Homogenization of the Cauchy problem for parabolic systems with periodic coefficients
Yu. M. Meshkova · St Petersburg Mathematical Journal · 2014
In $L_2(\mathbb {R}^d;\mathbb {C}^n)$, a class of matrix second order differential operators $\mathcal {B}_\varepsilon$ with rapidly oscillating coefficients (depending on $\mathbf {x}/\varepsilon$) is considered. For a fixed $s>0$ and small $\varepsilon >0$, approximation is found for the operator $\exp (-\mathcal {B}_\varepsilon s)$ in the $(L_2\to L_2)$- and $(L_2\to H^1)$-norm with an error term of order of $\varepsilon$. The results are applied to homogenization of solutions of the parabolic Cauchy problem.