Upper bounds for parabolic equations and the Landau equation

We consider a parabolic equation in nondivergence form, defined in the full space [0,∞)×Rd, with a power nonlinearity as the right-hand side. We obtain an upper bound for the solution in terms of a weighted control in Lp. This upper bound is applied to the homogeneous Landau equation with moderately soft potentials. We obtain an estimate in L∞(Rd) for the solution of the Landau equation, for positive time, which depends only on the mass, energy and entropy of the initial data.