Sub-potential lower bounds and Liouville’s type rigidity for parabolic De Giorgi classes

We discuss several structural properties of functions belonging to a parabolic energy class, reminiscent of the elliptic De Giorgi class. In earlier works, sub-potential lower bounds, giving insight into the structural behavior of elements of these classes, were established for the linear case: Here, we extend these results to the nonlinear one. By showing that subpotential lower bounds follow solely from the Harnack inequality, we show that positive solutions to Trudinger’s equation and elements of parabolic De Giorgi classes have a common lower bound. For both cases, we derive Liouville-type rigidity results in the parabolic setting.