Heat equation on the hypergraph containing vertices with given data

TL;DR

This paper studies the heat diffusion process on hypergraphs with fixed vertex data, providing an alternative proof of solution existence and analyzing solution dependence and long-term behavior.

Contribution

It offers a new proof approach for the solvability of heat equations on hypergraphs with fixed data, and examines solution stability and global dynamics.

Findings

Alternative proof of solvability for the hypergraph heat equation

Analysis of continuous dependence of solutions on data

Investigation of long-term behavior of solutions

Abstract

This paper is concerned with the Cauchy problem of a multivalued ordinary differential equation governed by the hypergraph Laplacian, which describes the diffusion of ``heat'' or ``particles'' on the vertices of hypergraph. We consider the case where the heat on several vertices are manipulated internally by the observer, namely, are fixed by some given functions. This situation can be reduced to a nonlinear evolution equation associated with a time-dependent subdifferential operator, whose solvability has been investigated in numerous previous researches. In this paper, however, we give an alternative proof of the solvability in order to avoid some complicated calculations arising from the chain rule for the time-dependent subdifferential. As for results which cannot be assured by the known abstract theory, we also discuss the continuous dependence of solution on the given data and the time-global behavior of solution.