Foundations of modern potential theory by N S Landkof

By N S Landkof

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17) . ) sign w(a, t + a)da, tER .

Denote by the set of all sp with the above mentioned properties. e . e, a E (O,at). 4. 1) has a unique weak solution. Proof. 1). 16) io io for any sp E . 5. Let 9 E Loo(QT). e. e. in QT. e. e, in (0, at) . 5. 10) . e, t E (0,T), n E N* . 18) . 15) is verified. e. e, in (0, T) . e. in QT. 4 - continued. 23) cp(at> t) = 0 t E (O,T). 23) belongs to

Then for t large enough we have = (0'* - M (h o (zo + 'Yo))) (zo + 'Yo). = 0 and t-+oo lim y (t) = 'Yo . then 'Yo = 0, otherwise, for t large enough we have > 0 and so Zo - 'Yo > O. We conclude that 3 lim y' (t) > 0, t-+oo which is absurd. 2) that t-+oo lim t-++oo where P (a) = 'UZo exp { -O'*a and _ p(a) ='Uzoexp { * -0' a- l lip (t) - pllu"'(o a -l J-L (s) ds } , a E (0, at). 1). In the same manner follows the conclusion for 0'* [] Va E 0, at , = O. 4. 1) (a,t) E Q, where Q = [0, at) x R. l are T-periodic with respect to the time t .

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