Revision of diffusion on metric graph

Dynamical processes along the edges of a graph with appropriate transmission conditions in the vertices appeared first around 1980. In the last forty years variety of new tools from analysis has been developed to deal with problems of that type firstly in Hilbert than in Banach space.
In this talk we present the summary of existing results on diffusion equation on the metric graph with Robin-type boundary conditions, namely: the existence, long time behaviour and convergence of singularly perturbed version of a problem. The considerations are conducted using a semigroup apparatus in L^1 space and the main results indicate how the graph structure influence the dynamics. As a motivation of theory serves the synaptic transmission model introduced by Aristizabal and Glavinovič in 2004. Presented results are based on the article:

M. Kramar-Fijavž and A. Puchalska, Semigroups for dynamical processes on metric graphs, Phil. Trans. R. Soc. A. 37820190619, arXiv:2006.03123