Background Image
Previous Page  2 / 17 Next Page
Information
Show Menu
Previous Page 2 / 17 Next Page
Page Background

a local energy tends to zero in time is identically zero. For arbitrary domains in the

case of operator with non-empty point spectrum we prove that there are smooth and

finite initial functions for which the local energy function does not decay. In the cases

of continuous and absolutely continuous spectrum of the Laplace operator we prove

the mean decay and the decay of the local energy function respectively.

Keywords

:

operator differential equation, hyperbolic problem, Dirichlet boundary

condition, stabilization, Laplace operator, spectrum.

Introduction.

Very often in mathematical physics arises the question

of large time behavior for the solutions of Cauchy problem for the non-

stationary operator equation

u

tt

+

Lu

= 0

, t >

0;

(1)

u

|

t

=0

=

f, u

t t

=0

=

g,

(2)

where

L

is a linear self-adjoint operator in a Hilbert space

H

. The interest of

mathematicians to this problem is natural because many important physical

problems leads to the Cauchy problem (1), (2). The examples of such

problems are acoustic and electromagnetic oscillations in homogeneous

and non-homogeneous media [1, 2]. Closely related operator equations

arises in the problem of small vibrations of an ideal non-homogeneous

fluid [3].

The general theory of the operator Cauchy problem (1), (2) in a Hilbert

space contains many results about solvability and a priory estimates for

solutions [4–9]. We will investigate qualitative properties of solutions of the

problem (1), (2) with special attention to the connection between spectral

properties of operator

L

and behavior of solutions for

t

→ ∞

.

Let us note that the existence results and qualitative properties of

solutions to the problem (1), (2) closely connected with the representation

formulas for solutions. So, one of the first results concerned to the case

of a bounded positive operator

L

[10] state that the solution of (1), (2)

represents by the series

u

(

t

) = cos(

Lt

)

f

+

sin(

Lt

)

L

g

, where

cos(

Lt

) =

X

k

=0

(

1)

k

t

2

k

L

k

(2

k

)!!

; sin(

Lt

) =

X

k

=0

(

1)

k

t

2

k

+1

L

k

+

1

2

(2

k

+ 1)!!

.

But more interesting due to physical and technical applications is the case

of unbounded operator

L

. Let

L

is an unbounded positive operator whose

domain

D

(

L

)

is dense in

H

. We suppose that the inverse operator

L

1

is bounded. We formulate the solvability result for the non-homogeneous

problem

Su

=

u

tt

+

Lu

=

h, t >

0;

(3)

u

|

t

=0

=

f, u

t

|

t

=0

=

g.

(4)

4

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