Homogenization of elliptic and parabolic equations with periodic coefficients in a bounded domain under the Neumann condition

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 second-order matrix elliptic differential operator $B_{N,\varepsilon}$, $0<\varepsilon\leqslant1$, under the Neumann boundary condition. The principal part of this operator is given in a factorized form. The operator includes first-order and zero-order terms. The coefficients of the operator $B_{N,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the generalized resolvent $(B_{N,\varepsilon}-\zeta Q_0(\cdot/\varepsilon))^{-1}$, where $Q_0$ is a periodic bounded and positive definite matrix-valued function, and $\zeta$ is a complex parameter. We obtain approximations of the generalized resolvent in the operator norm in $L_2(\mathcal{O};\mathbb{C}^n)$ and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev class $H^1(\mathcal{O};\mathbb{C}^n)$, with two-parametric (with respect to $\varepsilon$ and $\zeta$) error estimates. The results are applied to study the behavior of solutions of the initial boundary value problem with the Neumann condition for the parabolic equation $Q_0(\mathbf{x} / \varepsilon) \partial_t \mathbf{u}_\varepsilon(\mathbf{x},t) = -( B_{N,\varepsilon} \mathbf{u}_\varepsilon)(\mathbf{x},t)$ in the cylinder $\mathcal{O} \times (0,T)$, where $0 < T\leqslant\infty$