Upper bounds for parabolic equations and the Landau equation

We consider a parabolic equation in nondivergence form, defined in the full space $[0,\mathrm{\infty })\times {\mathbb{R}}^d$, with a power nonlinearity as the right hand side. We obtain an upper bound for the solution in terms of a weighted control in $L^p$. This upper bound is applied to the homogeneous Landau equation with moderately soft potentials. We obtain an estimate in $L^{\mathrm{\infty }}({\mathbb{R}}^d)$ for the solution of the Landau equation, for positive time, which depends only on the mass, energy and entropy of the initial data.