7 / 17
Now, we have an integral equality for an eigenfunction:
Z
Ω
|∇
v
|
2
dx
=
=
λ
Z
Ω
v
2
dx.
Therefore,
Z
Ω
|∇
v
|
2
+
λv
2
dx >
0;
Z
Ω
|∇
v
|
2
−
λv
2
dx
= 0
and by the absolute continuity of the Lebesgue integral there exists a
domain
Ω
0
is compact embedded to
Ω
such that for some
ε >
0
we have
the inequalities
Z
Ω
0
|∇
v
|
2
+
λv
2
dx
≥
ε
;
Z
Ω
0
|∇
v
|
2
−
λv
2
dx
≤
ε
2
.
(22)
Thus, by (21), (22) we obtain an inequality
E
Ω
0
(
t
)
≥
ε
4
for
t >
0
. Let us consider the solution
˜
u
of the problem (8)–(10) with initial
functions
f
= ˜
v
,
g
= 0
, where function
˜
v
∈ D
(Ω)
satisfy the inequality
k
v
−
˜
v
k
2
H
1
(Ω)
< ε/
16
.
Then
Z
Ω
0
(˜
u
2
t
+
|∇
˜
u
|
2
)
dx
≥
1
2
Z
Ω
0
u
2
t
+
|∇
u
|
2
dx
−
−
Z
Ω
0
(((
u
−
˜
u
)
t
)
2
+
|∇
(
u
−
˜
u
)
|
2
)
dx
≥
≥
ε
8
−
Z
Ω
0
(((
u
−
˜
u
)
t
)
2
+
|∇
(
u
−
˜
u
)
|
2
)
dx
≥
≥
ε
8
−
Z
Ω
(((
u
−
˜
u
)
t
)
2
+
|∇
(
u
−
˜
u
)
|
2
)
dx
=
=
ε
8
− k∇
(
v
−
˜
v
)
k
2
L
2
(Ω)
>
ε
8
−
ε
16
=
ε
16
>
0
, t >
0
.
The inequality (20) is proved.
I
Continuity of Spectrum and Mean Decay of Local Energy.
Let
Ω
be
an unbounded domain. In the case of
σ
p
(
L
) =
∅
we have the mean local
energy decay. A proof use the following theorem [1, Th. 9.15, 20].
Theorem 4.
Let real-valued function
m
(
z
)
∈
C
(
−∞
,
+
∞
)
,
m
(
z
) = 0
for
z
≤
0
,
var
[0
,
+
∞
)
m
(
z
)
<
∞
and
s
(
t
) =
∞
R
0
e
izt
dm
(
z
)
for
t >
0
. Then
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3
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