Homogenization of periodic parabolic systems in the $ L_2(\mathbb{R}^d)$-norm with the corrector taken into account

In $ L_2(\mathbb{R}^d;\mathbb{C}^n)$, consider a selfadjoint matrix second order elliptic differential operator $ \mathcal {B}_\varepsilon $, $ 0<\varepsilon \leq 1$. The principal part of the operator is given in a factorized form, the operator contains first and zero order terms. The operator $ \mathcal {B}_\varepsilon $ is positive definite, its coefficients are periodic and depend on $ \mathbf {x}/\varepsilon $. The behavior in the small period limit is studied for the operator exponential $ e^{-\mathcal {B}_\varepsilon t}$, $ t\geq 0$. The approximation in the $ (L_2\rightarrow L_2)$-operator norm with error estimate of order $ O(\varepsilon ^2)$ is obtained. The corrector is taken into account in this approximation. The results are applied to homogenization of the solutions for the Cauchy problem for parabolic systems.