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)Such a range is optimal for a Harnack estimate to hold.