A Harnack's inequality for mixed type evolution equations

TL;DR

This paper establishes a Harnack inequality for a class of mixed type evolution equations, including elliptic-parabolic and forward-backward parabolic equations, leading to regularity results like H"older continuity and maximum principles.

Contribution

It introduces a new homogeneous parabolic De Giorgi class for mixed type equations and proves local boundedness, Harnack inequality, and regularity results for functions in this class.

Findings

Proves local boundedness of solutions.

Establishes a Harnack inequality across interfaces where coefficients change sign.

Derives H"older continuity and maximum principles for solutions.

Abstract

We define a homogeneous parabolic De Giorgi classes of order 2 which suits a mixed type class of evolution equations whose simplest example is $\mu (x) \frac{\partial u}{\partial t} - \Delta u = 0$ where $\mu$ can be positive, null and negative, so in particular elliptic-parabolic and forward-backward parabolic equations are included. For functions belonging to this class we prove local boundedness and show a Harnack inequality which, as by-products, gives H\"older-continuity, in particular in the interface $I$ where $\mu$ change sign, and a maximum principle.