Asymptotic stability for a thermoelectromagnetic material
In this work we consider a linear thermoelectromagnetic material, whose behaviour is characterized by two rate-type equations for the heat flux and the electric current density. We derive the restrictions imposed by the laws of thermodynamics on the constitutive equations and introduce the free en...
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irk-123456789-1747612021-01-28T01:27:27Z Asymptotic stability for a thermoelectromagnetic material Amendola, G. In this work we consider a linear thermoelectromagnetic material, whose behaviour is characterized by two rate-type equations for the heat flux and the electric current density. We derive the restrictions imposed by the laws of thermodynamics on the constitutive equations and introduce the free energy which yields the existence of a domain of dependence. Uniqueness, existence and asymptotic stability theorems are then proved. 2001 Article Asymptotic stability for a thermoelectromagnetic material / G. Amendola // Нелінійні коливання. — 2001. — Т. 4, № 4. — С. 434-457 . — Бібліогр.: 8 назв. — англ. 1562-3076 AMS Subject Classification: 80A05, 74F05, 74F15 http://dspace.nbuv.gov.ua/handle/123456789/174761 en Нелінійні коливання Інститут математики НАН України |
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In this work we consider a linear thermoelectromagnetic material, whose behaviour is characterized by two rate-type equations for the heat flux and the electric current density. We derive the
restrictions imposed by the laws of thermodynamics on the constitutive equations and introduce
the free energy which yields the existence of a domain of dependence. Uniqueness, existence and
asymptotic stability theorems are then proved. |
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Article |
| author |
Amendola, G. |
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Amendola, G. Asymptotic stability for a thermoelectromagnetic material Нелінійні коливання |
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Amendola, G. |
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Amendola, G. |
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Asymptotic stability for a thermoelectromagnetic material |
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Asymptotic stability for a thermoelectromagnetic material |
| title_full |
Asymptotic stability for a thermoelectromagnetic material |
| title_fullStr |
Asymptotic stability for a thermoelectromagnetic material |
| title_full_unstemmed |
Asymptotic stability for a thermoelectromagnetic material |
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asymptotic stability for a thermoelectromagnetic material |
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Інститут математики НАН України |
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2001 |
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Asymptotic stability for a thermoelectromagnetic material / G. Amendola // Нелінійні коливання. — 2001. — Т. 4, № 4. — С. 434-457 . — Бібліогр.: 8 назв. — англ. |
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Нелінійні коливання |
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AT amendolag asymptoticstabilityforathermoelectromagneticmaterial |
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2025-07-15T11:55:30Z |
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Nonlinear Oscillations, Vol. 4, No. 4, 2001
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC
MATERIAL*
G. Amendola
Dipartimento di Matematica Applicata “U.Dini”, Facoltà di Ingegneria
via Diotisalvi 2, 56126-Pisa, Italy
In this work we consider a linear thermoelectromagnetic material, whose behaviour is charac-
terized by two rate-type equations for the heat flux and the electric current density. We derive the
restrictions imposed by the laws of thermodynamics on the constitutive equations and introduce
the free energy which yields the existence of a domain of dependence. Uniqueness, existence and
asymptotic stability theorems are then proved.
AMS Subject Classification: 80A05, 74F05, 74F15
1. Introduction
Electromagnetic materials in presence of thermal effects have been studied by Coleman
and Dill, who have also derived the restrictions which the laws of thermodynamics place on the
constitutive equations [1, 2]. Heat conduction in electromagnetic solids has been considered in
[3] to study its effects on the asymptotic behaviour of the solutions when constitutive equations
with memory have been supposed for the heat flux and for the electric current density.
In this work we are concerned with a linear thermoelectromagnetic solid, characterized by
a rate-type equation both for the heat flux and for the electric current density, while the simple
proportionality is assumed between the electric displacement, the magnetic induction and the
electric and magnetic fields, respectively.
After introducing in Sect. 2 Maxwell’s equations with the energy one, in Sect. 3 we consider
the thermodynamics of simple materials [4, 5], in order to derive the restrictions on the consti-
tutive equations and to introduce the free energy, which allows us to obtain, in the following
section, a domain of dependence. A theorem of uniqueness, existence and asymptotic stability
of solutions is then proved in the last section.
2. Basic Equations
We consider a thermoelectromagnetic solid B, which at time t occupies a bounded and
regular domain Ω in the three-dimensional Euclidean space R3. The position vector in Ω is
denoted by x and n is the unit outward normal to the smooth boundary ∂Ω.
We restrict our attention to the linear thermoelectromagnetic theory, where B is homoge-
neous, isotropic and conducting material characterized, in particular, by two rate-type equa-
tions both for the electric current density and for the heat flux. Thus, we assume for the electric
displacement D, the magnetic induction B, the rate at which heat is absorbed per unit volume
∗ Work perfomed under the auspices of C.N.R. and M.U.R.S.T..
434 c© G. Amendola, 2001
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 435
h the following constitutive equations:
D(x, t) = εE(x, t) + ϑ(x, t)a, B(x, t) = µH(x, t), (2.1)
h(x, t) = cϑ̇(x, t) + Θ0
[
A1 · Ḋ(x, t)/ε+ A2 · Ḃ(x, t)/µ
]
, (2.2)
where E and H are the electric and magnetic fields, ϑ is the temperature relative to the uniform
absolute reference temperature Θ0 in Ω, while for the electric current density J and for the heat
flux q [6] we suppose that the following equations
αJ̇(x, t) + J(x, t) = σE(x, t), τ q̇(x, t) + q(x, t) = −kg(x, t), (2.3)
hold, where g = ∇ϑ is the temperature gradient.
In these relations the dielectric constant ε, the permeability µ and the specific heat c are
positive constants as well as the other two parameters α and τ ; moreover, a, A1 and A2 denote
three constant vectors.
The fundamental system of thermoelectromagnetism consists of Maxwell’s equations to-
gether with the energy equation, i.e.,
∇×H(x, t) = Ḋ(x, t) + J(x, t) + Jf (x, t), ∇×E(x, t) = −Ḃ(x, t), (2.4)
∇ ·B(x, t) = 0, ∇ ·D(x, t) = ρ, (2.5)
h(x, t) = −∇ · q(x, t) + r(x, t), (2.6)
where ρ is the free charge density, Jf denotes a forced current density and is a known function
of x and t as well as r , which is the heat sources for unit volume; moreover, we have supposed
that the mass density is unitary in (2.6).
Sometimes we will understand the dependence on x.
3. Thermodynamic Restrictions on the Constitutive Equations
The assumed constitutive equations allow us to consider B as a simple material, whose
state is s = (E,H,J,q, ϑ), while the process of the material element is given by P (t) =
(Ė(t), Ḣ(t), ϑ̇(t),g(t)) and the response function is U(t) = (D(t),B(t),J(t), h(t),q(t)) for
every t ∈ [0, dP ) ⊂ R+, dP being the duration of the process.
The First Law of Thermodynamics for these materials is expressed by this equality∮
[h(t) + Ḋ(t) ·E(t) + Ḃ(t) ·H(t) + J(t) ·E(t)]dt = 0, (3.1)
which must hold for any cyclic process [1, 2], and, for smooth processes, yields the existence of
the internal energy e which satisfies
ė(t) = h(t) + Ḋ(t) ·E(t) + Ḃ(t) ·H(t) + J(t) ·E(t). (3.2)
436 G. AMENDOLA
The Second Law states that for any cyclic process the inequality [1, 2]∮ {
h(t)/[Θ0 + ϑ(t)] + q(t) · g(t)/[Θ0 + ϑ(t)]2
}
dt ≤ 0 (3.3)
holds, with the equality sign referring to reversible processes.
Also by (3.3) it is possible to derive, for smooth processes, another local relation,
η̇(t) ≥ h(t)/[Θ0 + ϑ(t)] + q(t) · g(t)/[Θ0 + ϑ(t)]2, (3.4)
where η denotes the entropy.
