5 / 17
Lemma 1.
Let
f
∈
o
H
1
(Ω)
,
g
∈
L
2
(Ω)
. Then the solution of the
problem
(8)
–
(10)
from the energy class can be written as Bochner – Stiltjes
integral
u
=
∞
Z
0
cos(
√
λt
)
dE
(
λ
)
f
+
∞
Z
0
sin(
√
λt
)
√
λ
dE
(
λ
)
g.
(13)
The integral (13) converges uniformly in
o
H
1
(Ω)
on
[0
, T
]
,
T >
0
. The
derivative
u
t
can be written as
u
t
=
−
∞
Z
0
√
λ
sin(
√
λt
)
dE
(
λ
)
f
+
∞
Z
0
cos(
√
λt
)
dE
(
λ
)
g,
(14)
the integral
(14)
converges uniformly in
L
2
(Ω)
on
[0
, T
]
.
Point Spectrum and Non-Decay of Local Energy.
In a bounded
domain
Ω
the spectrum of operator
L
is discrete. So, the solution of the
problem (8)–(10) represents by the series
u
(
t, x
) =
∞
X
j
=1
(
a
j
cos(
p
λ
j
t
) +
b
j
sin(
p
λ
j
t
)
p
λ
j
)
v
j
(
x
)
,
(15)
where
a
j
= (
f, v
j
)
L
2
(Ω)
,
b
j
= (
g, v
j
)
L
2
(Ω)
. Here
0
< λ
1
< λ
2
≤
. . .
,
lim
j
→∞
λ
j
= +
∞
, is the sequence of eigenvalues of the operator
L
,
{
v
j
}
is
the orthonormal basis of eigenfunctions in
L
2
(Ω)
.
Using the equality (15), we can prove that solution of the problem
(8)–(10) is an almost-periodic function with respect to
t
(
n
= 1
[16];
n
≥
2
[18, 19]).
Theorem 2.
Let
Ω
⊂
R
n
be a bounded domain and
lim
t
→∞
E
Ω
0
(
t
) = 0
for some domain
Ω
0
⊂
Ω
. Then
u
= 0
in
(0
,
∞
)
×
Ω
.
J
The equality
lim
t
→∞
E
Ω
0
(
t
) = 0
means that
lim
t
→∞
k∇
u
(
t, x
)
k
L
2
(Ω
0
)
= 0
.
Therefore,
0 = lim
t
→∞
Z
Ω
0
u
x
k
(
t, x
)
η
(
x
)
dx
=
= lim
t
→∞
∞
X
j
=1
a
j
cos(
p
λ
j
t
)+
b
j
sin(
p
λ
j
t
)
p
λ
j
Z
Ω
0
(
v
j
)
x
k
(
x
)
η
(
x
)
dx, k
= 1
, . . . , n,
for an arbitrary function
η
(
x
)
∈ D
(Ω
0
) =
o
C
∞
(Ω
0
)
. We have
0 = lim
T
→∞
1
T
T
Z
0
Z
Ω
0
u
x
k
(
t, x
)
η
(
x
)
dx
cos(
p
λ
m
t
)
dt
=
=
1
2
Z
Ω
0
X
λ
j
=
λ
m
a
j
(
v
j
)
x
k
(
x
)
η
(
x
)
dx
;
(16)
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3
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