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

摘要

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 Hölder continuity of solutions. These classes of singular equations include p-Laplacean type equations in the sub-critical range src="/content/6R455T7108347074/229_2009_317_Article_IEq1.gif" alt="$${1 and equations of the porous medium type in the sub-critical range src="/content/6R455T7108347074/229_2009_317_Article_IEq2.gif" alt="$${0 . Mathematical Subject Classification (2000) Primary 35K65 - 35B65 - Secondary 35B45 This work has been partially supported by I.M.A.T.I.-C.N.R.-Pavia. DiBenedetto’s work partially supported by NSF grant DMS-0652385.