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Example text

L~(t) l] is bounded Unfortunately, (for all g,m). on finite intervals to show global exis- (21) does not have a positive But the fact that the n o n - l l n e a r i t y us to show apriori boundedness equation we get (Hi) of T h e o r e m 2 holds, we need only of II~(t) 11 anyway. conserved energy is mild allows From the integral - II~(t) ll < I Ie- iAt @~ 28 fl + II ! - [te-iA (t-s) J(r I t 0 t Io l IJ(r _< I I%f + _< I(%l + (~(s) IIds Ir ~ IIr so by iteration Thus lie(t) II is b o u n d e d on bounded solution exists globally.

All paid of t h e s e as the d e t a i l s each as corollary to T h e o r e m in global C~ solution he p r o v e s that in S e c t i o n (this hold. are the has same 9 just then by difficult by the of A i n t o the result take assuming only way (H~) ~ the and we result one can is by h a v - if one has can get the estimates themselves out t h a t sine- m = 1,2,3,4. is an i n t e r e s t i n g since one if the its d e r i v a t i v e s is i m p l i e d This pointed spaces Co(R) , t h e n a more if J and of D o w e r s von W a h l energy and other conclusions the d o m a i n s W.

Solution ~(t) for finite) m ~ I, t~[O,Tm) there with For each set of the form {~ I I IAJ~I I ~ aj, j = 0 ..... m}, T can be chosen uniformly for ~o in the set. (b) In addition has the following to the hypotheses property: uously differentiable in (a) assume that If a solution with for each j < m,J ~ is j times ~(k) (t)~ D(A m-k) strongly contin- and Am-k~(k) (t) is con- tinuous for all k < j, then J(~(t)) is j times differentiable, dJJ(~(t))/dt j ~ D(A mrj-l) , and Am-j-ldJj(~(t))/dtJ is continuous.

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