11 / 17
= 1
,
2
, . . . , N
and it follows from (34) that for any
ε >
0
lim sup
t
→∞
1
t
t
Z
0
k
u
τ
k
2
L
2
(Ω
R
)
dτ
≤
ε
2
.
Now, we obtain
lim
t
→∞
1
t
t
Z
0
k
u
τ
k
2
L
2
(Ω
R
)
dτ
= 0
(35)
for any
R >
0
. Let us prove now that
lim
t
→∞
1
t
t
Z
0
k∇
u
k
2
L
2
(Ω
R
)
dτ
= 0
.
For any
R >
0
we define the space:
e
H
R
=
v
∈
H
1
(Ω
R
) :
v
|
Γ
R
= 0
,
where
Γ
R
= Γ
∩ {|
x
|
< R
}
. The space
e
H
R
is a Hilbert space with a scalar
product
(
v, w
)
e
H
R
=
Z
Ω
R
(
∇
v,
∇
w
)
dx.
Similarly we define the Hilbert space
e
H
=
{
v
∈
H
1
(Ω
R
)
for any
R >
0 :
v
|
Γ
= 0
,
Z
Ω
|∇
v
|
2
dx <
∞}
,
with a
scalar product
(
v, w
)
e
H
=
Z
Ω
(
∇
v,
∇
w
)
dx.
Let
R >
0
. It follows from (32) that the set of functions
{
u
(
t, x
)
}
,
t >
0
, is compact in the space
e
H
R
. Denote by
{
h
j,R
(
x
)
}
,
j
= 1
,
2
, . . .
, the
orthonormal basis in the space
e
H
R
. By the compactness criterion in the
space
e
H
R
with basis
{
h
j,R
}
for any
ε >
0
there exists
N >
0
such that
u
(
t, x
) =
N
X
j
=1
b
j,R
(
t
)
h
j,R
(
x
) +
∞
X
j
=
N
+1
b
j,R
(
t
)
h
j,R
(
x
)
,
and
k
∞
X
j
=
N
+1
b
j,R
(
t
)
h
j,R
(
x
)
k
e
H
R
< ε
for
t >
0
. Now we have
k
u
(
t, x
)
k
2
e
H
R
=
k
N
X
j
=1
b
j,R
h
j,R
k
2
e
H
R
+
k
∞
X
j
=
N
+1
b
j,R
h
j,R
k
2
e
H
R
<
<
N
X
j
=1
b
2
j,R
(
t
) +
ε
2
=
N
X
j
=1
(
u, h
j,R
)
2
e
H
R
+
ε
2
.
(36)
For all
υ
(
x
)
∈
e
H
we have an estimates
|
F
j
[
υ
]
|
= (
υ, h
j,R
)
e
H
R
≤
≤ k
υ
k
e
H
R
k
h
j,R
k
e
H
R
=
k
υ
k
e
H
R
≤ k
υ
k
e
H
,
j
= 1
,
2
, . . . , N
. It means that
the linear functional
F
j
[
υ
] = (
υ, h
j,R
)
e
H
R
is a bounded functional on
e
H
. By
the F. Riesz theorem there exist functions
˜
h
j,R
(
x
)
∈
e
H
,
j
= 1
,
2
, . . . , N
,
such that
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3
13