A potential well theory for the wave equation with nonlinear by Vitillaro E.

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A n d 2 li~ SKetch f$, the These which Schwarz integrations are statements derived by p a r t s w h i c h are and we the use = 0 the by making inequality Ne (3~) consequence repeated use of three of integration ma>'imum p r i n c i p l e . (over. e. t in [S,T] 7_ C3~) (2) For where ll'lI{ is henceforth product al 1 g ~ C~'t({RnX[S,T]) the norm denote on ~ . or, the Hi l h e r t h y . ~ ; L . Ne w i l l space often Lz ( ~ ( ) ' , t ) d ) ' ) write (''')t {or. which we the inner 38 As an example, j u s t s i m p l e s t of t o g i v e you the f l a v o r , the t h r e e i n e q u a l i t i e s .

And the Schroedinger e q u a t i o n . t h e c a s e where V = 0. He w i l l the literature, result integrable as does t h e quantum m e c h a n i c a l corresponding solution Our p~ of the general however, theorem final momentum - where of proof which solution of the until course it relies 50 This brings properties methods drift way. of ever of Nelson the has derivatives the final solving process? D~ the one and hope to d o this that analog the sample path probabilistic to o b t a i n in the stochastic showed a stochastic the equation backward he study by d i r e c t Schroedinger [ ].

W J{ an i n t e g r a t e d form of the pair. ( v , ~ ) the is. c o m p a t i b l e . Francesco Guerra has emphasized t o me the n a t u r a l n e s s of w r i t i n g (2~) 36 in the i n t e g r a t e d form above. Now f i x a finite equation i n i t i a l I will time i n t e r v a l v a l u e problem f o r now s k e t c h the p r o o f o f s o l u t i o n s to t h i s Part of of but the pair- ( v , ~ ) e q u a t i o n . The s o l u t i o n s w i l l they w i l l derivative posess s u f f i c i e n t t h e p o i n t here i s then is to s p e c i f y j u s t be weal<, s o l u t i o n s , and h a l f regularity a time for our p u r p o s e .

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