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)