xii INTRODUCTION η0 0, small. Then ∃(v0,v1) ˙ 1 × L2,λ(t) 0,t (0, 1),i0 {±1} such that (u(t),∂tu(t)) = i0 λ(t) 1 2 W x λ(t) , 0 +(v0,v1)+o(1) in ˙ 1 ×L2, where λ(t) = o(1 t). ii) Non-radial case. Assume that sup 0t1 ∇u(t) 2 + 1 2 ∂tu(t) 2 ∇W 2 + η0, η0 small. Then, after rotation, translation of R3, ∃(v0,v1) ˙ 1 × L2, i0 {±1}, l small, x(t) R3,λ(t) 0,t (0, 1) such that (u(t),∂tv(t)) = i0 λ(t) 1 2 Wl x−x(t) λ(t) , 0 , i0 λ(t) 3 2 ∂tWl x−x(t) λ(t) , 0 +(v0,v1)+o(1) in ˙ 1 ×L2, where λ(t) = o(1 t), limt↑1 x(t) 1−t = l1,e1 = (1, 0, 0), |l| 1 4 0 , and Wl(x, t) = W x1−tl 1−l2 , x2,x3 is the Lorentz transform of W . The final three chapters of the monograph are devoted to the proof of the soliton resolution conjecture for (NLW) in the radial case, by Duyckaerts-Kenig- Merle. This was obtained, for a particular sequence of times, in [30] and in [33] for general times. The result is: Theorem 9.1. Let u be a radial solution of (NLW). Then, one of the following holds: a) Type I blow-up: T+ and lim t↑T+ (u(t),∂tu(t)) ˙ 1 ×L2 = ∞. b) Type II blow-up: T+ and ∃(v0,v1) ˙ 1 × L2,J N\ {0} and ∀j {1,...,J} , ij {±1} and λj(t) 0 such that 0 λ1(t) λ2(t) · · · λJ(t) T+ t, and (u(t),∂tu(t)) = J j=1 ij λj(t) 1 2 W x λj(t) , 0 + (v0,v1) + o(1) in ˙ 1 × L2. c) T+ = and a solution vL of (LW), J N and for all j {1,...,J} , iJ {±1} , λj(t) 0 such that 0 λ1(t) λ2(t) · · · λJ(t) t, and (u(t),∂tu(t)) = ∑J j=1 ij λj(t) 1 2 W x λj(t) , 0 + (vL(t),∂tvL(t)) + o(1) in ˙ 1 × L2. Here a(t) b(t) means that a(t) b(t) 0, and (LW) denotes the linear wave equation. A fundamental new ingredient of the proof of Theorem 9.1 is the following dispersive property that all global in time radial solutions to (NLW) (other than 0, ±W up to scaling) must have: |x|R+|t| |∇x,tu(x, t)|2dx η, for some R 0,η 0 and all t 0 or all t 0. This is in Proposition 9.17. The proof is a consequence of the “channel of energy” property in Chapter 7. As far as exposition goes, most of the results mentioned above are proved in full, for others the proofs are merely sketched or ommitted completely. In the last two cases, appropriate references are given.
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