关于黎曼流形上的一个扩散方程的一些梯度估计

我们给出关于黎曼流形上的扩散方程θtu=Δu-▽φ·▽u（这里φ是一个C^2函数）的一些梯度估计。这推广了R.Hamilton和Qi S.Zhang关于热方程的一些梯度估计。     

Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation u_t(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^α(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.     

Second Order Estimates for Non-Concave Hessian Type Elliptic Equations on Riemannian Manifolds

In this paper, we derive second order estimates for a class of non-concave Hessian type elliptic equations on Riemannian manifolds. By applying a new method for $C^2$ estimates, we can weaken some conditions, which works for some non-concave equations. Gradient estimates are also obtained.     

Some Gradient Estimates and Liouville Properties of the Fast Diffusion Equation on Riemannian Manifolds*

In the paper, the authors provide a new proof and derive some new elliptic type (Hamilton type) gradient estimates for fast diffusion equations on a complete noncompact Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function. These results generalize earlier results in the literature. And some parabolic type Liouville theorems for ancient solutions are obtained.     

黎曼流形上不变集的分形维数估计

本文利用整体Lyapunov指数给出了光滑黎曼流形上C1-映射不变集的分形维数的上界.     

Local Aronson-Benilan Estimates for Porous Medium Equations under Ricci Flow

In this work we derive local gradient estimates of the Aronson-Benilan type for positive solutions of porous medium equations under Ricci flow with bounded Ricci curvature. As an application, we derive a Harnack type inequality.     

Decay Estimates for Solutions of Porous Medium Equations with Advection

In this paper, we show that bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form

$$ u_{t} + \operatorname{div}f(x,t,u) = \operatorname{div}\bigl( |u|^{\alpha } \nabla u\bigr), \quad x \in \mathbb{R}^{n} , \ t > 0, $$

where \(\alpha > 0 \) is constant, decrease to zero, under fairly broad conditions on the advection flux \(f\). Besides that, we derive a time decay rate for these solutions.

     

Logarithmic Gradient of the Heat Kernel on Riemannian Manifolds

Mathematical Notes -     

Stable Solutions of Elliptic Equations on Riemannian Manifolds

This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincaré inequality which allows us to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.     

黎曼流形上一类非线性抛物方程的Hamilton 梯度估计

本文我们得到了黎曼流形上一类非线性抛物方程的局部Hamilton梯度估计. 利用这个局部估计,我们得到了一个Harnack型不等式和一个Liouville型定理.     

Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds

This paper deals with the gradient estimates of the Hamilton type for the positive solutions to the following nonlinear diffusion equation:u_t = △u +▽φ·▽u+a(x)uln u + b(x)u on a complete noncompact Riemannian manifold with a Bakry-Emery Ricci curvature bounded below by- K(K ≥ 0),where φ is a C~2 function,a(x) and b(x) are C~1 functions with certain conditions.     

Characterizing Specific Riemannian Manifolds by Differential Equations

Some characterizations of certain rank-one symmetric Riemannian manifolds by the existence of nontrivial solutions to certain partial differential equations on Riemannian manifolds are surveyed.     

Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds

Let （M,g） be a complete non-compact Riemannian manifold with the m- dimensional Bakry-Emery 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 ut = △u - △↓ φ· △ ↓u - aulogu- bu,where φ is a C^2 function, and a ≠ 0 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）.     

Yamabe流下黎曼流形上热方程的梯度估计

研究了在Yamabe流下演化的一个完备非紧黎曼流形,对流形上热方程的正解给出了两种局部的梯度估计.作为应用,可以得到这个热方程的Harnack不等式.     

Estimates for the Energy Integral of Quasiregular Mappings on Riemannian Manifolds and Isoperimetry

The rate of growth of the energy integral of a quasiregular mapping is estimated in terms of a special isoperimetric condition on . The estimate leads to new Phragmen-Lindelof type theorems.     

黎曼流形上某些微分方程的通解

Let (Mⁿ,g) be a Riemannian manifold of dimension n and let ▽ denote the Riemannian connection defined by g. In this paper we study the following systems of differential equations.     

黎曼流形上一类指数调和型热方程的梯度估计

本文我们推到了黎曼流形上指数调和型热方程的一个Hamilton-Souplet-Zhang型梯度估计. 利用这个估计, 我们得到了一个Liouville型定理.     

紧致黎曼流形上一类非线性热方程的梯度估计

In this paper,we study gradient estimates for the nonlinear heat equation ut-△u =au log u,on compact Riemannian manifold with or without boundary.We get a Hamilton type gradient estimate for the positi...     

Gradient Estimates for Some Semi-Linear Hypoelliptic Equations

Given some smooth vector fields X ₁,X ₂,…,X _m on a compact manifold M, if they satisfy Hörmander’s condition, we establish global gradient estimates for the positive smooth solutions to the semi-linear hypoelliptic equations $$Lu+au\log u+bu=\partial_tu, \quad \mbox{on} \ M\times[0,\infty) $$ and $$Lu+au\log u+bu=0, \quad \mbox{on} \ M, $$ where a,b are constants, and $L=\sum_{i}X_{i}^{2}-X_{0}$ . We partially generalize the results of Cao and Yau (Math. Z. 211:485–504, 1992).