. OtZ (weakly) in L 2 (0, T; V). On the other hand, owing to part [ii), we have Ze' ~ u strongly in the space C([O,T);Hi(O)), such that Z = u - gD holds. 8[iii) guarantees OtU ~ 0. 10), we have ip = 0, gN(O) = 0, OtgN ~ 0, od ~ 0. 9 are satisfied. Consequently, the statement OtU ~ is guaranteed.

4 is such that u E W~(O, T; HI(O)). 13) we shall construct a function V 3 TJe(t) = ze(t) + 8(t) with TJe(t) ~ 0 for all t E [0, TJ (due to ao = r D we have V = HJ(O)). Then the desired function 0 ~ "I = u+8 will be obtained in passing to the limit as c ~ o. We define the function 8 by 8(t) = lot p(t') dt', where p = p(x, t) is the solution of the following auxiliary problem. Find p(t) E HI(O) with r p(t) = -')'A(t) < 0 on D, ( low p(t) =0 on aG, k(x, t) "V p(x, t) "Vv(x) dx = 0 Vv E HI(O\G), v = 0 on rD, aG.

13). With this choice we get + (JL (3~(z,,(t) -

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