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

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.

22 pages; 4 sections