We consider the problem
\[
(
P
)
{
u
t
=
(
u
m
)
x
x
,
u
(
x
,
0
)
=
u
0
(
x
)
with
x
∈
R
,
t
>
0
for
x
∈
R
\left ( {\text {P}} \right )\quad \quad \left \{ {\begin {array}{*{20}{c}} {{u_t} = {{({u^m})}_{xx}},} \\ {u(x,0) = {u_0}(x)} \\ \end {array} } \right .\quad \begin {array}{*{20}{c}} {{\text {with}}\,x \in {\mathbf {R}},\,t > 0} \\ {{\text {for}}\,x \in {\mathbf {R}}} \\ \end {array}
\]
where
m
>
1
m > 1
and
u
0
{u_0}
is a continuous, nonnegative function that vanishes outside an interval
(
a
,
b
)
(a,\,b)
and such that
(
u
0
m
−
1
)
≤
−
C
≤
0
(u_0^{m - 1}) \leq - C \leq 0
in
(
a
,
b
)
(a,\,b)
. Using a Trotter-Kato formula we show that the solution conserves the concavity in time: for every
t
>
0
,
u
(
x
,
t
)
t > 0,\,u(x,t)
vanishes outside an interval
Ω
(
t
)
=
(
ζ
1
(
t
)
,
ζ
2
(
t
)
)
\Omega (t) = ({}_{\zeta 1}(t),\,{}_{\zeta 2}(t))
and
\[
(
u
m
−
1
)
x
x
≤
−
C
1
+
C
(
m
(
m
+
1
)
/
(
m
−
1
)
)
t
{({u^{m - 1}})_{xx}} \leq - \frac {C} {{1 + C(m(m + 1)/(m - 1))t}}
\]
in
Ω
(
t
)
\Omega (t)
. Consequently the interfaces
x
=
ζ
i
(
t
)
x{ = _{\zeta i}}(t)
,
i
=
1
,
2
i = 1,\,2
, are concave curves. These results also give precise information about the large time behavior of solutions and interfaces.