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
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