On homogenization of the first initial-boundary value problem for periodic hyperbolic systems

Let $\mathcal{O}\subset {\mathbb{R}}^d$ a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};{\mathbb{C}}^n)$, we consider a self-adjoint matrix strongly elliptic second order differential operator $B_{D,\epsilon }$, $0<\varepsilon \leqslant 1$, with the Dirichlet boundary condition. The coefficients of the operator $B_{D,\epsilon }$ are periodic and depend on $\mathbf{x}\mathrm{/}\epsilon$. We are interested in the behavior of the operators $\cos (t{B}_{D,\epsilon }^{1\mathrm{/}2})$ and $B_{D,\varepsilon} ^{-1/2}\sin (t B_{D,\varepsilon} ^{1/2})$,$t\in$\mathbb{R}$$, in the small period limit. For these operators, approximations in the norm of operators acting from some subspace $\mathcal{H}$ of the Sobolev space $H^4(\mathcal{O};{\mathbb{C}}^n)$ to $L_2(\mathcal{O};{\mathbb{C}}^n)$ are found. Moreover, for $B_{D,\varepsilon} ^{-1/2}\sin (t B_{D,\varepsilon} ^{1/2})$, the approximation with the corrector in the norm of operators acting from $$\mathcal{H}$\subset H^4($\mathcal{O}$;$\mathbb{C}$^n)$to$H^1($\mathcal{O}$;$\mathbb{C}$^n)$ is obtained. The results are applied to homogenization for the solution of the first initial-boundary value problem for the hyperbolic equation $\partial ^2_t $\mathbf{u}$\varepsilon =-B{D,\varepsilon} $\mathbf{u}$_\varepsilon $.