Diffusion in networks with time-dependent transmission conditions

TL;DR

This paper investigates diffusion processes on networks with time-dependent transmission conditions and coefficients, establishing existence, uniqueness, and exponential convergence to equilibrium or special solutions.

Contribution

It introduces a framework for analyzing diffusion on networks with non-autonomous Kirchhoff conditions and time-dependent coefficients, proving key long-term behavior results.

Findings

Solutions exist and are unique under the given conditions.

Solutions converge exponentially fast to equilibrium when mass is conserved.

In some cases, solutions converge to a specific special solution.

Abstract

We study diffusion in a network which is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. We prove at first a result on existence and uniqueness using form methods. Our main results concern the long-term behavior of the solution. In the case when the conductivity and the diffusion coefficients match (so that mass is conserved) we show that the solution converges exponentially fast to an equilibrium. We also show convergence to a special solution in some other cases.