Homogenization of hyperbolic equations: operator estimates with correctors taken into account

An elliptic second-order differential operator $A_\varepsilon=b(\mathbf{D})^*g(\mathbf{x}/\varepsilon)b(\mathbf{D})$ on $L_2(\mathbb{R}^d)$ is considered, where $\varepsilon >0$, $g(\mathbf{x})$ is a positive definite and bounded matrix-valued function periodic with respect to some lattice, and $b(\mathbf{D})$ is a matrix first-order differential operator. Approximations for small $\varepsilon$ of the operator-functions $\cos(\tau A_\varepsilon^{1/2})$ and $A_\varepsilon^{-1/2} \sin (\tau A_\varepsilon^{1/2})$ in various operator norms are obtained. The results can be applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation $\partial^2_\tau \mathbf{u}_\varepsilon(\mathbf{x},\tau) = - A_\varepsilon \mathbf{u}_\varepsilon(\mathbf{x},\tau)$.