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Applicable Analysis
An International Journal
Volume 99, 2020 - Issue 9
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Articles

On homogenization of the first initial-boundary value problem for periodic hyperbolic systems

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Pages 1528-1563 | Received 07 Jul 2018, Accepted 21 Oct 2018, Published online: 09 Nov 2018
 

ABSTRACT

Let ORd be a bounded domain of class C3,1. In L2(O;Cn), we consider a self-adjoint matrix strongly elliptic second-order differential operator BD,ε, 0<ε1, with the Dirichlet boundary condition. The coefficients of the operator BD,ε are periodic and depend on x/ε. We are interested in the behavior of the operators cos(tBD,ε1/2) and BD,ε1/2sin(tBD,ε1/2), tR, in the small period limit. For these operators, approximations in the norm of operators acting from a certain subspace H of the Sobolev space H4(O;Cn) to L2(O;Cn) are found. Moreover, for BD,ε1/2sin(tBD,ε1/2), the approximation with the corrector in the norm of operators acting from HH4(O;Cn) to H1(O;Cn) is obtained. The results are applied to homogenization for the solution of the first initial-boundary value problem for the hyperbolic equation t2uε=BD,εuε.

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

Acknowledgments

The author is deeply grateful to T. A. Suslina for her attention to this work.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Research is supported by the Russian Science Foundation [grant number 14-21-00035].

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