On homogenization of the first initial-boundary value problem for periodic hyperbolic systems
Yu. M. Meshkova · Applicable Analysis · 2018
Let O⊂Rd be a bounded domain of class C3,1. In L2(O;Cn), we consider a self-adjoint matrix strongly elliptic second-order differential operator BD,ε, 0<ε⩽1, with the Dirichlet boundary condition. The coefficients of the operator BD,ε are periodic and depend on x/ε. We are interested in the behavior of the operators cos(tBD,ε1/2) and BD,ε−1/2sin(tBD,ε1/2), t∈R, in the small period limit. For these operators, approximations in the norm of operators acting from a certain subspace H of the Sobolev space H4(O;Cn) to L2(O;Cn) are found. Moreover, for BD,ε−1/2sin(tBD,ε1/2), the approximation with the corrector in the norm of operators acting from H⊂H4(O;Cn) to H1(O;Cn) is obtained. The results are applied to homogenization for the solution of the first initial-boundary value problem for the hyperbolic equation ∂t2uε=−BD,εuε.