1. THE LOCAL THEORY OF THE CAUCHY PROBLEM 9 Then, given ε 0, small, there exists M(ε) 0, such that, for all M ≥ M(ε), all (v0,v1) ∈ K we have that vM is globally defined (I = (−∞, +∞)) and sup τ∈(−∞,+∞) (vM(τ),∂tvM(τ)) ˙ 1 ×L2 ≤ ε. To prove this lemma, assume that we know the bound (ΨMv0, ΨMv1) ˙ 1 ×L2 ≤ C (v0,v1) ˙ 1 ×L2 . We then cover K by finitely many small balls, use the Perturbation Theorem, the “local theory of the Cauchy problem” Theorem and the fact that lim M→∞ {(ΨMv0, Ψmv1)} ˙ 1 ×L2 = 0 for fixed (v0,v1) ∈ ˙ 1 × L2. This last fact follows from the inequality above and the fact that it is true for v0,v1 ∈ C∞(R3), 0 which is dense in ˙ 1 × L2. Finally, we establish the bound: by scaling it suﬃces to prove it for M = 1. Clearly, Ψv1 L2 ≤ C v1 L2 . Finally, ∇ (Ψv0) = Ψ∇v0 + v0∇Ψ. The first term is clearly bounded in L2 by the L2 norm of ∇v0. For the second term, we use Hardy’s inequality |v0|2 |x|2 ≤ C |∇v0|2, together with the fact that ∇Ψ has compact support. We next turn to some variants on the notion of solution to (NLW). The one used above has its origin in [62], and is based on the space S(I) with norm v S(I) = v L8L8 I x . This notion of solution is particularly useful when dealing with Lorentz transformations, as we will see. The most common notion of solution is usually tied to the space L5Lx I 10 . Again, because of dealing with issues arising in the consideration of Lorentz transformations, a detailed comparison of both notions of solutions was carried out in [35]. We now proceed to this discussion. Because of the Strichartz estimate (S1), if Imax(u) = (T−(u0,u1),T+(u0,u1)) is the maximal interval of existence of u, then if I ⊂⊂ Imax(u), u L5 I L10 x ∞, as was mentioned earlier. Since by H¨ older inequality and Sobolev estimates we have u 8 L8L8 I x ≤ C sup t∈I (u(t),∂tu(t)) 3 ˙ 1 ×L2 · u 5 L5L10 I x , using also another Strichartz estimate (See Ginibre-Velo [45]): (S2) sup t (w(t),∂tw(t)) ˙ 1 ×L2 + w L8L8 t x + w L5L10 t x + D 1 2 w L4L4 t x ≤ C (w0,w1) ˙ 1 ×L2 + h L1L2 t x , for a solution of (LW), we see that the following variants of the finite blow-up and scattering criteria hold: (1.17) T+ ∞ =⇒ u L5 [0,T+] L10 x = ∞. and if u ∈ L5 [0,T+] L10 x +∞, then T+ = +∞ and u scatters to a linear solution as t → +∞. Remark 1.18. Let u be a solution of (NLW). Assume that (u0,u1) ∈ C∞(R3)× 0 C∞(R3). 0 Then, by Remark 1.15, u ∈ C∞ ( R3 × Imax ) and is a classical solution to ∂t 2 u − Δu = u5. Using this fact, a density argument, and the Perturbation The- orem, as in Remark 1.15, we can show that if (u0,u1) ∈ ˙ 1 × L2, the solution u solves ∂t 2 u − Δu = u5 in D ( R3 × Imax ) .

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