Waves and Diffusion on Metric Graphs with General Vertex Conditions

We prove well-posedness for very general linear wave- and diffusion equations on compact or non-compact metric graphs allowing various different 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 ${\mathrm{L}}^p({\mathbb{R}}_{+},{\mathbb{C}}^{\mathrm{\ell }})\times {\mathrm{L}}^p([0,1],{\mathbb{C}}^m)$ for $1\le p<+\mathrm{\infty }$.