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=
:
→
V
Lemma B.3.6
Assume
(
B.4b
)
with λ
0
,α>
0,
and that we are given w
J
,
→
V
∗
L
loc
(
L
loc
(
;
H
∗
)
s
:
J
and r
:
J
→ R
+
with I
loc
(
0
,
∞
)
:=
[
0
,
∞
)
;
V
)
∩
[
0
,
∞
)
and
w,w
∈
I
loc
(
0
,
∞
), w(t)
∈
K
for t>
0
,
(B.38a)
L
loc
(
0
,
r(t)
∈
∞
),
(B.38b)
L
loc
(
L
loc
(
;
V
∗
),
s(t)
=
p(t)
+
q(t)
;
p(t)
∈
[
0
,
∞
),
H
), q(t)
∈
[
0
,
∞
)
(B.39)
and such that
w
+
A
w,w
V
∗
,
V
≤
s(t),w(t)
V
∗
,
V
+
r(t).
(B.40)
Define
,
for T>
0
and for any decomposition
(
B.39
),
:=
√
α
X(T )
:=
w
L
∞
(J
;
H
)
,Y(T)
w
L
2
(J
;
V
)
,
(B.41)
2
R(T )
1
2
1
α
q
2
2
P(T)
:=
p
L
1
(J
;
H
)
,Q(T)
:=
w(
0
)
H
+
L
2
(J
;
V
∗
)
+
,
(B.42)
θ
:=
R(T )
sup
0
<θ<T
r(τ)
d
τ.
(B.43)
0
Then it holds
+
P(T)
2
Q(T )
2
1
2
max
{
X(T ),Y(T )
}≤
P(T)
+
(B.44)
R(T )
.
C
I(
0
,T )
≤
H
+
S(
0
,T )
+
w
w(
0
)
s
(B.45)
Proof
We integrate (
B.40
) over
J
and, by (
B.4b
) with
λ
=
0, obtain with (
B.39
) and
|
q,w
V
∗
,
V
|≤
q
∗
w
V
2
α
T
2
2
V
w(T )
H
+
w(τ)
d
τ
0
2
T
2
≤
w(
0
)
H
+
p(τ)
H
w(τ)
H
d
τ
0
2
T
+
q(τ)
∗
w(τ)
V
d
τ
+
2
R(T ).
0
Q(T )
2
. Now we argue as in the
proof of Lemma
B.3.4
to obtain (
B.44
). On the right hand side of (
B.44
), we may
take the infimum over all decompositions (
B.39
).
2
Y(T)
2
This implies
w(T )
+
≤
2
X(T )P(T )
+
e
−
λt
w(t)
, and
e
−
λt
s
,
Remark B.3.7
If in (
B.4b
)
λ>
0, we define
w(t)
=
s
=
p
=
e
−
λt
p
,
e
−
λt
q
,
e
−
2
λt
r
. The new quantities satisfy
w
+
A
w, w
V
∗
,
V
≤
s, w
V
∗
,
V
+
r,
q
=
r
=
(B.46)
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