Waves and Diffusion on Metric Graphs with General Vertex Conditions

TL;DR

This paper establishes well-posedness for a broad class of wave and diffusion equations on metric graphs with diverse vertex conditions, using operator semigroup theory to handle non-self-adjoint and non-local boundary conditions.

Contribution

It extends the theory of differential operators on metric graphs by proving well-posedness for equations with general, possibly non-local, vertex conditions using semigroup methods.

Findings

Proves generation of cosine families for a wide class of operators.

Shows well-posedness of wave and diffusion equations on metric graphs.

Handles non-self-adjoint and non-local boundary conditions.

Abstract

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 ${\mathrm{L}}^p({\mathbb{R}_+},{\mathbb{C}}^{\ell})\times{\mathrm{L}}^p([0,1],{\mathbb{C}}^m)$ for $1\le p<+\infty$.