12
SIGURD ANGENENT AND JOOST HULSHOF
5.2.
r
+(t) and
r
_(t). The functions
U±(Y, t)
are defined for ally 0, and we
know they provide sub or super solutions when 0
y:::;
2R(t)
1.
Thus the
'P±(r, t)
are sub and super solutions for 0
r :::;
2. We now consider 'P±(1,
t)
for large
t.
From
(5.2)
and 'lj;2(y)
=
(1 + o(1) )y3 log y for y / oo, we conclude
R4
'P±(1,
t)
=
u±(R(t)1, t)
=
2arctanR(t)1

RRt'I/J1(R1) ±
k
(logR)
2
'I/J2(R
1)
R4
=
7f
±
k (log R)2 'I/J2(R1)
R4
=
7f
±
(kl8 + o(1)) (log R)
2
R 3 log R
1
so that
(5.3)
R
'P±(1,
t)
=
1r
±
(kl8 + o(1)) log
11
R fort/ oo.
Next, we consider
Or'P±
for
r
2 and for large
t.
Again using (5.2) we find that
Or'P±(r, t)
=
R(t)18yu±(Y,
t)
(5.4)
=
R
1
{
U'(y)
RRt'I/J~
(y) ± k
(lo:~) 2
'1/J;(y)}
2R
r 3k R
2
r
R2 + r2 Rt(1 + o(1)) log
R
± (
8
+ o(l)) (logR)2
r
log fi·
If~
r
2 then logr
=
o(log R) so that
log~ =
(1
+ o(1)) log 11 R, and hence,
using ( 4.2) once again,
R R 3k R
Or'P± (r, t)
=
(2 + o(1))
2
+ (2 + o(1)) l
I
R
log 11
R
± (
+ o(1)) l
I
R
r
2
r
~1
8
~1
=(2
(
))R(~
1
±
(3kl8+o(1))r)
+ 0
1
r
2
+ log 1
I
R
whence
(5.5)
for ~
r
2, and fort/ oo.
LEMMA
5.1. For large enough t there exist unique r±(t)
E
(~,2)
such that
'P±(r±(t), t)
=
1r.
One has
1k6
+
o(1)
r±(t)
=
1 =f
1
I ()
=
1 + o(1).
og1 R
t
PROOF.
This follows immediately from (5.3) and (5.5). Indeed, these equa
tions imply that
Or'P±
2::
(2 + o(1))R for
~
r
2, while
'P± 
1r
=
±(k
+
o(1))RI(log1IR), so that
'P±
1r
must vanish at some
r±
=
1+o(1). But (5.5) im
plies
Or'P±
=
(2+o(1))R for
r
=
1+o(1), which then leads to the stated asymptotic
expression for r ± ( t). D