论文标题
抛物线方程的频率定理及其与惯性歧管的关系
Frequency theorem for parabolic equations and its relation to inertial manifolds theory
论文作者
论文摘要
我们获得了频率定理的版本（某些操作员不等式的解决性定理），该版本允许构建半连接抛物线方程的二次lyapunov功能。我们表明，C。Foias，R。Temam和G. R. Sell是在惯性歧管理论中使用的众所周知的光谱差距，是某种频率不平等的特殊情况，是在频率定理中产生的。特别是，这允许在我们相邻作品中提出的更一般的几何理论的背景下，为半几何理论构建半线性抛物线方程（包括一些非自治问题）的惯性歧管。该理论基于二次Lyapunov功能，并概括了R. A. Smith使用的频域方法。我们还讨论了频率不平等的最佳性及其与该领域的已知旧结果和最新结果的关系。
We obtain a version of the Frequency Theorem (a theorem on solvability of certain operator inequalities), which allows to construct quadratic Lyapunov functionals for semilinear parabolic equations. We show that the well-known Spectral Gap Condition, which was used in the theory of inertial manifolds by C. Foias, R. Temam and G. R. Sell, is a particular case of some frequency inequality, which arises within the Frequency Theorem. In particular, this allows to construct inertial manifolds for semilinear parabolic equations (including also some non-autonomous problems) in the context of a more general geometric theory developed in our adjacent works. This theory is based on quadratic Lyapunov functionals and generalizes the frequency-domain approach used by R. A. Smith. We also discuss the optimality of frequency inequalities and its relationship with known old and recent results in the field.
