# Boundary Stabilization of the Wave Equation with Time-Varying and Nonlinear Feedback

Authors: Jian-Sheng Tian; Wei Wang; Fei Xue; Pei-Yong Cong
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying and nonlinear term. By the Riemannian geometry methods and a suitable assumption of nonlinearity and the time-varying term, we obtain the uniform decay of the energy of the system.


---

## Body

## 1. Introduction

There are many results concerning the boundary stabilization of classical wave equations. See [1–6] for linear cases and [7–12] for nonlinear ones. The stability of the wave equation with variable coefficients has attracted much attention. See [13–23], and many others. In [20], by the methods in [11, 24], the authors study the stability of the wave equation with nonlinear term and time-varying term. However, under the condition the nonlinear term has upper bound and the time-varying term has lower bound, the stability of the wave equation was not studied in [20]. In this paper, our purpose is to study the stability of the wave equation under the condition the nonlinear term has upper bound and the time-varying term has lower bound.LetΩ be a bounded domain in R
n with smooth boundary Γ. It is assumed that Γ consists of two parts Γ
1 and Γ
2
(
Γ
=
Γ
1
∪
Γ
2
) with Γ
2
≠
∅
,
Γ
¯
1
∩
Γ
¯
2
=
∅. Define
(1)
A
u
=
-
div
⁡
A
(
x
)
∇
u
for
u
∈
H
1
(
Ω
)
,
where div
⁡ is the divergence operator of the standard metric of R
n; A
(
x
)
=
(
a
i
j
(
x
)
) is symmetric, positively definite matrices for each x
∈
R
n and a
i
j
(
x
) are smooth functions on R
n.We consider the stabilization of the wave equations with variable coefficients and time-varying delay in the dissipative boundary feedback:(2)
u
t
t
+
A
u
=
0
(
x
,
t
)
∈
Ω
×
(
0
,
+
∞
)
,
u
(
x
,
t
)
|
Γ
2
=
0
t
∈
(
0
,
+
∞
)
,
∂
u
(
x
,
t
)
∂
ν
A
+
ϕ
(
t
)
g
1
u
t
(
x
,
t
)
=
0
(
x
,
t
)
∈
Γ
1
×
∈
(
0
,
+
∞
)
,
u
(
x
,
0
)
=
u
0
(
x
)
,
u
t
(
x
,
0
)
=
u
1
(
x
)
x
∈
Ω
,
g
1
∈
C
(
R
) and there exists a positive constant c
1 such that
(3)
g
1
(
0
)
=
0
,
s
g
1
(
s
)
≤
|
s
|
2
for
s
∈
R
,
|
g
1
(
s
)
|
≥
c
1
|
s
|
for
|
s
|
>
1
,
and ϕ
(
t
)
∈
C
(
[
0
,
+
∞
)
) satisfies
(4)
ϕ
(
t
)
≥
ϕ
0
∀
t
≥
0
,
(5)
lim
⁡
t
-
>
+
∞
F
(
t
)
t
=
0
,
where ϕ
0 is a positive constant and F
(
t
)
=
max
⁡
0
≤
ρ
≤
t
ϕ
(
ρ
).∂
u
/
∂
ν
A is the conormal derivative
(6)
∂
u
∂
ν
A
=
〈
A
(
x
)
u
,
ν
〉
,
where 〈
·
,
·
〉 denotes the standard metric of the Euclidean space R
n and ν
(
x
) is the outside unit normal vector for each x
∈
Γ. Moreover, the initial data (
u
0
,
u
1
) belongs to a suitable space.Define the energy of the system (2) by
(7)
E
(
t
)
=
1
2
∫
Ω
(
u
t
2
+
∑
i
,
j
=
1
n
a
i
j
u
x
i
u
x
j
)
d
x
.We define(8)
g
=
A
-
1
(
x
)
for
x
∈
R
n
as a Riemannian metric on R
n and consider the couple (
R
n
,
g
) as a Riemannian manifold with an inner product:
(9)
〈
X
,
Y
〉
g
=
〈
A
-
1
(
x
)
X
,
Y
〉
,
|
X
|
g
2
=
〈
X
,
X
〉
g
X
,
Y
∈
R
x
n
.LetD
g denote the Levi-Civita connection of the metric g. For the variable coefficients, the main assumptions are as follows.Assumption A.
There exists a vector fieldH on Ω
¯ and a constant ρ
0
>
0 such that
(10)
D
g
H
(
X
,
X
)
≥
ρ
0
|
X
|
g
2
for
X
∈
R
x
n
,
x
∈
Ω
¯
.
Moreover we assume that
(11)
sup
⁡
x
∈
Ω
¯
div
⁡
H
<
inf
⁡
x
∈
Ω
¯
div
⁡
H
+
2
ρ
0
,
(12)
H
·
ν
≤
0
x
∈
Γ
2
,
H
·
ν
≥
δ
x
∈
Γ
1
,
where δ is a positive constant.Assumption (10) was introduced by [13] as a checkable assumption for the exact controllability of the wave equation with variable coefficients. For examples on the condition, see [13, 14].Based on Assumption (10), Assumption A was given by [19] to study the stabilization of the wave equation with variable coefficients and boundary condition of memory type.Define(13)
H
Γ
2
1
(
Ω
)
=
{
u
∈
H
1
(
Ω
)
∣
u
|
Γ
2
=
0
}
.
To obtain the stabilization of the system (2), we assume the system (2) is well-posed such that
(14)
u
∈
C
1
(
[
0
,
+
∞
)
,
L
2
(
Ω
)
)
∩
C
(
[
0
,
+
∞
)
,
H
Γ
2
1
(
Ω
)
)
.The main result of this paper is stated as follows.Theorem 1.
Let Assumption A holds true. Then there exist positive constantsC
,
C
2, such that
(15)
E
(
t
)
≤
C
1
h
(
C
2
E
(
0
)
t
)
+
C
1
F
(
t
)
t
E
(
0
)
,
t
>
0
.


