Vector-valued Heat Equations with Coupled, Dynamic Boundary Conditions

Abstract. Motivated by diffusion processes on metric graphs and open books, we consider an abstract setting for interface problems with quite general coupled boundary conditions, which we also allow to depend on time. Beside well-posedness, we discuss positivity, L∞-contractivity and further invariance properties. We show that the parabolic problem with time-dependent boundary conditions enjoy these properties if and only if so does its counterpart with time-independent boundary conditions. We also show the solution’s continuous dependence of on relevant parameter. Finally, we also briefly consider an alternative setting involving a different kind of time-dependent boundary condition.