A Harnack’s inequality and Hölder continuity for solutions of mixed type evolution equations

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 that 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 Hölder-continuity, in particular in the interface $I$ where $\mu$ change sign, and a maximum principle.