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Homogenization of the Cauchy problem for parabolic systems with periodic coefficients


Author: Yu. M. Meshkova
Translated by: the author
Original publication: Algebra i Analiz, tom 25 (2013), nomer 6.
Journal: St. Petersburg Math. J. 25 (2014), 981-1019
MSC (2010): Primary 35K46
DOI: https://doi.org/10.1090/S1061-0022-2014-01326-X
Published electronically: September 8, 2014
MathSciNet review: 3234842
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Abstract: In $ L_2(\mathbb{R}^d;\mathbb{C}^n)$, a class of matrix second order differential operators $ \mathcal {B}_\varepsilon $ with rapidly oscillating coefficients (depending on $ \mathbf {x}/\varepsilon $) is considered. For a fixed $ s>0$ and small $ \varepsilon >0$, approximation is found for the operator $ \exp (-\mathcal {B}_\varepsilon s)$ in the $ (L_2\to L_2)$- and $ (L_2\to H^1)$-norm with an error term of order of $ \varepsilon $. The results are applied to homogenization of solutions of the parabolic Cauchy problem.


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Additional Information

Yu. M. Meshkova
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: juliavmeshke@yandex.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01326-X
Keywords: Parabolic equation, Cauchy problem, homogenization, corrector
Received by editor(s): April 1, 2013
Published electronically: September 8, 2014
Additional Notes: Supported by the Ministry of education and science of Russian Federation, project 07.09.2012 no. 8501, 2012-1.5-12-000-1003-016
Article copyright: © Copyright 2014 American Mathematical Society