SINGULARITIES AT
t
= oo IN EQUIVARIANT HARMONIC MAP FLOW 7
3.2. Expansions for derivatives. In general asymptotic expansions f(y)
=
o(g(y)) may not always be differentiated, however, the expansions for X do with-
stand differentiation.
LEMMA
3.2. If v(y)
=
(C
+
o(1))yafor y /'
oo,
then, assuming a
=j:.
-1, -3,
!!:_X ( ) _ a+ 2 + o(1) a+1 d
d2
X ( ) _ a+ 2 + o(1) a
dy v y - (a+ 1)(a
+
3) y ' an dy2 v y - a+ 3 y
as y /'
oo.
If a=
-1,
then
d (
1 )
d2
!.C + o(1)
dyXv(y)=
2c+o(1)
logy, and dy2Xv(y)=
2
y
for y /'
oo.
PROOF.
The expansions for first derivatives follow directly by differentiating
the integrals which represent Xv(y). The expansion for the second order derivatives
are then obtained by using the differential equation Mu
=
v which u
=
Xv satisfies.
D
4. Construction of a sub and super solution.
4.1. Specification of 't/J1. In ( 2.4) we defined 't/J1 as a solution to M [ '¢1]
=
'1/Jo,
where '1/Jo(Y)
=
yU'(y). We imposed one boundary condition, '¢1(0)
=
0, but
otherwise left '¢1 unspecified. Thus '¢1 is determined upto a multiple of '¢0 (which
satisfies M['t/Jo]
=
0). Since 't/Jo is bounded (in fact, '1/Jo(Y) "" 2/y for y /'
oo)
any
choice of
'¢1
will satisfy the same asymptotic condition (2.6) at
oo.
We now make a specific choice of '¢1. First, let if;1
=
X['¢0
].
Then, in view
of the asymptotic behaviour of if;1 as y
-t
oo,
as well as the fact that if;1 is C
1
at
y
=
0, there will beaK 0 such that if;1(y)
~
-K't/Jo(Y) for
ally~
0. We choose
such a K and henceforth define
't/J1(y)
=
if;1(Y) +K't/Jo(y).
It
follows that there is a constant c 0 such that '¢1(y)
~
cy for all y
~
0.
4.2. The Ansatz. Let
2av f"(U;v)
2
~[v] =
R at -
M [v] -
RRtYUy
+
2y
2
v - RRtYVy,
so that (2.2) can be written as
~[v]
=
0. We now let
v
=
v1
+
v2
=
-RRt'¢1(Y)
+
v2(t, y)
with v2 undetermined for the moment, and compute
l
2
2av2 [
f"(U;v)
2
~
[v1 + v2
=
-R (RRt)t 't/J1 + R
£l-
M v2] +
2
v
ut
2y
+
(RRt)
2
y't/J~ (y)- RRtYV2,y
_ R3R ·'· R2av2 M[ ] f"(U;v)
2
-- tt'l-'1
+ --
V2
+
V
at 2y2
+
(RRt)
2
(y'¢~ (y)- 't/J1(y))- RRtYV2,y
i.e.
(4.1)
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