In this talk, I present a new method of proving global pointwise Harnack inequalities for posive solutions of parabolic equations, such as the classical heat equation or the heat equation including a potential, porous medium equation, and p-heat equation. My approach is based on a multipoint maximum principle argument, which neither require estimates involving fourth-order derivatives nor additional estimates as, for example, the Aronson-Bénilan estimate. We demonstrate our main techniques by providing a new proof of Li-Yau’s celebrated Harnack inequality for positive solutions of the linear heat equation with potentials, and improve/sharpen Li-Yau’s result in the case of the square potential.
This talk is based on joint work with Ben Andrews and Jessica Slegers.