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CHAPTER 1 The Local Theory of the Cauchy Problem As mentioned in the introduction, in this monograph we will concentrate on the focusing, energy critical nonlinear wave equation in 3 space dimensions. (NLW) ⎧ ⎪ ⎨ ⎪ ⎩ ∂2u t − Δu = u5,x ∈ R3,t ∈ R u|t=0 = u0 ∈ ˙ 1 (R3) ∂tu|t=0 = u1 ∈ L2 ( R3 ) . In this chapter we will review the “local theory of the Cauchy problem” for (NLW). Note that for this local theory there is no difference between the focusing (u5 in the right hand side) or defocusing (−u5 in the right-hand side) equations. The proofs presented in this chapter are from [59], [62] and [53]. See also the surveys [50] and [51]. In order to discuss the “local theory of the Cauchy problem” for (NLW), I will first review a few facts about the linear wave equation. (LW) ⎧ ⎪ ⎪ (∂2 t − Δ)w = h(x, t) ∈ R3 × R w|t=0 = w0 ∂tw|t=0 = w1. The Fourier method gives us a solution (1.1) w(t) = cos √ −Δt w0 + sin √ −Δt √ −Δ w1 + t 0 sin ( (t − t ) √ −Δ ) √ −Δ h(t )dt . which we denote as w(t) = S(t)(w0,w1) + D(t)(h). Here, ( cos (√ −Δt ) f ) (ξ) = cos (|ξ| t) f(ξ), where denotes the Fourier trans- form in R3, sin (√ −Δt ) f is defined similarly and 1 √ −Δ f (ξ) = f(ξ) |ξ| . One of the main properties of the linear wave equation is the finite speed of propagation: If supp (w0,w1) ∩ B(x0,a) = ∅ and supp h ∩ (∪0≤t≤aB(x0,a − t) × {t}) = ∅, then w ≡ 0 on ∪0≤t≤aB(x0,a − t) × {t}. In odd dimensions dimensions (like on R3), when h ≡ 0 there is a strength- ening of the finite speed of propagation, called the strong Huygens principle: if supp (w0,w1)ßB(x0,a),h≡0, then, for t 0, supp wß {x : t − a ≤|x − x0| ≤a + t}. (Note that, for even dimension, we only have supp wß {x : |x − x0| ≤ a + t}). The simplest way to prove these support facts is by examining the support properties of the forward fundamental solution of the wave equation (see [94]). 1 http://dx.doi.org/10.1090/cbms/122/01