14 1. THE LOCAL THEORY OF THE CAUCHY PROBLEM that (ul(0),∂tul(0)) C0 ( R3 ) × C0 ( R3 ) and ul satisfies ∂t 2 ul Δul = u5 in the classical sense so that, because of Lemma 1.21, ul is a solution of (NLW). In the general we will use Claim 1.19 to prove that ul is a solution of (NLW). Let ( u0,u1 k k )case, C0 (R3) × C0 (R3), be such that lim k→∞ ( u0,u1 k k ) (u0,u1) ˙ 1 ×L2 = 0. Let uk be the solution of (NLW) with initial data ( u0,u1 k k ) . Then, because u is global and scatters in both time directions, the Perturbation Theorem shows that uk is global for large k and (1.27) lim k→∞ uk u S(R)∩L5L10 t x = 0 Since uk u = cos −Δt (uk 0 u0) + sin ( (t s) −Δ ) −Δ ( uk 1 u1 ) + t 0 sin ( (t s) −Δ ) −Δ (uk)t(s) u5(s) ds, we deduce from Lemma 1.23 that sup t∈R ( uk l ul,∂tul k ∂tul ) ˙ 1 ×L2 k 0, Since uk l is a solution of (NLW) and uk l S(R) = uk S(R) , which is uniformly bounded by (1.27), we get by Claim 1.19 that ul is a solution of (NLW), concluding the proof. We next prove Lemma 1.25. Proof. Note that ul is well defined as an element of L8 loc (R4). We denote by (y, s) the space-time variables for u and (x, t) the space time variables for ul. These variables are related by (x, t) = Φl(y, s). We note that (1.28) |x|2 t2 = |y|2 s2 and (1.29) |s| + |y| Cl (|t| + |x|) , |t| + |x| Cl (|s| + |y|) , where Cl = 1+|l| 1−|l| . Step 1: Estimate at infinity. We prove that there exists B 0 and a scattering solution v of (NLW) such that (1.30) |x| |x| + B =⇒ ul(x, t) = v(x, t) Let A be a large constant. Denote by χA(y) = χ( y A ), where χ C∞(R3), χ(y) = 1 if |y| 1,χ(y) = 0 if |y| 1 2 . Let (u0, u1) = (χAu0,χAu1). Let δ be the small constant given by Remark 1.6. Choose A large, so that (u0, u1) ˙ 1 ×L2 δ. Let u be the solution of (NLW) with initial data (u0, u1) at s = 0. By Remark 1.6, u is a scattering solution of (NLW). By Lemma 1.26, the Lorentz transform ul of u is a scattering solution of
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