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in
(
t >
0)
×
Ω
, with the initial conditions
u
(0
, x
) =
f
(
x
);
u
t
(0
, x
) =
g
(
x
)
for
x
∈
Ω
(9)
and the boundary condition
u
= 0
(10)
on
(
t >
0)
×
Γ
. We assume the initial functions
f
∈
o
H
1
(Ω)
and
g
∈
L
2
(Ω)
are real-valued.
It is well known that a solution of the problem (8)–(10) satisfies the
energy conservation law
E
(
t
) =
k
u
t
k
2
L
2
(Ω)
+
k∇
u
k
2
L
2
(Ω)
=
E
(0)
(11)
for
t >
0
.
Below we study the connections between spectral properties of the
Laplace operator and the behavior for large values of time to solutions
of the problem (8)–(10). We investigate stabilization in time with special
regard to the properties of the local energy function
E
Ω
0
(
t
) =
k
u
t
k
2
L
2
(Ω
0
)
+
k∇
u
k
2
L
2
(Ω
0
)
(12)
where
Ω
0
⊂
Ω
is a bounded domain.
Let us note that many practically important problems deal with
unbounded domains
Ω
so the inverse operator
L
−
1
may be unbounded
in the main space
H
=
L
2
(Ω)
.
Solutions from the Energy Class and Spectral Representation.
We
consider solutions of the problem (8)–(10) from the energy class (see
[11]), that is a function
u
(
t, x
)
∈
C
([0
,
+
∞
);
o
H
1
(Ω))
such that
u
t
(
t, x
)
∈
∈
C
([0
,
+
∞
);
L
2
(Ω))
satisfying the initial conditions (9), the equality (11)
and the integral identity
T
Z
0
Z
Ω
((
∇
u,
∇
w
)
−
u
t
w
t
)
dt dx
=
Z
Ω
g
(
x
)
w
(0
, x
)
dx,
for all
w
∈
H
1
((0
, T
)
×
Ω)
satisfying
w
= 0
on
(0
, T
)
×
Γ
and
w
(
T, x
) = 0
,
T >
0
.
Consider a self-adjoint non-negative operator
L
:
D
(
L
)
→
L
2
(Ω)
generated in
L
2
(Ω)
space by differential expression
−
Δ
with Dirichlet
boundary condition. Using an estimates to solutions of elliptic boundary
value problems [14], we have the domain of operator
L
:
D
(
L
) =
o
H
1
(Ω)
∩
∩
H
2
(Ω
0
)
∩ {
Δ
u
∈
L
2
(Ω)
}
,
here
Ω
0
⊂
Ω
is an arbitrary bounded domain.
Let
{
E
(
λ
)
}
,
−∞
< λ <
+
∞
, be a family of spectral projectors associated
with operator
L
[15].
6
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3