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Now, we have an integral equality for an eigenfunction:

Z

Ω

|∇

v

|

2

dx

=

=

λ

Z

Ω

v

2

dx.

Therefore,

Z

Ω

|∇

v

|

2

+

λv

2

dx >

0;

Z

Ω

|∇

v

|

2

λv

2

dx

= 0

and by the absolute continuity of the Lebesgue integral there exists a

domain

Ω

0

is compact embedded to

Ω

such that for some

ε >

0

we have

the inequalities

Z

Ω

0

|∇

v

|

2

+

λv

2

dx

ε

;

Z

Ω

0

|∇

v

|

2

λv

2

dx

ε

2

.

(22)

Thus, by (21), (22) we obtain an inequality

E

Ω

0

(

t

)

ε

4

for

t >

0

. Let us consider the solution

˜

u

of the problem (8)–(10) with initial

functions

f

= ˜

v

,

g

= 0

, where function

˜

v

∈ D

(Ω)

satisfy the inequality

k

v

˜

v

k

2

H

1

(Ω)

< ε/

16

.

Then

Z

Ω

0

u

2

t

+

|∇

˜

u

|

2

)

dx

1

2

Z

Ω

0

u

2

t

+

|∇

u

|

2

dx

Z

Ω

0

(((

u

˜

u

)

t

)

2

+

|∇

(

u

˜

u

)

|

2

)

dx

ε

8

Z

Ω

0

(((

u

˜

u

)

t

)

2

+

|∇

(

u

˜

u

)

|

2

)

dx

ε

8

Z

Ω

(((

u

˜

u

)

t

)

2

+

|∇

(

u

˜

u

)

|

2

)

dx

=

=

ε

8

− k∇

(

v

˜

v

)

k

2

L

2

(Ω)

>

ε

8

ε

16

=

ε

16

>

0

, t >

0

.

The inequality (20) is proved.

I

Continuity of Spectrum and Mean Decay of Local Energy.

Let

Ω

be

an unbounded domain. In the case of

σ

p

(

L

) =

we have the mean local

energy decay. A proof use the following theorem [1, Th. 9.15, 20].

Theorem 4.

Let real-valued function

m

(

z

)

C

(

−∞

,

+

)

,

m

(

z

) = 0

for

z

0

,

var

[0

,

+

)

m

(

z

)

<

and

s

(

t

) =

R

0

e

izt

dm

(

z

)

for

t >

0

. Then

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

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