Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates
Yu. M. Meshkova, Tatiana Aleksandrovna Suslina · St Petersburg Mathematical Journal · 2018
Let $\mathcal {O}\subset \mathbb {R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal {O};\mathbb {C}^n)$, a selfadjoint matrix second order elliptic differential operator $B_{D,\varepsilon }$, $00$, is studied as $\varepsilon \rightarrow 0$. Approximations for the exponential $e^{-B_{D,\varepsilon }t}$ are obtained in the operator norm on $L_2(\mathcal {O};\mathbb {C}^n)$ and in the norm of operators acting from $L_2(\mathcal {O};\mathbb {C}^n)$ to the Sobolev space $H^1(\mathcal {O};\mathbb {C}^n)$. The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.