A multi-point maximum principle to prove the global Harnack inequality for the doubly nonlinear heat equation

Abstract

This work aims to present a new proof of the global pointwise Harnack inequality satisfied by positive viscosity solutions of the doubly nonlinear heat equation on ${\mathbb{R}}^d$. Our approach is based on a multi-point maximum principle argument introduced by the authors in a recent article [Calc. Var., 2026]. Required for the proof is a new variation of the maximum principle for semicontinuous functions, which is a standard tool in the analysis of viscosity solutions. In particular, our techniques provide a new proof of the Harnack inequalities satisfied by positive solutions of the porous medium equation and the $p$-heat equation, which were derived by Auchmuty and Bao [Proc. Amer. Math. Soc., 1994] using results of Aronson and Bénilan [C. R. Acad. Sci. Paris, 1979] and Esteban and Vázquez [Arch. Rational Mech. Anal., 1988]. The results contained in this article have also been discussed in the third author's doctoral thesis.