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)∂u∂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 changes sign, and a maximum principle.