A Harnack's inequality for 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 $μ(x) \frac{\partial u}{\partial t} - Δu = 0$ where $μ$ can be positive, null and negative, so in particular 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 $μ$ change sign, and a maximum principle.

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