Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 59, pp. 1–18.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

STABILIZATION OF THE CRITICAL NONLINEAR

KLEIN-GORDON EQUATION WITH

VARIABLE COEFFICIENTS ON R3

SONG-REN FU, ZHEN-HU NING

Abstract. We prove the exponential stability of the defocusing critical semi-
linear wave equation with variable coefficients and locally distributed damping

on R3. The construction of the variable coefficients is almost equivalent to the

geometric control condition. We develop the traditional Morawetz estimates
and the compactness-uniqueness arguments for the semilinear wave equation

to prove the unique continuation result. The observability inequality and the

exponential stability are obtained subsequently.

1. Introduction

In this article, we consider the defocusing critical nonlinear Klein-Gordon equa-
tion

utt − divA(x)∇u+ a(x)ut + u+ u5 = 0, (x, t) ∈ R3 × (0,+∞),

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ R3,
(1.1)

where A(x) = {aij(x)}3ij=1 is a positive definite matrix such that

aij(x) ∈W 2,∞(R3) for i, j = 1, 2, 3.

Let the damping term a(x) be a real nonnegative function of class W 2,∞(R3), and
let the initial data (u0, u1) ∈ H1(R3)× L2(R3).

The Klein-Gordon equation is the basic equation in relativistic quantum mechan-
ics and in quantum field theory. It is a special relativistic form of the Schrödinger
equation, which describes particles with zero spin. The studying of Klein-Gordon
equations with nonlinear perturbation is essential in both physics and mathematics.
In particular, the stability of the semilinear Klein-Gordon equations have attracted
much attention, but the critical case hard to study. See for instance [25] and refer-
ences therein.

1.1. Notation and statement of the problem. Let O be the origin in the space
R3, and r(x) = |x| be the Euclidean norm of x ∈ R3. Let 〈·, ·〉, div, ∇, ∆, and
I3 = (δij)3×3 be the standard inner product, divergence operator, gradient operator,
Laplace operator, and the unit matrix in R3, respectively.

2020 Mathematics Subject Classification. 93B05, 93C20, 35G16, 35L72, 35L15.
Key words and phrases. Critical semilinear wave equation; variable coefficients; stability;

Morawetz estimates; Riemannian geometry; unique continuation.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted May 11, 2022. Published August 5, 2022.

1



2 S.-R. FU, Z.-H. NING EJDE-2022/59

We define

g = A−1(x) = G(x) for x ∈ R3

as a Riemannian metric on R3. Thus, we consider (R3, g) as a Riemannian manifold
and

〈X,Y 〉g = 〈A−1(x)X,Y 〉, |X|2g = 〈X,X〉g, X, Y ∈ R3
x, x ∈ R3, (1.2)

where R3
x is the tangential space at x ∈ R3. We assume that there exist positive

constants m1,m2 such that

m1|X|2 ≤ 〈A(x)X,X〉 = |X|2g ≤ m2|X|2 for X ∈ R3
x, x ∈ R3. (1.3)

Let D be the Levi-Civita connection in the metric g, and H be a vector field. We
will use many times that H(u) = 〈H,∇gu〉g. The covariant differential DH of the
vector field H is a tensor field of rank 2, and

DH(X,Y )(x) = 〈DYH,X〉g(x) X,Y ∈ R3
x, x ∈ R3. (1.4)

For a given y > 0, we define the ball

B(y) = {x ∈ R3 : |x| ≤ y}. (1.5)

We also set divg, ∇g, and ∆g the divergence operator, e gradient operator, and
Laplace-Beltrami operator in the metric g, respectively.

We define the energy functional associated with (1.1) as

E(t) =
1

2

∫
R3

(u2
t + |∇gu|2g + u2)dx+

1

6

∫
R3

u6dx. (1.6)

In this article we consider mainly the exponential decay of E(t).
We say that a subdomain ω ⊂ Ω satisfies the Geometric Control Condition

(GCC) if each unit geodesic initiated from Ω enters ω before a finite time T . In
particular, if ω is the boundary Γ of Ω, then the Geometric Control Condition states
that each geodesic initiated from Ω mush hit the boundary Γ in time less than T .

1.2. Previous results. There is a large number of results for the wave equa-
tions with either locally distributed damping or suitable boundary dissipation.
For stability results of linear wave equations in compact domains, we refer to
[15, 26, 27, 33, 34, 43]. For the linear wave equations in non-compact domains,
we refer to [2, 3, 27, 30, 31, 42].

A lot of contributions to the stability analysis of the nonlinear wave equations
arose subsequently. We mention that [8] concerned the wave equation on compact
surfaces with nonlinear locally distributed damping, described by

utt −∆gu+ a(x)h(ut) = 0.

The authors obtained the stability result that E(t) ≤ S(t)(t/T0−1) for fixed T0 > 0
under some assumptions on the function h and on the compact domain, where
S(t) vanishes as t tends to infinity. Moreover, the energy decays exponentially
with respect to the initial energy if the feedback h is linear. Later, in [1, 8] the
authors studied the well-posedness and sharp uniform decay rates of the energy
related to the Klein-Gordon equation. This is done subject to a nonlinear and
locally distributed damping, posed in a complete and noncompact n dimensional
Riemannian manifold (M, g) without boundary, utt−∆u+u+a(x)h(ut) = 0. They
obtained the exponential stability result under some suitable assumptions on h, a
and the geometric conditions of (M, g).



