G-convergence of periodic parabolic operators with a small parameter by the time derivative - Sidenko, N. R. (Ukrains’kyi Matematychnyi Zhurnal, 1993)
In this paper, we consider a sequence P^k of divergent parabolic operators of the second order, which are periodic in time with period T = const, and a sequence P^k_ψ of shifts of these operators by an arbitrary periodic vector function ψ X = {L^2((0, T) × Ω)}^n where Ω is a bounded Lipschitz domain in the space R^n. The compactness of the family {P_Ψ^k ¦ Ψ X, k ℕ} in k with respect to strong G-convergence, the convergence of arbitrary solutions of the equations with the operator P^k_ψ, and the local character of the strong G-convergence in Ω are proved under the assumptions that the matrix of coefficients of L^2 is uniformly elliptic and bounded and that their time derivatives are uniformly bounded in the space L^2(Ω; L^2(0,T)).