Linear evolution equations are considered usually for the time variable being defined on an interval where typically initial conditions or time periodicity of solutions is required to single out certain solutions. Here, we would like to make a point of allowing time to be defined on a metric graph or network where on the branching points coupling conditions are imposed such that time can have ramifications and even loops. This not only generalizes the classical setting and allows for more freedom in the modeling of coupled and interacting systems of evolution equations, but it also provides a unified framework for initial value and time-periodic problems. For these time-graph Cauchy problems questions of well-posedness and regularity of solutions for parabolic problems are studied along with the question of which time-graph Cauchy problems cannot be reduced to an iteratively solvable sequence of Cauchy problems on intervals. Based on two different approaches—an application of the Kalton–Weis theorem on the sum of closed operators and an explicit computation of a Green’s function—we present the main well-posedness and regularity results. We further study some qualitative properties of solutions. While we mainly focus on parabolic problems, we also explain how other Cauchy problems can be studied along the same lines. This is exemplified by discussing coupled systems with constraints that are non-local in time akin to periodicity.

通常将线性演化方程式考虑为在一定间隔上定义的时间变量，在该间隔上通常需要初始条件或解的时间周期来挑选某些解。在这里，我们要提出一个允许在度量图或网络上定义时间的观点，在度量图或网络上，在分支点上施加耦合条件，以使时间可能产生分支甚至是循环。这不仅概括了经典的设置，并为耦合耦合的相互作用方程组的建模提供了更大的自由度，而且还为初始值和时间周期问题提供了统一的框架。对于这些时间图柯西问题，研究了抛物线问题的适定性和解的正则性问题，以及哪些时间图柯西问题不能简化为区间上的柯西问题的迭代可解序列的问题。基于两种不同的方法-闭合算子之和上的Kalton-Weis定理的应用以及Green函数的显式计算，我们给出了主要的适定性和规律性结果。我们进一步研究溶液的一些定性性质。尽管我们主要关注抛物线问题，但我们也解释了如何按照相同思路研究其他柯西问题。这可以通过讨论耦合系统来说明，该耦合系统在时间上类似于周期性在时间上不是局部的。