G-convergence of periodic parabolic operators with a small parameter by the time derivative

Abstract

In this paper, we consider a sequenceP ^k of divergent parabolic operators of the second order, which are periodic in time with periodT=const, and a sequenceP ^k_Ψ of shifts of these operators by an arbitrary periodic vector function Ψ εX=L²((0,T) × Ω)ⁿ where Ω is a bounded Lipschitz domain in the space ℝⁿ. The compactness of the family {P ^k_Ψ ¦ Ψ εX, k ε ℕ ink with respect to strongG-convergence, the convergence of arbitrary solutions of the equations with the operatorP ^k_Ψ , and the local character of the strongG-convergence in Ω are proved under the assumptions that the matrix of coefficients ofL ² is uniformly elliptic and bounded and that their time derivatives are uniformly bounded in the space L²(Ω;L²(0,T)).