ∇v∗1(x) + iωρϑ̂(x, ω)φ∗1(x) + iωα∇û(x, ω) · ∇φ∗1(x)− q̂(x, ω) · ∇φ∗1(x) +
+iωτ q̂(x, ω) · p∗1(x) + q̂(x, ω) · p∗1(x)− kϑ̂(x, ω)∇ · p∗1(x)
}
dx =
=
∫
Ω
[f̂(x, ω)v∗1(x) + ĝ(x, ω)φ∗1(x) + l̂(x, ω) · p∗1(x)]dx. (3.4)
We observe that the problem (2.1) – (2.3) with (2.5), (2.6), in terms of Fourier’s transforms,
is expressed by the system
−ω2û(x, ω) + γω2∆û(x, ω) + ∆2û(x, ω) + α∆ϑ̂(x, ω) = f̂(x, ω), (3.5)
iωρϑ̂(x, ω)− iωα∆û(x, ω) +∇ · q̂(x, ω) = ĝ(x, ω), (3.6)
iωτ q̂(x, ω) + q̂(x, ω) + k∇ϑ̂(x, ω) = l̂(x, ω) ∀x ∈ Ω, (3.7)
û(x, ω) = 0,
∂û(x, ω)
∂ν
= 0, ϑ̂(x, ω) = 0 ∀x ∈ Γ. (3.8)
The dependence on x is sometimes understood and not written.
Thus, we can give the following definition.
Definition 3.2. A triplet (û(ω), ϑ̂(ω), q̂(ω)) ∈ H2
0(Ω) ×H1
0 (Ω) × L2(Ω) with ω ∈ R is called
a weak solution to the problem (3.5) – (3.8) with (f̂(ω), ĝ(ω), l̂(ω)) ∈ [L2(Ω)]3 if it satisfies (3.4)
for every triplet (v1, φ1,p1) ∈ H2
0(Ω)×H1
0 (Ω)×H1(Ω).
Let us put
I(ω) =
∫
Ω
[
ω2
(
|û|2 + |∇û|2
)
+ |∆û|2 +
∣∣∣ϑ̂∣∣∣2 +
∣∣∣∇ϑ̂∣∣∣2 + |q̂|2
]
dx; (3.9)
we have the following result.
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EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY FOR A THERMOELASTIC PLATE 151
Theorem 3.1. If (û(ω), ϑ̂(ω), q̂(ω)) ∈ H2
0(Ω) × H1
0 (Ω) × L2(Ω) is a weak solution to the
problem (3.5) – (3.8) with (f̂(ω), ĝ(ω), l̂(ω)) ∈ [L2(Ω)]3, then there exists a positive coefficient
δ(ω), depending on the material constants and Ω, such that
I(ω) ≤ δ2(ω)
∫
Ω
(∣∣∣f̂ ∣∣∣2 + |ĝ|2 +
∣∣∣̂l∣∣∣2) dx, (3.10)
where ω ∈ R.
Proof. Let us consider the system (3.5) – (3.7), where we first suppose ω 6= 0.
Multiplying (3.5) by û∗ we get a relation where we can integrate by parts taking account of
the boundary conditions (3.8); thus, it assumes the following form
−ω2
∫
Ω
|û|2 dx + γ
∫
Ω
|∇û|2 dx
+
∫
Ω
|∆û|2 dx− α
∫
Ω
∇ϑ̂ · ∇û∗dx =
∫
Ω
f̂ û∗dx. (3.11)
Analogously, from (3.6) multiplied by û∗ and ϑ̂∗ we get
iω
ρ∫
Ω
ϑ̂û∗dx + α
∫
Ω
|∇û|2 dx
− ∫
Ω
q̂ · ∇û∗dx =
∫
Ω
ĝû∗dx, (3.12)
iω
ρ∫
Ω
∣∣∣ϑ̂∣∣∣2 dx + α
∫
Ω
∇û · ∇ϑ̂∗dx
− ∫
Ω
q̂ · ∇ϑ̂∗dx =
∫
Ω
ĝϑ̂∗dx. (3.13)
Then, in a similar manner the scalar products of (3.7) by q̂∗, ∇ϑ̂∗ and ∇û∗ yield
(1 + iωτ)
∫
Ω
|q̂|2 dx + k
∫
Ω
∇ϑ̂ · q̂∗dx =
∫
Ω
l̂ · q̂∗dx, (3.14)
(1 + iωτ)
∫
Ω
q̂ · ∇ϑ̂∗dx + k
