Parabolic models for chemotaxis on weighted networks

TL;DR

This paper studies the Keller-Segel chemotaxis model on weighted networks, proving global existence of solutions for both parabolic and parabolic-elliptic cases using heat kernel estimates on graphs.

Contribution

It introduces transition conditions at network vertices and employs explicit heat kernel formulas to establish global solutions for chemotaxis models on weighted networks.

Findings

Global existence of solutions for both models

Use of heat kernel estimates on weighted graphs

Explicit fundamental solution formulas for heat equations

Abstract

In this work we consider the Keller-Segel model for chemotaxis on networks, both in the doubly parabolic case and in the parabolic-elliptic one. Introducing appropriate transition conditions at vertices, we prove the existence of a time global and spatially continuous solution for each of the two systems. The main tool is the use of the explicit formula for the fundamental solution of the heat equation on a weighted graph and of the corresponding sharp estimates.