10 / 17
Applying Friedrichs inequality with an arbitrary
R >
0
we obtain an
estimate
k
u
k
L
2
(Ω
R
)
≤
C
(
R
)
k∇
u
k
L
2
(Ω
R
)
for
t >
0
. Thence, holds the
following inequality:
k∇
u
k
H
1
(Ω
R
)
+
k
u
t
k
H
1
(Ω
R
)
≤
C
(
R
)
.
(32)
So, the set of functions
{
u
t
(
t, x
)
}
and
{∇
u
(
t, x
)
}
,
t >
0
, are compact in
L
2
(Ω
R
)
for any
R >
0
. Let us prove that
lim
t
→∞
1
t
t
Z
0
k
u
τ
k
2
L
2
(Ω
R
)
dτ
= 0
.
By the estimate (32) the set of functions
{
u
t
(
t, x
)
}
,
t >
0
is compact in
L
2
(Ω
R
)
. Let
{
h
j,R
(
x
)
}
,
j
= 1
,
2
, . . .
,
x
∈
Ω
R
be an orthonormal basis in the
L
2
(Ω
R
)
space. We continue the functions
h
j,R
by zero to
Ω
\
Ω
R
. Denote
the continued functions as
h
j,R
too. Then
u
t
(
t, x
) =
∞
X
j
=1
c
j,R
(
t
)
h
j,R
(
x
)
for
t >
0
. We have
lim
N
→∞
k
u
t
−
N
X
j
=1
c
j,R
(
t
)
h
j,R
(
x
)
k
L
2
(Ω
R
)
= 0
for
t >
0
. By the compactness criterion [21, P. 247, Th. 3] in the space
L
2
(Ω
R
)
with basis
{
h
j,R
}
for all
ε >
0
there exists
N >
0
such that
u
t
(
t, x
) =
N
X
j
=1
c
j,R
(
t
)
h
j,R
(
x
) +
∞
X
j
=
N
+1
c
j,R
(
t
)
h
j,R
(
x
)
for
t >
0
,
x
∈
Ω
R
and
k
∞
X
j
=
N
+1
c
j,R
h
j,R
k
L
2
(Ω
R
)
< ε
,
t >
0
.
Thence,
k
u
t
k
2
L
2
(Ω
R
)
=
k
N
X
j
=1
c
j,R
h
j,R
k
2
L
2
(Ω
R
)
+
k
∞
X
j
=
N
+1
c
j,R
h
j,R
k
2
L
2
(Ω
R
)
=
=
N
X
j
=1
c
2
j,R
(
t
) +
k
∞
X
j
=
N
+1
c
j,R
h
j,R
k
2
L
2
(Ω)
<
N
X
j
=1
c
2
j,R
(
t
) +
ε
2
.
(33)
Integrate (33), we obtain
lim
t
→∞
1
t
t
Z
0
k
u
τ
k
2
L
2
(Ω
R
)
dτ <
N
X
j
=1
lim
t
→∞
1
t
t
Z
0
c
2
j,R
dτ
+
ε
2
=
=
N
X
j
=1
lim
t
→∞
1
t
t
Z
0
(
u
τ
, h
j,R
)
2
L
2
(Ω
R
)
dτ
+
ε
2
.
(34)
By the equality (29) we have
lim
t
→∞
1
t
t
Z
0
(
u
τ
, h
j,R
)
2
L
2
(Ω)
dτ
= 0
for
j
=
12
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3