G-convergence of parabolic operators

The asymptotic behavior (as h→+∞) for a sequence of initial-boundary value problems of the form uh′-div(ah(x,t,Duh)) = f in Ω×]0,T[, uh(0) = u0, uhεLp(0,T;W01,p(Ω)), where Ω is an open bounded set in RN, T is a positive real number and 2≤p<∞, is examined. The maps ah are assumed to be monotone and to satisfy certain boundedness and coerciveness assumptions uniformly with respect to h. The existence of a subsequence still denoted by {ah} and a map a with the same qualitative properties as the maps {ah} such that, h→∞, uh→u weakly in Lp(0,T;W01/p(Ω)) and ah(x,t,Duh)→a(x,t,Du) weakly in Lp′(0,T;Lp′(Ω;RN)), is presented.