1. THE LOCAL THEORY OF THE CAUCHY PROBLEM 5 that the strong Huygens principle may not hold for (ut,∂tu). A similar remark applies to two solutions whose initial data agree on B(x0,a). We next turn to a perturbation theorem which is an important ingredient in what follows. In its proof we will need an inhomogenous Strichartz estimate, which, for the inhomogenous problem improves on the previous estimate. The current version is due to Taggart [99], but many other authors have contributed to this. (See also [41], [102]). The perturbation theorem originates in [12]. Theorem 1.11. ([99]) Let β = θ 2 , 0 θ 1. Let q be defined by 1 q = 1−θ 8 + θ 4 . Choose θ so close to 1 that q 6, 4 q 1 + 1 8 . Define q by 1 2 = 1 q + 1 q . Then, Dβ t 0 sin ( (t − t ) √ −Δ ) √ −Δ h(t )dt L q I Lx q ≤ C Dβh Lq I Lxq , and t 0 sin ( (t − t ) √ −Δ ) √ −Δ h(t )dt S(I) ≤ C Dβh Lq I Lxq . Theorem 1.12 (Perturbation Theorem). Let I ⊂ R be a time interval, t0 ∈ I, (u0,u1) ∈ ˙ 1 × L2,M,A,A 0 be given. Let u be defined on R3 × I satisfy sup t∈I (u(t),∂tu(t)) ˙ 1 ×L2 ≤ A u S(I) ≤ M D 1 2 u W (I ) ∞, for each I ⊂⊂ I. Assume that (∂t − Δ) (u) = −F (u) + e(x, t) ∈ R3 × I, (u0 − u(t0),u1 − ∂tu(t0) ˙ 1 ×L2 ≤ A , and that D 1 2 e L 4 3 I Lx3 4 + S(t − t0) (u0 − u(t0),u1 − ∂tu(t0)) S(I) ≤ ε. Then, there exists ε0 = ε0(M, A , A) such that there exists a solution of ( ∂t 2 − Δ ) (u) = −F (u) in R3 × I, with (u(t0),∂tu(t0)) = (u0,u1), for 0 ε ε0, with u S(I) ≤ C(M, A, A ) and for all t ∈ I, (u(t),∂tu(t)) − (u(t) − ∂tu(t)) ˙ 1 ×L2 ≤ C(M, A, A ) [A + εα] , 0 α 1. It suﬃces to consider t0 = 0,I = [0,L],L ∞ and to assume that u already exists to conclude a-priori estimates for it. After that an application of Theorem 1.4 concludes the proof. The first remark is that D 1 2 u W (I) ≤ M, M = M(M, A). To see this, split I = ∪α j=1 Ij,γ = γ(M), so that on each Ij we have u S(Ij) ≤ η, where η is to be determined. Let Ij 0 = [aj 0 , bj 0 ], so that the integral equation gives u(t) = S(t) (u(aj 0 ),∂tu(aj 0 )) + t t0 sin ( (t − t ) √ −Δ ) √ −Δ e(t )dt − t aj 0 sin ( (t − t ) √ −Δ ) √ −Δ F (u)dt .

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