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(

u,

q

)

L

2

(Ω)

=

Lu,

Lq

L

2

(Ω)

=

=

Z

0

λ

cos(

λt

)

d

(

E

(

λ

)

f, q

) +

Z

0

λ

sin(

λt

)

d

(

E

(

λ

)

g, q

) =

=

Z

0

cos

ztdm

3

(

z

) +

Z

0

sin

ztdm

4

(

z

)

.

(27)

Here we have the equalities

dm

1

(

z

) =

zd

(

E

(

z

2

)

f, q

);

dm

2

(

z

) =

d

(

E

(

z

2

)

g, q

);

dm

3

(

z

) =

z

2

d

(

E

(

z

2

)

f, q

);

dm

4

(

z

) =

zd

(

E

(

z

2

)

g, q

)

.

The operator

L

is a positive operator with

σ

p

(

L

) =

, so

m

j

(

z

)

are

continuous functions for

−∞

< z <

+

and

m

j

(

z

) = 0

for

z <

0

. By

the inequalities (25) and the relation [17, Ch. 9, Par. 1, Pt. 128, Eq. (12)] we

obtain that

var

[0

,

+

)

m

j

(

z

)

<

, j

= 1

, . . . ,

4

.

(28)

It follows from (23)–(26) that for all

q

(

x

)

∈ D

(Ω)

lim

t

→∞

1

t

t

Z

0

(

u

τ

(

τ, x

)

, q

(

x

))

2

L

2

(Ω)

= 0;

(29)

lim

t

→∞

1

t

t

Z

0

(

u

(

τ, x

)

,

q

(

x

))

2

L

2

(Ω)

= 0

.

(30)

Consider the closure of this equalities on

q

in

L

2

(Ω)

, we obtain that (29) is

valid for all functions

q

L

2

(Ω)

and (30) — for all

q

L

2

(Ω)

H

1

R

)

for any

R >

0

, satisfying the condition

q

= 0

on

Γ

and such that

R

Ω

|∇

q

|

2

dx <

. Furthermore, we have

k

Δ

u

k

2

L

2

(Ω)

+

k∇

u

t

k

2

L

2

(Ω)

=

=

Z

0

λ

2

cos

2

(

λt

)

d

(

E

(

λ

)

f, f

) +

Z

0

λ

sin

2

(

λt

)

d

(

E

(

λ

)

g, g

)+

+

Z

0

λ

2

sin

2

λtd

(

E

(

λ

)

f, f

) +

Z

0

λ

cos

2

λtd

(

E

(

λ

)

g, g

) =

=

Z

0

λ

2

d

(

E

(

λ

)

f, f

) +

Z

0

λd

(

E

(

λ

)

g, g

) =

=

k

Δ

f

k

2

L

2

(Ω)

+

k∇

g

k

2

L

2

(Ω)

.

(31)

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

11