## 2. Basic Inequality of the System

In this section we work onΩ with two metrics at the same time, the standard dot metric 〈
·
,
·
〉 and the Riemannian metric g
=
〈
·
,
·
〉
g given by (8).Iff
∈
C
1
(
R
n
), we define the gradient ∇
g
f of f in the Riemannian metric g, via the Riesz representation theorem, by
(16)
X
(
f
)
=
〈
∇
g
f
,
X
〉
g
,
where X is any vector field on (
R
n
,
g
). The following lemma provides further relations between the two metrics; see [13] in Lemma 2.1.Lemma 2.
Letx
=
(
x
1
,
…
,
x
n
) be the natural coordinate system in R
n. Let f, h be functions and let H, X be vector fields. Then(a)
(17)
〈
H
(
x
)
,
A
(
x
)
X
(
x
)
〉
g
=
〈
H
(
x
)
,
X
(
x
)
〉
,
x
∈
R
n
;
(b)
(18)
∇
g
f
=
∑
i
=
1
n
(
∑
j
=
1
n
a
i
j
(
x
)
f
x
j
)
∂
∂
x
i
=
A
(
x
)
∇
f
,
x
∈
R
n
,
where∇
f is the gradient of f in the standard metric;
(c)
(19)
∇
g
f
(
h
)
=
〈
∇
g
f
,
∇
g
h
〉
g
=
〈
∇
f
,
A
(
x
)
∇
h
〉
,
x
∈
R
n
,
where the matrixA
(
x
) is given in formula (1).To prove Theorem1, we still need several lemmas further. Define
(20)
E
0
(
t
)
=
1
2
∫
Ω
(
u
t
2
+
|
∇
g
u
|
g
2
)
d
x
.
Then, we have
(21)
E
(
t
)
=
E
0
(
t
)
+
ξ
∫
t
-
τ
(
t
)
t
∫
Γ
1
u
t
2
(
x
,
ρ
)
d
Γ
d
ρ
.Lemma 3.
Let(
u
) be the solution of system (2). Then there exists a constant C
1 such that
(22)
E
(
0
)
-
E
(
T
)
=
C
1
∫
0
T
∫
Γ
1
ϕ
(
t
)
u
t
(
x
,
t
)
g
1
(
u
t
(
x
,
t
)
)
d
Γ
d
t
,
where T
≥
0. The assertion (22) implies that E
(
t
) is decreasing.Proof.
Differentiating (7), we obtain
(23)
E
′
(
t
)
=
∫
Ω
(
u
t
u
t
t
+
∇
g
u
·
∇
u
t
)
d
x
=
∫
Γ
1
ϕ
(
t
)
u
t
(
x
,
t
)
g
1
(
u
t
(
x
,
t
)
)
d
Γ
.
Then the inequality (22) holds true.


