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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中文翻译：

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Hilbert空间中具有周期系数的常微分算子的浮球问题和中心流形减少

摘要：考虑了具有周期算子系数的一阶微分方程在一对希尔伯特空间中的作用。此设置对圆柱体中具有周期系数的椭圆方程和时间周期系数的抛物线方程都进行建模。我们的主要结果是构造点投影仪，将系统进行光谱拆分，将其分解为具有常数系数的常微分方程的有限维系统和无穷维部分，这些部分的解在某种意义上具有更好的性能。这补充了分别由P. Kuchment和SA Nazarov获得的圆柱体周期性次椭圆问题和准圆柱体椭圆问题的渐近结果。作为应用，给出了具有周期系数的希尔伯特空间中一类非线性常微分方程的中心流形归约。该结果概括了A. Mielke探索的具有恒定系数的已知情况。

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