Gradient estimates for a class of elliptic and parabolic equations on Riemannian manifolds

TL;DR

This paper establishes gradient estimates for solutions to nonlinear elliptic and parabolic equations on Riemannian manifolds with Ricci curvature bounds, extending classical results by allowing variable coefficients.

Contribution

It provides new gradient estimates for nonlinear equations with variable coefficients on Riemannian manifolds, broadening the scope of classical estimates.

Findings

Gradient estimates for elliptic equations with variable coefficients.

Gradient estimates for parabolic equations with variable coefficients.

Extension of classical estimates to non-constant coefficient cases.

Abstract

Let $(N, g)$ be a complete noncompact Riemannian manifold with Ricci curvature bounded from below. In this paper, we study the gradient estimates of positive solutions to a class of nonlinear elliptic equations $$\Delta u(x)+a(x)u(x)\log u(x)+b(x)u(x)=0$$ on $N$ where $a(x)$ is $C^{2}$-smooth while $b(x)$ is $C^{1}$ and its parabolic counterparts $$(\Delta-\frac{\partial}{\partial t})u(x,t)+a(x,t)u(x,t)\log u(x,t) + b(x,t)u(x,t)=0$$ on $N\times[0, \infty)$ where $a(x,t)$ and $b(x,t)$ are $C^{2} $ with respect to $x\in N$ while are $C^{1}$ with respect to the time $t$. In contrast with lots of similar results, here we do not assume the coefficients of equations are constant, so our results can be viewed as extensions to several classical estimates.