[Author]
 Meshkova, Yu. M.

[Title]
 Homogenization with the corrector for periodic parabolic systems in the
 $L_2(\mathbb{R}^d)$-norm

[AMS Subj-class]
 35B27 Homogenization; equations in media with periodic structure
 35K45 Initial value problems for second-order parabolic systems

[Keywords]
 periodic differential operators, parabolic systems, homogenization,
 operator error estimates

[Abstract]
 In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a self-adjoint matrix 
 elliptic second order differential operator $\mathcal{B}_\varepsilon$, 
 $0<\varepsilon \leqslant 1$. The principal part of the operator is given 
 in a factorized form, the operator contains first and zero order terms. 
 The operator $\mathcal{B}_\varepsilon$ is positive definite, its coefficients 
 are periodic and depend on $\mathbf{x}/\varepsilon$. We study the behavior of 
 the operator exponential $e^{-\mathcal{B}_\varepsilon t}$, $t\geqslant 0$, in 
 the small period limit. Approximation in the $(L_2\rightarrow L_2)$-operator 
 norm with error estimate of order $O(\varepsilon ^2)$ is obtained. In this 
 approximation, the corrector is taken into account. The results are applied 
 to homogenization of the solutions for the parabolic Cauchy problem. 

[Comments]
 LaTeX, Russian, 69 pp.
 Postal address: 
 Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29B,
 Saint Petersburg 199178 Russia.

[Contact e-mail]
 y.meshkova@spbu.ru