Global gradient estimates for nonlinear parabolic operators

¹ Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, 6009 Crawley, Australia.
² Department of Mathematics, School of Science, Wuhan University of Technology, 122 Luoshi Road, 430070 Hubei, Wuhan, P.R. China.

^* Corresponding author: serena.dipierro@uwa.edu.au

Received: 6 September 2020
Accepted: 4 February 2021

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.