Background Image
Previous Page  8 / 17 Next Page
Information
Show Menu
Previous Page 8 / 17 Next Page
Page Background

lim

t

→∞

1

t

t

Z

0

|

s

(

τ

)

|

2

= 0

.

(23)

Now we prove the main result of this section.

Theorem 5.

Let

σ

p

(

L

) =

. Then for all

f

o

H

1

(Ω)

,

g

L

2

(Ω)

and

all bounded domains

Ω

0

Ω

lim

t

→∞

1

t

t

Z

0

E

Ω

0

(

τ

)

= 0

.

(24)

J

It is sufficient to prove the relation (24) for

f, g

∈ D

(Ω)

. For any

f

o

H

1

(Ω)

,

g

L

2

(Ω)

and arbitrary

ε >

0

there exist

˜

f ,

˜

g

∈ D

(Ω)

such

that

k

f

˜

f

k

H

1

(Ω)

< ε

,

k

g

˜

g

k

L

2

(Ω)

< ε

. Therefore, for

Ω

0

= Ω

R

= Ω

∩{|

x

|

<

< R

}

we have

E

Ω

R

(

t

) =

Z

Ω

R

u

2

t

+

|∇

u

|

2

dx

2

Z

Ω

R

˜

u

2

t

+

|∇

˜

u

|

2

dx

+

+2

Z

Ω

((

u

˜

u

)

t

)

2

+

|∇

(

u

˜

u

)

|

2

dx

= 2

Z

Ω

R

˜

u

2

t

+

|∇

˜

u

|

2

dx

+

+2

k

f

˜

f

k

2

H

1

(Ω)

+

k

g

˜

g

k

2

L

2

(Ω)

<

2

Z

Ω

R

˜

u

2

t

+

|∇

˜

u

|

2

dx

+ 4

ε

2

for

t >

0

. Now, to prove (24) suppose that

f, g

∈ D

(Ω)

. Note that

f, g

∈ D

(Ω)

D

(

L

p

)

,

p

= 1

,

2

, . . .

, where

D

(

L

p

)

is a domain of the

p

-th

power of the Laplace operator

L

=

Δ

with Dirichlet boundary condition.

Thence [17, Ch. 9, Par. 1] for

f, g

∈ D

(Ω)

we have the inequalities

Z

0

λ

2

p

d

(

E

(

λ

)

f, f

)

<

;

Z

0

λ

2

p

d

(

E

(

λ

)

g, g

)

<

, p

= 0

,

1

,

2

, . . .

(25)

Now, by the operator calculus for self-adjoint operators [17, Ch. 9, Par. 1]

for

q

(

x

)

∈ D

(Ω)

and

t >

0

we obtain the relations

(

u

t

, q

)

L

2

(Ω)

=

Z

0

sin(

λt

)

λd

(

E

(

λ

)

f, q

) +

Z

0

cos(

λt

)

d

(

E

(

λ

)

g, q

) =

=

Z

0

sin

ztdm

1

(

z

) +

Z

0

cos

ztdm

2

(

z

);

(26)

10

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