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx =
∫
Ω
l̂ · ∇ϑ̂∗dx, (3.15)
(1 + iωτ)
∫
Ω
q̂ · ∇û∗dx + k
∫
Ω
∇ϑ̂ · ∇û∗dx =
∫
Ω
l̂ · ∇û∗dx. (3.16)
First, we consider the imaginary part of (3.11)
Im
∫
Ω
∇ϑ̂ · ∇û∗dx = − 1
α
Im
∫
Ω
f̂ û∗dx, (3.17)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
152 G. AMENDOLA
which allows us to write the real part of (3.13) as follows:
Re
∫
Ω
q̂ · ∇ϑ̂∗dx = −Re
∫
Ω
ĝϑ̂∗dx− Im
∫
Ω
f̂ωû∗dx; (3.18)
thus, from the real part of (3.14), we have
∫
Ω
|q̂|2 dx = Re
∫
Ω
l̂ · q̂∗dx + k
Re
∫
Ω
ĝϑ̂∗dx + Im
∫
Ω
f̂ωû∗dx
. (3.19)
Then, we take the imaginary part of (3.15), which, on account of (3.18), becomes
Im
∫
Ω
q̂ · ∇ϑ̂∗dx = Im
∫
Ω
l̂ · ∇ϑ̂∗dx + ωτ
Re
∫
Ω
ĝϑ̂∗dx + Im
∫
Ω
f̂ωû∗dx
(3.20)
and use it, together with (3.18), to derive from the real part of (3.15) the following relation:
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx =
1
k
Re
∫
Ω
l̂ · ∇ϑ̂∗dx+ωτ Im
∫
Ω
l̂ · ∇ϑ̂∗dx +
+(1 + ω2τ2)
Re
∫
Ω
ĝϑ̂∗dx + Im
∫
Ω
f̂ωû∗dx
. (3.21)
To estimate the other terms of (3.9), some other relations must be determined.
We begin with the imaginary part of (3.13), which, by virtue of (3.20), yields
ωαRe
∫
Ω
∇û · ∇ϑ̂∗dx = Im
∫
Ω
ĝϑ̂∗dx + Im
∫
Ω
l̂ · ∇ϑ̂∗dx +
+ωτ
Re
∫
Ω
ĝϑ̂∗dx + Im
∫
Ω
f̂ωû∗dx
− ωρ ∫
Ω
∣∣∣ϑ̂∣∣∣2 dx. (3.22)
Then, subtracting the real part of (3.16), multiplied by ωτ , from the imaginary part of (3.16), on
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY FOR A THERMOELASTIC PLATE 153
account of (3.17) and (3.22), we get
Im
∫
Ω
q̂ · ∇û∗dx =
1
1 + ω2τ2
Im
∫
Ω
l̂ · ∇û∗dx− ωτ Re
∫
Ω
l̂ · ∇û∗dx +
+
k
α
(1 + ω2τ2) Im
∫
Ω
f̂ û∗dx+
τk
α
Im
∫
Ω
ĝϑ̂∗dx + Im
∫
Ω
l̂ · ∇ϑ̂∗dx +
+ωτ Re
∫
Ω
ĝϑ̂∗dx− ωρ
∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
. (3.23)
Adding the imaginary part of (3.16), multiplied again by ωτ , to its real part and using (3.17)
and (3.22), we obtain
Re
∫
Ω
q̂ · ∇û∗dx =
1
1 + ω2τ2
Re
∫
Ω
l̂ · ∇û∗dx + ωτ Im
∫
Ω
l̂ · ∇û∗dx− k
ωα
×
×
Im
∫
Ω
ĝϑ̂∗dx + Im
∫
Ω
l̂ · ∇ϑ̂∗dx + ωτ Re
∫
Ω
ĝϑ̂∗dx− ωρ
∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
.
We now observe that∫
Ω
|û|2 dx ≤ λu(Ω)
∫
Ω
|∇û|2 dx,
∫
Ω
∣∣∣ϑ̂∣∣∣2 dx ≤ λϑ(Ω)
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx (3.24)
by virtue of Poincaré’s inequality, where λu and λϑ are positive constants depending on the
domain Ω.