We are concerned with the linear theory, therefore we must consider the approximation of
the Second Law; thus, (3.3) and (3.4) become [7]∮
{h(t)[Θ0 − ϑ(t)] + q(t) · g(t)} dt/Θ2
0 ≤ 0, (3.5)
η̇(t) ≥ {h(t)[Θ0 − ϑ(t)] + q(t) · g(t)} /Θ2
0. (3.6)
We first examine (3.5), which, eliminating Θ0h(t) by use of (3.1) and integrating on any
cycle, yields∮ {
h(t)ϑ(t)−Θ0[Ė(t) ·D(t) + Ḣ(t) ·B(t)− J(t) ·E(t)]− q(t) · g(t)
}
dt ≥ 0. (3.7)
If we substitute into (3.7) the constitutive equations (2.1), (2.2) and the two relations, we
can derive from (2.3) for E and g on supposing σ 6= 0 and k 6= 0, we get an inequality, which,
integrated on any cycle, reduces to∮ {[
(A1 − a) · Ė(t) + A2 · Ḣ(t)
]
ϑ(t) +
1
σ
J2(t) +
1
kΘ0
q2(t)
}
dt ≥ 0. (3.8)
From this inequality, taking account of the independence of Ė, Ḣ and g, it follows that
A1 = a, A2 = 0, σ > 0, k > 0. (3.9)
In particular, the expression (2.2), using (3.9)1,2 and (2.1), becomes
h(x, t) =
(
c+ Θ0a
2/ε
)
ϑ̇(x, t) + Θ0a · Ė(x, t). (3.10)
Let us introduce the pseudo-free energy
ψ(x, t) = e(x, t)−Θ0η(x, t). (3.11)
Substituting into (3.6) the expression of h derived by (3.2) and using (3.11) yields
ψ̇(t) ≤ [h(t)ϑ(t)− q(t) · g(t)]/Θ0 + Ḋ(t) ·E(t) + Ḃ(t) ·H(t) + J(t) ·E(t). (3.12)
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 437
We introduce the following functional
ψ̃(x, t) =
1
2
[
1
ε
D2(x, t) +
1
µ
B2(x, t) +
α
σ
J2(x, t) +
τ
kΘ0
q2(x, t) +
c
Θ0
ϑ2(x, t)
]
. (3.13)
We observe that this free energy ψ̃ satisfies (3.12). In fact, if we differentiate ψ̃ with respect
to time, taking account of (2.1) and (3.10), eliminating J̇ and q̇ with (2.3), we get
.
ψ̃(t) =
1
Θ0
[h(t)ϑ(t)− q(t) · g(t)] + Ḋ(t) ·E(t)
+ Ḃ(t) ·H(t) + J(t) ·E(t)− 1
kΘ0
q2(t)− 1
σ
J2(t), (3.14)
which satisfies (3.12).
4. Domain of Dependence Inequality
The system of equations of the evolutive problem of the thermoelectromagnetic solid B
consists of (2.4) and (2.6), which, on account of (2.1) and (3.10), can be put in the following
form:
∇×H(x, t) = εĖ(x, t) + ϑ̇(x, t)a + J(x, t) + Jf (x, t), (4.1)
∇×E(x, t) = −µḢ(x, t), (4.2)
(
c+ Θ0a
2/ε
)
ϑ̇(x, t) + Θ0a · Ė(x, t) = −∇ · q(x, t) + r(x, t), (4.3)
together with (2.3) and (2.5)1, which becomes
∇ ·H(x, t) = 0 (4.4)
for any (x, t) ∈ Ω×R+. We observe that Eq. (2.5)2, ε∇·E+∇ϑ ·a = ρ, allows us to determine
ρ.
To this system we must add the initial conditions
E(x, 0) = E0(x), H(x, 0) = H0(x), J(x, 0) = J0(x),
q(x, 0) = q0(x), ϑ(, 0) = ϑ0(x) ∀x ∈ Ω (4.5)
and the boundary conditions
E(x, t)× n = 0, ϑ(x, t) = 0 ∀(x, t) ∈ ∂Ω×R+. (4.6)
Compatibility of the boundary conditions and the initial ones is assumed.
We dente by P this initial-boundary-value problem.
438 G. AMENDOLA
Theorem 4.1. The free energy defined by (3.13) satisfies
−[E(t)×H(t) + ϑ(t)q(t)/Θ0] · u ≤ χψ̃(t), (4.7)
where u is a unit vector and
χ = 2
{(
k
τc
)1/2
+
1
εµ
[
ε1/2+ | a |
(
Θ0
c
)1/2
]
µ1/2
}
. (4.8)
Proof. To prove (4.7) it is enough to consider (3.13), from which it follows that
1
ε
| D |≤
[
2
ε
ψ̃
]1/2
,
1
µ
| B |≤
[
2
ε
ψ̃
]1/2
, | q |≤
[
2k
τ
Θ0ψ̃
]1/2
, | ϑ |≤
[
2
c
Θ0ψ̃
]1/2
, (4.9)
and to derive from (2.1) the expressions of E and H, which, taking account of (4.9)1,2, yield
| E |≤ 21/2
ε
[
ε1/2+ | a |
(
Θ0
c
)1/2
]
ψ̃1/2, | H |≤
[
2
µ
ψ̃
]1/2
. (4.10)
These inequalities give (4.7) at once.
We are now in a position to show the existence of a domain of dependence introducing the
total energy
E (D , t) =
∫
D
ψ̃(x, t)dx ∀D ⊂ Ω. (4.11)
Theorem 4.2. The total energy E (D , t) satisfies the following inequality
E(B(x0, ρ), T ) ≤ E(B(x0, ρ+ χT ), 0)
+
T∫
0
∫
Ω∩B(x0,ρ+χ(T−t))
[
1
Θ0
r(x, t)ϑ(x, t)− Jf (x, t) ·E(x, t)
]
dxdt (4.12)
for any fixed (x0, T ) ∈ Ω×R+, with B(x0, ρ) = {x ∈ Ω : |x− x0| ≤ ρ} and χ given by (4.8).
Proof. Let
Eφ(Ω, t) =
∫
Ω
ψ̃(x, t)φ(x, t)dx, (4.13)
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 439
where φ(x, t) ∈ C∞0 (R3,R+); its derivative with respect to time, taking account of (3.14) and
eliminating h, Ḃ and Ḋ by the use of (2.6) and (2.4), can be written as
Ėφ(Ω, t) =
∫
Ω
{
1
Θ0
[rϑ−∇ · (ϑq)]− Jf ·E +∇×H ·E−∇×E ·H
}
φdx
−
∫
Ω
(
1
σ
J2 +
1
kΘ0
q2
)
φdx +
∫
Ω
ψ̃φ̇dx. (4.14)
This equality, using the identity ∇× E ·H−∇×H · E = ∇ · (E×H) and taking account
of the boundary conditions (4.6) and (3.9)3,4, gives the following inequality:
Ėφ(Ω, t) ≤
∫
Ω
(rϑ/Θ0 − Jf ·E)φdx +
∫
Ω
[
(ϑq/Θ0 + E×H) · ∇φ+ ψ̃φ̇
]
dx
−
∫
∂Ω
(ϑq/Θ0 + E×H) · nφda. (4.15)
We now put
φ(x, t) = φδ(x, t) = φδ(|x− x0| − ρ− χ(T − t)) = φδ(y) =
{
1 ∀ y ≤ −δ,
0 ∀ y > δ,
(4.16)
where φδ ∈ C∞(R) is a nonnegative, monotonic decreasing function, δ > 0, ρ > 0 and χ is
given by (4.8), for any fixed point x0 ∈ Ω and T ∈ R+ with t ∈ (0, T ).
