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F

j

[

υ

] =

υ,

˜

h

j,R

e

H

=

υ,

˜

h

j,R

L

2

(Ω)

=

Z

Ω

υ,

˜

h

j,R

dx.

(37)

It follows from (30) and (37) that

lim

t

→∞

1

t

t

Z

0

(

u, h

j,R

)

2

e

H

R

= lim

t

→∞

1

t

t

Z

0

u,

˜

h

j,R

2

e

H

dt

=

= lim

t

→∞

1

t

t

Z

0

u,

˜

h

j,R

2

L

2

(Ω)

= 0

(38)

for

j

= 1

,

2

, . . . , N

. Using (36) and (38), we obtain for any

ε >

0

the

relation

lim sup

t

→∞

1

t

t

Z

0

k∇

u

k

2

L

2

R

)

= lim sup

t

→∞

1

t

t

Z

0

k

u

k

2

e

H

R

dt

6

6

N

X

j

=1

 

lim

t

→∞

1

t

t

Z

0

(

u,

˜

h

j,R

)

2

e

H

dt

 

+

ε

2

=

ε

2

.

So, we have the equality

lim

t

→∞

1

t

t

Z

0

k∇

u

(

τ, x

)

k

2

L

2

R

)

= 0

(39)

for any

R >

0

. Now, for any bounded

Ω

0

Ω

we take

R

sufficiently large

such that

Ω

0

Ω

R

. Combining the relations (35) and (39), we obtain (24).

Theorem 5 is proved.

I

In [1] were considered the unbounded domains

Ω

with compact

boundaries, for which

σ

p

(

L

) =

. It was proved [1, Lemma 9.17] that

for any bounded domain

Ω

0

Ω

the following relation holds:

lim inf

t

→∞

E

Ω

0

(

t

) = 0

.

(40)

It is readily seen that (40) follows from (24).

Absolute Continuity of Spectrum and Decay of Local Energy.

The

spectrum

σ

(

L

)

of self-adjoint operator

L

:

D

(

L

)

H

is absolutely

continuous

σ

(

L

) =

σ

ac

(

L

)

if

(

E

(

λ

)

h, h

)

is an absolutely continuous

function of

λ

for all

h

H

[15].

14

ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3