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

Abstract

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, consider a self-adjoint matrix second order elliptic differential operator $\mathcal{B}_\varepsilon$, $0<\varepsilon \leqslant 1$. The principal part of the operator is given in a factorised form, the operator contains first and zero order terms. The operator $\mathcal{B}_\varepsilon$ is positive definite, its coefficients are periodic and depend on $\mathbf{x}/\varepsilon$. We study the behaviour in the small period limit of the operator exponential $e^{-\mathcal{B}_\varepsilon t}$, $t\geqslant 0$. 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.