By Michael Reed

ISBN-10: 0387076174

ISBN-13: 9780387076171

ISBN-10: 3540076174

ISBN-13: 9783540076179

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**Example text**

Bm'Q) (w)~t { (30) 2 where or m i = m,e i=l - d ~ 1 is a z e r o . We may always or o n e , assume and where m I = maxj w mj. stands Now, for either for m ~ 5, w e m u s t 2 m - m. , ] j # I. Thus, we can just use (26) P il(Bmlw)'''(Bm'w) (Q)~i[~ ! IIBm1*ll P < c llBm§ in c a s e u , , u 2, m > 5. IBml to conclude - For (27), m = 2,3,4 or both. we For just check example, 1 each when of the p o s s i b i l i t i e s m = 3, t h e r e are t h r e e : 2 If (B2w) (Bw)w] i II < 2 -- (B2w) (Bw) II kinds of terms: by 2 I I (Bw) (Bw) (Bw)wr I 2 _< l (21) i lw[l < rlB'wll ]iB~w]r il~'wrl = (26), 2 -- m using < [IB~wil l[Bwlr = m 4 8 - 2 by C27) 2 I I (Bw) 211211Bwl I= I lwi [.

N ~ 3, we N e v e r t h e l e s s the hypotheses of T h e o r e m 6 are satisfied for each m as can be easily checked by m a k i n g similar calculations to those in Example 1 and in part C of Section 2. Thus we can apply T h e o r e m 6 and Sobolev's lemma as above to conclude that if the initial data is smooth enough, then the solution continues to have the same degree of smoothness in the interval where it exists and satisfies the a p p r o p r i a t e d i f f e r e n t i a l equation in a classical sense.

Is differentiable is twice continuously We now repeat the ar- by hypothesis differentiable) is three times strongly differentiable since we now to conclude that and so forth. I Notice that the interval on which the solution exists depends on m. In particular, have slightly T m may go to zero as stronger estimates m > and smoothness. Theorem 9 and smoothness) (global existence of part As in Section and apriori boundedness then we get global existence the hypotheses ~ . ,m (Hj) is replaced by (H~) IIAJJ(~)II ~c(II~II .....

### Abstract Non Linear Wave Equations by Michael Reed

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