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 established. These classes of singular equations include the p-Laplacean equation and equations of the porous medium type. Key novel points include 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 2N/(N+1)<p<2. Such a range is optimal for a Harnack estimate to hold.