Abstract
In L2(Rd;Cn), we consider a wide class of matrix elliptic operators Aε of order 2p (where p≥2) with periodic rapidly oscillating coefficients (depending on x/ε). Here ε > 0 is a small parameter. We study the behavior of the operator exponent e-Aετ for τ > 0 and small ε. We show that the operator e-Aετ converges as ε → 0 in the operator norm in L2(Rd;Cn) to the exponent e-A0τ of the effective operator A0. Also we obtain an approximation of the operator exponent e-Aετ in the norm of operators acting from L2(Rd;Cn) to the Sobolev space Hp(Rd; Cn). We derive estimates of errors of these approximations depending on two parameters: ε and τ. For a fixed τ > 0 the errors have the exact order O(ε). We use the results to study the behavior of a solution of the Cauchy problem for the parabolic equation ∂τuε(x,τ)= -(Aε uε)(x,τ)+F(x,τ) in Rd.