Rational Energy Decay Rate of a Wave Equation: The Case of Dimension ≥ 2 ()
1. Introduction
Let
be a bounded domain of
with boundary
, and let
denote the outward unit normal vector to
. Given a point
, set
,
and assume that
on
. We are going to study the long time behavior of the solutions of the following system:
(1)
Its decay rate has been investigated by various techniques in the past; see, e.g., [1] and [2]. Our method will be based on a theorem of Haraux; see [3].
First, we study the well-posedness of (1). We set
and we introduce the Hilbert space
with the inner product
Proposition 1.1. The system (1) is well-posed in H.
Proof. Let us introduce the operators
with
Setting
with
we have
and a simple computation shows that
Using the techniques in ( [4], Page 141), we get
, and then applying Theorem 1.2.3 in ( [5], Page 3), we conclude that the operator
generates a
semigroup of contraction
. □
The main purpose of this paper is to prove the following result concerning the energy of the solutions.
Theorem 1.2. Let us define the energy by the formula
and assume the following assumption on the potential energy of smooth solutions:
with suitable constants
, and
. If
, then there exists a constant M such that
for every
.
We prove Theorem 1.2 by the multiplier method in the following two sections, first for
and then for
.
2. Proof of Theorem 1.2 for n = 2
Taking the derivative of
, we obtain
, so that the energy is a decreasing function.
Since
, we can consider the energy of higher order
. We multiply the equality
by
, then we integrate by parts for t with
, and finally we use Rellich’s formula for
to obtain the following equality:
Since
in this section, we have
Now we majorize all the terms on the right hand side of the above equality:
We note that by the Cauchy-Schwarz inequality we have
for all
, so that
Using the inequality
hence we obtain the estimate
with some constants A and C , and this implies the following relations:
In passing, we have obtained the estimate
Using the assumption on potential energy and the above inequalities we obtain with some constant M that
for all
. Now applying Haraux’s theorem (see [2] or [3]) we conclude that
for all
.
3. Proof of Theorem 1.2 for n ≥ 3
For
we have to modify the proof of the case
because one of the terms in Rellich’s formula does not vanish any more.
Taking the derivative of
, we have
: so the energy is a decreasing function. We note that
, so we can consider the energy of high order:
. So we begin by the equality
, that we multiply by
, then we integrate by parts for t, with
, and we use Rellich’s formula for
, to obtain
(2)
Now we majorize all terms on the right hand side of the above equality:
We note that by the Cauchy-Schwarz inequality we have
so that
Using the inequality
hence we obtain the inequality
for some constants A and C, and therefore
In passing, we have obtained the estimate
Using the assumption on the potential energy and the above inequalities hence we infer that
for all
, with some constant M. Now by applying Haraux’s theorem we conclude that
4. Conclusion
Under some a priori assumptions on the potential energy, we have obtained a polynomial decay rate of the solutions of the wave equation with dynamic boundary feedback by the multiplier method.