Variational and Semigroup Methods for Waves and Diffusion in Networks

TL;DR

This paper develops a mathematical framework using variational and semigroup methods to analyze wave and diffusion equations on networks, establishing well-posedness, spectral connections, and asymptotic behaviors.

Contribution

It introduces a novel combination of variational and semigroup techniques to study PDEs on networks, extending existing spectral analysis and asymptotic results.

Findings

Established well-posedness of wave and diffusion equations in network contexts.

Connected the spectrum of the generator to network structure.

Described the long-term behavior of solutions.

Abstract

We study diffusion and wave equations in networks. Combining semigroup and variational methods we obtain well-posedness and many nice properties of the solutions in general L^p -context. Following earlier articles of other authors, we discuss how the spectrum of the generator can be connected to the structure of the network. We conclude by describing asymptotic behavior of solutions to the diffusion problem.