EJDE-2022/59 STABILIZATION OF A KLEIN-GORDON EQUATION 3

For the long time behavior of the nonlinear wave equations in compact spaces, we
refer to [7, 9, 20, 22, 24, 44, 46]. For the nonlinear wave equations in noncompact
spaces, we refer to [3, 10, 20, 28, 29, 38, 39, 40, 45, 48]. We note that most of
the noncompact spaces concerned in literature are either the whole spaces Rn or
domains outside a convex obstacle. The geometric control condition (GCC) is
always used as a necessary assumption to get the stability results.

For the energy subcritical semilinear wave equations, we point out that [20]
studied the exponential stability of the semilinear wave equation with a damping
effective in a zone satisfying the geometric control condition only. The nonlinear-
ity is assumed to be subcritical, defocusing, and analytic. The new contribution
compared to previous results, is their proof of a unique continuation result in large
time for some undamped equation. For the stabilization of the subcritical semilinear
wave equations, we refer to [3, 7, 10, 48] and references therein.

We know that the global well-posedness and the stability results related to the
subcritical nonlinear wave-type equations are easier than the critical ones. In [25],
exponential stability of the critical semilinear Klein-Gordon equation was proved on
a 3-D compact manifold with small initial data. They posed a geometric assumption
slightly stronger than the classical GCC. The smallness of the initial data in the
norm L2 × H−1 was assumed in order to avoid the missing unique continuation
theorem:
u = 0 is the unique strong solution in the energy space of

utt −∆u+ u+ |u|4u = 0 in M × (0, T ),

ut = 0 in ω × (0, T ),
(1.7)

where ω is an open subset of M satisfies the GCC in a given time T0 > 0. For
general case, we do not clearly know how to eliminate the smallness of the initial
data. In this article, because of the complexity of the critical case, the unique
continuation property of the energy critical semilinear wave equations is difficult to
obtain. Comparing to the previous results, a stronger assumption on A(x) is as-
sumed to obtain a unique continuation result (see Assumption and Proposition 3.1).
Therefore, the exponential stability can be achieved by developing the traditional
Morawetz estimates and the compactness-uniqueness arguments for the semilinear
equation.

1.3. Main assumptions and main result. We use the following assumption in
this article:

(A1) There exists a constant 0 < δ ≤ 1 such that

〈
(
(1− δ)A(x)− r

2

∂A(x)

∂r

)
X,X〉 ≥ 0 for X ∈ R3

x, x ∈ R3. (1.8)

We will give an example that satisfies (1.8) and will show the relationship between
(A1) and the Geometric Control Condition in the appendix.

Condition (1.8) below which seems strong, is used to guarantee the classical
Morawetz mutiplier H = x = r ∂∂r works in the metric g. That is, we have

DH(X,X) ≥ δ|X|2g.

More precisely, such a technical assumption is helpful to obtain the the following
unique continuation result:



4 S.-R. FU, Z.-H. NING EJDE-2022/59

u = 0 is the only solution to

utt − divA(x)∇u+ u+ u5 = 0, (x, t) ∈ R3 × (0, T ),

ut = 0, (x, t) ∈ (R3\B(R0))× (0, T ).
(1.9)

Generally, the unique continuation property for the critical semilinear wave equa-
tions is still an open problem, even in compact spaces.

For the global well-posedness of (1.1), we assume that

(A2) System (1.1) admits a unique solution such that

u ∈ C1(0,∞;L2(R3)) ∩ C(0,∞;H1(R3)).

Remark 1.1. The global existence results for critical wave equations are complex.
Fortunately, the powerful Strichartz estimate, as a space-time estimate, offers us
an effective tool to handle the critical case. In general, for lower regularity initial
data (u0, u1) ∈ H1 × L2, we have

u(t) ∈ C([0,+∞);H1) ∩ L5
tL

10
x ([0, T )× R3), for all T < +∞.

Here we list some more references on this topic. For Cauchy problem, global exis-
tence of C2-solutions in dimension n = 3 was first obtained by Rauch [32], assuming
the initial energy to be small. Later, the global existence results have improved in
many subsequent papers: [4, 13, 14, 16, 21, 36, 37]. Now, the global well-posedness
of the energy critical defocusing wave equations are classical. We refer to [47] for the
critical wave equations with variable coefficients on R3, and to [25] for the critical
Klein-Gordon equations on 3-D compact Riemannian manifolds.

The main result in this article reads as follows.

Theorem 1.2. Suppose that (A1), (A2) hold. Let E(0) ≤ E0 and

a(x) ≥ a0, x ∈ R3\B(R0), (1.10)

where E0, a0, R0 are positive constants. Then there exist positive constants C1, C2,
which are dependent on E(0), such that

E(t) ≤ C1e
−C2tE(0), ∀t > 0. (1.11)

2. Multiplier identities and key lemmas

Here we establish several geometric multiplier identities, which are useful for the
unique continuation results.

