Weighted Aronson-Bénilan estimates and Harnack inequalities for slow diffusion equations with a nonlinear forcing term

TL;DR

This paper introduces new mathematical estimates for solutions to nonlinear slow diffusion equations on weighted manifolds.

Contribution

The paper provides novel Aronson-Bénilan and Li-Yau type gradient estimates with time-variable coefficients for nonlinear slow diffusion equations.

Findings

New gradient estimates unify and improve previous results on slow diffusion equations.

The estimates incorporate time-variable coefficients and Harnack quantities.

The results lead to parabolic Harnack inequalities and global bounds.

Abstract

We formulate and prove new Aronson-Bénilan and Li-Yau type gradient estimates for positive solutions to nonlinear slow diffusion equations. The framework is that of a smooth metric measure space (i.e., a weighted manifold) and the estimates make use of a range of Harnack quantities with suitable time-variable coefficients. The proofs exploit the intricate relation between geometry, nonlinearity and dynamics of the equation and the results extend, unify and improve various earlier estimates on slow diffusion equations. A number of important corollaries and implications, notably, to parabolic Harnack inequalities and global bounds are presented and discussed.