2 1. THE LOCAL THEORY OF THE CAUCHY PROBLEM An important estimate for the (LW) is the Strichartz estimate, which is a dispersive equation analog of the Sobolev estimates used in elliptic equations: (S1) sup t (w(t),∂tw(t)) ˙ 1 ×L2 + w L8 x,t + w L5L10 t x + D 1 2 w L4 x,t ≤ C (w0,w1) ˙ 1 ×L2 + D 1 2 h L 4 3 x,t . (See Ginibre-Velo [45] and Keel-Tao [49]). Another important ingredient for the “local theory of the Cauchy problem” for (NLW) is the chain rule for fractional derivatives (See [63]). (Here and above (Dαf) (ξ) = |ξ|α f(ξ)). Chain Rule: F ∈ C2,F (0) = 0,F (0) = 0 |F (a + b)| ≤ C [|F (a)| + |F (b)|] |F (a + b)| ≤ C [|F (a)| + |F (b)|]. Then, for 0 α 1, DαF (u) Lx p ≤ C F (u) Lx1 p Dαu Lx2 p , 1 p = 1 p1 + 1 p2 , Dα [F (u) − F (v)] Lx p ≤ C F (u) Lx1 p + F (v) Lx1 p × Dα(u − v) Lp2 x + C F (u) Lr1 x + F (v) Lr1 x × Dαu Lx2 r + Dαv Lx2 r u − v Lx3r , 1 p = 1 r1 + 1 r2 + 1 r3 , 1 p = 1 p1 + 1 p2 . In the local theory of the Cauchy problem for (NLW), it is convenient to intro- duce, for a time interval I and a function v(x, t), the norms v S(I) = v L8L8(R3) I and v W (I) = v L4L4(R3) I . With this notation we have the following consequence of the Chain Rule, which will be very useful for us: Let F (u) = u5. Then, (1.2) D 1 2 F (u) L 4 3 I Lx 4 3 ≤ C u 4 S(I) D 1 2 u W (I) , D 1 2 (F (u) − F (v)) L 4 3 I Lx 4 3 ≤ C F (u) L2L2 I x + F (v) L2L2 I x × D 1 2 (u − v) W (I) + C F (u) L 8 3 I Lx 8 3 + F (v) L 8 3 I Lx3 8 × D 1 2 u W (I) + D 1 2 v W (I) · u − v S(I) . With F (u) = u5 as before, we now have: Definition 1.3. Let 0 ∈ I. We say that u is a solution of (NLW) in I if (u, ∂tu) ∈ C I ˙ 1 × L2 , D 1 2 u ∈ W (I),u ∈ S(I), (u, ∂tu)|t=0 = (u0,u1) and the integral equation u(t) = S(t)(u0,u1) + t 0 sin ( (t − s) √ −Δ ) √ −Δ F (u(s)) ds. holds. The main result in the “local theory of the Cauchy problem” is: Theorem 1.4 (Pecher [87], Ginibre-Soffer-Velo [44], Lindblad-Sogge [74], etc.). Assume that (u0,u1) ∈ ˙ 1 × L2, 0 ∈ ◦ I, (u0,u1) ˙ 1 ×L2 ≤ A. Then, there exists δ = δ(A) 0,C = C(A) 0 such that if S(t) (u0,u1) S(I) δ, there exists a

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