This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by
u
t
t
−
Δ
M
u
+
a
(
x
)
g
(
u
t
)
=
0
on \thinspace
M
×
]
0
,
∞
[
,
\begin{equation} \left . \begin {array}{l} u_{tt} - \Delta _{\mathcal {M}}u+ a(x) g(u_{t})=0 \; \text {on \thinspace }\mathcal {M}\times \left ] 0,\infty \right [ , \end{array} \right . \nonumber \end{equation}
where
M
⊂
R
3
\mathcal {M}\subset \mathbb {R}^3
is a smooth oriented embedded compact surface without boundary. Denoting by
g
\mathbf {g}
the Riemannian metric induced on
M
\mathcal {M}
by
R
3
\mathbb {R}^3
, we prove that for each
ϵ
>
0
\epsilon > 0
, there exist an open subset
V
⊂
M
V \subset \mathcal M
and a smooth function
f
:
M
→
R
f:\mathcal M \rightarrow \mathbb R
such that
m
e
a
s
(
V
)
≥
m
e
a
s
(
M
)
−
ϵ
meas(V)\geq meas(\mathcal M)-\epsilon
,
H
e
s
s
f
≈
g
Hess f \approx \mathbf {g}
on
V
V
and
inf
x
∈
V
|
∇
f
(
x
)
|
>
0
\underset {x\in V}\inf |\nabla f(x)|>0
.
In addition, we prove that if
a
(
x
)
≥
a
0
>
0
a(x) \geq a_0> 0
on an open subset
M
∗
⊂
M
\mathcal {M}{\ast } \subset \mathcal M
which contains
M
∖
V
\mathcal {M}\backslash V
and if
g
g
is a monotonic increasing function such that
k
|
s
|
≤
|
g
(
s
)
|
≤
K
|
s
|
k |s| \leq |g(s)| \leq K |s|
for all
|
s
|
≥
1
|s| \geq 1
, then uniform and optimal decay rates of the energy hold.