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 $$\partial^2_\tau \mathbf{u}_\varepsilon(\mathbf{x},\tau) = - A_\varepsilon \mathbf{u}_\varepsilon(\mathbf{x},\tau)$$ .