50 2. SPECIAL AND LIMIT SOLUTIONSPROOF. By Lemma 2.28, we may let Co be an upper bound for l(-iP^2 )tl·
Choose t 1 E (0, T) so that C 0 (T - t 1 ) < D^2 /8. Then one has
7
"lj;(x*(t), t)^2 2 D^2 - Co(T - t) > SD^2for all t E [t 1 , T). Now let x 1 be the unique solution of "lj;(x1, t1)^2 = ~ D^2 in
the interval [x*(t 1 ), l]. Lemma 2.28 implies for all t E [t1, T) that"lj;(x1, t)^2 :S 3 2 7 2 ( ( ) )2
4 D + Co(t - t1) < SD < "lj! x* t , t.Thus one has x*(t) < x 1 < 1 for all t E [t1, T), and hence -iPs < 0 and -iPss < 0
on the interval [x1, 1] for all t E [t1, T). It follows that the metric distance
di(t) ~ s(l, t) - s(x1, t)· from ( x1, t) to the pole P + is decreasing in time. Indeed,
d ( ) 1
1
dt di t = xi Tds n-iPss < 0.Next let x2 E (x1, 1) be defined by "lj;(x2, t1)^2 = iD^2. Then fort E [t1, T)
one hasand
"lj;(x1, t)^2 2 3 2 5 2
4D - Co(T- t1) >SD.
Thus "lj;(x 1 , t)^2 - "lj;(x2, t)^2 2 D^2 /8. Hence by crudely estimating the quantity
"lj;(x1, t) + "lj;(x2, t) from below by"lj;(x2, t) 2 V~D^2 - Co(T - t 1 ) 2 /15274 = D/2,
we obtain
D^2 /8"lj;(x1, t) - "lj;(x2, t) 2 D/ 2 = D/4.
Lemma 2.34 implies that for x E [x2, 1), one has- "'' 'l's> "lj;(x1, t) - "lj!(x, t) > D /4 -, u i:
s(x1, t) - s(x, t) s(x1, t1) - s(l, t1) · ·
At this point we once again consider the quantity a defined in (2.53). Wefound that L(a) = 0, where Lis the differential operator
L = 8t - a; -(n - 4); as+ 4(n - 1) ~~.
One can compute that the quantity u ~ "lj!'fl satisfies
"lj! n - l
L(u) = (4- 'r/) ;us+ ---;j}2(4"lj;;- 'r!)un - l
= (4 - 'r/)"lj;'fl-^2 "lj;; + ---;j;2(4"lj;; - 'r/)U.