We have
φ
′
δ(y) ≤ 0, φ̇δ(x, t) = φ
′
δ(y)χ, ∇φδ(x, t) = φ
′
δ(y)u(x), (4.17)
where u = ∇ |x− x0| is a unit vector.
Substituting (4.17)2,3 into (4.15) and using (4.17)1, (4.7) and (4.6) yield
4.18Ėφδ(Ω, t) ≤
∫
Ω
[r(x, t)ϑ(x, t)/Θ0 − Jf (x, t) ·E(x, t)]φδ(x, t)dx, (4.18)
which, integrated over (0, T ), gives
Eφδ(Ω, T )− Eφδ(Ω, 0) ≤
T∫
0
∫
Ω
[r(x, t)ϑ(x, t)/Θ0 − Jf (x, t) ·E(x, t)]φδ(x, t)dxdt. (4.19)
Whence, the limit as δ → 0 yields (4.12), because φδ(x, t) tends to the characteristic function of
B(x0, ρ+ χ(T − t)).
440 G. AMENDOLA
5. Existence, Uniqueness and Asymptotic Stability
The problem P, expressed by the system of equations (4.1) – (4.4) with (2.3) to which are
associated the initial-boundary conditions (4.5), (4.6), can be transformed into an equivalent
one with zero initial data, we shall denote by P′.
For this purpose, we put
E(x, t) = Ẽ(x, t) + l(x, t), H(x, t) = H̃(x, t) + p(x, t), J(x, t) = J̃(x, t) + w(x, t),
q(x, t) = q̃(x, t) + z(x, t), ϑ(x, t) = ϑ̃(x, t) + γ(x, t),
where (Ẽ, H̃, J̃, q̃, ϑ̃) and (l,p,w, z, γ) belong to the function spaces of the solution (E,H,J,q, ϑ)
and hence, in particular, satisfy the boundary conditions (4.6); moreover, taking in mind (2.5),
we suppose that (l,p, γ) satisfies ∇ · p = 0 and ε∇ · l = −∇γ · a and we impose the following
initial conditions:
l(x, 0) = E0(x), p(x, 0) = H0(x), w(x, 0) = J0(x), z(x, 0) = q0(x), γ(x, 0) = ϑ0(x).
Then, we see that (Ẽ, H̃, J̃, q̃, ϑ̃) satisfies the system of equations of P, where the sources Jf
and r must be changed and other sources must be introduced in the other equations; all these
sources are expressed by
F(x, t) = Jf (x, t) + εl̇(x, t) + γ̇(x, t)a + w(x, t)−∇× p(x, t),
G(x, t) = −µṗ(x, t)−∇× l(x, t),
R(x, t) = r(x, t)−Θ0l̇(x, t) · a−
(
c+ Θ0a
2/ε
)
γ̇(x, t)−∇ · z(x, t),
I(x, t) = −αẇ(x, t)−w(x, t) + σl(x, t),
Q(x, t) = −rż(x, t)− z(x, t)− k∇γ(x, t).
We observe that the hypothesis that l and γ satisfy (2.5)2 without ρ implies that Ẽ and ϑ̃
satisfy (2.5)2 and give ρ without any new source.
Using the same notation (E,H,J,q, ϑ), without ∼, the problem P′ is the following
−εĖ(x, t)− ϑ̇(x, t)a +∇×H(x, t)− J(x, t) = F(x, t), (5.1)
µḢ(x, t) +∇×E(x, t) = G(x, t), (5.2)
Θ0a · Ė(x, t) +
(
c+ Θ0a
2/ε
)
ϑ̇(x, t) +∇ · q(x, t) = R(x, t), (5.3)
αJ̇(x, t) + J(x, t)− σE(x, t) = I(x, t), (5.4)
τ q̇(x, t) + q(x, t) + k∇ϑ(x, t) = Q(x, t), (5.5)
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 441
with (4.4), (4.5), which now become
E0(x) = 0, H0(x) = 0, J0(x) = 0, q0(x) = 0, ϑ0(x) = 0, (5.6)
and the boundary conditions (4.6).
Taking account of the Maxwell equation (4.4) and the boundary conditions (4.6), we intro-
duce the function spaces
I (Ω) =
H(x) ∈ L2(Ω) :
∫
Ω
H · ∇ϕ dx = 0 ∀ϕ ∈ C∞0 (Ω)
,
H1
E(Ω) =
{
E(x) ∈ L2(Ω) : ∇×E ∈ L2, E(x)× n |∂Ω= 0
}
,
H1
H(Ω) =
{
H(x) ∈ H1(Ω) : ∇ ·H = 0, H(x) · n |∂Ω= 0
}
,
H1
ϑ(Ω) =
{
ϑ(x) ∈ H1(Ω) : ϑ(x) |∂Ω= 0
}
,
H (Ω,R+) = L2(R+;H1
E(Ω))× L2(R+;H1
H(Ω))× L2(R+;L2(Ω))
× L2(R+;L2(Ω))× L2(R+;H1
ϑ(Ω)),
W (Ω,R+) = H1(R+;L2(Ω))×H1(R+;L2(Ω))×H1(R+;L2(Ω))
×H1(R+;L2(Ω))×
{
H1(R+;L2(Ω)) ∩ L2(R+;H1
ϑ(Ω))
}
,
V (Ω,R+) = L2(R+;L2(Ω))× L2(R+; I (Ω))× L2(R+;L2(Ω))
× L2(R+;L2(Ω))× L2(R+;L2(Ω)),
which are Hilbert’s spaces with the usual scalar products.
The sources belong to V (Ω,R+), modified in the following manner
V ′(Ω,R+) =
{
(F,G, I,Q, R) ∈ V (Ω,R+) :
∂n+1
∂tn+1
(F,G, I,Q, R) ∈ V (Ω,R+),
[
∂n
∂tn
(F,G, I,Q, R)
]
t=0
= 0 (n = 0, 1, 2, 3)
}
,
where the last conditions on the initial values of the derivatives of the new sources are satisfied
by a suitable choice of the corresponding derivatives of l, p, w, z and γ.
442 G. AMENDOLA
Definition 5.1. A 5-tuple (E,H,J,q, ϑ) ∈ H (Ω,R+) is called weak solution of the problem
P ′ with sources (F,G, I,Q, R) ∈ V ′(Ω,R+) if
+∞∫
0
∫
Ω
[εE(x, t) + ϑ(x, t)a] · ė(x, t)− µH(x, t) · ḣ(x, t)− αJ(x, t) · u̇(x, t)
− τq(x, t) · v̇(x, t)−Θ0
[(
c/Θ0 + a2/ε
)
ϑ(x, t) + a ·E(x, t)
]
β̇(x, t)
+∇×E(x, t) · h(x, t) + [∇×H(x, t)− J(x, t)] · e(x, t)− q(x, t) · ∇β(x, t)
+ [J(x, t)− σE(x, t)] · u(x, t) +[q(x, t) + k∇ϑ(x, t)] · v(x, t)} dxdt
=
+∞∫
0
∫
Ω
[F(x, t) · e(x, t) + G(x, t) · h(x, t) + I(x, t) · u(x, t)
+ Q(x, t) · v(x, t) +R(x, t)β(x, t)]dxdt (5.7)
for all (e,h,u,v, β) ∈ W (Ω,R+) such that e(x, 0) = 0, h(x, 0) = 0, u(x, 0) = 0, v(x, 0) = 0,
β(x, 0) = 0.
