Waves and diffusion on metric graphs with general vertex conditions

We prove well-posedness for general linear wave- and diffusion equations on compact or non-compact metric graphs allowing various conditions in the vertices. More precisely, using the theory of strongly continuous operator semigroups we show that a large class of (not necessarily self-adjoint) second order differential operators with general (possibly non-local) boundary conditions generate cosine families, hence also analytic semigroups, on $ {\mathrm{L}}^p({\mathbb{R}_+}, {\mathbb{C}}^{\ell})\times{\mathrm{L}}^p([0, 1], {\mathbb{C}}^m) $ for $ 1\le p<+\infty $.