Note on quantitative homogenization results for parabolic systems in Rd

In L-2(R-d; C-n), we consider a semigroup e(-tA epsilon), t >= 0, generated by a matrix elliptic second- order differential operator A(epsilon) >= 0. Coefficients of A(epsilon) are periodic. depend on X/epsilon, and oscillate rapidly as epsilon -> 0. Approximations for e(-tA epsilon )were obtained by Suslina (Funktsional Analiz i ego Prilozhen 38(4):86-90, 2004) and Suslina (Math Model Nat Phenom 5(4):390-447, 2010) via the spectral method and by Zhikov and Pastukhova (Russ J Math Phys 13(2):224-237, 2006) via the shift method. In the present note, we give another short proof based on the contour integral representation for the semigroup and approximations for the resolvent with two-parametric error estimates obtained by Suslina (2015).