Let us introduce the time-Fourier transform of the causal extension on R of a function
g : R+ → Rn,
ĝ(ω) =
+∞∫
−∞
g(t) exp[−iωt]dt. (5.8)
If g and g′ ∈ L2(R+), then the Fourier transforms ĝ and ĝ′ ∈ L2(R) and we have
ĝ′(ω) = iωĝ(ω)− g(0), g(0) =
1
π
+∞∫
−∞
ĝ(ω)dω. (5.9)
We denote by Ĥ (Ω,R), Ŵ (Ω,R) and V̂ ′(Ω,R) the spaces of the Fourier transforms with
respect to time of the corresponding functions of H (Ω,R+), W (Ω,R+) and V ′(Ω,R+). Be-
tween each pair of these spaces there exists an isomorphism by virtue of Plancherel’s theorem,
which allows us to define in a natural way the scalar products in Ĥ (Ω,R), Ŵ (Ω,R), V̂ ′(Ω,R)
and to transform our problem as follows.
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 443
By use of Plancherel’s theorem and (5.9) with zero initial data, (5.7) yields
1
2π
+∞∫
−∞
∫
Ω
({
−iω[εÊ(x, ω) + ϑ̂(x, ω)a] +∇× Ĥ(x, ω) + Ĵ(x, ω)
}
· ê∗(x, ω)
+ [iωµĤ(x, ω) +∇× Ê(x, ω)] · ĥ∗(x, ω) + [(1 + iωα)Ĵ(x, ω)
− σÊ(x, ω)] · û∗(x, ω) + [(1 + iωτ)q̂(x, ω) + k∇ϑ̂(x, ω)] · v̂∗(x, ω)
+ iωΘ0[(c/Θ0 + a2/ε)ϑ̂(x, ω) + a · Ê(x, ω)]β̂∗(x, ω)− q̂(x, ω)
·∇β̂∗(x, ω)
)
dxdω =
1
2π
+∞∫
−∞
∫
Ω
[F̂(x, ω) · ê∗(x, ω) + Ĝ(x, ω) · ĥ∗(x, ω)
+ Î(x, ω) · û∗(x, ω) + Q̂(x, ω) · v̂∗(x, ω) + R̂(x, ω)β̂∗(x, ω)]dxdω (5.10)
for any (ê, ĥ, û, v̂, β̂) ∈ Ŵ (Ω,R).
Here ∗ denotes the complex conjugate.
Remark 5.1. Let (Ê, Ĥ, Ĵ, q̂, ϑ̂) be the Fourier transform of a weak solution of P′, as a
consequence of (5.10) it follows that it is a weak solution of the problem we can derive by the
formal application of the Fourier transfom with respect to time to the problem P′, i.e.,
−iω[εÊ(x, ω) + ϑ̂(x, ω)a]) +∇× Ĥ(x, ω)− Ĵ(x, ω) = F̂(x, ω), (5.11)
iωµĤ(x, ω) +∇× Ê(x, ω) = Ĝ(x, ω), (5.12)
iω
[
Θ0a · Ê(x, ω) +
(
c+ Θ0a
2/ε
)
ϑ̂(x, ω)
]
+∇ · q̂(x, ω) = R̂(x, ω), (5.13)
(1 + iωα)Ĵ(x, ω)− σÊ(x, ω) = Î(x, ω), (5.14)
(1 + iωτ)q̂(x, ω) + k∇ϑ̂(x, ω) = Q̂(x, ω), (5.15)
with
Ê(x, ω)× n = 0, ϑ̂(x, ω) = 0 (5.16)
for all ω ∈ R.
We observe that these boundary conditions, the first of which yields H · n = 0 too, and the
assumed hypothesis on the greater regularities of ϑ ∈ H1
ϑ(Ω) and H ∈ H1
H(Ω) with Ω simply
connected [8] yield the following inequalities:∫
Ω
∣∣∣ϑ̂∣∣∣2 dx ≤ βϑ(Ω)
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx, ∫
Ω
∣∣∣Ĥ∣∣∣2 dx ≤ βH(Ω)
∫
Ω
∣∣∣∇× Ĥ
∣∣∣2 dx, (5.17)
444 G. AMENDOLA
where βϑ(Ω) and βH(Ω) are constant and depend only on the domain Ω.
We denote by Λ[(Ê, Ĥ, Ĵ, q̂, ϑ̂), (ê, ĥ, û, v̂, β̂)] and by 〈(F̂, Ĝ, Î, Q̂, R̂), (ê, ĥ, û, v̂, β̂)〉 the first
and the second integral of (5.10), respectively.
Theorem 5.1. For any (F̂, Ĝ, Î, Q̂, R̂) ∈ V̂ ′(Ω,R) there exists a unique solution (Ê, Ĥ, Ĵ, q̂, ϑ̂) ∈
Ĥ (Ω,R) such that
Λ[(Ê, Ĥ, Ĵ, q̂, ϑ̂), (ê, ĥ, û, v̂, β̂)] = 〈(F̂, Ĝ, Î, Q̂, R̂), (ê, ĥ, û, v̂, β̂)〉 (5.18)
for any (ê, ĥ, û, v̂, β̂) ∈ Ŵ (Ω,R).
Proof. Uniqueness. To show the uniqueness of the solution, we prove that the homo-
geneous system obtained by (5.11) – (5.15), where the right-hand side of each equation is put
equal to zero, has only the zero solution for all ω ∈ R.
Thus, keeping always in mind the system so modified, if we consider the inner product of
the conjugate of (5.12) with Ĥ and integrate over Ω we have
−iωµ
∫
Ω
∣∣∣Ĥ∣∣∣2 dx +
∫
Ω
∇× Ê∗ · Ĥdx = 0, (5.19)
whose real part gives
Re
∫
Ω
∇× Ê∗ · Ĥdx = 0. (5.20)
From (5.11), the inner product with Ê∗, taking account of the identity∇× Ê∗ · Ĥ−∇× Ĥ ·
Ê∗ = ∇ · (Ê∗ × Ĥ) and (5.16)1, yields
−iωε
∫
Ω
∣∣∣Ê∣∣∣2 dx− iωa · ∫
Ω
ϑ̂Ê∗dx +
∫
Ω
∇× Ê∗ · Ĥdx−
∫
Ω
Ĵ · Ê∗dx = 0, (5.21)
from which and (5.20) it follows that
ω Ima ·
∫
Ω
ϑ̂Ê∗dx = Re
∫
Ω
Ĵ · Ê∗dx. (5.22)
The real part of
1
σ
(1− iωα)
∫
Ω
∣∣∣Ĵ∣∣∣2 dx− ∫
Ω
Ê∗ · Ĵdx = 0, (5.23)
derived by the inner product of the conjugate of (5.14) with Ĵ, together with (5.22) give
ω Ima ·
∫
Ω
ϑ̂Ê∗dx =
1
σ
∫
Ω
∣∣∣Ĵ∣∣∣2 dx. (5.24)
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 445
From (5.13), multiplying by ϑ̂∗, upon an integration by parts and on account of (5.16)2, it
follows that
iω
Θ0a ·
∫
Ω
Êϑ̂∗dx +
∫
Ω
(
c+ Θ0a
2/ε
) ∣∣∣ϑ̂∣∣∣2 dx
− ∫
Ω
q̂ · ∇ϑ̂∗dx = 0, (5.25)
whence
ω Ima ·
∫
Ω
Êϑ̂∗dx = − 1
Θ0
Re
∫
Ω
q̂ · ∇ϑ̂∗dx. (5.26)
The real part of
1
k
(1− iωτ)
∫
Ω
|q̂|2 dx +
∫
Ω
∇ϑ̂∗ · q̂dx = 0, (5.27)
obtained by taking the inner product of the conjugate of (5.15) with q̂, allows us to rewrite
(5.26) as
ω Ima ·
∫
Ω
Êϑ̂∗dx =
1
kΘ0
∫
Ω
|q̂|2 dx. (5.28)
Adding (5.24) and (5.28) we get
1
σ
∫
Ω
∣∣∣Ĵ∣∣∣2 dx +
1
kΘ0
∫
Ω
|q̂|2 dx = 0, (5.29)
from which, taking account of (3.9)3,4, we have
∫
Ω
∣∣∣Ĵ∣∣∣2 dx = 0,
∫
Ω
|q̂|2 dx = 0. (5.30)
Then, from (5.14) and (5.15) it follows that
∫
Ω
∣∣∣Ê∣∣∣2 dx = 0,
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx = 0 (5.31)
and from (5.17), or (5.13) and (5.12), analogous results hold for ϑ̂ and Ĥ.