Hence we can increase the following term
− ρ
α
ω2 Re
∫
Ω
ϑ̂û∗dx ≤ ω2
∫
Ω
|û|2 dx
1/2 ρ2
α2
∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
1/2
≤
≤ ω2
2
∫
Ω
|∇û|2 dx +
ρ2
α2
λu(Ω)
∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
,
which appears in the imaginary part of (3.12) multiplied by ω/α, whence, using (3.23), (3.24)2
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
154 G. AMENDOLA
and (3.21), we have the following inequality:
∫
Ω
|ω∇û|2 dx ≤ 1
kα2
{
2k2 + ω2ρλϑ(Ω)
[
ρλu(Ω)
(
1 + ω2τ2
)
− 2τk
]}
×
×
(
Im
∫
Ω
f̂ωû∗dx +
ωτ
1 + ω2τ2
Im
∫
Ω
l̂ · ∇ϑ̂∗dx
)
+
2
α
Im
∫
Ω
ĝωû∗dx +
+
2
α
1
1 + ω2τ2
Im
∫
Ω
l̂ · ω∇û∗dx− ωτ
(
Re
∫
Ω
l̂ · ω∇û∗dx −
− k
α
Im
∫
Ω
ĝϑ̂∗dx
)+
ρλϑ(Ω)
kα2
ω2
1 + ω2τ2
[ρλu(Ω)(1 + ω2τ2)−
− 2τk] Re
∫
Ω
l̂ · ∇ϑ̂∗dx +
ω2τ2
kα2
{
2k2
1 + ω2τ2
+
ρλϑ(Ω)
τ2
[ρλu(Ω)×
× (1 + ω2τ2)− 2τk]
}
Re
∫
Ω
ĝϑ̂∗dx. (3.25)
Finally, the real part of (3.11), taking into account (3.22) divided by ω, (3.24), (3.25) and
(3.21), gives
∫
Ω
|∆û|2 dx ≤ 1
ω
Re
∫
Ω
f̂ωû∗dx +
2
α
[γ + λu(Ω)]
{
Im
∫
Ω
ĝωû∗dx+
1
1 + ω2τ2
×
×
[
Im
∫
Ω
l̂ · ω∇û∗dx− ωτ Re
∫
Ω
l̂ · ω∇û∗dx
]}
+
{
τ +
1
k
(
ρλϑ(Ω)(1 + ω2τ2) +
+
γ + λu(Ω)
α2
{
2k2 + ρλϑ(Ω)ω2[ρλu(Ω)(1 + ω2τ2)− 2τk]
})}
Im
∫
Ω
f̂ωû∗dx +
+
{
τ +
1
k
(
ρλϑ(Ω)(1 + ω2τ2) + [γ + λu(Ω)]
ω2τ2
α2
{
2k2
1 + ω2τ2
+
ρλϑ(Ω)
τ2
×
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EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY FOR A THERMOELASTIC PLATE 155
× [ρλu(Ω)(1 + ω2τ2)− 2τk]
})}
Re
∫
Ω
ĝϑ̂∗dx +
1
ω
{
1 + [γ + λu(Ω)]
2k
α2
×
× ω2τ2
1 + ω2τ2
}
Im
∫
Ω
ĝϑ̂∗dx +
ρλϑ(Ω)
k
{
1 +
γ + λu(Ω)
α2
ω2
1 + ω2τ2
[ρλu(Ω)×
× (1 + ω2τ2)− 2τk]
}
Re
∫
Ω
l̂ · ∇ϑ̂∗dx +
1
ω
{
1 +
ω2τ
k
(
ρλϑ(Ω) +
+
γ + λu(Ω)
α2
1
1 + ω2τ2
{
2k2 + ρλϑ(Ω)ω2[ρλu(Ω)
(
1 + ω2τ2
)
−
− 2τk]}
)}
Im
∫
Ω
l̂ · ∇ϑ̂∗dx. (3.26)
We can now increase (3.9), using (3.19), (3.21), (3.25), (3.26), together with (3.24), to obtain
I(ω) ≤ [1 + λu(Ω)]
∫
Ω
|ω∇û|2 dx +
∫
Ω
|∆û|2 dx + [1 + λϑ(Ω)]
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx +
+
∫
Ω
|q̂|2 dx ≤ 1
ω
Re
∫
Ω
f̂ωû∗dx + (c1
{
2k + c4ω
2[c3(1 + τ2ω2)− 2τk]
}
+
+ τ + k + c2(1 + τ2ω2)) Im
∫
Ω
f̂ωû∗dx + 2c1α Im
∫
Ω
ĝωû∗dx +
(
c1τ
2ω2 ×
×
{
2k
1 + τ2ω2
+
c4
τ2
[c3(1 + τ2ω2)− 2τk]
}
+ τ + k + c2(1 + τ2ω2)
)
×
× Re
∫
Ω
ĝϑ̂∗dx +
1
ω
(
1 +
2c1kτω
2
1 + τ2ω2
)
Im
∫
Ω
ĝϑ̂∗dx + Re
∫
Ω
l̂ · q∗dx +
+
(
c1c4ω
2
1 + τ2ω2
[c3(1 + τ2ω2)− 2τk] + c2
)
Re
∫
Ω
l̂ · ∇ϑ̂∗dx +
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156 G. AMENDOLA
+
1
ω
(
1 + τω2
(
c1
{
2k + c4ω
2[c3(1 + τ2ω2)− 2τk]
} 1
1 + τ2ω2
+ c2
))
×
× Im
∫
Ω
l̂ · ∇ϑ̂∗dx +
2c1α
1 + τ2ω2
−τωRe
∫
Ω
l̂ · ω∇û∗dx +
+ Im
∫
Ω
l̂ · ω∇û∗dx
, (3.27)
where we have introduced the following positive constants:
c1 =
1 + γ + 2λu(Ω)
α2
, c2 =
1 + (1 + ρ)λϑ(Ω)
k
, c3 = ρλu(Ω), c4 =
ρλϑ(Ω)
k
.
