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To see what = that is i n v o l v e d is t h e solution ~(t) E D ( A 2 ) , ~ ' (t) ~ D(A) We must show that ~(t) 9 D ( A 2 ) , J(~(t)) u ( t ) E D ( B 3) - J(~(t))) = <0,- sum of three terms and of w h i c h inequality we have I lu(tl~ Cuct+~hl "H- uCtl ) - u t r <_ el IBuCt) I1~ I [ uCt+hl h - u(t) of p a r t and ut(t) s D ( B 2 ) . u(t+h) 3 _ u(t) h one let us is s t r o n g l y for J, Sobolev are - see Section > is <0, u(t)2(u,(,t+h). - u(t) )> h Thus, be the l (I]~) is s i m i l a r .

One global One then (36) using the in a w e a k sense. applies most tions. Therefore, = f(x) ut(x,o) = g(x) the the regularized inequality) one on D(A) = D ( B 2) ~ sections r We w i l l be clear Define take from Fn(X) are in the go o v e r the a way that approach -uP shows that u(t) of S t r a u s s satisfies which to r e a l - v a l u e d func- onl[: 0 (B L 2 (R n) D(B) of r e a l - v a l u e d and of c o u r s e introduction. initial the a r g u m e n t se- A + ~~ spaces to this to be the then argument section are of the r e g u l a r i z e d side attention for this There all use b a s i c a l l y {Un(t) } b y a c o m p a c t n e s s By a l i m i t i n g a r g u m e n t our L 2 ( R n) past hand (36) in such Un(t) the and given side follow D(B) = e -tA all n.

B of Section Note the h y p o t h e s e s ~o G D = ~ (r), the of C o r o l l a r y integral the solution of T the family t is u n i f o r m l y is g l o b a l 2, we can c o n c l u d e equa- {Mr} w h i c h and since is now equicontinuous. for all the cases w h e r e we got global that 2 interval w h i c h has the same bound. ( - T,T), is i n d e p e n d e n t same p r o o f works ( - T,T) for every on the w h o l e on all of ~ of strong interval (35) in the case that M t m = o, considered is u n i f o r m l y continuous existence in part for each - t, but only equicontinuous estimate corresponding to 55 on finite Finally, we make two remarks.

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