Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations

Forward, backward and elliptic Harnack inequalities for non–negative solutions of a class of singular, quasilinear, parabolic equations, are es-tablished. These classes of singular equations include the p–Laplacean equation and equations of the porous medium type. Key novel points in-clude form of a Harnack estimate backward in time, that has never been observed before, and measure theoretical proofs, as opposed to comparison principles. These Harnack estimates are established in the super–critical range (1.5) below. Such a range is optimal for a Harnack estimate to hold.