Putting
γ1(ω) =
1
| ω |
+ c1[2k + c4ω
2 | c3(1 + τ2ω2)− 2τk |] + τ + k + c2(1 + τ2ω2),
γ2(ω) = 2c1 | α |,
γ3(ω) = c1τ
2ω2
[
2k
1 + τ2ω2
+
c4
τ2
| c3(1 + τ2ω2)− 2τk |
]
+ τ + k +
+ c2(1 + τ2ω2) +
1
| ω |
(
1 +
2c1kτω
2
1 + τ2ω2
)
,
γ4(ω) = 1,
γ5(ω) =
c1c4ω
2
1 + τ2ω2
| c3(1 + τ2ω2)− 2τk | +c2 +
1
| ω |
(
1 + τω2
{
c1[2k +
+ c4ω
2 | c3(1 + τ2ω2)− 2τk |] 1
1 + τ2ω2
+ c2
})
,
γ6(ω) =
2c1 | α |
1 + τ2ω2
(1 + τ | ω |),
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EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY FOR A THERMOELASTIC PLATE 157
the inequality (3.27) can be written as follows:
I(ω) ≤γ(ω)
(
|
∫
Ω
f̂ωû∗dx | + |
∫
Ω
ĝωû∗dx | +
∫
Ω
ĝϑ̂∗dx+ |
∫
Ω
l̂ · q̂∗dx | +
+ |
∫
Ω
l̂ · ∇ϑ̂∗dx | + |
∫
Ω
l̂ · ω∇û∗dx |
)
, (3.28)
where the positive function γ(ω) is given by
γ(ω) = max {γi(ω), i = 1, 2, ..., 6} .
From (3.28) we have
I(ω) ≤ γ(ω)
[∫
Ω
| f̂ |2 dx
1
2
∫
Ω
|ωû|2 dx
1
2
+
∫
Ω
|ĝ|2 dx
1
2
∫
Ω
|ωû|2 dx
1
2
+
+
∫
Ω
|ĝ|2 dx
1
2
∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
1
2
+
∫
Ω
| l̂ |2 dx
1
2
∫
Ω
|q̂|2 dx
1
2
+
+
∫
Ω
| l̂ |2 dx
1
2
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx
1
2
+
∫
Ω
| l̂ |2 dx
1
2
∫
Ω
|ω∇û|2 dx
1
2]
≤
≤ δ(ω)
[∫
Ω
(∣∣∣f̂ ∣∣∣2 + |ĝ|2 +
∣∣∣̂l∣∣∣2) dx] 1
2
[∫
Ω
(
|ωû|2 + |ω∇û|2 + |∆û|2 +
+
∣∣∣ϑ̂∣∣∣2 +
∣∣∣∇ϑ̂∣∣∣2 + |q̂|2
)
dx
] 1
2
and hence
I
1
2 (ω) ≤ δ(ω)
∫
Ω
(∣∣∣f̂ ∣∣∣2 + |ĝ|2 +
∣∣∣̂l∣∣∣2) dx
1
2
,
whence (3.10) follows.
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158 G. AMENDOLA
We now consider the case ω = 0, where the problem (3.5) – (3.8) becomes
∆2û(x, 0) + α∆ϑ̂(x, 0) = f̂(x, 0), (3.29)
∇ · q̂(x, 0) = ĝ(x, 0), (3.30)
q̂(x, 0) + k∇ϑ̂(x, 0) = l̂(x, 0) ∀x ∈ Ω, (3.31)
û(x, 0) = 0,
∂û(x, 0)
∂ν
= 0, ϑ̂(x, 0) = 0 ∀x ∈ Γ. (3.32)
It is enough to multiply (3.29) by û∗(x, 0) and (3.30) by ϑ̂∗(x, 0), to take the inner products
of (3.31) with q̂∗(x, 0) and ∇ϑ̂∗(x, 0) and consider their real parts to obtain∫
Ω
|∆û|2 dx = Re
∫
Ω
f̂ û∗dx + αRe
∫
Ω
∇ϑ̂ · ∇û∗dx, (3.33)
Re
∫
Ω
q̂ · ∇ϑ̂∗dx = −Re
∫
Ω
ĝϑ̂∗dx, (3.34)
∫
Ω
|q̂|2 dx = Re
∫
Ω
l̂ · q̂∗dx− kRe
∫
Ω
∇ϑ̂ · q̂∗dx, (3.35)
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx =
1
k
Re
∫
Ω
l̂ · ∇ϑ̂∗dx−Re
∫
Ω
q̂ · ∇ϑ̂∗dx
. (3.36)
We now recall that if û ∈ H2
0(Ω) then ‖ û ‖ + ‖ ∇û ‖≤ C ‖ ∆û ‖ [9], where C is a
constant, whence we can consider∫
Ω
|û|2 dx ≤ C2
∫
Ω
|∆û|2 dx,
∫
Ω
|∇û|2 dx ≤ C2
∫
Ω
|∆û|2 dx
together with (3.24).
