Global gradient estimates for nonlinear parabolic operators

TL;DR

This paper derives gradient estimates for nonlinear parabolic equations on Riemannian manifolds, extending known results to more general settings including sourcing terms and variable diffusion coefficients.

Contribution

It provides new gradient estimates for nonlinear parabolic operators on manifolds, incorporating complex geometric and data-dependent factors.

Findings

Gradient estimates depend on structural constants and geometry.

Universal interior estimates are established independent of parabolic data.

Results extend to equations with sourcing terms and general diffusion coefficients.

Abstract

We consider a parabolic equation driven by a nonlinear diffusive operator and we obtain a gradient estimate in the domain where the equation takes place. This estimate depends on the structural constants of the equation, on the geometry of the ambient space and on the initial and boundary data. As a byproduct, one easily obtains a universal interior estimate, not depending on the parabolic data. The setting taken into account includes sourcing terms and general diffusion coefficients. The results are new, to the best of our knowledge, even in the Euclidean setting, though we treat here also the case of a complete Riemannian manifold.