Let
g
g
be a nonincreasing, odd
C
1
{C^1}
function and
l
>
0
l > 0
. We establish that for any solution
u
∈
C
(
R
;
H
0
1
(
0
,
l
)
)
u \in C({\mathbf {R}};H_0^1(0,l))
of the equation
u
t
t
−
u
x
x
+
g
(
u
)
=
0
{u_{tt}} - {u_{xx}} + g(u) = 0
and any
x
0
∈
]
0
,
l
[
{x_0} \in ]0,l[
, the function
t
↦
u
(
t
,
x
0
)
t \mapsto u(t,{x_0})
satisfies the following alternative: either
u
(
t
,
x
0
)
=
0
,
∀
t
∈
R
u(t,{x_0}) = 0,\forall t \in {\mathbf {R}}
, or
∀
a
∈
R
\forall a \in {\mathbf {R}}
, there exist
t
1
{t_1}
and
t
2
{t_2}
in
[
a
,
a
+
2
l
]
[a,a + 2l]
such that
u
(
t
1
,
x
0
)
>
0
u({t_1},{x_0}) > 0
and
u
(
t
2
,
x
0
)
>
0
u({t_2},{x_0}) > 0
. We study the structure of the set of points satisfying the first possibility. We give analogous results for
u
x
{u_x}
and for some other homogeneous boundary conditions.