4 1. THE LOCAL THEORY OF THE CAUCHY PROBLEM Choose now tn T+(u0,u1). We can then show, using the integral equation that for n large, S(t tn) (u(tn),∂tu(tn)) S(tn,T+(u0,u1)) ε 2 , since u(t) = S(t tn)u(tn) + t tn sin ((t−t ) −Δ ) −Δ F (u)dt . But then, for some ε0 0, small S(t tn) (u(tn),∂tu(tn)) S(tn,T+((u0,u1))+ε0) ε, which by Theorem 1.4 contradicts the definition of T+(u0,u1). Remark 1.9 (Scattering). If T+(u0,u1) = +∞ and M = u S(0,+∞) ∞, then u scatters at +∞, i.e., ∃(u+,u+) 0 1 ˙ 1 × L2 so that (u(−,t) ∂tu(−,t)) ( S(t)(u+,u+),∂tS(t)(u+,u+) 0 1 0 1 ) ˙ 1 ×L2 t→+∞ 0. In fact, to see this, using the integral equation as before, we have that sup t∈[0,+∞) (u(t),∂tu(t)) ˙ 1 ×L2 + D 1 2 u W (0,+∞) + u S(0,+∞) C(M). But then, since u(t) = cos t −Δu0 + t 0 sin(t t ) −Δ −Δ F (u)(t )dt + sin t −Δ −Δ u1 = cos(t −Δ)u0 + sin(t −Δ) −Δ t 0 cos(−t −Δ)F (u)(t )dt + cos(t −Δ) t 0 sin(−t −Δ) −Δ F (u)(t )dt + sin(t −Δ) −Δ u1, ∂tu(t) = −Δ sin(t −Δ)u0 + cos t −Δu1 + cos(t −Δ) t 0 cos −t −Δ F (u)(t )dt sin(t −Δ) t 0 sin −t −Δ F (u)(t )dt u+ 0 = u0 + +∞ 0 sin(−t −Δ) −Δ F (u)(t )dt , u1 + = u1 + +∞ 0 cos −t −Δ F (u)(t )dt does the job in light of Strichartz estimates and the resulting estimates for F (u). Remark 1.10. Since the Theorem above is obtained by the contraction map- ping theorem, the solution can be constructed by Picard iteration. Let u(0)(t) = S(t)(u0,u1), and u(n+1)(t) = S(t)(u0,u1) + t 0 sin ((t−t ) −Δ ) −Δ F ( u(n)(t ) ) dt . Then, as n ∞, u(n)(t) converges to the solution u in the time interval I. Thus, if supp(u0,u1) B(x0,a) = ∅, then supp (u(t),∂tu(t)) ( R3 × I ) \ ∪0≤t≤a B(x0,a t) × {t}, by an inductive application of the linear finite speed of propagation. Note
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