Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions

TL;DR

This paper investigates the asymptotic behavior of solutions to singularly perturbed hyperbolic equations on metric graphs, focusing on how solutions degenerate and vary across different network edges as perturbation parameters tend to zero.

Contribution

It develops a boundary layer method to construct and justify complete asymptotic expansions for solutions on complex metric graphs with varying degeneration rates.

Findings

Asymptotic expansions are successfully constructed for solutions.

Degeneration of hyperbolic equations is characterized on different graph edges.

Method provides a detailed understanding of solution behavior in singular perturbation limits.

Abstract

We are interested in evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic expansions of solutions are constructed and justified.