A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology
A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology
复制标题
DOI：
10.1016/j.jde.2021.09.019
复制
发表时间：
2021-10-04
影响因子：
2.4
通讯作者：
Rost, Gergely
中科院分区：
文献类型：
--
作者：
Balazs, Istvan;Getto, Philipp;Rost, Gergely
我们分析了一个微分方程系统的状态依赖延迟（SD-DDE）从细胞生物学，其中的延迟被隐式定义为的时间时，一个常微分方程的解决方案，参数化的SD-DDE状态，满足一个阈值。我们表明，该系统是适定的，解决方案定义一个连续的半流的Lipschitz函数的状态空间。此外，我们建立了一个相关的系统的凸紧集，是不变的时间t-映射有限时间。众所周知，由于时滞的状态依赖性，适定性的充分必要条件可以与泛函几乎是局部Lipschitz有关，这大致意味着局部Lipschitz域的限制Lipschitz函数，我们的方法涉及这样的条件。为了实现透明度和更广泛的适用性，我们阐述了一般类的两个组件的功能微分方程系统，其中包含SD-DDE从细胞生物学和制定我们的结果也为这一类。(c)2021年，任作家。爱思唯尔公司出版这是一个在CC BY-NC-ND许可证下的开放获取文章（http：//creativecommons.org/licenses/by-nc-nd/4.0/）。
We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class. (c) 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).