Abstract
In L2(\({\mathbb{R}}\) d; \({\mathbb{C}}\) n), we consider a wide class of matrix elliptic operators \(\mathcal{A}\) ε 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 exponential \({e}^{{-\mathcal{A}}_{\varepsilon }\tau }\) for τ > 0 and small ε. It is shown that the operator \({e}^{{-\mathcal{A}}_{\varepsilon }\tau }\) converges as ε → 0 in the operator norm in L2(\({\mathbb{R}}\) d; \({\mathbb{C}}\) n) to the exponential \({e}^{-{\mathcal{A}}^{0}\tau }\) of the effective operator \(\mathcal{A}\) 0. Also we obtain an approximation of the operator exponential \({e}^{{-\mathcal{A}}_{\varepsilon }\tau }\) in the norm of operators acting from L2(\({\mathbb{R}}\) d; \({\mathbb{C}}\) n) to the Sobolev space Hp(\({\mathbb{R}}\) d; \({\mathbb{C}}\) n). We derive error estimates for these approximations depending on two parameters: ε and τ. For a fixed τ > 0, the errors are of the sharp order O(ε). The results are applied to study the behavior of the solution of the Cauchy problem for the parabolic equation ∂τuε(x, τ) = −(\(\mathcal{A}\) εuε)(x, τ) + F(x, τ) in \({\mathbb{R}}\) d.
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 67, No. 1, Spectral Analysis, 2021.
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Miloslova, A.A., Suslina, T.A. Homogenization of the Higher-Order Parabolic Equations with Periodic Coefficients. J Math Sci 277, 959–1023 (2023). https://doi.org/10.1007/s10958-023-06894-0
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DOI: https://doi.org/10.1007/s10958-023-06894-0