Thus, it follows that
I0 =
∫
Ω
(
|û|2 + |∇û|2 + |∆û|2 +
∣∣∣ϑ̂∣∣∣2 +
∣∣∣∇ϑ̂∣∣∣2 + |q̂|2
)
dx ≤
≤ (1 + 2C2)
∫
Ω
|∆û|2 dx + [1 + λϑ(Ω)]
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx +
∫
Ω
|q̂|2 dx, (3.37)
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EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY FOR A THERMOELASTIC PLATE 159
where all the functions must be considered in (x, 0).
We observe that the last term in (3.33), which appears in (3.37) multiplied by (1 + 2C2), can
be increased as follows:
(1 + 2C2)αRe
∫
Ω
∇ϑ̂ · ∇û∗dx ≤ (1 + 2C2) | αRe
∫
Ω
∇ϑ̂ · ∇û∗dx |≤
≤ 1
2
α2(1 + 2C2)2
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx +
∫
Ω
|∇û|2 dx
, (3.38)
where the last integral is present in the expression of I0 too.
Then, substituting into (3.37) the inequality derived from (3.33) by using (3.38) and the two
relations (3.35) and (3.36), after eliminating their last integrals by using (3.34), the inequality
(3.37) can be put in the following form:
I0 ≤ 2(1 + 2C2) Re
∫
Ω
f̂ û∗dx +
1
k
{
2[1 + λϑ(Ω)] + α2(1 + 2C2)2 +
+ 2k2
}
Re
∫
Ω
ĝϑ̂∗dx +
1
k
{
2[1 + λϑ(Ω)] + α2(1 + 2C2)2
}
Re
∫
Ω
l̂ · ∇ϑ̂∗dx +
+ 2 Re
∫
Ω
l̂ · q̂∗dx. (3.39)
Therefore, denoting by m the maximum of the coefficients of the four integrals at the right-
hand side of (3.39) and proceeding as we have already done to derive (3.10), we see that if
ω = 0, (3.10) must be replaced by
I0 ≤ M
∫
Ω
(∣∣∣f̂(x, 0)
∣∣∣2 + |ĝ(x, 0)|2 +
∣∣∣̂l(x, 0)
∣∣∣2) dx, (3.40)
where M = 16m2. This proves the theorem.
We observe that δ(ω) tends to infinity as ω4 if ω approaches infinity. Therefore, we introduce
the following space:
W(Ω,R+) =
{
(f, g, l) ∈ [Q(Ω,R+)]3 :
∂n+1
∂tn+1
(f, g, l) ∈ [Q(Ω,R+)]3,
[
∂n
∂tn
(f, g, l)
]
t=0
= 0 (n = 0, 1, 2, 3)
}
.
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160 G. AMENDOLA
Theorem 3.2. If the sources (f, g, l) ∈ W(Ω,R+), then the inverse Fourier transforms of
(û, ϑ̂, q̂) exist and are L2-functions.
Proof. As we have already observed, in the inequality (3.10) of Theorem 3.1, δ depends on
ω in such a way as to assure that the integral of the right-hand side over R exists if (f̂ , ĝ, l̂) ∈
∈ Ŵ(Ω,R). That is, we have
+∞∫
−∞
∫
Ω
(∣∣∣δ(ω)f̂(x, ω)
∣∣∣2 + |δ(ω)ĝ(x, ω)|2 +
∣∣∣δ(ω)̂l(x, ω)
∣∣∣2) dxdω < +∞;
therefore, it follows that I(ω) is also integrable over R and hence Plancherel’s theorem assures
the existence of the inverse transforms of (û, ϑ̂, q̂). This completes the proof of the theorem.
