A priori gradient bounds for fully nonlinear parabolic equations and applications to porous medium models

Abstract

We prove a priori gradient bounds for classical solutions of the fully nonlinear parabolic equation $$u_t=F(D^{2}u,Du,u,x,t).$$ The domain is the torus {\mathbb{T}}^{d} of dimension $d\ge 1$. Up to the price of technicalities, our work can be extended to the case of bounded domains or the case of the whole space ${\mathbb{R}}^d$. Several applications are given, including the standard porous medium equation.

12 pages