Existence. To prove the existence we consider the following lemmas.
446 G. AMENDOLA
Lemma 5.1. For all ω ∈ R, any weak solution to the problem (5.11) – (5.16) satisfies the
following inequality
G(ω) ≤ ν2(ω)
∫
Ω
(∣∣∣F̂∣∣∣2 +
∣∣∣Ĝ∣∣∣2 +
∣∣∣̂I∣∣∣2 +
∣∣∣Q̂∣∣∣2 +
∣∣∣R̂∣∣∣2) dx, (5.32)
where
G(ω) =
∫
Ω
(∣∣∣Ê∣∣∣2 +
∣∣∣Ĥ∣∣∣2 +
∣∣∣Ĵ∣∣∣2
+ |q̂|2 +
∣∣∣ϑ̂∣∣∣2 +
∣∣∣∇× Ê
∣∣∣2 +
∣∣∣∇× Ĥ
∣∣∣2 +
∣∣∣∇ϑ̂∣∣∣2)dx (5.33)
and ν(ω) is a positive function of ω, Ω and the material constants.
Proof. Let us integrate over Ω the relations derived by taking the inner products of (5.11)
by Ê∗, aϑ̂∗ and ∇× Ĥ∗, of the conjugate of (5.12) by Ĥ and ∇× Ê, of (5.14) by Ê∗ and of the
conjugate of the same (5.14) by Ĵ,∇×Ĥ and aϑ̂, of (5.15) by∇ϑ̂∗, of the conjugate of the same
(5.15) by q̂ and multiplying (5.13) by ϑ̂∗.
Thus, from (5.11), with two integrations similar to the one made to derive (5.21), we get
− iωε
∫
Ω
∣∣∣Ê∣∣∣2 dx− iωa · ∫
Ω
ϑ̂Ê∗dx +
∫
Ω
Ĥ · ∇ × Ê∗dx
−
∫
Ω
Ĵ · Ê∗dx =
∫
Ω
F̂ · Ê∗dx, (5.34)
− iωεa ·
∫
Ω
Êϑ̂∗dx−iωa2
∫
Ω
∣∣∣ϑ̂∣∣∣2 dx + a ·
∫
Ω
∇× Ĥϑ̂∗dx
− a ·
∫
Ω
Ĵϑ∗dx =
∫
Ω
F̂ · aϑ̂∗dx, (5.35)
− iωε
∫
Ω
∇× Ê · Ĥ∗dx−iωa ·
∫
Ω
ϑ̂∇× Ĥ∗dx +
∫
Ω
∣∣∣∇× Ĥ
∣∣∣2 dx
−
∫
Ω
Ĵ · ∇ × Ĥ∗dx =
∫
Ω
F̂ · ∇ × Ĥ∗dx; (5.36)
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 447
from (5.12) we have
−iωµ
∫
Ω
∣∣∣Ĥ∣∣∣2 dx +
∫
Ω
∇× Ê∗ · Ĥdx =
∫
Ω
Ĝ∗ · Ĥdx, (5.37)
−iωµ
∫
Ω
Ĥ∗ · ∇ × Êdx +
∫
Ω
∣∣∣∇× Ê
∣∣∣2 dx =
∫
Ω
Ĝ∗ · ∇ × Êdx; (5.38)
from (5.13), with an integration by parts taking account of (5.16)2, we obtain
iωΘ0a ·
∫
Ω
Êϑ̂∗dx + iω
(
c+ Θ0a
2/ε
) ∫
Ω
∣∣∣ϑ̂∣∣∣2 dx− ∫
Ω
q̂ · ∇ϑ̂∗dx =
∫
Ω
R̂ϑ̂∗dx. (5.39)
Moreover, (5.14) yields
(1− iωα)
∫
Ω
∣∣∣Ĵ∣∣∣2 dx− σ ∫
Ω
Ê∗ · Ĵdx =
∫
Ω
Î∗ · Ĵdx, (5.40)
(1 + iωα)
∫
Ω
Ĵ · Ê∗dx− σ
∫
Ω
∣∣∣Ê∣∣∣2 dx =
∫
Ω
Î · Ê∗dx, (5.41)
(1− iωα)
∫
Ω
Ĵ∗ · ∇ × Ĥdx− σ
∫
Ω
∇× Ê∗ · Ĥdx =
∫
Ω
Î∗ · ∇ × Ĥdx, (5.42)
(1− iωα)a ·
∫
Ω
Ĵ∗ϑ̂dx− σa ·
∫
Ω
Ê∗ϑdx =
∫
Ω
Î∗ · aϑ̂dx, (5.43)
in the last relation by one we have integrated as for (5.21).
Finally, (5.15) gives
(1− iωτ)
∫
Ω
|q̂|2 dx + k
∫
Ω
∇ϑ̂∗ · q̂dx =
∫
Ω
Q̂∗ · q̂dx, (5.44)
(1 + iωτ)
∫
Ω
q̂ · ∇ϑ̂∗dx + k
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx =
∫
Ω
Q̂ · ∇ϑ̂∗dx. (5.45)
We first consider that the real and the imaginary parts of (5.44)
Re
∫
Ω
q̂ · ∇ϑ̂∗dx =
1
k
Re
∫
Ω
Q̂∗ · q̂dx−
∫
Ω
|q̂|2 dx
, (5.46)
Im
∫
Ω
q̂ · ∇ϑ̂∗dx =
1
k
Im
∫
Ω
Q̂∗ · q̂dx + ωτ
∫
Ω
|q̂|2 dx
, (5.47)
448 G. AMENDOLA
from which, subtracting (5.47) multiplied by ωτ from (5.46) and taking account of
Re
∫
Ω
q̂ · ∇ϑ̂∗dx− ωτ Im
∫
Ω
q̂ · ∇ϑ̂∗dx = Re
∫
Ω
Q̂ · ∇ϑ̂∗dx− k
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx, (5.48)
the real part of (5.45), we obtain
k
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx− 1
k
(1 + τ2ω2)
∫
Ω
|q̂|2 dx = Re
∫
Ω
Q̂ · ∇ϑ̂∗dx
+
1
k
ωτ Im
∫
Ω
Q̂∗ · q̂dx− Re
∫
Ω
Q̂∗ · q̂dx
. (5.49)
Analogously, from the real and the imaginary parts of (5.40),∫
Ω
∣∣∣Ĵ∣∣∣2 dx− σRe
∫
Ω
Ê∗ · Ĵdx = Re
∫
Ω
Î∗ · Ĵdx, (5.50)
−ωα
∫
Ω
∣∣∣Ĵ∣∣∣2 dx− σ Im
∫
Ω
Ê∗ · Ĵdx = Im
∫
Ω
Î∗ · Ĵdx, (5.51)
and the real part of (5.41),
−Re
∫
Ω
Ĵ · Ê∗dx + ωα Im
∫
Ω
Ĵ · Ê∗dx = −σ
∫
Ω
∣∣∣Ê∣∣∣2 dx− Re
∫
Ω
Î · Ê∗dx, (5.52)
we have
σ
∫
Ω
∣∣∣Ê∣∣∣2 dx− 1
σ
(1 + α2ω2)
∫
Ω
∣∣∣Ĵ∣∣∣2 dx = −Re
∫
Ω
Î · Ê∗dx
− 1
σ
Re
∫
Ω
Î∗ · Ĵdx− ωα Im
∫
Ω
Î∗ · Ĵdx
. (5.53)
A useful relation follows at once from the real part of (5.37),
Re
∫
Ω
∇× Ê∗ · Ĥdx = Re
∫
Ω
Ĝ∗ · Ĥdx. (5.54)
If we consider (5.41) again, adding its real part (5.52) to the imaginary one, multiplied by
ωα, we obtain the relation
Re
∫
Ω
Ĵ · Ê∗dx =
1
1 + α2ω2
σ ∫
Ω
∣∣∣Ê∣∣∣2 dx + Re
∫
Ω
Î · Ê∗dx + ωα Im
∫
Ω
Î · Ê∗dx
, (5.55)
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 449
which, together with (5.54), allows us to write the real part of (5.34) in as:
ω Ima ·
∫
Ω
ϑ̂Ê∗dx = Re
∫
Ω
F̂ · Ê∗dx− Re
∫
Ω
Ĝ∗ · Ĥdx
+
1
1 + α2ω2
σ ∫
Ω
∣∣∣Ê∣∣∣2 dx + Re
∫
Ω
Î · Ê∗dx + ωα Im
∫
Ω
Î · Ê∗dx
. (5.56)
A different expression for the left-hand side of (5.56) can be derived from the real part of (5.39),
