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Example text

Then for each the h y p o t h e s e s of T h e o r e m 9 for And suppose~ that J satisfies condition Jm for each ~oE /~D(AJ), the equation (6) has a unique global solution so that ~(t) is infinitely often s t r o n g l y d i f f e r e n t i a b l e and each d e r i v a t i v e ~ ~ is in D(AJ). First we will apply these ideas to utt - ~u + m 2 = - u 3 u(x,o) = f(x) ut(x,o) = g(x) (25) We use exactly the same set up as in Dart A of Section 2. We need just verify the higher order estimates and the h y p o t h e s e s on J in part b of T h e o r e m 6.

U(t) )> h Thus, be the l (I]~) is s i m i l a r . J m is a l s o quotient 2 i can be written = m 1 B as t h o u g h The s I]Am-l~ll2flAm~l] < c -- treating + m 2 I I ~ <_ C[ ]Bm~+lul l~l IBm2+lul I~I IBms+lul I 2 -- so t h e h y p o t h e s e s + m Of t h e Then, (Bm~u)(Bm2u)(Bm3u) again l to a s u m o f t e r m s the e s t i m a t e : I - u , and is c o n t i n u o u s .

Constants involved Secondly, all of this discussion 15 continuously (which may depend on t). in the following In all the examples on initial data. ~, can conclude that m IIAJ(\$(t) if ( - T,T) (35) depends on T. or by using the trick of T h e o r e m (in the cases where - (somewhat vague) in Section we have an We summarize theorem. 2, the solutions depend 5. W e a k Solutions In t h i s section we will show utt has g l o b a l veral the weak proofs same solutions problem. One global One then (36) using the in a w e a k sense.