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We define a homogeneous De Giorgi class of order $p = 2$ that contains thesolutions of evolution equations of the types $\rho (x,t) u_t + A u = 0$ and$(\rho (x,t) u)_t + A u = 0$, where $\rho > 0$ almost everywhere and $A$ is asuitable elliptic operator. For functions belonging to this class we prove aninhomogeneous Harnack inequality, i.e. a Harnack inequality that takes intoaccount the mean value of $\rho$ in different regions of $\Omega \times (0,T)$.As a consequence we get a Harnack inequality for solutions, and in these caseonly for solutions, of elliptic-parabolic equations and of forward-backwardparabolic equations. As a byproduct one obtains H\"older continuity forsolutions of a subclass the first equation: in particular the solutions of thissubclass are H\"older continuous in the interface where $\rho$ changes itssign.

An inhomogeneous Harnack inequality for parabolic equations and applications to elliptic-parabolic and forward-backward parabolic equations