Lemma 2.1. Let Ω ⊂ R3 be a bounded domain with smooth boundary ∂Ω. Let
ν(x) be the unit normal vector of ∂Ω, pointing outside on Ω. Suppose that u(x, t)
is a solution of the equation

utt − divA(x)∇u+ a(x)ut + u+ u5 = 0, (x, t) ∈ Ω× (0,+∞). (2.1)

Let H be a C1 vector field defined on R3. Then∫ T

0

∫
∂Ω

〈∇gu, ν〉H(u)dΓdt+
1

2

∫ T

0

∫
∂Ω

(u2
t − |∇gu|

2
g − u

2 − 1

3
u6)H · νdΓdt

=

∫
Ω

utH(u) dx
∣∣∣T
0

+

∫ T

0

∫
Ω

DH(∇gu,∇gu) dx dt+

∫ T

0

∫
Ω

a(x)utH(u) dx dt

+
1

2

∫ T

0

∫
Ω

(u2
t − |∇gu|2g − u2 − 1

3
u6) divH dx dt.

(2.2)



EJDE-2022/59 STABILIZATION OF A KLEIN-GORDON EQUATION 5

Moreover, if we assume that P ∈ C2(R3), then∫ T

0

∫
Ω

(u2
t − |∇gu|2g − u2 − u6)P dx dt

=

∫
Ω

Puut dx
∣∣∣T
0

+
1

2

∫ T

0

∫
∂Ω

u2〈∇gP, ν〉dΓdt−
∫ T

0

∫
∂Ω

Pu〈∇gu, ν〉dΓdt

− 1

2

∫ T

0

∫
Ω

u2(divA(x)∇P ) dx dt+
1

2

∫
Ω

a(x)Pu2 dx
∣∣T
0
.

(2.3)

Proof. Note that

∇gu(H(u)) = ∇gu〈∇gu,H〉g = D2u(H,∇gu) +DH(∇gu,∇gu)

= D2u(∇gu,H) +DH(∇gu,∇gu)

=
1

2
H(|∇gu|2g) +DH(∇gu,∇gu)

= DH(∇gu,∇gu) +
1

2
div(|∇gu|2gH)− 1

2
|∇gu|2g divH.

(2.4)

Hence, we have

(divA(x)∇u)H(u) = div(H(u)∇gu)−∇gu(H(u))

= div(H(u)∇gu)−DH(∇gu,∇gu)− 1

2
div(|∇gu|2gH)

+
1

2
|∇gu|2g divH.

(2.5)

We multiply the wave equation (2.1) by H(u) and integrate over Ω×(0, T ) to obtain

(divA(x)∇u)(H(u))

= (utt + a(x)ut + u+ u5)H(u)

= (utH(u))t −
1

2
H(u2

t ) +
1

2
H(u2) +

1

6
H(u6) + a(x)utH(u)

= (utH(u))t −
1

2
div(u2

tH) +
1

2
u2
t divH+

1

2
div(u2H)

− 1

2
u2 divH+

1

6
div(u6H)− 1

6
u6 divH+ a(x)utH(u).

(2.6)

From this and (2.5), the equality (2.2) follows from Green’s formula.
Similarly, we multiply the wave equation (2.1) by Pu and integrate over Ω ×

(0, T ). Note that

0 = (utt − divA(x)∇u+ a(x)ut + u+ u5)Pu

= (utPu)t − Pu2
t − div(Pu∇gu) + P |∇gu|2g

+
1

2
∇gP (u2) + Pu2 + Pu6 + Pa(x)uut

= (utPu)t − Pu2
t − div(Pu∇gu) + P |∇gu|2g

+
1

2
div(u2∇gP )− 1

2
u2 divA(x)∇P + Pu2 + Pu6 +

1

2
(Pa(x)u2)t .

(2.7)

Then equality (2.3) follows from Green’s formula. �



6 S.-R. FU, Z.-H. NING EJDE-2022/59

Lemma 2.2. Let u(x, t) be a solution of (1.1). Then

E(t)
∣∣T
0

= −
∫ T

0

∫
R3

a(x)u2
t dx dt, (2.8)

which implies E(t) is decreasing.

Proof. Multiply the first equation in (1.1) by ut and integrate over R3× (0, T ), the
equality (2.8) holds immediately. �

3. Unique continuation

In this section, we prove two unique continuation results, which are crucial for
the compactness-uniqueness arguments.

Lemma 3.1. There exists a constant C > 0 such that∫
R3

w2

r2
dx ≤ C

∫
R3

|∇w|2 dx (3.1)

for all w ∈ H1(R3).

Proof. Note that

div
(w2

r

∂

∂r

)
= w2 div

(1

r

∂

∂r

)
+

2

r
wwr =

1

r2
w2 +

2

r
wwr. (3.2)

Integrating (3.2) over R3 yields∫
R3

1

r2
w2dx = −

∫
R3

2

r
wwrdx, (3.3)

which implies (3.1). �

Lemma 3.2. Let E0 be a positive constant. Assume that E(0) ≤ E0 and

f(u) = u2
t + u2 + |∇gu|2g + u6.