## 3. Proofs of Theorem1

From Proposition 2.1 in [13], we have the following identities.Lemma 4.
Suppose thatu
(
x
,
t
) solves equation u
t
t
+
A
u
=
0, (
x
,
t
)
∈
Ω
×
(
0
,
+
∞
) and that H is a vector field defined on Ω
¯. Then, for T
≥
0,
(24)
∫
0
T
∫
Γ
∂
u
∂
ν
A
H
(
u
)
d
Γ
d
t
+
1
2
∫
0
T
∫
Γ
(
u
t
2
-
|
∇
g
u
|
g
2
)
H
·
ν
d
Γ
d
t
=
(
u
t
,
H
(
u
)
)
|
0
T
+
∫
0
T
∫
Ω
D
g
H
(
∇
g
u
,
∇
g
u
)
d
x
d
t
+
1
2
∫
0
T
∫
Ω
(
u
t
2
-
|
∇
g
u
|
g
2
)
div
⁡
H
d
x
d
t
.
Moreover, assume thatP
∈
C
1
(
Ω
¯
). Then
(25)
∫
0
T
∫
Ω
(
u
t
2
-
|
∇
g
u
|
g
2
)
P
d
x
d
t
=
(
u
t
,
u
P
)
|
0
T
+
1
2
∫
0
T
∫
Ω
∇
g
P
(
u
2
)
d
x
d
t
-
∫
0
T
∫
Γ
P
u
∂
u
∂
ν
A
d
Γ
d
t
.Lemma 5.
Suppose that all assumptions in Theorem1 hold true. Let u be the solution of the system (2). Then there exist positive constants C
,
T
0 for which
(26)
E
(
T
)
≤
C
T
∫
0
T
∫
Γ
1
(
u
t
2
+
(
∂
u
∂
ν
A
)
2
)
d
Γ
d
t
,
where T
≥
T
0.Proof.
We letθ be a positive constant satisfying
(27)
1
2
sup
⁡
x
∈
Ω
¯
div
⁡
H
<
θ
<
1
2
inf
⁡
x
∈
Ω
¯
div
⁡
H
+
ρ
0
.
Set
(28)
H
=
H
,
P
=
θ
-
ρ
0
.
Substituting the identity (25) into the identity (24), we obtain
(29)
Π
Γ
=
(
u
t
,
H
(
u
)
+
P
u
)
|
0
T
+
∫
0
T
∫
Ω
(
D
g
H
(
∇
g
u
,
∇
g
u
)
-
ρ
0
|
∇
g
u
|
g
2
)
d
x
d
t
+
∫
0
T
∫
Ω
‍
(
(
1
2
div
⁡
H
+
ρ
0
-
θ
)
u
t
2
+
(
θ
-
1
2
div
⁡
H
)
|
∇
g
u
|
g
2
)
d
x
d
t
,
where
(30)
Π
Γ
=
∫
0
T
∫
Γ
∂
u
∂
ν
A
(
H
(
u
)
+
u
P
)
d
Γ
d
t
+
1
2
∫
0
T
∫
Γ
(
u
t
2
-
|
∇
g
u
|
g
2
)
H
·
ν
d
Γ
d
t
.
DecomposeΠ
Γ as
(31)
Π
Γ
=
Π
Γ
1
+
Π
Γ
2
.
Since u
|
Γ
2
=
0, we obtain ∇
Γ
u
|
Γ
2
=
0; that is,
(32)
∇
g
u
=
∂
u
∂
ν
A
ν
A
|
ν
A
|
g
2
for
x
∈
Γ
2
.
Similarly, we have
(33)
H
(
u
)
=
〈
H
,
∇
g
u
〉
g
=
∂
u
∂
ν
A
H
·
ν
|
ν
A
|
g
2
for
x
∈
Γ
2
.
Using the formulas (32) and (33) in the formula (30) on the portion Γ
2, with (12), we obtain
(34)
Π
Γ
2
=
1
2
∫
0
T
∫
Γ
2
(
∂
u
∂
ν
A
)
2
H
·
ν
|
ν
A
|
g
2
d
Γ
d
t
≤
0
.
From (12), we have
(35)
Π
Γ
1
=
∫
0
T
∫
Γ
1
∂
u
∂
ν
A
(
H
(
u
)
+
u
P
)
d
Γ
d
t
+
1
2
∫
0
T
∫
Γ
1
(
u
t
2
