u(t) weakly in V uniformly in t E [0, TJ, hence (U(t2) - u(td , <'&)v = i t2 t, (u(s), <'&)v ds for all <,& E V and for t l ,t2 E [O ,Tj. e. t E [O ,Tj in V. 6. The sequence {uZ'} converges to Ut strongly in £2( [0, T j; H).

T. Banks, H. Tran, and S. Wynne Integrating from 0 to t, we obtain Ilu~112 + Ilu~112 + CD fat Ilu;x(s)112 ds:S Ilu7'1I 2+ IIuOxl12 +K fat (1Iu;(s)112 + Ilu~(s)1I2) ds + 5C D1 faT IIF(s)II~. ds +5c~CDI faT IJ(sW ds + lOc2 cicD1T + 5c2CDI(CIMo + C2)2T +5(Cl/fC D1 (C1 (Mo + 1) + C 2 j2T 3 . Applying Gronwall's inequality we can conclude that the sequences {llu~)112} and {llu~1I2} are bounded. T)) independent of m such that IluZ'(t)112+ Ilu;;' (t)1I2 + CD fat Ilu~(s)112ds :S K (18) o for each t E [0, Tj.

We then apply the Banach-Alaoglu Theorem to obtain the desired results in the lemma. 4. The set {u m moreover, } is an equicontinuous and bounded subset of C([O, T ]; V) ) uTn(t) -> u(t) uniformly in t E [0, T], i. , urn -> u weakly in G w ([0, T]; V). weakly in V Proof. 3. To prove the equicontinuity, we have um(t + ~t) - um(t) = for t, t + ~t E [0, T]. 1 , we obtain ° < j < ~tl / 2(T K t+L'>t t Ilu~'(s)llvds + Gi/ K)1 /2. Thus, for any E > and t E [0, T], :3 b(E,t) = (E/K(T + GDl )? such that It' - tl < b implies IluTn(t') - um(t)11 < E.