# Harnack and pointwise estimates for degenerate or singular parabolic equations

## TL;DR

This paper reviews the development of Harnack inequalities for nonlinear parabolic equations, covering classical, linear, and recent degenerate/singular cases, emphasizing geometric implications and proof techniques.

## Contribution

It provides a comprehensive overview of Harnack inequalities for nonlinear parabolic equations, including new developments for degenerate and singular cases, with detailed proofs and methodological insights.

## Key findings

- Harnack inequalities hold for a broad class of nonlinear parabolic equations.
- Intrinsic Harnack inequalities adapt to degenerate and singular cases.
- Expansion of positivity method is key in proving these inequalities.

## Abstract

In this paper we give both an historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical consequences. The linear parabolic Harnack inequality of Moser is discussed extensively, together with its link to two-sided kernel estimates and to the Li-Yau differential Harnack inequality. Then we overview the more recent developements of the theory for nonlinear degenerate/singular equations, highlighting the differences with the quadratic case and introducing the so-called intrinsic Harnack inequalities. Finally, we provide complete proofs of the Harnack inequalities in some paramount case to introduce the reader to the expansion of positivity method.