强耦合网络方程的适定性与对称性
Well-Posedness and Symmetries of Strongly Coupled Network Equations
摘要：
我们考虑有限网络边缘上的扩散过程，并考虑不同的、可能非邻边之间的反馈效应。这概括了文献中常见的设置，其中唯一考虑的相互作用发生在边界处，即。例如，在网络的节点中。我们讨论了相应的初值问题的适定性以及解的压缩性和正性。最后，我们讨论了定性性质，可以制定在状态空间的线性子空间的不变性，即。例如，相关物理系统的对称性。神经生物学模型以及线性薛定谔方程系统的量子图的应用进行了讨论。
英文摘要：
We consider a diffusion process on the edges of a finite network and allow for feedback effects between different, possibly non-adjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i. e., in the nodes of the network. We discuss well-posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions. Finally, we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space, i. e., of symmetries of the associated physical system. Applications to a neurobiological model as well as to a system of linear Schroedinger equations on a quantum graph are discussed.
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