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