Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates

Abstract

Let $\mathcal{O}\subset {\mathbb{R}}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};{\mathbb{C}}^n)$, we consider a selfadjoint matrix second order elliptic differential operator $B_{D,\epsilon }$, $0<\epsilon ⩽1$, with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator $B_{D,\epsilon }$ is positive definite; its coefficients are periodic and depend on $\mathbf{x}/\epsilon$. We study the behavior of the operator exponential $e^{-B_{D,\epsilon }t}$, $t>0$, as $\epsilon \to 0$. We obtain approximations for the exponential $e^{-B_{D,\epsilon }t}$ in the operator norm on $L_2(\mathcal{O};{\mathbb{C}}^n)$ and in the norm of operators acting from $L_2(\mathcal{O};{\mathbb{C}}^n)$ to the Sobolev space $H^1(\mathcal{O};{\mathbb{C}}^n)$. The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.

38 pages. arXiv admin note: text overlap with arXiv:1702.00550, arXiv:1503.05892