10
PROOF. We have
SIGURD ANGENENT AND JOOST HULSHOF
R4
lv2(y,t)l::::;
Ck(logR)2y3L(y).
Since
-RRt
"'R2/I
log Rl and
yL(y) ::::;
C7j;l(y), we get
R2
Ck Ck
lv2(y,t)1::::; Ck(logR) 2yL(y)::::;
llogRI(-RRt)7/J1(Y)::::; llogRiv1(y,t).
Since R(t)--- 0 as t / oo, we get lv21::::;
~v 1
for large enough t. D
Proposition 4.3 therefore applies. Together with proposition 4.2 we find that
l l
2
OV2
(4.4)
~
[v1 + v2
=
-M [v2 +
R
Bt-
RRtYV2,y
+ T1 + T2 +
T3
R4
2
8v2
CR4
::::; -k
(log
R)2 yL(y)
+
R
Bt -
RRtYV2,y
+ (log
R)2 yL(y)
R4
::::; (C-
k)
(log R)
2
yL(y)
+
T5
+ T7
where
20V2
T5
=
R
Bt
and T7
=
-RRtYV2,y·
4.5.
Estimation of T6 and
T7. For T6 we compute
I
2 OV21
I
2 () ( R
4
)
I
R
Bt
=
R
8t (log R)2 7/J2 (y)
= (
4 + O((log R)-
1))
(!gl~)~
I7/J2(Y)I
R6
::::; C
(log
R)3 y3 L(y)
R4
::::; C
(log
R)3 yL(y)
Next, we deal with T7. We have
IY7/J~(y)l
::::;
Cy3L(y),
which implies
IT71 ::::;
I-RRtYV2,yl
CR2
R4
::::; I log Rl (log
R)2
y3L(y)
CR4
::::; I log Rl3
yL(y).
In combination with ( 4.4) we therefore find that
C
R4
~[v1+v2]::::;(C+
llogRI-k)(logR)2yL(y).
This finally leads to the following result.
THEOREM 4.5. If k 0 is large enough, and if R(t) satisfies -Rt
=
-2 ;~~t(~J ~),
then a t
0
exists such that
Previous Page Next Page