12 / 17
F
j
[
υ
] =
υ,
˜
h
j,R
e
H
=
∇
υ,
∇
˜
h
j,R
L
2
(Ω)
=
Z
Ω
∇
υ,
∇
˜
h
j,R
dx.
(37)
It follows from (30) and (37) that
lim
t
→∞
1
t
t
Z
0
(
u, h
j,R
)
2
e
H
R
dτ
= lim
t
→∞
1
t
t
Z
0
u,
˜
h
j,R
2
e
H
dt
=
= lim
t
→∞
1
t
t
Z
0
∇
u,
∇
˜
h
j,R
2
L
2
(Ω)
dτ
= 0
(38)
for
j
= 1
,
2
, . . . , N
. Using (36) and (38), we obtain for any
ε >
0
the
relation
lim sup
t
→∞
1
t
t
Z
0
k∇
u
k
2
L
2
(Ω
R
)
dτ
= lim sup
t
→∞
1
t
t
Z
0
k
u
k
2
e
H
R
dt
6
6
N
X
j
=1
lim
t
→∞
1
t
t
Z
0
(
u,
˜
h
j,R
)
2
e
H
dt
+
ε
2
=
ε
2
.
So, we have the equality
lim
t
→∞
1
t
t
Z
0
k∇
u
(
τ, x
)
k
2
L
2
(Ω
R
)
dτ
= 0
(39)
for any
R >
0
. Now, for any bounded
Ω
0
⊂
Ω
we take
R
sufficiently large
such that
Ω
0
⊂
Ω
R
. Combining the relations (35) and (39), we obtain (24).
Theorem 5 is proved.
I
In [1] were considered the unbounded domains
Ω
with compact
boundaries, for which
σ
p
(
L
) =
∅
. It was proved [1, Lemma 9.17] that
for any bounded domain
Ω
0
⊂
Ω
the following relation holds:
lim inf
t
→∞
E
Ω
0
(
t
) = 0
.
(40)
It is readily seen that (40) follows from (24).
Absolute Continuity of Spectrum and Decay of Local Energy.
The
spectrum
σ
(
L
)
of self-adjoint operator
L
:
D
(
L
)
→
H
is absolutely
continuous
σ
(
L
) =
σ
ac
(
L
)
if
(
E
(
λ
)
h, h
)
is an absolutely continuous
function of
λ
for all
h
∈
H
[15].
14
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3