Then

lim inf
y→∞

∫
|x|=y

rf(u)dΓ = 0. (3.4)

Proof. Suppose that (3.4) is not true. Then there exist positive constants M and
β such that ∫

|x|=y
f(u)dΓ ≥ β

y
, y ≥M. (3.5)

Note that ∫
R3

f(u)dx =

∫ ∞
0

∫
|x|=y

f(u)dΓdy

=
(∫ M

0

+

∫ ∞
M

) ∫
|x|=y

f(u)dΓdy

≥
∫ M

0

∫
|x|=y

f(u)dΓdy +

∫ ∞
M

β

y
dy = +∞,

(3.6)

which contradicts ∫
R3

(u2
t + u2 + |∇gu|2g + u6)dx ≤ 6E0 < +∞. (3.7)

�



EJDE-2022/59 STABILIZATION OF A KLEIN-GORDON EQUATION 7

Proposition 3.3. Let (A1), (A2) hold and let R0 > 0 be the constant given in
Theorem 1.2. Then there exists a constant T0 > 0 such that for any T > T0, the
only solution (u, ut) ∈ C([0, T ], H1(R3)× L2(R3)) to the system

utt − divA(x)∇u+ u+ u5 = 0, (x, t) ∈ R3 × (0, T ),

ut = 0, (x, t) ∈ (R3\B(R0))× (0, T ),
(3.8)

is u ≡ 0.

Proof. Letting a(x) ≡ 0, it follows from (2.8) that

E(t) = E(0), ∀ t ≥ 0. (3.9)

Let φ ∈ C∞(R3) be a nonnegative cut-off function such that

φ = 1, x ∈ R3\B(R0 + 1) and φ = 0, x ∈ B(R0). (3.10)

Let Ω = B(y) with a radius y > 0, H = x, and P ∈ C2(R3) ∩W 1,∞(R3). Notice
that x = r ∂∂r , it follows from (3.4) that

lim inf
y→∞

∫
∂B(y)

[〈∇gu, ν〉H(u) +
1

2
(u2
t − |∇gu|2g − u2 − 1

3
u6)〈H, ν〉]dΓ

≤ lim inf
y→∞

∫
|x|=y

r(u2
t + |∇gu|2g + u2 + u6)dΓ = 0,

(3.11)

and

lim inf
y→∞

∫
∂B(y)

[
1

2
u2〈∇gP, ν〉 − Pu〈∇gu, ν〉]dΓ

≤ ||P ||W 1,∞(R3) lim inf
y→∞

∫
|x|=y

[u2 + (
1

r
u2 + r|∇gu|2g)]dΓ = 0.

(3.12)

Let P = φ, a(x) ≡ 0 and Ω = B(y) in (2.3). Let y → +∞, it follows from (2.3)
and (3.12) that∫ T

0

∫
R3

(|∇gu|2g + u2 + u6)P dx dt ≤ CE(0) + C

∫ T

0

∫
B(R0+1)

u2 dx dt. (3.13)

From (3.1), we have∫ T

0

∫
B(R0+1)

u2 dx dt ≤ C(R0)

∫ T

0

∫
R3

|∇gu|2g dx dt. (3.14)

Thus, we have∫ T

0

∫
R3

u2 dx dt ≤ CE(0) + C(R0)

∫ T

0

∫
R3

|∇gu|2g dx dt. (3.15)



8 S.-R. FU, Z.-H. NING EJDE-2022/59

Let H = x, a(x) ≡ 0 and Ω = B(y) in (2.2). Let y → +∞, it follows from (2.2),
(5.6), and (3.11) that

0 =

∫
R3

utH(u) dx
∣∣∣T
0

+

∫ T

0

∫
R3

DH(∇gu,∇gu) dx dt+

∫ T

0

∫
R3

a(x)utH(u) dx dt

+
3

2

∫ T

0

∫
R3

(u2
t − |∇gu|2g − u2 − 1

3
u6) dx dt

≥
∫
R3

utH(u) dx
∣∣∣T
0

+ δ

∫ T

0

∫
R3

|∇gu|2g dx dt+

∫ T

0

∫
R3

a(x)utH(u) dx dt

+
3

2

∫ T

0

∫
R3

(u2
t − |∇gu|2g − u2 − u6) dx dt+

∫ T

0

∫
R3

u6 dx dt.

(3.16)
Again let Ω = B(y) and a(x) = 0 in (2.3). Combining (2.3) with (3.16) and letting
y → +∞, we obtain∫ T

0

∫
R3

[
(
3

2
− P )u2

t + (P − 3

2
+ δ)|∇gu|2g + (P − 3

2
)u2 + (P − 1

2
)u6
]
dx dt

≤ −
∫
R3

[Puut + utH(u)] dx
∣∣∣T
0

+
1

2

∫ T

0

∫
R3

u2 div(A(x)∇P ) dx dt.

(3.17)

We denote

δc =
δ

1 + C(R0)
< 1, (3.18)

where C(R0) is given by (3.15).
Taking P = 3−δc

2 , a(x) ≡ 0 in (3.17), we have∫ T

0

∫
R3

[1
2
δcu

2
t + δ1|∇gu|2g −

1

2
δcu

2 + δ2u
6
]
dx dt ≤ CE(0), (3.19)

where δ1 = δ − 1
2δc = δ(1+2C(R0))

2(1+C(R0)) and δ2 = 1 − 1
2δc = 2(1+C(R0))−δ

2(1+C(R0)) > 0. On the

other hand, by (3.15), for δ0 > 0, we have

δ(1 + δ0)

2(1 + C(R0))

∫ T

0

∫
R3

u2 dx dt ≤ CE(0) +
δ(1 + δ0)C(R0)

2(1 + C(R0))

∫ T

0

∫
R3

|∇gu|2g dx dt.