Corollary 3.1. Under the hypotheses of Theorem 3.2, if we consider two solutions of our
problem, (û(i), ϑ̂(i), q̂(i)), each of which corresponds to two given source fields (f̂ (i), ĝ(i), l̂(i)),
i = 1, 2, we have∥∥∥(û(1) − û(2),ϑ̂(1) − ϑ̂(2), q̂(1) − q̂(2))
∥∥∥2
≤
≤ 1
2π
+∞∫
−∞
∫
Ω
δ2(ω)
(∣∣∣f̂ (1) − f̂ (2)
∣∣∣2 +
∣∣∣ĝ(1) − ĝ(2)
∣∣∣2 +
∣∣∣̂l(1) − l̂(2)
∣∣∣2) dxdω. (3.41)
This result follows at once from the linearity of (3.5) – (3.7) and from Theorem 3.1.
Theorem 3.3. For any fixed ω ∈ R and every (f̂ , ĝ, l̂) ∈ [L2(Ω)]3, the system (3.5) – (3.8)
admits at most only one solution (û, ϑ̂, q̂) ∈ H2
0(Ω)×H1
0 (Ω)× L2(Ω).
Proof. This uniqueness theorem is a consequence of Theorem 3.1, since it is equivalent
to establishing that the homogeneous system given by (3.5) – (3.7), with the homogeneous
boundary conditions (3.8), has only the zero solution in H2
0(Ω) ×H1
0 (Ω) × L2(Ω). Inequalities
(3.10) and (3.40) assure the uniqueness for every ω ∈ R, which proves the theorem.
Theorem 3.4. For any triplet (f, g, l) ∈ W(Ω,R+) there exists a solution
(u, ϑ,q) ∈ U(Ω,R+)× T (Ω,R+)×Q(Ω,R+)
of the problem (2.1) – (2.3) with (2.5), (2.6) in the sense of Definition 3.1.
Proof. In order to prove the existence of a solution to our problem, we show that the set
A =
{
(f̂ , ĝ, l̂) ∈ Ŵ(Ω,R) : there exists (û, ϑ̂, q̂) ∈ Û(Ω,R)× T̂ (Ω,R) ×
× Q̂(Ω,R) which satisfies (3.3) ∀(v̂, φ̂, p̂) ∈ Û(Ω,R)× F̂(Ω,R)×
×V̂(Ω,R)
}
is dense and closed in Ŵ(Ω,R).
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EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY FOR A THERMOELASTIC PLATE 161
Denoting by ξ[(û, ϑ̂, q̂), (v̂, φ̂, p̂)] the expression in the left-hand side of (3.3), we can write
this identity as follows:
ξ[(û, ϑ̂, q̂), (v̂, φ̂, p̂)] =
1
2π
〈(f̂ , ĝ, l̂), (v̂, φ̂, p̂)〉. (3.42)
To prove that A is dense, we denote by A its closure in Ŵ(Ω,R) and suppose that there
exists (f̂ (0), ĝ(0), l̂(0)) ∈ Ŵ(Ω,R)\A and (f̂ (0), ĝ(0), l̂(0)) 6= 0. Thus, the Hahn – Banach theorem
states that there exists (v̂(0), φ̂(0), p̂(0)) ∈ Û(Ω,R)× F̂(Ω,R)× V̂(Ω,R) such that
〈(f̂ (0), ĝ(0), l̂(0)), (v̂(0), φ̂(0), p̂(0))〉 6= 0, 〈(f̂ , ĝ, l̂), (v̂(0), φ̂(0), p̂(0))〉 = 0 ∀(f̂ , ĝ, l̂) ∈ A. (3.43)
Conditions (3.43)2 and (3.42) yield
ξ[(û, ϑ̂, q̂), (v̂(0), φ̂(0), p̂(0))] = 0 ∀(û, ϑ̂, q̂) ∈ Û(Ω,R)× T̂ (Ω,R)× Q̂(Ω,R)
from which, therefore, with the same technique used to prove the uniqueness theorem, we find
that
(v̂(0), φ̂(0), p̂(0)) = 0,
against (3.43)1. Hence, the set A is dense.
To prove that A is closed, let us consider, for every (f̂ , ĝ, l̂) ∈ Ŵ(Ω,R), a sequence{
(f̂ (n), ĝ(n), l̂(n)) ∈ A, n = 1, 2, ...
}
convergent to (f̂ , ĝ, l̂) and the sequence of the corresponding solutions (û(n), ϑ̂(n), q̂(n)) ∈
∈ Û(Ω,R)× T̂ (Ω,R)× Q̂(Ω,R). Using (3.41) of Corollary 3.1, we have∥∥∥(û(n) − û(m), ϑ̂(n) − ϑ̂(m), q̂(n) − q̂(m))
∥∥∥2
≤
≤ 1
2π
+∞∫
−∞
∫
Ω
δ2(ω)
(∣∣∣f̂ (n) − f̂ (m)
∣∣∣2 +
∣∣∣ĝ(n) − ĝ(m)
∣∣∣2 +
∣∣∣̂l(n) − l̂(m)
∣∣∣2) dxdω,
and hence it follows that the sequence
{
(û(n), ϑ̂(n), q̂(n)), n = 1, 2, ...
