A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class

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. The functions belonging to this class are local bounded and satisfy a Harnack type inequality. Interesting by-products are H\"older-continuity, at least in the ``evolutionary'' part of $\Omega$ and in particular in the interface $I$ where $\mu$ change sign, and an interesting maximum principle.