Taking δ0 = 1, we have

δ(1 + δ0)

2(1 + C(R0))
− 1

2
δc =

1

2
δc and δ1 −

δ(1 + δ0)C(R0)

2(1 + C(R0))
=

1

2
δc. (3.20)

Thus, with (3.17)-(3.20), we conclude that∫ T

0

E(t)dt ≤ CE(0), (3.21)

which implies (T −C)E(0) ≤ 0. Therefore, the assertion (3.8) holds and the proof
is complete. �

The following proposition has a similar proof the one above.



EJDE-2022/59 STABILIZATION OF A KLEIN-GORDON EQUATION 9

Proposition 3.4. Let (A1), (A2) hold and let R0 > 0 be the constant given in
Theorem 1.2. Then there exists a constant T0 > 0 such that for any T > T0, the
only solution (u, ut) ∈ C([0, T ], H1(R3)× L2(R3)) to the system

utt − divA(x)∇u+ u = 0, (x, t) ∈ R3 × (0, T ),

ut = 0, (x, t) ∈ (R3\B(R0))× (0, T ),
(3.22)

is u ≡ 0.

4. Proofs of the main theorem

Lemma 4.1. Let (A1), (A2) hold, and u(x, t) solve system (1.1). then

E(0) ≤ C
∫ T

0

∫
R3

a(x)u2
t dx dt+ C

∫ T

0

∫
B(R0)

u2 dx dt (4.1)

holds for sufficiently large T .

Proof. Recall that a(x) ≥ a0 for x ∈ R3\B(R0), then there exists a small constant
ε0 > 0 such that

a(x) ≥ a0

2
, x ∈ R3\B(R0 − 2ε0). (4.2)

Let b(z) be a smooth nonnegative function on [0,+∞) satisfying

b(z) = 1, 0 ≤ z ≤ R0 − ε0, b(z) = 0, z ≥ R0. (4.3)

Let H(x) be a vector field on B(R0) satisfying

H(x) = b(r)x, x ∈ B(R0).

It follows from (5.6) that

DH(X,X) ≥ δ|X|2g for X ∈ R3
x, x ∈ B(R0 − ε0),

divH = 3 for x ∈ B(R0 − ε0).
(4.4)

Let H = H and Ω = B(R0) in (2.2). Then

0 ≥
∫

Ω

utH(u) dx
∣∣T
0

+ δ

∫ T

0

∫
B(R0−ε0)

|∇gu|2g dx dt

− C
∫ T

0

∫
B(R0)\B(R0−ε0)

|∇gu|2g dx dt+

∫ T

0

∫
B(R0)

a(x)utH(u) dx dt

+
1

2

∫ T

0

∫
B(R0)

(u2
t − |∇gu|

2
g − u

2 − 1

3
u6) divH dxdt

=

∫
Ω

utH(u) dx
∣∣∣T
0

+ δ

∫ T

0

∫
B(R0−ε0)

|∇gu|2g dx dt

− C
∫ T

0

∫
B(R0)\B(R0−ε0)

|∇gu|2g dx dt+

∫ T

0

∫
B(R0)

a(x)utH(u) dx dt

+

∫ T

0

∫
B(R0)

[
1

3
u6 +

1

2
(u2
t − |∇gu|2g − u2 − u6)] divH dxdt.

(4.5)



10 S.-R. FU, Z.-H. NING EJDE-2022/59

Let P = (divH − b(r)δ)/2 and Ω = B(R0) in (2.3). Substituting (2.3) into (4.5),
we obtain∫

B(R0)

ut(H(u) + Pu) dx
∣∣∣T
0
− 1

2

∫ T

0

∫
B(R0)

u2(divA(x)∇P ) dx dt

+
1

2

∫
B(R0)

a(x)Pu2 dx
∣∣∣T
0

+

∫ T

0

∫
B(R0)

a(x)utH(u) dx dt

+
δ

2

∫ T

0

∫
B(R0−ε0)

(u2
t + |∇gu|2g + u2 + u6)dx

≤ C
∫ T

0

∫
B(R0)\B(R0−ε0)

(u2 + |∇gu|2g + u6) dx dt

+ C

∫ T

0

∫
B(R0−ε0)

u2 dx dt+ C

∫ T

0

∫
B(R0)

a(x)u2
t dx dt.

(4.6)

Therefore, ∫ T

0

∫
B(R0−ε0)

(
u2
t + |∇gu|2g + u2 +

1

3
u6
)
dx dt

≤ C(E(0) + E(T )) +

∫ T

0

∫
B(R0)

a(x)(Cεu
2
t + ε|∇gu|2g) dx dt

+ C

∫ T

0

∫
B(R0)\B(R0−ε0)

(u2 + |∇gu|2g + u6) dx dt

+ C

∫ T

0

∫
B(R0−ε0)

u2 dx dt+ C

∫ T

0

∫
B(R0)

a(x)u2
t dx dt.

