Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations

Abstract

A two-parameter family of Harnack type inequalities for non-negative solutions of a class of singular, quasilinear, homogeneous parabolic equations is established, and it is shown that such estimates imply the Holder continuity of solutions. These classes of singular equations include p-Laplacean type equations in the sub-critical range \({1 < p \le\frac{2N}{N+1}}\) and equations of the porous medium type in the sub-critical range \({0 < m \le\frac{(N-2)_+}{N}}\).