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Applying Friedrichs inequality with an arbitrary

R >

0

we obtain an

estimate

k

u

k

L

2

R

)

C

(

R

)

k∇

u

k

L

2

R

)

for

t >

0

. Thence, holds the

following inequality:

k∇

u

k

H

1

R

)

+

k

u

t

k

H

1

R

)

C

(

R

)

.

(32)

So, the set of functions

{

u

t

(

t, x

)

}

and

{∇

u

(

t, x

)

}

,

t >

0

, are compact in

L

2

R

)

for any

R >

0

. Let us prove that

lim

t

→∞

1

t

t

Z

0

k

u

τ

k

2

L

2

R

)

= 0

.

By the estimate (32) the set of functions

{

u

t

(

t, x

)

}

,

t >

0

is compact in

L

2

R

)

. Let

{

h

j,R

(

x

)

}

,

j

= 1

,

2

, . . .

,

x

Ω

R

be an orthonormal basis in the

L

2

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

R

)

= 0

for

t >

0

. By the compactness criterion [21, P. 247, Th. 3] in the space

L

2

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

R

)

< ε

,

t >

0

.

Thence,

k

u

t

k

2

L

2

R

)

=

k

N

X

j

=1

c

j,R

h

j,R

k

2

L

2

R

)

+

k

X

j

=

N

+1

c

j,R

h

j,R

k

2

L

2

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

R

)

dτ <

N

X

j

=1

 

lim

t

→∞

1

t

t

Z

0

c

2

j,R

 

+

ε

2

=

=

N

X

j

=1

 

lim

t

→∞

1

t

t

Z

0

(

u

τ

, h

j,R

)

2

L

2

R

)

 

+

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)

= 0

for

j

=

12

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