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

We define a homogeneous parabolic De Giorgi classes of order 2 which suits a mixed type class of evolution equations whose simplest example is\ud$\mu (x) \frac{\partial u}{\partial t} - \Delta u = 0$ where $\mu$ can be positive, null and negative, so that\udelliptic-parabolic and forward-backward parabolic equations are included.\udFor functions belonging to this class we prove local boundedness and show a Harnack inequality which, as by-products, gives\udH\"older-continuity, in particular in the interface $I$ where $\mu$ change sign, and a maximum principle. \