Background Image
Previous Page  5 / 17 Next Page
Information
Show Menu
Previous Page 5 / 17 Next Page
Page Background

Lemma 1.

Let

f

o

H

1

(Ω)

,

g

L

2

(Ω)

. Then the solution of the

problem

(8)

(10)

from the energy class can be written as Bochner – Stiltjes

integral

u

=

Z

0

cos(

λt

)

dE

(

λ

)

f

+

Z

0

sin(

λt

)

λ

dE

(

λ

)

g.

(13)

The integral (13) converges uniformly in

o

H

1

(Ω)

on

[0

, T

]

,

T >

0

. The

derivative

u

t

can be written as

u

t

=

Z

0

λ

sin(

λt

)

dE

(

λ

)

f

+

Z

0

cos(

λt

)

dE

(

λ

)

g,

(14)

the integral

(14)

converges uniformly in

L

2

(Ω)

on

[0

, T

]

.

Point Spectrum and Non-Decay of Local Energy.

In a bounded

domain

Ω

the spectrum of operator

L

is discrete. So, the solution of the

problem (8)–(10) represents by the series

u

(

t, x

) =

X

j

=1

(

a

j

cos(

p

λ

j

t

) +

b

j

sin(

p

λ

j

t

)

p

λ

j

)

v

j

(

x

)

,

(15)

where

a

j

= (

f, v

j

)

L

2

(Ω)

,

b

j

= (

g, v

j

)

L

2

(Ω)

. Here

0

< λ

1

< λ

2

. . .

,

lim

j

→∞

λ

j

= +

, is the sequence of eigenvalues of the operator

L

,

{

v

j

}

is

the orthonormal basis of eigenfunctions in

L

2

(Ω)

.

Using the equality (15), we can prove that solution of the problem

(8)–(10) is an almost-periodic function with respect to

t

(

n

= 1

[16];

n

2

[18, 19]).

Theorem 2.

Let

Ω

R

n

be a bounded domain and

lim

t

→∞

E

Ω

0

(

t

) = 0

for some domain

Ω

0

Ω

. Then

u

= 0

in

(0

,

)

×

Ω

.

J

The equality

lim

t

→∞

E

Ω

0

(

t

) = 0

means that

lim

t

→∞

k∇

u

(

t, x

)

k

L

2

0

)

= 0

.

Therefore,

0 = lim

t

→∞

Z

Ω

0

u

x

k

(

t, x

)

η

(

x

)

dx

=

= lim

t

→∞

X

j

=1

a

j

cos(

p

λ

j

t

)+

b

j

sin(

p

λ

j

t

)

p

λ

j

Z

Ω

0

(

v

j

)

x

k

(

x

)

η

(

x

)

dx, k

= 1

, . . . , n,

for an arbitrary function

η

(

x

)

∈ D

0

) =

o

C

0

)

. We have

0 = lim

T

→∞

1

T

T

Z

0

Z

Ω

0

u

x

k

(

t, x

)

η

(

x

)

dx

cos(

p

λ

m

t

)

dt

=

=

1

2

Z

Ω

0

X

λ

j

=

λ

m

a

j

(

v

j

)

x

k

(

x

)

η

(

x

)

dx

;

(16)

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

7