Algebra i analiz St. Petersburg Math. J. Tom 25 (2013),  6 Vol. 25 (2014), No. 6, Pages 981–1019 S 1061-0022(2014)01326-X Article electronically published on September 8, 2014 HOMOGENIZATION OF THE CAUCHY PROBLEM FOR PARABOLIC SYSTEMS WITH PERIODIC COEFFICIENTS YU. M. MESHKOVA Abstract. In L 2 (R d ; C n ), a class of matrix second order differential operators B ε with rapidly oscillating coefficients (depending on x/ε) is considered. For a fixed s> 0 and small ε> 0, approximation is found for the operator exp(-B ε s) in the (L 2 → L 2 )- and (L 2 → H 1 )-norm with an error term of order of ε. The results are applied to homogenization of solutions of the parabolic Cauchy problem. Introduction 0.1. In this paper, we deal with homogenization theory for periodic differential opera- tors (DO’s). A broad literature is devoted to homogenization problems (see, for example, [ZhKO, BaPa, BeLP]). We rely on the operator-theoretic (spectral) approach to homoge- nization problems. This approach was developed in the papers [BSu1, BSu2, BSu3, BSu4] by Birman and Suslina. 0.2. We study homogenization in the small period limit ε → 0 for the following Cauchy problem: (0.1) ρ(ε −1 x)∂ s u ε (x,s)= − p B ε u ε (x,s)+ F(x,s); ρ(ε −1 x)u ε (x, 0) = φ(x). Here φ ∈ L 2 (R d ; C n ) and F ∈ L p ((0,T ); L 2 (R d ; C n )) for some p. The solution u ε (x,s) is a C n -valued function of x ∈ R d and s ≥ 0; p B ε is a matrix elliptic second order DO acting in L 2 (R d ; C n ). A measurable (n × n)-matrix-valued function ρ(x) is assumed to be bounded, uniformly positive definite, and periodic relative to some lattice Γ ⊂ R d . Let Ω be the cell of the lattice Γ. We use the notation ϕ ε (x)= ϕ(ε −1 x), where ϕ(x) is a measurable Γ-periodic function in R d . The principal part p A ε of the operator p B ε is given in a factorized form (0.2) p A ε = b(D) ∗ g ε (x)b(D), where b(D) is a matrix homogeneous first order DO and g(x) is a Γ-periodic, bounded, and positive definite matrix-valued function in R d . (The precise assumptions on b(D) and g(x) are given below, see §4.) Homogenization problems for the operator (0.2) were analyzed in detail in [BSu1, BSu2, BSu3, BSu4]. Now we study more general operators p B ε that include first and zero order terms: (0.3) p B ε u = p A ε u + d  j=1 ( a ε j (x)D j u + D j (a ε j (x)) ∗ u ) + Q ε (x)u + λu. 2010 Mathematics Subject Classification. Primary 35K46. Key words and phrases. Parabolic equation, Cauchy problem, homogenization, corrector. Supported by the Ministry of education and science of Russian Federation, project 07.09.2012 no. 8501, 2012-1.5-12-000-1003-016. c 2014 American Mathematical Society 981 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use