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Let

H

1

be a Hilbert space of measurable functions

u

(

t

) : [0

, T

]

H

with a

scalar product

(

u, v

)

H

1

=

T

Z

0

(

u

(

t

)

, v

(

t

))

H

dt

and the norm

k

u

k

2

H

1

= (

u, u

)

H

1

.

We say that the function

u

(

t

)

has a derivative on

[0

, T

]

if the following

representation holds:

u

(

t

) =

u

(

t

0

) +

t

Z

t

0

v

(

τ

)

dt, t

0

[0

, T

]

.

(5)

Here the function

v

(

t

)

H

1

gives for almost all

t

[0

, T

]

the value of

the derivative

du/dt

. Consider an operator

Au

= (

Su, u

(0)

, u

t

(0))

.

The

operator

A

defines on the domain

D

(

A

)

of all

u

H

1

such that functions

du/dt

,

Lu

,

Ldu/dt

belong to

H

1

and continuous; the functions

d

2

u

dt

2

,

L

d

2

u

dt

2

belong to

H

1

and piecewise continuous. The image of operator

R

(

A

)

is a

linear manifold in a Hilbert space

W

=

H

1

×

D

(

L

)

×

H

where

D

(

L

)

is a Hilbert space of elements

u

=

L

1

2

ψ

,

ψ

H

with the scalar product

(

u, v

)

D

(

L

)

=

Lu,

Lv

. It is possible to consider the solution of the

problem (3), (4) as a solution of the operator equation

Au

= (

h, f, g

)

,

where

(

h, f, g

)

W

.

Theorem 1.

The operator

A

admits a closure

A

,

R

(

A

) =

R

(

A

) =

W

.

There exists a bounded inverse operator

A

1

on

W

and the problem

Au

= (

h, f, g

)

(6)

has the unique solution for all

h

H

1

,

f

D

(

L

)

,

g

H

[8].

It is possible to prove that the solution of the operator equation (6) is

a weak solution of the problem (3), (4). It means that

u

H

1

,

u

t

H

1

,

Lu

H

1

,

u

(0) =

f

and the following integral identity holds:

T

Z

0

(

u

t

, w

t

)

Lu,

Lw dt

+ (

g, w

(0)) = 0

(7)

for all

w

H

1

,

w

t

H

1

,

Lw

H

1

,

w

(

T

) = 0

. The solution from such

class is unique.

Below we consider the case of hyperbolic problems (1), (2). The main

example of an elliptic second-order operator

L

is the Laplace operator

Lu

=

Δ

u

. Let

Ω

R

n

,

n

2

, be an an arbitrary (may be unbounded)

domain with smooth boundary

Γ

. We consider the mixed problem for the

wave equation

u

tt

Δ

u

= 0

(8)

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

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