By Michael Reed

ISBN-10: 3540076174

ISBN-13: 9783540076179

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

**Example text**

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 ..... IIAJ-X~II)IIAJ~II Let ~ o E D(A m) and suppose that on any finite the solution $(t), (6) is global in t. I I~(t) I I is bounded. Further, Then the solution if J satisfies m times strongly differentiable interval of existence of ~(t) of condition Jm then ~(t) is for all t and satisfies dd--•j-•(t) ( D(A m-j) Proof The idea is the same as in Theorem existence showed I I~(t) I I is apriori bounded.

The Let hyDotheses ~ = __. F i r s t 2 Adding the - ~)ll and 1 differentiation. ,. (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[~ ! __

Bounded to coupling below lose by These and coupled constant, equations Since (F 4) (F 2) w i t h 8 is H e r m e t i a n As u s u a l we r e w r i t e u(t) (F3) d (for one u(t) denotes have Let ~ and 8 be space dimension) function = u(x,t) as on R 2. is an R - v a l u e d the dot p r o d u c t a conserved not much use. energy in C 2 o f but ~(t) it is not So w e d o n ' t have anythimg defining = Re(u(t)) u(t) on the r i g h t will remain as a first d--t ~(t) coupled - igSu(t)~(t) be c o m p l e x - v a l u e d , u(t) and r e w r i t i n g ex- = ~(t).