# Talk by Rico Zacher

On March 21st, 2022, Prof. Dr. Rico Zacher (University Ulm) gave a talk about **"Li-Yau inequalities for general non-local diffusion equations via reduction to the heat kernel" **as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.

## Abstract

I will present a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. The approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we obtain a Li-Yau inequality for positive solutions $u$ to the fractional (in space) heat equation of the form $(-\Delta)^{\beta/2}(\log u)\le C/t$, where $\beta\in (0,2)$. I will also show that this Li-Yau inequality allows to derive a Harnack inequality. The general result is further illustrated with an example in the discrete setting by proving a sharp Li-Yau inequality for diffusion on a complete graph. This is joint work with Frederic Weber (Münster).

Slides of the talk (PDF 299 KB)