  We consider a diffusion problem on a network on whose nodes we impose
Dirichlet and generalized, non-local Kirchhoff-type conditions. We prove
well-posedness of the associated initial value problem, and we exploit the
theory of sub-Markovian and ultracontractive semigroups in order to obtain
upper Gaussian estimates for the integral kernel. We conclude that the same
diffusion problem is governed by an analytic semigroup acting on all $L^p$-type
spaces as well as on suitable spaces of continuous functions. Stability and
spectral issues are also discussed. As an application we discuss a system of
semilinear equations on a network related to potential transmission problems
arising in neurobiology.
