On the uniqueness for the heat equation with density on infinite graphs

TL;DR

This paper investigates the uniqueness of solutions to heat equations with density on infinite weighted graphs, distinguishing between cases with bounded and vanishing densities, and identifying different uniqueness classes.

Contribution

It provides a novel analysis of how density bounds affect the uniqueness of heat equation solutions on infinite graphs, highlighting two distinct scenarios.

Findings

Uniqueness depends on whether the density is bounded below by a positive constant.

Different classes of uniqueness are identified for bounded and vanishing densities.

The study advances understanding of heat equations on infinite weighted graphs.

Abstract

We study the uniqueness of solutions to a class of heat equations with positive density posed on infinite weighted graphs. We separately consider the case when the density is bounded from below by a positive constant and the case of possibly vanishing density, showing that these two scenarios lead to two different classes of uniqueness.