(4.7)

Taking ε sufficiently small, we have∫ T

0

∫
B(R0−ε0)

(
u2
t + |∇gu|2g + u2 +

1

3
u6
)
dx dt

≤ C(E(0) + E(T )) + C

∫ T

0

∫
B(R0)

a(x)u2
t dx dt+ C

∫ T

0

∫
B(R0−ε0)

u2 dx dt

+ C

∫ T

0

∫
B(R0)\B(R0−ε0)

(u2 + |∇gu|2g + u6) dx dt.

(4.8)

Therefore,∫ T

0

∫
R3

(
u2
t + |∇gu|2g + u2 +

1

3
u6
)
dx dt

≤ C(E(0) + E(T )) + C

∫ T

0

∫
R3

a(x)u2
t dx dt

+ C

∫ T

0

∫
R3\B(R0−ε0)

(u2 + |∇gu|2g + u6) dx dt+ C

∫ T

0

∫
B(R0−ε0)

u2 dx dt.

(4.9)
Let w(z) be a smooth nonnegative function on [0,+∞) satisfying

w(z) = 0, 0 ≤ z ≤ R0 − 2ε0 and w(z) = 1, z ≥ R0 − ε0.



EJDE-2022/59 STABILIZATION OF A KLEIN-GORDON EQUATION 11

Let P = w(r) and Ω = B(y) in (2.3). Let y → +∞, it follows from (2.3) and
Lemma 3.2 that∫ T

0

∫
R3

(u2
t − |∇gu|2g − u2 − u6)P dx dt

= (ut, uP )
∣∣T
0
− 1

2

∫ T

0

∫
R3

u2 divA(x)∇P dx dt+
1

2

∫
R3

a(x)Pu2 dx
∣∣∣T
0
.

(4.10)

From (4.2), we obtain∫ T

0

∫
R3

(|∇gu|2g + u2 + u6)P dx dt

≤ C(E(0) + E(T )) + C

∫ T

0

∫
R3

a(x)u2
t dx dt

+ C

∫ T

0

∫
B(R0−ε0)\B(R0−2ε0)

u2 dx dt.

(4.11)

Substituting (4.11) into (4.9) yields∫ T

0

∫
R3

(u2
t + u2 + |∇gu|2g +

1

3
u6) dx dt

≤ C(E(0) + E(T )) + C

∫ T

0

∫
R3

a(x)u2
t dx dt+ C

∫ T

0

∫
B(R0−ε0)

u2 dx dt.

(4.12)

With (2.8), we deduce that

CE(T ) = CE(0)− C
∫ T

0

∫
R3

a(x)u2
t dx dt, (4.13)

and

4CE(0) =

∫ 4C

0

E(t)dt−
∫ 4C

0

(E(t)− E(0))dt

≤
∫ 4C

0

E(t)dt+ 4C

∫ 4C

0

∫
R3

a(x)u2
t dx dt.

(4.14)

Inserting (4.13) and (4.14) into (4.12), taking T > 4C, we have

E(0) ≤ C
∫ T

0

∫
R3

a(x)u2
t dx dt+ C

∫ T

0

∫
B(R0−ε0)

u2 dx dt. (4.15)

The proof is complete. �

Lemma 4.2 (Observability inequality). Let (A1), (A2) hold. Let u(x, t) solve
system (1.1). Then for any E(0) ≤ E0 <∞,

E(0) ≤ C(E0, T )

∫ T

0

∫
R3

a(x)u2
t dx dt, (4.16)

for sufficiently large T .

Proof. We apply the compactness-uniqueness arguments to prove the conclusion.
It follows from (4.1) that

E(0) ≤ C
∫ T

0

∫
R3

a(x)u2
t dx dt+ C

∫ T

0

∫
B(R0)

u2 dx dt. (4.17)



12 S.-R. FU, Z.-H. NING EJDE-2022/59

By contradiction. Suppose that estimate (4.16) does not hold, then there exists a
sequence {uk}∞k=1 such that

Ek(0) ≤ E0, (4.18)

where

Ek(t) =
1

2

∫
R3

(u2
kt + u2

k + |∇guk|2g)dx+
1

6

∫
R3

u6
kdx,

and ∫ T

0

∫
B(R0)

u2
k dx dt ≥ k

∫ T

0

∫
R3

a(x)u2
kt dx dt. (4.19)

From (2.8), we have

Ek(t) ≤ E0, 0 ≤ t ≤ T, (4.20)

and ∫ T

0

Ek(t)dt ≤ TE0.

Therefore, there exists û and a subset of {uk}∞k=1, still denoted by {uk}∞k=1, such
that

uk → û weakly in H1(R3 × (0, T )), (4.21)

uk → û strongly in L2(B(R0)× (0, T )), (4.22)

Case a: ∫ T

0

∫
B(R0)

û2 dx dt > 0. (4.23)

Note that H1(R3) ↪→ L6(R3) and L6(R3) is the dual space of L6/5(R3). It follows
from (4.20) that

{u5
k} is bounded in L∞([0, T ], L6/5(R3)). (4.24)

Then

{u5
k} is bounded in L6/5(R3 × (0, T )), (4.25)

which implies

u5
k → û5 weakly in L6/5(R3 × (0, T )). (4.26)

It follows from (4.19) that

a(x)ût = 0, (x, t) ∈ R3 × (0, T ).

