论文标题
非线性半群，用于希尔伯特空间，惯性歧管和维度估计的延迟方程
Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension estimates
论文作者
论文摘要
我们将$ \ mathbb {r}^{n} $中的非自主非线性延迟方程式的适当性研究为适当的希尔伯特空间中的进化方程。我们为大量此类方程式提供了解决求解操作员（非自治案例）或非线性半群（自主案例）的结构。对于某些延迟的PDE，可以轻松地扩展主要想法。我们的方法的局限性较小，并且比基于增生操作员理论的某些先前已知的构造和解决操作员的某些以前已知的结构要少得多。在自主情况下，我们还研究了这些半群的不同性能，以便使用希尔伯特空间几何形状应用各种维度估计。但是，获得延迟方程的有效尺寸估计是一个非平凡的问题，我们通过标量延迟方程来解释它。我们还讨论了有关惯性流形及其延迟方程的构造有关的相邻结果。
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.