}
is a Cauchy sequence
and
lim
n→+∞
(û(n), ϑ̂(n), q̂(n)) = (û, ϑ̂, q̂, ) ∈ Û(Ω,R)× T̂ (Ω,R)× Q̂(Ω,R)
for the completeness of the space.
Then, we consider the sequence of identities, obtained by substituting into (3.3) each soluti-
on with the corresponding triplet of sources of the two sequences now introduced; the limit of
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162 G. AMENDOLA
these identities as n → +∞ yields an analogous identity for the limits (û, ϑ̂, q̂) and (f̂ , ĝ, l̂) and
hence (f̂ , ĝ, l̂) ∈ A.
The application of the Plancherel theorem allows us to complete the proof of the existence
of the solution to our problem. This ends the proof of the theorem.
4. Thermodynamic restrictions and free energy. In this last section we examine the restricti-
ons placed by the thermodynamic principles on the material constants which characterize the
behaviour of the thin homogeneous, isotropic, thermoelastic plate, we have considered in the
previous sections; moreover we give an explicit representation of a pseudo free energy.
With the notation already introduced in Section 2, under the hypotheses of small deformati-
ons and small variations of the temperature with respect to the given reference configuration
and to the absolute temperature Θ0, we assume the following constitutive equations for the
mean stress tensor T and the rate at which heat is absorbed for a unit volum h:
T(x, t) = −a∇[∆u(x, t)] + b∇utt(x, t)− εg(x, t), (4.1)
ρ0h(x, t) = Θ0β∆ut(x, t) + ρ0cϑt(x, t), (4.2)
where g=∇ϑ, ρ0 is the mass density, c is the heat capacity and a, b, ε, β are constitutive constants.
The fundamental system of the linear theory of thermoelasticity, when Cattaneo – Maxwell’s
equation is assumed as the relation between the heat flux and the temperature gradient, is
ρ0utt(x, t) = ∇ ·T(x, t) + ρ0f(x, t), (4.3)
ρ0h(x, t) = −∇ · q(x, t) + ρ0g(x, t), (4.4)
τqt(x, t) + q(x, t) = −k∇ϑ(x, t), (4.5)
where we have introduced the body force f and the heat source g, k and τ(> 0) being two
constants.
We observe that the constitutive equations (4.1), (4.2) characterize a thermodynamic system
when the state is σ(x) = (∇ut(x),∆u(x), ϑ(x),q(x)) at x ∈ Ω and the thermokinetic process of
duration dP ∈ R+ is a piecewise continuous map defined on [0, dP ) by P (x, t) = (∇[∆u(x, t)],
∇utt(x, t),∆ut(x, t), ϑt(x, t),g(x, t)). If we introduce a state-transition function ρ̃ : Σx×Πx →
→ Σx which assigns to the initial state σi, of the space Σx at the point x, and the process P , of
the thermokinetic process space Πx, the final state σf , that is σf = ρ̃(σi, P ), we can consider
Σx
σ0
= {σ ∈ Σx : ∃ P ∈ Πx, σ = ρ̃(σ0, P )}, the subset of the states which can be obtained
from a fixed state σ0 with a process P . The assumed constitutive equations are functions of
(σ, P ), that is, we have T = T̃(σ, P ), h = h̃(P ), q = q̃(σ, P ).
Thus, we recall the definition of a cycle constant on Γ [9], which is a pair (σ(x), P (x)) such
that ρ̃(σ0(x), P (x)) = σ0(x) ∀x ∈ Ω and σ(x, t) = ρ̃(σ0(x), P[0,t)(x)) is constant ∀x ∈ Γ and
t ∈ [0, dP ), P[0,t) being the restriction of P to [0, t) ⊂ [0, dP ).
Now, we give the expressions of the two law of thermodynamics [10], the first of which yields
the existence of the internal energy ẽ : Σx
σ0
→ R such that
ρ0ẽt(σ(x, t)) = ρ0h̃(P (x, t)) + T̃(σ(x, t), P (x, t)) · ∇ut(x, t) (4.6)
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EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY FOR A THERMOELASTIC PLATE 163
for continuous processes, while the second one states that the inequality
∮ ∫
Ω
{
ρ0h̃(P (x, t))
Θ0 + ϑ(x, t)
+
q̃(σ(x, t), P (x, t)) · g(x, t)
[Θ0 + ϑ(x, t)]2
}
dxdt ≤ 0 (4.7)
holds for an isolated material Ω for every pair (σ0(x), P (x)) which defines cycle constant on Γ,
the equality sign referring to reversible processes.