-
|
∇
g
u
|
g
2
)
H
·
ν
d
Γ
d
t
≤
C
ε
∫
0
T
∫
Γ
1
(
∂
u
∂
ν
A
)
2
d
Γ
d
t
+
ε
∫
0
T
∫
Γ
1
(
u
2
+
|
∇
g
u
|
g
2
)
d
Γ
d
t
+
∫
0
T
∫
Γ
1
(
C
u
t
2
-
δ
|
∇
g
u
|
g
2
)
d
Γ
d
t
≤
C
∫
0
T
∫
Γ
1
(
∂
u
∂
ν
A
)
2
d
Γ
d
t
+
ε
E
(
t
)
+
C
∫
0
T
‍
∫
Γ
1
‍
u
t
2
d
Γ
d
t
.
Substituting the formulas (34) and (35) into the formula (29), with (27), we obtain
(36)
∫
0
T
E
(
t
)
d
t
≤
C
(
E
(
0
)
+
E
(
T
)
)
+
C
∫
0
T
∫
Γ
1
(
u
t
2
+
(
∂
u
∂
ν
A
)
2
)
d
Γ
d
t
.
It follows from (22) that
(37)
∫
0
T
E
(
t
)
d
t
≥
T
E
(
T
)
.
Substituting the formulas (22) and (37) into the formula (36), the inequality (26) holds.Proof of Theorem1.
SinceE
(
t
) is decreasing, with (4) and (26), for sufficiently large T, we have
(38)
E
(
T
)
≤
C
T
∫
0
T
∫
Γ
1
(
ϕ
2
(
t
)
g
2
(
u
t
)
+
u
t
2
)
d
Γ
d
t
≤
C
T
{
∫
0
T
∫
Γ
1
(
u
t
2
+
g
2
(
u
t
)
)
d
Γ
d
t
+
F
(
T
)
∫
0
T
‍
∫
Γ
1
‍
u
t
ϕ
(
t
)
g
(
u
t
)
d
Γ
d
t
}
≤
C
T
{
∫
0
T
‍
∫
{
x
∈
Γ
1
,
|
u
t
|
≤
1
}
‍
h
(
u
t
g
(
u
t
)
)
d
Γ
d
t
+
C
F
(
T
)
E
(
0
)
}
≤
C
T
∫
0
T
‍
∫
Γ
1
‍
h
(
u
t
g
(
u
t
)
)
d
Γ
d
t
+
C
F
(
T
)
T
E
(
0
)
≤
C
meas
⁡
(
Γ
1
)
h
(
∫
0
T
‍
∫
Γ
1
‍
u
t
g
(
u
t
)
d
Γ
d
t
T
·
meas
⁡
(
Γ
1
)
)
+
C
F
(
T
)
T
E
(
0
)
≤
C
1
h
(
C
2
E
(
0
)
T
)
+
C
1
F
(
T
)
T
E
(
0
)
.
Note that E
(
t
) is decreasing; the estimate (15) holds.


## 4. Application of the System (2)

Nonlinear feedback describes a property of a physical system; that is, the response by the physical system to an applied force is nonlinear in its effect. One of the applications of the system (2) is in sound waves, where the system (2) describes the reflection of sound in heterogeneous materials at surfaces of some materials with nonlinearity of interest in engineering practice. Theorem 1 indicates that the energy of the sound waves with the reflection of sound at surfaces in heterogeneous materials at surfaces of some materials with nonlinearity is uniform decay under a suitable assumption of the nonlinearity.



---

*Source: 176583-2014-06-29.xml*