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6 1. THE LOCAL THEORY OF THE CAUCHY PROBLEM By Strichartz estimates, we get: sup t∈Ij 0 (u(t),∂tu(t)) ˙ 1 ×L2 + D 1 2 u W (Ij0) ≤ CA + C D 1 2 e L 4 3 I j 0 Lx 4 3 + C D 1 2 F (u) L 4 3 I j 0 Lx3 4 ≤ CA + Cε + C u 4 S(Ij 0 ) D 1 2 u W (Ij0) If we then choose η so that Cη4 ≤ 1 2 , the claim follows. Choose now β, q, q as in the inhomogenous Strichartz estimate. Note the fol- lowing facts: Dβf L q I Lx q ≤ C f 1−θ S(I) · D 1 2 f θ W (I) . (by interpolation in x, H¨ older in t) and |f|4 Dβf Lq I Lx q ≤ C f 4 S(I) Dβf LqLxq I , which follows from H¨ older. Note that we then have Dβu LqLxq I ≤ M, and that by our hypothesis and the first inequality we have: DβS(t) ((u0 − u(0),u1 − ∂tu(0)) LqLxq I ≤ ε , where ε ≤ Mεα,α = 1 − θ. Now, write u = u + w, so that w verifies ⎧ ⎪ ⎪ (∂t 2 − Δw) = − (F (u + w) − F (u)) − e w|t=0 = u0 − u(0) ∂tw|t=0 = u1 − ∂tu(0) Split now I = ∪J j=1 Ij,J = J(M, η), so that on each Ij we have u S(Ij) + Dβu Lq Ij Lxq ≤ η, where η 0 is to be chosen. Set Ij = [aj,aj+1),a0 = 0,aj+1 = L. The integral equation on Ij gives: w(t) = S(t − aj) (w(aj),∂tw(aj)) − t aj sin ( (t − t ) √ −Δ ) √ −Δ e(t )dt − t aj sin ( (t − t ) √ −Δ ) √ −Δ [F (u + w) − F (u)] dt . Applying the Strichartz estimates, the interpolation inequality, and the improved inhomogenous Strichartz estimates, we see that w S(Ij) + Dβw L q I j Lx q ≤ S(t − aj) (w(aj),∂tw(aj)) S(Ij) + DβS (t − aj) (w(aj),∂tw(aj)) L q I j Lx q + Cε +C Dβ [F (u + w) − F (u)] L q I j Lx q .