Contemporary Developments in Continuum Mechanics and Partial by G. M. de La Penha, L. A. J. Medeiros

By G. M. de La Penha, L. A. J. Medeiros

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M. J. Functional A n a l . T. Equations 3,1 7 3 in One (1973). M. the C a u c h y P r o b l e m f o r Arch. 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.

M. de La Penha. A. G. S. Departamento de Matemgtica Universidade de Brasilia Brasilia, DF - Brasil 1. Introduction We consider the Cauchy problem -a +U at n C A . I a x . Is are constant J u and f x = (xl, are column vectors o f ... ,xn) E the time. n, R Hermitian symmetric matrices, k components, is tlie space variable and t E IR is We use the Radon transform to study the behavior of the solution times. 1 and Corollary 3 . 1 through b . 1) (regarded as unperturbed) with a perturbed equation au + at E(x) - 11 aU C A.

_ e n_ e r_ g y_of f _ Ei[f], bhere Then, l i m Ilu(t, E ker A m, B, of as t + m . any s o l u t i o n Uo(t)f that is: converges slowly t o Then, B, then 5 . The p e r t u r b e d e q u a t i o n and t h e wave o p e r a t o r s . W e now c o n s i d e r t h e p e r t u r b e d Cauchy problem A V I L A and D . G . S. 3) ( i )i t i s c l e a r that the /I - 1 1 , d e f i n e d by E(x)f(x)-f(x)dx. €lo = L2(Rn) denote the H i l b e r t space H 1) * I ) , norm b;. = IE(x)-II = m. norm i s e q u i v a l e n t t o t h e norm II f l l x; C’I k we d e f i n e t h e o p e r a t o r A~ = E ( X ) -1 A , It follows that -iAE = D(A).

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