9 / 17
(
∇
u,
∇
q
)
L
2
(Ω)
=
√
Lu,
√
Lq
L
2
(Ω)
=
=
∞
Z
0
λ
cos(
√
λt
)
d
(
E
(
λ
)
f, q
) +
∞
Z
0
√
λ
sin(
√
λt
)
d
(
E
(
λ
)
g, q
) =
=
∞
Z
0
cos
ztdm
3
(
z
) +
∞
Z
0
sin
ztdm
4
(
z
)
.
(27)
Here we have the equalities
dm
1
(
z
) =
−
zd
(
E
(
z
2
)
f, q
);
dm
2
(
z
) =
d
(
E
(
z
2
)
g, q
);
dm
3
(
z
) =
z
2
d
(
E
(
z
2
)
f, q
);
dm
4
(
z
) =
zd
(
E
(
z
2
)
g, q
)
.
The operator
L
is a positive operator with
σ
p
(
L
) =
∅
, so
m
j
(
z
)
are
continuous functions for
−∞
< z <
+
∞
and
m
j
(
z
) = 0
for
z <
0
. By
the inequalities (25) and the relation [17, Ch. 9, Par. 1, Pt. 128, Eq. (12)] we
obtain that
var
[0
,
+
∞
)
m
j
(
z
)
<
∞
, j
= 1
, . . . ,
4
.
(28)
It follows from (23)–(26) that for all
q
(
x
)
∈ D
(Ω)
lim
t
→∞
1
t
t
Z
0
(
u
τ
(
τ, x
)
, q
(
x
))
2
L
2
(Ω)
dτ
= 0;
(29)
lim
t
→∞
1
t
t
Z
0
(
∇
u
(
τ, x
)
,
∇
q
(
x
))
2
L
2
(Ω)
dτ
= 0
.
(30)
Consider the closure of this equalities on
q
in
L
2
(Ω)
, we obtain that (29) is
valid for all functions
q
∈
L
2
(Ω)
and (30) — for all
q
∈
L
2
(Ω)
∩
H
1
(Ω
R
)
for any
R >
0
, satisfying the condition
q
= 0
on
Γ
and such that
R
Ω
|∇
q
|
2
dx <
∞
. Furthermore, we have
k
Δ
u
k
2
L
2
(Ω)
+
k∇
u
t
k
2
L
2
(Ω)
=
=
∞
Z
0
λ
2
cos
2
(
√
λt
)
d
(
E
(
λ
)
f, f
) +
∞
Z
0
λ
sin
2
(
√
λt
)
d
(
E
(
λ
)
g, g
)+
+
∞
Z
0
λ
2
sin
2
√
λtd
(
E
(
λ
)
f, f
) +
∞
Z
0
λ
cos
2
√
λtd
(
E
(
λ
)
g, g
) =
=
∞
Z
0
λ
2
d
(
E
(
λ
)
f, f
) +
∞
Z
0
λd
(
E
(
λ
)
g, g
) =
=
k
Δ
f
k
2
L
2
(Ω)
+
k∇
g
k
2
L
2
(Ω)
.
(31)
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3
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