Therefore, with (4.21) and (4.26), we obtain

ûtt − divA(x)∇û+ û+ û5 = 0, (x, t) ∈ R3 × (0, T ),

ût = 0, (x, t) ∈ (R3\B(R0))× (0, T ).
(4.27)

It follows from Proposition 3.3 that

û(x, t) ≡ 0, (x, t) ∈ R3 × (0, T ), (4.28)

which contradicts (4.23).

Case b:

û(x, t) ≡ 0 (x, t) ∈ B(R0)× (0, T ). (4.29)

We denote

vk = uk/
√
ck for k ≥ 1, (4.30)



EJDE-2022/59 STABILIZATION OF A KLEIN-GORDON EQUATION 13

where

ck =

∫ T

0

∫
B(R0)

u2
k dx dt. (4.31)

Then vk satisfies

vktt − divA(x)∇vk + a(x)vkt + vk + u4
kvk = 0, (x, t) ∈ R3 × (0, T ), (4.32)∫ T

0

∫
B(R0)

v2
k dx dt = 1. (4.33)

It follows from (4.19) that

1 ≥ k
∫ T

0

∫
R3

a(x)v2
kt dx dt. (4.34)

From this and (4.17), we have

Êk(0) ≤ 1 +
1

k
≤ 2, (4.35)

where

Êk(t) =
1

2

∫
R3

(v2
kt + v2

k + |∇gvk|2g)dx+
1

6

∫
R3

u4
kv

2
kdx.

Hence, there exists a v̂ and a subsequence of {vk}∞k=1, still denoted by {vk}∞k=1,
such that

vk → v̂ weakly in H1(R3 × (0, T )),

vk → v̂ strongly in L2(B(R0)× (0, T )).
(4.36)

Collecting (2.8), (4.30), and (4.35), we obtain

Êk(t) ≤ Êk(0) ≤ 2, ∀0 ≤ t ≤ T. (4.37)

Notice that H1(R3) ↪→ L6(R3). Therefore {vk} are bounded in L∞([0, T ], L6(R3)).
Hence, we have∫ T

0

∫
R3

∣∣u4
kvk
∣∣6/5 dx dt = c

12/5
k

∫ T

0

∫
R3

v6
k dx dt ≤ c

12/5
k C(T ). (4.38)

We combine (4.29) with (4.31) to obtain

lim
k→+∞

∫ T

0

∫
R3

∣∣u4
kvk
∣∣6/5 dx dt = 0. (4.39)

By (4.34) and (4.36), we have

a(x)v̂t = 0, (x, t) ∈ R3 × (0, T ).

Therefore, from (4.32) and (4.39) it follows that

v̂tt − divA(x)∇v̂ + v̂ = 0, (x, t) ∈ R3 × (0, T ),

v̂t = 0, (x, t) ∈ (R3\B(R0))× (0, T ).
(4.40)

The following holds by Proposition 3.4,

v̂ ≡ 0, (x, t) ∈ R3 × (0, T ). (4.41)

Then it follows from (4.33) that∫ T

0

∫
B(R0)

v̂2 dx dt = 1, (4.42)

which contradicts (4.41). The proof is complete. �



14 S.-R. FU, Z.-H. NING EJDE-2022/59

Proof of Theorem 1.2. From (2.8) and (4.16), we obtain

E(0) ≤ C(E0, T )(E(0)− E(T )).

Then

E(T ) ≤ C(E0, T )− 1

C(E0, T )
E(0),

which implies E(t) is of exponential decay. �

5. Appendix: Comments on assumption (A1)

As an example, (A1) is satisfied by the functionA(x) = diag{α1(x), α2(x), α3(x)},
where αi(x) are all smooth positive functions on R3, for 1 ≤ i ≤ 3. Assume that,
for 1 ≤ i ≤ 3,

0 < m1 ≤ αi(x) ≤ m2 < +∞, x ∈ R3, (5.1)

(1− δ)αi(x)− r(x)

2

∂αi(x)

∂r
≥ 0, x ∈ R3. (5.2)

Then

m1|X|2 ≤ 〈A(x)X,X〉 ≤ m2|X|2, X ∈ R3
x, x ∈ R3, (5.3)〈(

(1− δ)A(x)− r(x)

2

∂A(x)

∂r

)
X,X

〉
≥ 0, X ∈ R3

x, x ∈ R3. (5.4)

It is easy to see that the standard unit matrix, I3 = (δij)1≤i,j≤3, satisfies (1.8).

Another example satisfying (5.2) is αi(x) = e−r
2

.
In the following, let Ω ⊂ R3 be a bounded domain with a smooth boundary ∂Ω,

we will show the relationship between (A1) and the geometric control condition
(GCC). The proof is similar to the one for [31].

Proposition 5.1. Let H(x) = x. Then

DH(X,X) =
〈(
G(x) +

r(x)

2

∂G(x)

∂r

)
X,X

〉
, X ∈ R3

x, x ∈ R3. (5.5)

Proof. Let x ∈ R3, X =
∑3
i=1Xi

∂
∂xi
∈ R3

x. Note that

H(x) =

3∑
i=1

xi
∂

∂xi
.

