希尔伯特空间中的时滞方程非线性半群、惯性流形与维数估计
Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension estimates
AI总结：
本文在希尔伯特空间中研究时滞方程，构造非线性半群与求解算子，方法更初等，并探讨维数估计与惯性流形。
摘要：
我们研究在$\mathbb{R}^{n}$中非自治非线性时滞方程在适当希尔伯特空间中的适定性，将其视为演化方程。我们为一大类此类方程给出了求解算子（非自治情形）或非线性半群（自治情形）的构造。主要思想可轻松扩展到某些带时滞的偏微分方程。与先前基于增算子理论构造此类半群和求解算子的方法相比，我们的方法限制更少且更为初等。在自治情形下，我们还研究这些半群的可微性性质，以便利用希尔伯特空间几何应用各种维数估计。然而，获得时滞方程的有效维数估计是一个非平凡问题，我们通过一个标量时滞方程加以说明。我们还讨论了与惯性流形及其对时滞方程构造相关的邻近结果。
英文摘要：
We study the well-posedness of nonautonomous nonlinear delay equations in $\mathbb{R}^{n}$ as evolutionary equations in a proper Hilbert space. We present a construction of solving operators (nonautonomous case) or nonlinear semigroups (autonomous case) for a large class of such equations. The main idea can be easily extended for certain PDEs with delay. Our approach has lesser limitations and much more elementary than some previously known constructions of such semigroups and solving operators based on the theory of accretive operators. In the autonomous case we also study differentiability properties of these semigroups in order to apply various dimension estimates using the Hilbert space geometry. However, obtaining effective dimension estimates for delay equations is a nontrivial problem and we explain it by means of a scalar delay equation. We also discuss our adjacent results concerned with inertial manifolds and their construction for delay equations.
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