# Michael Reed's Abstract Non Linear Wave Equations PDF

By Michael Reed

ISBN-10: 0387076174

ISBN-13: 9780387076171

ISBN-10: 3540076174

ISBN-13: 9783540076179

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**Additional info for Abstract Non Linear Wave Equations**

**Example text**

This poses no d i f f i c u l t y in one or two d i m i n s i o n s since in those cases I lul Ip ~ el IBu[ I2 for all p < -. for p ~ 6 and in four d i m e n s i o n s But in three d i m e n s i o n ~ this only holds for p ~ 4. However we can also use the estimate I IBn~ I i C[ JBn~ I (26) 2 which is valid for part C). n m< 4 (see the Sobolev inequalities in Section 2, In the following we will use just I [Bn~ (26) and I~ i el [Bn~ Thus the proof we give s i m u l t a n e o u s l y handles the cases (27) n = 1,2,3,4.

In discussed shows that example large the blow up of the L 2 - n o r m in Section locally, 2, part not because E, is caused in the n o n - g l o b a l by the function it fails to d e c a y sufficiently get- fast at infinity. Now, data. we treat Whenever the q u e s t i o n of c o n t i n u o u s we have the h y p o t h e s i s dependence (H~) on the initial of T h e o r e m i, the~, v according to the corollary, ~Cr) we can at least some interval non-linear = {~o~J11~oll <_ r solve the ( - T(r), on each ball integral T(r)).

To begin with, we suppose D(B) ~ L2(Rn), and A= i (o -B 2 I) o just as in part A. 11 In this last computation we have again for ease of exposition treated B as though it acts by differentiation. Note that~the next to last step is the only place where we use the fact that the dimension because we needed the Sobolev is < 4, inequality < KKIBu I{,llBu~ll~ Since the extra hypothesis show that tence. [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.

### Abstract Non Linear Wave Equations by Michael Reed

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