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

We prove a priori gradient bounds for classical solutions of the fully nonlinear parabolic equation $u_{t}=F(D^2u,D u,u,x,t).$ The domain is the torus {\mathbb{T}}^{d} of dimension $d\ge1$. 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.

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