Resumen de A new approach to the expansion of positivity set of non-negative solutions to certain singular parabolic partial differential equations

Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri

• Let be a non-negative solution to a singular parabolic equation of -Laplacian type () or porous-medium type (). If is bounded below on a ball by a positive number , for times comparable to and , then it is bounded below by , for some , on a larger ball, say , for comparable times. This fact, stated quantitatively in this paper, is referred to as the ``spreading of positivity'' of solutions of such singular equations and is at the heart of any form of Harnack inequality. The proof of such a ``spreading of positivity'' effect, first given in 1992, is rather involved and not intuitive. Here we give a new proof, which is more direct, being based on geometrical ideas.