Dynamics of Partial Differential Equations

Volume 13 (2016)

Number 1

Dedicated to emeritus associate editor Shiyi Chen

Synchronization in coupled second order in time infinite-dimensional models

Pages: 1 – 29

DOI: http://dx.doi.org/10.4310/DPDE.2016.v13.n1.a1

Author

Igor Chueshov (School of Mathematics and Informatics, Karazin Kharkov National University, Kharkov, Ukraine)

Abstract

We study asymptotic synchronization at the level of global attractors in a class of coupled second order in time models which arises in dissipative wave and elastic structure dynamics. Under some conditions we prove that this synchronization arises in the infinite coupling intensity limit and show that for identical subsystems this phenomenon appears for finite intensities. Our argument involves a method based on “compensated” compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, Karman and Berger plate models with different types of boundary conditions. Our results can be also applied to the nonlinear wave equations in an arbitrary dimension. We consider synchronization in sine-Gordon type models which describes distributed Josephson junctions.

Keywords

synchronization, wave dynamics, global attractor, upper semicontinuity

2010 Mathematics Subject Classification

34D06, 35B41, 37L30

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Published 29 March 2016