Suppose
L
is the generator of a continuous contraction semigroup
P
t
on
E
and
f
:
R
0
+
→
E
is a
C
1
curve in
E
. Verify that the unique solution of the inhomogeneous equation
d
d
t
u
(
t
)
=
f
(
t
)
+
L
u
(
t
)
u
(
0
)
=
x
∈
d
o
m
L
is given by
Duhamel's formula
:
u
(
t
)
=
P
t
x
+
∫
0
t
P
t
−
s
f
(
s
)
d
s
.
Put
h
(
t
)
:
=
∫
0
t
P
t
−
s
f
(
s
)
d
s
, then
u
(
t
)
=
P
t
x
+
h
(
t
)
solves the inhomogeneous equation iff
h
′
(
t
)
=
L
h
(
t
)
+
f
(
t
)
. Thus we need to show that
h
is a differentiable
d
o
m
L
-valued curve and its derivative is given by
L
h
(
t
)
+
f
(
t
)
. Now
h
(
t
)
=
∫
0
t
P
t
−
s
f
(
s
)
d
s
=
∫
0
t
P
s
f
(
t
−
s
)
d
s
and thus by the proof of
proposition
h
is a differentiable
d
o
m
L
-valued curve. Moreover
h
(
t
+
r
)
−
h
(
t
)
=
∫
0
t
+
r
P
t
+
r
−
s
f
(
s
)
d
s
−
∫
0
t
P
t
−
s
f
(
s
)
d
s
=
(
P
r
−
1
)
∫
0
t
P
t
−
s
f
(
s
)
d
s
+
P
r
∫
t
t
+
r
P
t
−
s
f
(
s
)
d
s
and since
h
(
t
)
∈
d
o
m
(
L
)
we conclude that:
h
′
(
t
)
=
L
h
(
t
)
+
f
(
t
)
.
Uniqueness: If
u
,
v
are different solutions, then
w
:
=
u
−
v
solves the homogeneous equation
w
′
(
t
)
=
L
w
(
t
)
and
w
(
0
)
=
0
, i.e. by uniqueness of solutions of the homogeneous equation:
w
=
0
.