Background Image
Previous Page  11 / 17 Next Page
Information
Show Menu
Previous Page 11 / 17 Next Page
Page Background

= 1

,

2

, . . . , N

and it follows from (34) that for any

ε >

0

lim sup

t

→∞

1

t

t

Z

0

k

u

τ

k

2

L

2

R

)

ε

2

.

Now, we obtain

lim

t

→∞

1

t

t

Z

0

k

u

τ

k

2

L

2

R

)

= 0

(35)

for any

R >

0

. Let us prove now that

lim

t

→∞

1

t

t

Z

0

k∇

u

k

2

L

2

R

)

= 0

.

For any

R >

0

we define the space:

e

H

R

=

v

H

1

R

) :

v

|

Γ

R

= 0

,

where

Γ

R

= Γ

∩ {|

x

|

< R

}

. The space

e

H

R

is a Hilbert space with a scalar

product

(

v, w

)

e

H

R

=

Z

Ω

R

(

v,

w

)

dx.

Similarly we define the Hilbert space

e

H

=

{

v

H

1

R

)

for any

R >

0 :

v

|

Γ

= 0

,

Z

Ω

|∇

v

|

2

dx <

∞}

,

with a

scalar product

(

v, w

)

e

H

=

Z

Ω

(

v,

w

)

dx.

Let

R >

0

. It follows from (32) that the set of functions

{

u

(

t, x

)

}

,

t >

0

, is compact in the space

e

H

R

. Denote by

{

h

j,R

(

x

)

}

,

j

= 1

,

2

, . . .

, the

orthonormal basis in the space

e

H

R

. By the compactness criterion in the

space

e

H

R

with basis

{

h

j,R

}

for any

ε >

0

there exists

N >

0

such that

u

(

t, x

) =

N

X

j

=1

b

j,R

(

t

)

h

j,R

(

x

) +

X

j

=

N

+1

b

j,R

(

t

)

h

j,R

(

x

)

,

and

k

X

j

=

N

+1

b

j,R

(

t

)

h

j,R

(

x

)

k

e

H

R

< ε

for

t >

0

. Now we have

k

u

(

t, x

)

k

2

e

H

R

=

k

N

X

j

=1

b

j,R

h

j,R

k

2

e

H

R

+

k

X

j

=

N

+1

b

j,R

h

j,R

k

2

e

H

R

<

<

N

X

j

=1

b

2

j,R

(

t

) +

ε

2

=

N

X

j

=1

(

u, h

j,R

)

2

e

H

R

+

ε

2

.

(36)

For all

υ

(

x

)

e

H

we have an estimates

|

F

j

[

υ

]

|

= (

υ, h

j,R

)

e

H

R

≤ k

υ

k

e

H

R

k

h

j,R

k

e

H

R

=

k

υ

k

e

H

R

≤ k

υ

k

e

H

,

j

= 1

,

2

, . . . , N

. It means that

the linear functional

F

j

[

υ

] = (

υ, h

j,R

)

e

H

R

is a bounded functional on

e

H

. By

the F. Riesz theorem there exist functions

˜

h

j,R

(

x

)

e

H

,

j

= 1

,

2

, . . . , N

,

such that

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

13