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

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 $B_{D,\varepsilon}$, $0<\varepsilon\leqslant1$, 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 $B_{D,\varepsilon}$ is positive definite; its coefficients are periodic and depend on $\mathbf{x}/\varepsilon$. We study the behavior of the operator exponential $e^{-B_{D,\varepsilon}t}$, $t>0$, 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.