SINGULARITIES AT
t
=
= IN EQUIVARIANT HARMONIC MAP FLOW 13
6.
Proof of Theorem
1.2
6.1.
Every solution becomes singular. We consider a solution VJ(r, t) of
(1.2) whose initial data satisfy (1.4) and (1.5). We will assume in addition that
(6.1) OrVJ(O, 0) 0, and hence VJ(r, 0) 2 or
for all
r E
[0,
1] and some small enough
o
0.
We may do this without loss of
generality, since the strong maximum principle will force any solution VJ(r, t) of
(1.2) which satisfies (1.4) to satisfy (6.1) immediately fort
0.
So if our chosen
initial function VJ(r,
0)
does not satisfy (6.1), then we replace it with VJ(r, t) for any
small
t
0.
Choose t1
0
so large that r+(t) 21/2 fort 2 t1.
LEMMA 6.1. There is an E
E
(0,
~)
such that VJ(r,O) 2 'P+(Er, h) for all r
E
[0, 1].
Moreover, for all t
2 0
one has
(6.2)
PROOF. This follows immediately from (6.1) and the fact that for all
t
0 one
can find a constant C(t) oo such that 'P+(r, t)
~
C(t)r holds for all r E [0, 1].
We observe that {!(r, t) = 'P+(er, l} +
e2t)
is a subsolution of (1.2). Also, it
follows from r +(t)
~
E for all t 2 t1that {1(1, t) = 'P+(E, t1 +
e2t)
1. Hence
the Maximum Principle implies 'P 2
P
for all r E [0, 1] and t 2 0, as claimed. D
We improve the previous lemma by showing that one can take
E
arbitrarily
close toE= 1, possibly at the expense of increasing
l}.
Let() E (0, 1) be given, and
choose t2 0 such that
r
+ (t)
()
for all
t
2 t2.
LEMMA 6.2. For large enough t3 2 l}one has 'P+(er, t3) 2 'P+(()r, t2) for all
r
E
[0,
1).
Furthermore
VJ(r, t4 + t) 2 'P+(()r, t2 + ()
2
t)
for all
t
0, where t4 = (t3-
t1)/e2.
PROOF. Since r +(t2)
()
we have 'P+(()r, t2) 'P+((), t2) n. On the other
hand, for any r 0 one has limt/= 'P+(Er, t) =
7f
with uniform convergence on
any interval o
~
r
~
1. Thus for any o 0 there will be a
t
=
t(
o) 0 such
that 'P+(er, t) 2 'P+(()r, t2) holds for r E [8, 1). On the short interval [0, 8) one has
'P+(()r, t2)
~
Cr for some fixed C oo. Hence, if tis chosen large enough one will
also have 'P+(Er, t) 2 'P+(()r, t2) for r E [0, 8].
It
follows from (6.2), i.e. VJ(r,t) 2 'P+(er,t
1
+e2t)
and 'P+(er,t3) 2 'P+(()r,t
2
)
that, with t4 = ( t3 - t1) /
E2,
one has
VJ(r, t4) 2 'P+(()r, h) for r E [0, 1].
Since {!(r, t) = 'P+(()r, t
2
+()2t) is a subsolution for (1.2) which satisfies cp(1, t) =
'P+((),t
2
+()2
t)
1r
(because() r+(t
2
+() 2
t)) for all t 0, the Maximum Principle
implies cp(r, t)
~
VJ(r, t4 + t) for all r E [0, 1), t 2 0. D
Inspection of the subsolution 'P+ reveals that for large enough t there is a
unique R(t) = (1 + o(1))R(t) such that 'P+(R(t), t) =
7f
/2. Since R(t) and R(t)
differ by o(R(t)), the asymptotic expansion for R(t) also applies to R(t), i.e. R(t) =
exp ( -2y't + o( y't) ).
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