Since we are concerned with a linear theory, we must derive an approximation of the second
law; therefore, neglecting the terms of order greater than two, (4.7) assumes the form
1
Θ2
0
∮ ∫
Ω
{
ρ0h̃(P (x, t))[Θ0 − ϑ(x, t)] + q̃(σ(x, t), P (x, t)) · g(x, t)
}
dxdt ≤
≤ 1
Θ2
0
∮ ∫
Γ
F̃(σ(x, t), P (x, t)) · ν(x)dΓdt, (4.8)
where we have added to the right-hand side the surface integral in order to consider the global
formulation of the second law in agreement with the existence of the flux F.
From the inequality (4.8), under the hypotheses that the material, we are considering, is
self-consistent, that is, when the constitutive equations, relative to x ∈ Ω and t ∈ R+, do not
depend upon fields outside Ω at time t, it follows that
1
Θ2
0
∮ {
ρ0h̃(P (x, t))[Θ0 − ϑ(x, t)] + q̃(σ(x, t), P (x, t)) · g(x, t) −
−∇ · F̃(σ(x, t), P (x, t))
}
dt ≤ 0,
for any x ∈ Ω, since (4.8) must hold for any subbody of Ω.
Furthermore, as a consequence of the second law, it is possible to show [11] the existence of
the entropy η̂ : Σx
σ0
→ R for any x ∈ Ω such that
η̃t(σ(x, t)) ≥ 1
ρ0Θ2
0
{
ρ0h̃(P (x, t))[Θ0 − ϑ(x, t)] +
+ q̃(σ(x, t), P (x, t)) · g(x, t)−∇ · F̃(σ(x, t), P (x, t))
}
(4.9)
for any smooth process.
In order to obtain consequences of the laws of thermodynamics on the material constants,
we observe that (4.8), elimintaing ρ0Θ0h by means of (4.6) and integrating on a cycle, reduces
to
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164 G. AMENDOLA
∮
[ρ0h̃(P (t))ϑ(t) + Θ0T̃(σ(t), P (t)) · ∇ut(t)− q̃(σ(t), P (t)) · g(t) +
+∇ · F̃(σ(t), P (t))]dt ≥ 0, (4.10)
where the dependence on x is understood.
Substituting the expression (4.2) of h, assuming for the heat flux F the following form:
F(x, t) = Θ0[a∆u(x, t)∇ut(x, t)− βϑ(x, t)∇ut(x, t)],
and using (4.5), from (4.10) we get∮ {
ρ0cϑt(t)ϑ(t) + Θ0[b∇utt(x, t)− εg(x, t)] · ∇ut(t) +
1
k
[τqt(t) +
+ q(t)] · q(t) + Θ0a∆ut(t)∆u(t)− βΘ0g(t) · ∇ut(t)]
}
dt ≥ 0.
Since the integral is taken on a cycle, this inequality reduces to∮ [
1
k
q2(t)−Θ0(ε+ β)g(t) · ∇ut(t)]
]
dt ≥ 0,
which holds for every g; therefore, we have the restrictions
ε+ β = 0, k > 0. (4.11)
Finally, we introduce the following approximate pseudo-free energy:
ψ(x, t) = e(x, t)−Θ0η(x, t),
which, using (4.6) to eliminate ρ0Θ0h, allows us to transform (4.9) as follows:
ρ0ψt(t) ≤
ρ0
Θ0
h(t)ϑ(t) + T(t) · ∇ut(t)−
1
Θ0
q(t) · g(t) +
1
Θ0
∇ · F(t). (4.12)
Substituting (4.1), (4.2) and eliminating g by using of (4.5), (4.12) yields
ρ0ψt(t) ≤
d
dt
1
2
{
a[∆u(t)]2 + b[∇ut(t)]2 +
ρ0c
Θ0
ϑ2(t) +
τ
kΘ0
q2(t)
}
+
1
kΘ0
q2(t), (4.13)
whence we can assume
ρ0ψ(∇ut,∆u, ϑ,q) ≤ 1
2
{
a[∆u(t)]2 + b[∇ut(t)]2 +
ρ0c
Θ0
ϑ2(t) +
τ
kΘ0
q2(t)
}
,
which satisfies (4.13) on account of (4.11)2.
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EXISTENCE, UNIQUENESS, AND ASYMPTOTIC STABILITY FOR A THERMOELASTIC PLATE 165
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Received 13.02.2003
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