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