Existence and explicit formula for a semigroup related to some network problems with unbounded edges

TL;DR

This paper establishes the existence of a C0-semigroup for a network problem with unbounded edges and derives an explicit formula for it using the method of characteristics and Laplace transform techniques.

Contribution

It provides the first explicit formula for the semigroup associated with network problems with unbounded edges, extending the theoretical understanding of such systems.

Findings

Existence of a C0-semigroup for the problem

Explicit formula for the semigroup derived

Method of characteristics applied successfully

Abstract

In this paper we consider an initial-boundary value problem related to some network dynamics where the underlying graph has unbounded edges. We show that there exists a C0-semigroup for this problem using a general result from the literature. We also find an explicit formula for this semigroup. This is achieved using the method of characteristics and then showing that the Laplace transform of the solution is equal to the resolvent operator of the generator.