Upper bound on heat kernels of finite particle systems of Keller-Segel type

TL;DR

This paper establishes an upper bound on the heat kernel for finite particle systems modeled after Keller-Segel equations, providing insights into their blow-up behavior and critical phenomena in two dimensions.

Contribution

It introduces a novel upper bound for the heat kernel of Keller-Segel particle systems by linking them to non-local operators, advancing understanding of their critical behavior.

Findings

Derived an upper bound on the heat kernel for Keller-Segel particle systems.

Connected Keller-Segel particles to non-local operators to analyze blow-up effects.

Provided insights into the critical behavior of the system in two dimensions.

Abstract

We obtain an upper bound on the heat kernel of the Keller-Segel finite particle system that exhibits blow up effects. The proof exploits a connection between Keller-Segel finite particles and certain non-local operators. The latter allows to address some aspects of the critical behaviour of the Keller-Segel system resulting from its two-dimensionality.