270 9. TYPE I SINGULARITIES
We claim there is a constant r < oo depending only on g (0) such that
A
-----<r.A+μ+ 11 + p -
Indeed, since the left-hand side is scale invariant, we may assume without
loss of generality thatmin 11 ( x, 0) 2 -1.
xEM3
Let A = e^6. If -3A :S A :S 3A, then
A 3A
-----<-
A+μ+ 11 + p - c.
If not, then by Theorem 9.4 we have either 11 2 - A and A > 3A or else
11 < -A and 3 1 111 :S R < A; in either case,A A
- ---< <3.
A + μ + II+ p A/3 + p -
Now for E > 0 to be chosen later, we may by the claim above choose Tc
large enough so that when Tc :St< T, we have
A E
(1 - E) p < (1 - E) pI' < ---
A + μ + II+ p - - 2 (T - t)
and hence
d E 1 - E μ^2 + 112
dt log :SEA - 2 (T - t) - - 2- A + μ + 11 + P ·
- ---< <3.
By hypothesis, for every c > 0 there are 8 ( c) > 0 and Tc < T such that
M (t) tJ. X (t) for Tc :St < T, whereX (t) ~ {IP': (T - t) A (IP') 2 c > 0 and μ (IP')^2 + 11 (IP')^2 :S 8. A (IP')^2 }is the avoidance set discussed in Section 3 of Chapter 4. Choose C ( E, c) < oo
large enough so that M (T) EiC (T), where T =max {Tc, Tc} andf JC (t) ~ {IP': (T - tr (A (IP') - II (IP')) < c}.
(A (IP')+μ (IP')+ II (IP')+ p)l-c -
Notice that JC (t) is convex. We next claim that when c = 1/2, there is E > 0
small enough that 1t log <I> :S 0 whenever T :S t < T and M (t) E JC (t) \X (t).
This claim implies that M (t) remains in JC (t) \X (t), hence that <I> :S C. By
Theorem 4.10, this implies that Fis bounded above, becauseIR~l 2 = ~ [(A -μ)2 +(A - 11)2 + (μ - 11)2] :S (A - 11)2.
To prove the claim, notice that M (t) tJ. X (t) only if A (T - t) < c or
μ^2 + 112 > 8A^2. In the first case, we haved E 1 - E μ^2 + 112 ( 1 )
dt log <I> :SEA - 2 (T - t) - - 2- A + μ + 11 + p :SE A - 2 (T - t) < O.