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 $B_{D,\varepsilon}$, $0 0$, as $\varepsilon\rightarrow 0$. We obtain approximations for the exponential $e^{-B_{D,\varepsilon}t}$ 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.