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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