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 $\partial_\tau^2 { \mathbf u}_\varepsilon = - { \mathcal A}_\varepsilon { \mathbf u}_\varepsilon + { \mathbf F}$.
Categories: math.AP