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3, we can g u a r a n t e e that if the solution u starts out uO ~Co(R), vo&C (R)) then on some finite t interval the solution w i l l be twice c o n t i n u o u s l y differentiable. Further, another result of section 4, it w i l l have compact support. using These r e g u l a r i t y statements don't affect the ideas below, they just a l l o w us to integrate by parts w i t h impunity. If uo and v o are r e a l - v a l u e d then u will be r e a l - v a l u e d and E(t) = { (Vu) 2 + m2u 2 + ut2 2 u p+l } dx p+l R is the c o n s e r v e d energy.

H(t) exists of g l o b a l where o n t. [- T , T ] solution and therefore proof not Substituting on second we have in t h r e e d i - 3. Smoothness of Solutions As we have remarked before, not completely ple we would smooth satisfactory the existence theorems hypotheses 1 are For exam- like to know that if we choose the data ah time zero to be (say C = ), then the solution of the equation in Section from a classical point of view. in the classical are needed. sense. (i) will stay smooth and satisfy Essentially In our examples A = if~ the powers of A, Where I 1 ~-m 2 act like powers of the Laplacian two kinds of further .

F is nice, - 59 - E n(O) = }I IBfll z. + +I Ig]122 + IGn(f)dx converges as n > ~ to a number E{O) Therefore K }l IBfll 2 the numbers is constant 2 n {En(O) } are uniformly hounded. But since En(t) in t for each n, this means that there is a constant C so that (43) Let E (t) < C n for all t and n. S(r) be the ball in R n of radius r and choose r o so that the sup- ports of f and g lie in Then by (43) and S(ro). Let ~ T,T] be a given finite interval. (40) I fUn(t) I I2 _< / 2C SO Un(t) are a uniformly with values in equicontinuous L~(S(r ~ + T)).

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