which, on account of (5.46), can be written as follows:
ω Ima ·
∫
Ω
ϑ̂Ê∗dx =
1
Θ0
Re
∫
Ω
R̂ϑ̂∗dx +
1
k
Re
∫
Ω
Q̂∗q̂dx−
∫
Ω
|q̂|2 dx
. (5.57)
Finally, subtracting (5.57) from (5.56), we get
σ
∫
Ω
∣∣∣Ê∣∣∣2dx +
1
kΘ0
(1 + α2ω2)
∫
Ω
|q̂|2 dx
= (1 + α2ω2)
1
kΘ0
Re
∫
Ω
Q̂∗ · q̂dx + kRe
∫
Ω
R̂ϑ̂∗dx
− Re
∫
Ω
F̂ · Ê∗dx
+ Re
∫
Ω
Ĝ∗ · Ĥdx
− Re
∫
Ω
Î · Ê∗dx− ωα Im
∫
Ω
Î · Ê∗dx. (5.58)
This relation, if we subtract (5.53), yields
1
σ
∫
Ω
∣∣∣Ĵ∣∣∣2 dx +
1
kΘ0
∫
Ω
|q̂|2 dx =
1
kΘ0
Re
∫
Ω
Q̂∗ · q̂dx + kRe
∫
Ω
R̂ϑ̂∗dx
+
1
1 + α2ω2
1
σ
Re
∫
Ω
Î∗ · Ĵdx− ωα
Im
∫
Ω
Î · Ê∗dx +
1
σ
Im
∫
Ω
Î∗ · Ĵdx
− Re
∫
Ω
F̂ · Ê∗dx + Re
∫
Ω
Ĝ∗ · Ĥdx (5.59)
and, moreover, gives an increase for
1
kΘ0
∫
Ω
|q̂|2 dx, which allows us to derive from (5.49) the
450 G. AMENDOLA
inequality
k
Θ0
∫
Ω
∣∣∣∇ϑ̂∣∣∣2 dx ≤ 1
Θ0
Re
∫
Ω
Q̂ · ∇ϑ̂∗dx+ω
τ
k
Im
∫
Ω
Q̂∗ · q̂dx
+ωτ Re
∫
Ω
Q̂∗ · q̂dx
+ (1 + τ2ω2)
1
Θ0
Re
∫
Ω
R̂ϑ̂∗dx + Re
∫
Ω
Ĝ∗ · Ĥdx
−Re
∫
Ω
F̂ · Ê∗dx
− 1 + τ2ω2
1 + α2ω2
Re
∫
Ω
Î · Ê∗dx + ωα Im
∫
Ω
Î · Ê∗dx
. (5.60)
It remains to consider the real parts of (5.38) and (5.36), that is
1
µ
∫
Ω
∣∣∣∇× Ê
∣∣∣2 dx =
1
µ
Re
∫
Ω
Ĝ∗ · ∇ × Êdx− ω Im
∫
Ω
∇× Ê · Ĥ∗dx, (5.61)
1
ε
∫
Ω
∣∣∣∇× Ĥ
∣∣∣2 dx =
1
ε
Re
∫
Ω
F̂ · ∇ × Ĥ∗dx + Re
∫
Ω
Ĵ · ∇ × Ĥ∗dx
−ω Ima ·
∫
Ω
ϑ̂∇× Ĥ∗dx
− ω Im
∫
Ω
∇× Ê · Ĥ∗dx, (5.62)
where only the last three terms must be evaluated.
To do this we first consider the imaginary part of (5.39), which, on account of (5.47), assumes
the following form:
ωRea ·
∫
Ω
Êϑ̂∗dx =
1
Θ0
Im
∫
Ω
R̂ϑ̂∗dx +
1
k
Im
∫
Ω
Q̂∗ · q̂dx
+ ω
τ
kΘ0
∫
Ω
|q̂|2 dx−
(
c
Θ0
+
a2
ε
)∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
. (5.63)
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 451
The imaginary part of (5.34), together with (5.51) and (5.63), yields the required quantity,
−ω Im
∫
Ω
∇× Ê · Ĥ∗dx = ω
Im
∫
Ω
F̂ · Ê∗dx− 1
σ
Im
∫
Ω
Î∗ · Ĵdx
+
1
Θ0
(
Im
∫
Ω
R̂ϑ̂∗dx +
1
k
Im
∫
Ω
Q̂∗ · q̂dx
)]
+ ω2
ε∫
Ω
∣∣∣Ê∣∣∣2 dx
+
τ
kΘ0
∫
Ω
|q̂|2 dx−
(
c
Θ0
+
a2
ε
)∫
Ω
∣∣∣ϑ̂∣∣∣2 dx− α
σ
∫
Ω
∣∣∣Ĵ∣∣∣2 dx
. (5.64)
Then, from (5.43), adding its imaginary part to the real one multiplied by ωα, we obtain a
relation, which, after introducing (5.57) and (5.63), gives
ω
ε
Ima ·
∫
Ω
Ĵϑ̂∗dx =
1
1 + α2ω2
σ
ε
ω2α
(
c
Θ0
+
a2
ε
)∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
+ (1− ω2ατ)
1
kΘ0
∫
Ω
|q̂|2 dx− ω α
Θ0
Im
∫
Ω
R̂ϑ̂∗dx +
1
k
Im
∫
Ω
Q̂∗ · q̂dx
− ω2α
σ
Re
∫
Ω
Î∗ · aϑ̂dx− 1
Θ0
Re
∫
Ω
R̂ϑ̂∗dx +
1
k
Re
∫
Ω
Q̂∗ · q̂dx
−ω
σ
Im
∫
Ω
Î∗ · aϑ̂dx
. (5.65)
This expression, together with (5.63), allows us to evaluate, from the imaginary part of
(5.35), the other quantity
−ω
ε
Ima ·
∫
Ω
ϑ̂∇× Ĥ∗dx =
ω
ε
Im
∫
Ω
F̂ · aϑ̂∗dx− 1
1 + α2ω2
1
ε
σ
Θ0
Re
∫
Ω
R̂ϑ̂∗dx
+
1
k
Re
∫
Ω
Q̂∗ · q̂dx
+ ω
ωαRe
∫
Ω
Î∗ · aϑ̂dx + Im
∫
Ω
Î∗ · aϑ̂dx
+
(
1− ασ
ε
1
1 + α2ω2
)
ω
kΘ0
Im
∫
Ω
Q̂∗ · q̂dx + k Im
∫
Ω
R̂ϑ̂∗dx
452 G. AMENDOLA
+ ω2
[
1
1 + α2ω2
ασ
ε
(
c
Θ0
+
a2
ε
)
− c
Θ0
] ∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
+
[
τω2
(
1− 1
1 + α2ω2
ασ
ε
)
+
1
1 + α2ω2
σ
ε
]
1
kΘ0
∫
Ω
|q̂|2 dx. (5.66)
Finally, for the third quantity we must derive, consider (5.42). Subtracting from its real part
the imaginary one multiplied by ωα and taking into account (5.54) and (5.64), we get
1
ε
Re
∫
Ω
Ĵ·∇ × Ĥ∗dx =
1
1 + α2ω2
1
ε
σRe
∫
Ω
Ĝ∗ · Ĥdx + Re
∫
Ω
Î∗ · ∇ × Ĥdx
− ωα Im
∫
Ω
Î∗ · ∇ × Ĥdx− ωασ
Im
∫
Ω
F̂ · Ê∗dx− 1
σ
Im
∫
Ω
Î∗ · Ĵdx+
+
1
kΘ0
Im
∫
Ω
Q̂∗ · q̂dx + k Im
∫
Ω
R̂ϑ̂∗dx
+ ω2ασ
α
σ
∫
Ω
∣∣∣Ĵ∣∣∣2 dx
+
(
c
Θ0
+
a2
ε
)∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
−
ε∫
Ω
∣∣∣Ê∣∣∣2 dx +
τ
kΘ0
∫
Ω
|q̂|2 dx
. (5.67)
Thus, we have an estimate for all the quantities of (5.33), which, taking account of (5.17),
can be increased as follows:
ξG(ω) ≤
∫
Ω
{[
σ
∣∣∣Ê∣∣∣2 +
1
kΘ0
(1 + α2ω2) |q̂|2
](
1
σ
∣∣∣Ĵ∣∣∣2 +
1
kΘ0
|q̂|2
)
+
1
µ
∣∣∣∇× Ê
∣∣∣2
+ [1 + εβH(Ω)]
1
ε
∣∣∣∇× Ĥ
∣∣∣2 dx +
[
1 +
Θ0
k
βϑ(Ω)
]
k
Θ0
∣∣∣∇ϑ̂∣∣∣2} , (5.68)
where ξ = min {σ, 1, 1/σ, 1/kΘ0, 1/µ, 1/ε, k/Θ0} .
