A periodic parabolic Cauchy problem: Homogenization with corrector

A wide class of matrix elliptic second-order differential operators $\mathcal{A}=\mathcal{A}(\mathbf{x},\mathbf{D})$ with periodic coefficients, acting in $L_2(\mathbb{R}^d;\mathbb{C}^n)$, is studied. The operator $\mathcal{A}$ is assumed to admit a factorization of the form $\mathcal{A}=\mathcal{X}^*\mathcal{X}$, where $\mathcal{X}$ is a homogeneous first-order differential operator. An approximation for the operator exponential $e^{-\mathcal{A}\tau}$ as $\tau\rightarrow\infty$ in the $(L_2(\mathbb{R}^d;\mathbb{C}^n))$-operator norm is obtained, with error estimate of the order of $\tau^{-1}$. In the approximation, a corrector is taken into account. The result is applied to the study of homogenization for solutions of the Cauchy problem $\partial_\tau\mathbf{u}_\varepsilon= -\mathcal{A}_\varepsilon\mathbf{u}_\varepsilon$, where $\mathcal{A}_\varepsilon=\mathcal{A}(\mathbf{x}/\varepsilon,\mathbf{D})$. An approximation with corrector for $\mathbf{u}_\varepsilon$ in the $(L_2(\mathbb{R}^d;\mathbb{C}^n))$-norm is obtained for fixed $\tau>0$, with error estimate of the order of $\varepsilon^2$.