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

Abstract

In this paper, we consider a sequence $\mathcal{P}^k$ of divergent parabolic operators of the second order, which are periodic in time with period $T = \text{const}$, and a sequence $\mathcal{P}^k_{\psi}$ of shifts of these operators by an arbitrary periodic vector function $ \psi \in X = \{L^2((0, T) \times \Omega)\}^n$ where $\Omega$ is a bounded Lipschitz domain in the space $\mathbb{R}^n$. The compactness of the family $\{P_{Ψ^k} ¦ Ψ \in X, k \in ℕ\}$ in $k$ with respect to strong $G$-convergence, the convergence of arbitrary solutions of the equations with the operator $\mathcal{P}^k_{\psi}$, 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 $L^2(Ω; L^2(0,T))$.