Approximations of the operator exponential in a periodic diffusion problem with drift

A Cauchy problem for a parabolic diffusion equation with 1-periodic coefficients containing first order terms is studied. For the corresponding semigroup we construct approximations in the $L^2$-operator norm on sections $t=\mathrm{const}$ of order $O(t^{-m/2})$ as $t\to\infty$ for $ m=1$ or $m=2$. The spectral method based on the Bloch representation of an operator with periodic coefficients is used.
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