Institute of Mathematics of Polish Academy of Sciences

Abstract: The De Giorgi-Nash-Moser framework provides a powerful regularity theory for elliptic and parabolic equations. Applying these techniques to evolutionary partial integro-differential equations presents major hurdles, as nonlocal operators do not readily align with local estimates. In this talk, we present a weak Harnack inequality for supersolutions to a class of evolutionary partial integro-differential equations. Because the problem lacks a natural scaling, identifying the appropriate geometry constitutes a primary challenge. Furthermore, we address recent results on the strong Harnack inequality for globally positive local solutions to a general class of time-nonlocal subdiffusion equations. While the classical Harnack inequality holds for globally nonnegative local solutions in purely time-dependent settings, it is known to fail in spatial dimensions greater than one. Our results settle this question for the one-dimensional spatial case, filling a critical gap in the literature and illustrating the decisive role that spatial dimension plays in the validity of the Harnack estimate. It is a joint work with Adam Kubica and Rico Zacher.