INTRODUCTION xi As was mentioned earlier there are examples of type II solutions, constructed in [69] and [71] and of generalized type II solutions, such as W , or more generally the solitary waves, i.e., Lorentz transforms Q l of non-zero solutions to ΔQ+Q5 = 0 (this class of Q is denoted by Σ), as well as constructions in [29] with complicated asymptotics. In [36], Duyckaerts-Kenig-Merle developed a very general compact- ness argument, which connects extended type II solutions and solutions with the compactness property (see Theorem 6.12). In the case of NLW, a more detailed result is possible, Theorem 6.13, obtained in [36]. A simple version of this is: Theorem 6.14. Let u be an extended type (II) solution of (NLW). Then there exists Q Σ, l R3, l 1 and sequences {tn} [0,T+(u)), {xn} R3, {λn} R+,tn T+(u) and such that λnu 1 2 (xn + λnx, tn) , λn∂tu 3 2 (xn + λnx, tn) n ( Q l (0),∂tQl(0) ) , where the convergence is weak in ˙ 1 × L2. Q l (x, t) = Q x + 1 l2 1 1 l2 1 l · x t 1 t2 l , l = l . Theorem 6.14 is proved in Chapter 6. We regard the result in Theorem 6.13 of Chapter 6 as a first step towards the proof of a full decomposition, for u an extended type II solution, u(x, tn) = J j=1 1 λj,n 1 2 Qj lj x xj,n λj,n , 0 + v (x, tn) + on(1) ∂tu(x, tn) = J j=1 1 λ 3 2 j,n ∂tQj lj x xj,n λj,n , 0 + ∂tv (x, tn) + on(1), where v is a radiation term (a solution of the linear equation), on(1) goes to 0 in ˙ 1 × L2, lj 1, lj R3,Qj Σ, {xj,n} R3, {λj,n} 0, and tn T+(u), which is a special case of the soliton resolution conjecture for (NLW), which says that any extended type II solution u of (NLW) can be written, as t T+(u) as a sum of decoupled solitary waves and a radiation term, plus a term that goes to 0 in ˙ 1 ×L2. The proof of this conjecture is our final goal in this direction. In Chapter 7, we discuss “channels of energy” and “outer energy lower bounds” for the linear wave equation, see Corollary 7.6 and Proposition 7.9, which were discovered in [31], [32], and which are our main tools in passing from weak convergence results, of the type in Theorem 6.13 and Theorem 6.14, to strong convergence results. The first such result is discussed in Chapter 8. It shows the universality of the construction in [69], [71], as well as establishing a case of the soliton resolution conjecture discussed earlier, when T+(u) ∞, under an additional smallness hypothesis. This result is from [31] and [32]. Theorem 8.1. Assume that u is a type II solution, T+ = 1. i) If u is radial and sup 0t1 ∇u(t)≤ ∇W + η0,
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