SINGULARITIES AT

t =

oo

IN EQUIVARIANT HARMONIC MAP FLOW

11

is a subsolution for

(2.1)

fort

2:

to, while

will be a supersolution for

(2.1)

fort

2:

to.

Unfortunately the sub and super solution provided by this theorem are ordered

in the wrong way: the subsolution lies

above

the supersolution and it is impossible

to conclude that there is a solution between them.

5. The sub and supersolutions in the r variable

5.1. The functions

'P±·

We choose sufficiently large

k,

and define

u±(y, t)

as

above in Theorem 4.5. As always,

R(t)

will be a solution of (2.9), or, equivalently,

(4.2). To fix our choice of R we prescribe the initial condition

(5.1)

R(O)

= p,

for some fixed p E (0, 1). We define

While these functions are sub and supersolutions for

t

2:

to,

for some

t

0

oo,

they do not satisfy the boundary condition

'P

=

1r

at

r

=

1.

Indeed we have ob-

tained the differential equation (2.9) by imposing this boundary condition on the

first two terms U(y)

+

v

1

(y, t) which make up

U±.

We will now use the invariance

of the Harmonic Map Flow equation under the parabolic similarity transforma-

tion

p(r, t)

t-t

p(Br,(Pt)

to turn

'P±

into sub and super soutions which satisfy the

boundary conditions.

q;

Subso/ution

R -----\-

I I

The formal solution

r_(t) r

FIGURE 5.1. An unfortunate ordering of a sub and supersolution