Gradient Systems on Networks

TL;DR

This paper studies a class of linear differential operators with nonlinear boundary conditions on networks, establishing well-posedness and invariance for associated nonlinear diffusion problems using Lyapunov functions.

Contribution

It introduces a framework for analyzing nonlinear boundary conditions on network-based differential operators, extending classical quantum graph results.

Findings

Proved well-posedness of the nonlinear diffusion problem.

Established invariance properties using Lyapunov functions.

Extended analysis to nonlinear boundary conditions on networks.

Abstract

We consider a class of linear differential operators acting on vector-valued function spaces with general coupled boundary conditions. Unlike in the more usual case of so-called quantum graphs, the boundary conditions can be nonlinear. After introducing a suitable Lyapunov function we prove well-posedness and invariance results for the corresponding nonlinear diffusion problem.