Vector-valued heat equations and networks with coupled dynamic boundary conditions

TL;DR

This paper studies vector-valued heat equations with coupled dynamic boundary conditions, establishing well-posedness, invariance properties, and continuous dependence on boundary conditions in a general framework inspired by diffusion on complex spaces.

Contribution

It introduces a broad abstract setting for interface problems with dynamic boundary conditions, analyzing key properties and their equivalence to static boundary cases.

Findings

Well-posedness of the parabolic problem with dynamic boundary conditions

Positivity and invariance properties are equivalent to those with static boundary conditions

Continuous dependence of solutions on boundary conditions

Abstract

Motivated by diffusion processes on metric graphs and ramified spaces, we consider an abstract setting for interface problems with coupled dynamic boundary conditions belonging to a quite general class. Beside well-posedness, we discuss positivity, $L^\infty$-contractivity and further invariance properties. We show that the parabolic problem with dynamic boundary conditions enjoy these properties if and only if so does its counterpart with time-independent boundary conditions. Furthermore, we prove continuous dependence of the solution to the parabolic problem on the boundary conditions in the considered class.