A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology

Abstract

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.