Homogenization of hyperbolic equations with periodic coefficients in ${\mathbb R}^d$: sharpness of the results

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, a selfadjoint strongly elliptic second order differential operator ${\mathcal A}_\varepsilon$ is considered. It is assumed that the coefficients of the operator ${\mathcal A}_\varepsilon$ are periodic and depend on ${\mathbf x}/\varepsilon$, where $\varepsilon >0$ is a small parameter. We find approximations for the operators $\cos ( {\mathcal A}_\varepsilon^{1/2}\tau)$ and ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}\tau)$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d)$ to $L_2({\mathbb R}^d)$ (with suitable $s$). We also find approximation with corrector for the operator ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}\tau)$ in the $(H^s \to H^1)$-norm. The question about the sharpness of the results with respect to the type of the operator norm and with respect to the dependence of estimates on $\tau$ is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation $\partial_\tau^2 {\mathbf u}_\varepsilon = - {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$.