Then

DH(X,X) =

3∑
i,j,k=1

〈
D ∂

∂xi

(
xk

∂

∂xk

)
,
∂

∂xj

〉
g
XiXj

=

3∑
i,j=1

gijXiXj +

3∑
i,j,k=1

xk
〈
D ∂

∂xi

∂

∂xk
,
∂

∂xj

〉
g
XiXj

= |X|2g +

3∑
i,j,k=1

xk
〈
D ∂

∂xk

∂

∂xi
,
∂

∂xj

〉
g
XiXj

= |X|2g +

3∑
i,j,k=1

xk
2

∂gij
∂xk

XiXj =
〈(
G(x) +

r(x)

2

∂G(x)

∂r

)
X,X

〉
. �



EJDE-2022/59 STABILIZATION OF A KLEIN-GORDON EQUATION 15

Proposition 5.2. Let (A1) hold and let H(x) = x. Then

DH(X,X) ≥ δ|X|2g, X ∈ R3
x, x ∈ R3. (5.6)

Proof. Let x ∈ R3, X,Y ∈ R3
x and Y = G(x)X. We deduce that

0 ≤ Y T
(
(1− δ)A(x)− r

2

∂A(x)

∂r

)
Y

= 〈G(x)
(
(1− δ)A(x)− r

2

∂A(x)

∂r

)
G(x)X,X〉

=
〈(

(1− δ)G(x) +
r

2

∂(G(x))

∂r

)
X,X

〉
.

(5.7)

Inequality (5.6) follows from (5.5). �

Proposition 5.3. Let (A1) hold. Then, for any x ∈ Ω and any unit-speed geodesic
γ(t) starting from x, if γ(t) ∈ Ω for 0 ≤ t ≤ t0, then

t0 ≤
2

δ
sup{|H|g(x) : x ∈ Ω}.

Proof. Note that |γ′(t)|g = 1 and Dγ′(t)γ
′(t) = 0. From (5.6), we deduce that

〈H, γ′(t)〉g
∣∣t0
0

=

∫ t0

0

γ′(t)〈H, γ′(t)〉gdt =

∫ t0

0

DH(γ′(t), γ′(t))dt ≥ δt0. (5.8)

The proof is complete. �

Let S(r) be the sphere in R3 with a radius r. Then

〈X, ∂
∂r
〉 = 0, for X ∈ S(r)x, x ∈ R3\O,

where S(r)x is the tangential space of S(r) at x. The following lemma shows that
GCC may not hold if A(x) satisfies (5.9) and (5.10) below.

Proposition 5.4. Assume that

A(x)
∂

∂r
=

∂

∂r
, x ∈ R3, (5.9)

〈(A(x)− r

2

∂A(x)

∂r
)X,X〉 = 0 for X ∈ S(R1)x, |x| = R1. (5.10)

where R1 is a positive constant. Then, for any x ∈ S(R1) and any unit-speed
geodesic γ(t) starting from x with γ′(0) ∈ S(R1)x, we have

γ(t) ∈ S(R1), ∀ t ≥ 0.

Proof. Note that

G(x)
∂

∂r
=

∂

∂r
, x ∈ R3.

Therefore,

D(r
∂

∂r
) = D(rDr) = Dr ⊗Dr + rD2r.

By a proof similar to the one of Proposition 5.2, we obtain

D(rDr)(X,X) = 0, X ∈ S(R1)x, |x| = R1.

Then

D2r(X,X) = 0, X ∈ S(R1)x, |x| = R1.



16 S.-R. FU, Z.-H. NING EJDE-2022/59

Let ĝ be a Riemannian metric induced by g in S(R1) and D̂ be the associated
Levi-Civita connection. Let γ̂(t) be a unit-speed geodesic of (S(R1), ĝ) starting
from x ∈ S(R1), then

〈γ̂′(t), ∂
∂r
〉g = 0, D̂γ̂′(t)γ̂

′(t) = 0, ∀t ≥ 0.

Therefore,

Dγ̂′(t)γ̂
′(t) = D̂γ̂′(t)γ̂

′(t) + 〈Dγ̂′(t)γ̂
′(t),

∂

∂r
〉g
∂

∂r

= D̂γ̂′(t)γ̂
′(t)−D2r(γ̂′(t), γ̂′(t))

∂

∂r
= 0,

(5.11)

which implies γ̂(t) is also a geodesic of (R3, g). Then

γ(t) = γ̂(t) ∈ S(R1), ∀t ≥ 0,

for unit-speed geodesic γ(t) of (R3, g) satisfying γ(0) = γ̂(0) and γ′(0) = γ̂′(0). The
proof is complete. �

Acknowledgments. The authors would like to thank the anonymous referees for
their careful reading of the manuscript, and for their constructive comments.

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Song-Ren Fu

Key Laboratory of Systems and Control, Institute of Systems Science, Academy of
Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Email address: songrenfu@amss.ac.cn

Zhen-Hu Ning

Faculty of Information Technology, Beijing University of Technology, Beijing, 100124,

China
Email address: nzh41034@163.com


	1. Introduction
	1.1. Notation and statement of the problem
	1.2. Previous results
	1.3. Main assumptions and main result

	2. Multiplier identities and key lemmas
	3. Unique continuation
	4. Proofs of the main theorem
	5. Appendix: Comments on assumption (A1)
	Acknowledgments

	References