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

Delio Mugnolo

#### 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 enjoys these properties if and only if its counterpart with time-independent boundary conditions does also. Furthermore, we prove continuous dependence of the solution to the parabolic problem on the boundary conditions in the considered class.

#### Article information

Source
Adv. Differential Equations, Volume 15, Number 11/12 (2010), 1125-1160.

Dates
First available in Project Euclid: 18 December 2012