HARNACK TYPE INEQUALITIES FOR SOME DOUBLY NONLINEAR SINGULAR PARABOLIC EQUATIONS

We prove Harnack type inequalities for a wide class of parabolic doubly nonlinear equations including $u_t = div(|u|^{m−1}|Du|^{p−2}Du)$. We will distinguish between the supercritical range $3 −rac{p}{N} < p + m < 3$ and the subcritical $2 < p +m ≤ 3 −rac{p}{N}$ range. Our results extend similar estimates holding for general equations having the same structure as the parabolic p-Laplace or the porous medium equation and recently collected in the monograph " Harnack’s Inequality for Degenerate and SingularParabolic Equations" by E. DiBenedetto, U. Gianazza and V. Vespri,