Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates

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Let Oᵈ be a bounded domain of class C¹,¹. In L₂(O;Cⁿ), we consider a selfadjoint matrix second order elliptic differential operator BD,ε, 0<ε1, with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator BD,ε is positive definite; its coefficients are periodic and depend on x/ε. We study the behavior of the operator exponential e-BD,εt, t>0, as ε→ 0. We obtain approximations for the exponential e-BD,εt in the operator norm on L₂(O;Cⁿ) and in the norm of operators acting from L₂(O;Cⁿ) to the Sobolev space H¹(O;Cⁿ). The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.