By Courant R.

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B. 40 B A I L L O N and J . M . CHADAM b e g i n with t h e p h y s i c a l l y r e l e v a n t conserved q u a n t i t i e s o f c h a r g e and e n e r g y and t h e n s y s t e m a t i c a l l y b o o t - s t r a p o u r way t o the xm-norm. 4031 one h a s 4 3x = c o n s t a n t . I $ ( x , t ) 1 2 @ -b( x , t ) d F r o m t h i s one h a s t h e second l i n e f o l l o w i n g f r o m H s l d e r ' s i n e q u a l i t y and t h e l a s t l i n e from S o b o l e v i n e q u a l i t i e s and t h e f a c t t h a t li$(t)ii 2 i s conserved.

And G l a s s e y , Rat. T. Mech. , 54 (1974). O n the M a x w e l l - D i r a c E q u a t i o n s w i t h Z e r o M a g n e t i c F i e l d and T h e i r S o l u t i o n i n T w o Space D i m e n s i o n s , 53, 495 ( 1 9 7 6 ) . J. M a t h . Anal. and A p p l i c . B. M. Fukuda, I. and Tsutsumi, M. - CHADAM On the Yukawa-Coupled Klein-Gordon-Schroedinger Equations i n Three Space Dimensions, Proc. , 5 1 , C 51 Fukuda, I. and Tsutsumi, M. - Schr8dinger Equations, 11, C 61 Segal, I . E . - 402, (1975).

U(O, *)] J J , j=l, ... , k , u(t,x) of such t h a t 4. G. S. 32 COSTA Energy decav. A solution non-static if with f i n i t e energy i s c a l l e d Uo(t)f E ( k e r A)'. 4, i t i s n o t h a r d t o s e e t h a t t h i s is t h e c a s e i f P o ( ~ ) ? ( s , , u )E 0. e. (JJ)P(S-V . ( o ) t , w ) . J J So There e x i s t s a s e t C (ker A ) I , dense i n such t h a t Uo(t)So = so f o r each f 1x1 5 a l t l - R , for all E So, It1 t E R; Uo(t)f R/a, 2 v a n i s h e s i n some d o u b l e - where a = a ( f ) > 0, R = R ( f ) > 0.