Gradient estimates for a nonlinear diffusion equation on complete manifolds

TL;DR

This paper establishes localized gradient estimates for positive solutions to a nonlinear diffusion equation on complete manifolds with bounded below Bakry-Émery Ricci curvature, extending previous results in the field.

Contribution

It provides a generalized Hamilton-type gradient estimate for a class of nonlinear diffusion equations on non-compact manifolds with curvature bounds.

Findings

Derived localized gradient estimates for solutions.

Extended previous results to more general curvature conditions.

Applicable to nonlinear diffusion equations with specific parameters.

Abstract

Let $(M,g)$ be a complete non-compact Riemannian manifold with the $m$-dimensional Bakry-\'{E}mery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation \[ u_t=\Delta u-\nabla\phi\cdot\nabla u-au\log u-bu, \] where $\phi$ is a $C^2$ function, and $a\neq0$ and $b$ are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).