Averaging of higher-order parabolic equations with periodic coefficients

A. A. Miloslova ; T. A. Suslina

Contemporary Mathematics. Fundamental Directions, Partial Differential Equations, Tome 67 (2021) no. 1, pp. 130-191

Résumé

In $L_2(\mathbb{R}^d;\mathbb{C}^n),$ we consider a wide class of matrix elliptic operators ${\mathcal A}_\varepsilon$ of order $2p$ (where $p \geqslant 2$) with periodic rapidly oscillating coefficients (depending on ${\mathbf x}/\varepsilon$). Here $\varepsilon >0$ is a small parameter. We study the behavior of the operator exponent $e^{- {\mathcal A}_\varepsilon \tau}$ for $\tau>0$ and small $\varepsilon.$ We show that the operator $e^{- {\mathcal A}_\varepsilon \tau}$ converges as $\varepsilon \to 0$ in the operator norm in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ to the exponent $e^{- {\mathcal A}^0 \tau}$ of the effective operator ${\mathcal A}^0.$ Also we obtain an approximation of the operator exponent $e^{- {\mathcal A}_\varepsilon \tau}$ in the norm of operators acting from $L_2(\mathbb{R}^d;\mathbb{C}^n)$ to the Sobolev space $H^p(\mathbb{R}^d;\mathbb{C}^n).$ We derive estimates of errors of these approximations depending on two parameters: $\varepsilon$ и $\tau.$ For a fixed $\tau>0$ the errors have the exact order $O(\varepsilon).$ We use the results to study the behavior of a solution of the Cauchy problem for the parabolic equation $\partial_\tau \mathbf{u}_\varepsilon(\mathbf{x},\tau) = -({\mathcal A}_\varepsilon \mathbf{u}_\varepsilon)(\mathbf{x},\tau) + \mathbf{F}(\mathbf{x}, \tau)$ in $\mathbb{R}^d.$