Gradient estimates for a nonlinear parabolic equation and Liouville theorems

TL;DR

This paper derives gradient estimates for positive solutions to a nonlinear parabolic equation on metric measure spaces, leading to Liouville theorems and applications to Yamabe-type problems on Riemannian manifolds.

Contribution

It provides new local gradient estimates for nonlinear parabolic equations on metric measure spaces, enabling nonexistence results and applications to geometric problems.

Findings

Gradient estimates established for solutions on metric measure spaces

Conditions identified for nonexistence of nontrivial solutions

Applications to Yamabe-type problems on manifolds

Abstract

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.