Harnack inequality for parabolic equations with coefficients depending on time

Abstract

Abstract We define a homogeneous De Giorgi class of order p = 2 {p=2} that contains the solutions of evolution equations of the types ξ ⁢ ( x , t ) ⁢ u t + A ⁢ u = 0 {\xi(x,t)u_{t}+Au=0} and ( ξ ⁢ ( x , t ) ⁢ u ) t + A ⁢ u = 0 {(\xi(x,t)u)_{t}+Au=0} , where ξ > 0 {\xi>0} almost everywhere and A is a suitable elliptic operator. For functions belonging to this class, we prove a Harnack inequality. As a byproduct, one can obtain Hölder continuity for solutions of a subclass of the first equation.