# Abstract Non Linear Wave Equations by Michael Reed

o (32) e-iA(t2 - tl):pt1_____9 Pt2 if -T < t2 --< t, _< o and J : Pt Then, if Proof ) Pt for ~ o e Po' we have #(t) E Pt for all t ~ (-T,T) t E (-T,T). We just use the same proof as for Corollary cept that we take for ~(T,e~o) ~(t) on satisfy and all (-T,T) which ~(t) ~ Pt for each 1 of T h e o r e m the set of continuous W-valued 1 exfunctions ~(o) = ~0' sup I ]~(t) - e -It 9 A~o l [ ~ %e(-T,T) t ~ (-T,T).

11~o~ - Let ~o ' ~o + f Then satisfies 9 D and define the integral equation initial data, we have, lL _< I I ~"o*" - ~"~'1 I + + ~11 ~o" on D. z,,o) The proof is almost trivial. Oi(t) (H~) of Theo- and t ~ ( - T,T) where K depends on T and D but not on for each t Proof set in ~ , ~o s D, Mt~ ~ exists and I IMt~oll for all closed) la(~,. (s)) - J(~:, (s)) 1 Ids t oC(I I,~,. (s)I I, I1~, 2 - 11~, ,. (s) (s) ll) r -) (s) llas I Ids SO (34) Corollary , i iol(t) - ~2 (t) ll --< I1~'~--0 O~JII e tc(~'K) Assume hypotheses (Hn).