We consider the relations (5.58), (5.59) and (5.60) together with (5.61) and (5.62), which
must be substituted into this inequality. We observe that in (5.64), (5.66) and (5.67), to be
considered for (5.61) and (5.62), there are some negative terms which can be neglected, but
there are the following positive terms
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 453
{
[2 + εβH(Ω)]τω2 + [1 + εβH(Ω)]
(
τω2 +
σ
ε
1
1 + α2ω2
)} 1
kΘ0
∫
Ω
|q̂|2dx
+[2 + εβH(Ω)]
ε
σ
ω2
σ ∫
Ω
∣∣∣Ê∣∣∣2 dx
+
σα2
ε
[1 + εβH(Ω)]
ω2
1 + α2ω2
1
σ
∫
Ω
∣∣∣Ĵ∣∣∣2 dx
+ 2
Θ0
k
[1 + εβH(Ω)]
ασ
ε
(
c
Θ0
+
a2
ε
)
ω2
1 + α2ω2
k
Θ0
∫
Ω
∣∣∣ϑ̂∣∣∣2 dx
,
whose four expressions, in parenthesis and containing the four integrals, can be increased by
the right-hand sides of (5.59), (5.58), (5.59), (5.17)1 and (5.60), respectively.
To simplify the result, it is useful to put
β1 = [1 + εβH(Ω)]σ/ε, β2 = [2 + εβH(Ω)] ε/σ, β3 = [3 + 2εβH(Ω)] τ,
β4 = 1 + Θ0βϑ(Ω)/k, β5 = 2
(
c/Θ0 + a2/ε
)
αΘ0βϑ(Ω)/k,
with which we define
c1 = 2 + β1, c2 = α2 + β2 + β3, c3 = β2α
2, c4 = β1β5,
c5 = β2σ/ε− β1α, c6 = β1/σ, c7 = 1 + β1
and finally we introduce
d1 = c1 + β4, d2 = c2 + β4τ, d3 = kc1 + β4/Θ0, d4 = kc2 + β4τ
2/Θ0,
d5 = c5 − c6ασ, d6 = c2 + β4τ
2, d7 = β4τ + c6ε.
Thus, (5.68) becomes
ξG(ω) ≤
(
−
[
d1 +
(
d2 + c4
1 + τ2ω2
1 + α2ω2
)
ω2 + c3ω
4
]
Re
∫
Ω
F̂ · Ê∗dx
+ c5
ω
1 + α2ω2
Im
∫
Ω
F̂ · Ê∗dx
)
+
(
c6 Re
∫
Ω
F̂ · ∇ × Ĥ∗dx
)
+
(
c6ω Im
∫
Ω
F̂ · aϑ̂∗dx
)
+
([
d1 +
(
d2 + c4
1 + τ2ω2
1 + α2ω2
)
ω2 + c3ω
4 + β1
1
1 + α2ω2
]
Re
∫
Ω
Ĝ∗ · Ĥdx
)
454 G. AMENDOLA
+
(
1
µ
Re
∫
Ω
Ĝ∗ · ∇ × Êdx
)
+
([
d3 +
(
d4 +
c4
Θ0
1 + τ2ω2
1 + α2ω2
)
ω2 + kc3ω
4
− β1
Θ0
1
1 + α2ω2
]
Re
∫
Ω
R̂ϑ̂∗dx +
1
Θ0
(
c6ε+ d5
1
1 + α2ω2
)
ω Im
∫
Ω
R̂ϑ̂∗dx
)
−
([
1 + β2ω
2 +
(
β4 + c4
ω2
1 + α2ω2
)
1 + τ2ω2
1 + α2ω2
]
Re
∫
Ω
Î · Ê∗dx +
[
1 + β2ω
2
+
(
β4 + c4
ω2
1 + α2ω2
)
1 + τ2ω2
1 + α2ω2
+ (c7 + β3ω
2)
1
1 + α2ω2
]
αω Im
∫
Ω
Î · Ê∗dx
)
+
(
1
σ
1
1 + α2ω2
(c7 + β3ω
2) Re
∫
Ω
Î∗ · Ĵdx− [c5 + (c7 + β3ω
2)α]ω Im
∫
Ω
Î∗ · Ĵdx
)
+
(
c6
1
1 + α2ω2
Re
∫
Ω
Î∗ · ∇ × Ĥdx− αω Im
∫
Ω
Î∗ · ∇ × Ĥdx
)
−
(
c6
ω
1 + α2ω2
αωRe
∫
Ω
Î∗ · aϑ̂dx + Im
∫
Ω
Î∗ · aϑ̂dx
)
+
(
1
kΘ0
[
c1 + d6ω
2 +
(
c3 + c4τ
2 1
1 + α2ω2
)
ω4 − β1
1
1 + α2ω2
]
Re
∫
Ω
Q̂∗ · q̂dx
+
[
d7 + (d5 + c4τω
2)
1
1 + α2ω2
]
ω Im Q̂∗ · q̂dx
})
+
(
1
Θ0
(
β4 + c4
ω2
1 + α2ω2
)
Re
∫
Ω
Q̂ · ∇ϑ̂∗dx
)
. (5.69)
If we denote by the λi,j(ω), i = 1, 2, ..., 12; j = 1, 2, the coefficients of the real (j = 1) and
the imaginary (j = 2) parts of the twelve different integrals in (5.69) and consider
λ(ω) =
1
ξ
max {|λi,1(ω)|+ |λi,2(ω)| , i = 1, 2, . . . , 12} , (5.70)
from (5.69) with applications of Schwarz’s inequality it follows that
G(ω) ≤ 12λ(ω)max {1, |a|}
∫
Ω
(∣∣∣F̂∣∣∣2 +
∣∣∣Ĝ∣∣∣2 +
∣∣∣̂I∣∣∣2 +
∣∣∣Q̂∣∣∣2 +
∣∣∣R̂∣∣∣2) dx
1/2
G1/2(ω) (5.71)
and hence (5.32).
