Floquet problem and center manifold reduction for ordinary differential operators with periodic coefficients in Hilbert spaces

Kozlov, V.; Taskinen, J.

doi:10.1090/spmj/1660

Floquet problem and center manifold reduction for ordinary differential operators with periodic coefficients in Hilbert spaces
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by V. Kozlov and J. Taskinen

St. Petersburg Math. J. 32, 531-550

DOI: https://doi.org/10.1090/spmj/1660

Published electronically: May 11, 2021

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Abstract:

A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite-dimensional system of ordinary differential equations with constant coefficients and an infinite-dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders and for elliptic problems in quasicylinders obtained by P. Kuchment and S. A. Nazarov, respectively.

As an application, a center manifold reduction is presented for a class of nonlinear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients explored by A. Mielke.

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