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 455
Lemma 5.2. If the sources (F,G, I,Q, R) ∈ V ′(Ω,R+), the hypotheses there stated assure
that the inverse Fourier transforms of (Ê, Ĥ, Ĵ, q̂, ϑ̂) ∈ Ĥ (Ω,R) exist and are L2-functions with
zero initial data.
Proof. Since the 5-tuple (F,G, I,Q, R) ∈ V ′(Ω,R+), the hypotheses assumed on the
sources together with the form of ν(ω), which is a continuous function of ω ∈ R and ap-
proaches infinity as ω4, allow us to integrate over R the right-hand side of (5.32), i.e.,
+∞∫
−∞
∫
Ω
ν2(ω)
(∣∣∣F̂∣∣∣2 +
∣∣∣Ĝ∣∣∣2 +
∣∣∣̂I∣∣∣2 +
∣∣∣Q̂∣∣∣2 +
∣∣∣R̂∣∣∣2) dxdω < +∞. (5.72)
Thus, from the inequality (5.32) it follows that
+∞∫
−∞
G(ω)dω = 2π‖(Ê(x, ω), Ĥ(x, ω), Ĵ(x, ω), q̂(x, ω), ϑ̂(x, ω))‖2
Ĥ
≤
+∞∫
−∞
∫
Ω
ν2(ω)
(∣∣∣F̂∣∣∣2 +
∣∣∣Ĝ∣∣∣2 +
∣∣∣̂I∣∣∣2 +
∣∣∣Q̂∣∣∣2 +
∣∣∣R̂∣∣∣2) dxdω, (5.73)
whence, by virtue of Plancherel’s theorem, the inverse Fourier transforms of (Ê, Ĥ, Ĵ, q̂, ϑ̂)
exist.
From the linearity of system (5.11) – (5.15) and inequality (5.32) of Lemma 5.1 we have this
corollary.
Corollary 5.1. Let (F̂(i), Ĝ(i), Î(i), Q̂(i), R̂(i)), i = 1, 2, be two source fields. The correspond-
ing solutions of our problem (Ê(i), Ĥ(i), Ĵ(i), q̂(i), ϑ̂(i)) satisfy
2π
∥∥∥(Ê(1) − Ê(2), Ĥ(1) − Ĥ(2), Ĵ(1) − Ĵ(2), q̂(1) − q̂(2), ϑ̂(1) − ϑ̂(2))
∥∥∥2
Ĥ
≤
+∞∫
−∞
∫
Ω
ν2(ω)
(∣∣∣F̂(1) − F̂(2)
∣∣∣2 +
∣∣∣Ĝ(1) − Ĝ(2)
∣∣∣2 +
∣∣∣̂I(1) − Î(2)
∣∣∣2
+
∣∣∣Q̂(1) − Q̂(2)
∣∣∣2 +
∣∣∣R̂(1) − R̂(2)
∣∣∣2) dxdω. (5.74)
Lemma 5.3. The subset
Sa =
{
(F̂, Ĝ, Î, Q̂, R̂) ∈ V̂ ′(Ω,R) : ∃ (Ê, Ĥ, Ĵ, q̂, ϑ̂) ∈ Ĥ (Ω,R)
which satisfies (5.10) ∀(ê, ĥ, û, v̂, β̂) ∈ Ŵ (Ω,R)
}
(5.75)
is dense and closed in V̂ ′(Ω,R).
456 G. AMENDOLA
Proof. To prove that Sa is dense we suppose that there is a nonzero element (F̂0, Ĝ0, Î0,
Q̂0, R̂0) ∈ V̂ ′(Ω,R)\S̄a, S̄a being the closure of Sa in V̂ ′(Ω,R). We use the Hahn – Banach
theorem, which states that there exists (ê0, ĥ0, û0, v̂0, β̂0) ∈ Ŵ (Ω,R) which satisfies these
relations
〈(F̂0, Ĝ0, Î0, Q̂0, R̂0), (ê0, ĥ0, û0, v̂0, β̂0)〉 6= 0,
〈(F̂, Ĝ, Î, Q̂, R̂), (ê0, ĥ0, û0, v̂0, β̂0)〉 = 0 ∀(F̂, Ĝ, Î, Q̂, R̂) ∈ Sa. (5.76)
The second condition is equivalent to
Λ[(Ê, Ĥ, Ĵ, q̂, ϑ̂), (ê0, ĥ0, û0, v̂0, β̂0)] = 0 ∀(Ê, Ĥ, Ĵ, q̂, ϑ̂) ∈ Ĥ (Ω,R) (5.77)
because of (5.18), whence it follows that
(ê0, ĥ0, û0, v̂0, β̂0) = 0, (5.78)
which does not satisfies (5.76). Hence, Sa is dense in V̂ ′(Ω,R).
To show the closure of Sa in V̂ ′(Ω,R) we consider a sequence of sources, which is denoted
by
{
(F̂(n), Ĝ(n), Î(n), Q̂(n), R̂(n)) ∈ Sa, n = 1, 2, ...
}
and assumed convergent to (F̂, Ĝ, Î, Q̂,
R̂) ∈ V̂ ′(Ω,R) .
Denoting by (Ê(n), Ĥ(n), Ĵ(n), q̂(n), ϑ̂(n)) ∈ Ĥ (Ω,R) the corresponding solutions, we con-
sider (5.74) of Corollary 5.1, which gives∥∥∥(Ê(n) − Ê(m), Ĥ(n) − Ĥ(m), Ĵ(n) − Ĵ(m), q̂(n) − q̂(m), ϑ̂(n) − ϑ̂(m))
∥∥∥2
Ĥ
≤ 1
2π
+∞∫
−∞
∫
Ω
ν2(ω)
(∣∣∣F̂(n) − F̂(m)
∣∣∣2 +
∣∣∣Ĝ(n) − Ĝ(m)
∣∣∣2 +
∣∣∣̂I(n) − Î(m)
∣∣∣2
+
∣∣∣Q̂(n) − Q̂(m)
∣∣∣2 +
∣∣∣R̂(n) − R̂(m)
∣∣∣2) dxdω, (5.79)
whence it follows that
{
(Ê(n), Ĥ(n), Ĵ(n), q̂(n), ϑ̂(n)), n = 1, 2, ...
}
is a Cauchy sequence and the
completeness of the space gives
lim
n→+∞
(Ê(n), Ĥ(n), Ĵ(n), q̂(n), ϑ̂(n)) = (Ê, Ĥ, Ĵ, q̂, ϑ̂) ∈ Ĥ (Ω,R). (5.80)
Thus, it is enough to consider the sequence of identities obtained by substituting the so-
lutions (Ê(n), Ĥ(n), Ĵ(n), q̂(n), ϑ̂(n)) and the sources (F̂(n), Ĝ(n), Î(n), Q̂(n), R̂(n)) into (5.18); the
limit of these identities as n → +∞, consists of a similar identity in terms of the limits (Ê, Ĥ, Ĵ,
q̂, ϑ̂) and (F̂, Ĝ, Î, Q̂, R̂); therefore, we conclude that (F̂, Ĝ, Î, Q̂, R̂) ∈ Sa.
The application of the Plancherel theorem yields the existence of (E,H,J,q, ϑ) ∈ H (Ω,R+),
which is the solution of our problem.
Thus, the proof of Theorem 5.1 is complete.
ASYMPTOTIC STABILITY FOR A THERMOELECTROMAGNETIC MATERIAL 457
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3. Amendola G. “Linear stability for a thermoelectromagnetic material with memory,” Quart. Appl. Math., 1,
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4. Noll W. “A new mathematical theory of simple materials,” Arch. Ration. Mech. and Anal., 48, 1 – 50 (1972).
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Received 17.10.2001
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