Differential Harnack estimates for backward heat equations with potentials under the Ricci flow

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.     

Gradient estimates for the heat equation under the Ricci flow

The paper considers a manifold M evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on M. Among other results, we prove Li-Yau-type inequalities in this context. We consider both the case where M is a complete manifold without boundary and the case where M is a compact manifold with boundary. Applications of our results include Harnack inequalities for the heat equation on M.     

Interpolating Between Li-Yau's and Hamilton's Harnack Inequalities on a Surface

We establish a one-parameter family of Harnack inequalities connecting Li and Yau's differential Harnack inequality for the heat equation to Hamilton's Harnack inequality for the Ricci flow on a 2-dimensional manifold with positive scalar curvature.     

A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow

We establish a point-wise gradient estimate for all positive solutions of the conjugate heat equation. This contrasts to Perelman's point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman's estimate, the most general form of our gradient estimate does not require any curvature assumption. Moreover, assuming only lower bound on the Ricci curvature, we also prove a localized gradient estimate similar to the Li-Yau estimate for the linear Schrödinger heat equation. The main difference with the linear case is that no assumptions on the derivatives of the potential (scalar curvature) are needed. A classical Harnack inequality follows.     

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.     

Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds

Let (M,g) be a complete noncompact Riemannian manifold with the m-dimensional Bakry-Émery Ricci curvature bounded below. In this paper, we give a local Li-Yau type gradient estimate for the positive solutions to a general nonlinear parabolic equation

ut=Δu−∇?⋅∇u−aulogu−qu     

Harnack inequality for the linearized parabolic Monge-Ampère equation

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation

on parabolic sections associated with , under the assumption that the Monge-Ampère measure generated by satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group , where denotes the group of all invertible affine transformations on .

     

Some gradient estimates and Harnack inequalities for nonlinear parabolic equations on Riemannian manifolds

In this paper, by the method of J. F. Li and X. J. Xu (Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456–4491 ), we shall consider the nonlinear parabolic equation on Riemannian manifolds with , . First of all, we shall derive the corresponding Li–Xu type gradient estimates of the positive solutions for . As applications, we deduce Liouville type theorem and Harnack inequality for some special cases. Besides, when , our results are different from Li and Yau's results. We also extend the results of J. F. Li and X. J. Xu, and the results of Y. Yang.     

Gradient estimates and Harnack inequalities for diffusion equation on Riemannian manifolds

We derive the gradient estimates and Harnack inequalities for positive solutions of the diffusion equation u _t = Δu ^m on Riemannian manifolds. Then, we prove a Liouville type theorem.     

完备Khler流形上Khler-Ricci流的局部Harnack估计

本文首先给出非正规化Khler-Ricci流下曲率的发展方程,然后得到了关于曲率的Harnack量在满足曲率局部条件下所产生的一个特殊项CNS.通过对CNS的估计,得到了完备Khler流形上关于Khler-Ricci流的局部Harnack不等式.最后,作为主要定理的应用,我们将结果推广到数量曲率的情形.     

非线性抛物方程在非线性边界条件下解的爆破时间的下界

研究了非线性抛物方程在非线性边界条件下的解的爆破问题,通过构造一个能量表达式,运用微分不等式的方法,得到该能量表达式所满足的微分不等式,然后通过积分得到当爆破发生时解在非线性边界条件下的爆破时间的下界.     

Harnack Estimates for Weak Solutions to a Singular Parabolic Equation

By an interpolation method, an intrinsic Harnack estimate and some supestimates are established for nonnegative solutions to a general singular parabolic equation.     

Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds

In this paper, we derive a local gradient estimate for the positive solution to the following parabolic equation

Gradient Estimates for a Nonlinear Heat Equation Under Finsler-geometric Flow

This paper considers a compact Finsler manifold $(M^n, F(t), m)$ evolving under a Finsler-geometric flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation $$\partial_{t}u(x,t)=\Delta_{m} u(x,t),~~~~~~~~~~(x,t)\in M\times[0,T],$$where $\Delta_{m}$ is the Finsler-Laplacian. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Our results generalize and correct the work of S. Lakzian, who established similar results for the Finsler-Ricci flow. Our results are also natural extension of similar results on Riemannian-geometric flow, previously studied by J. Sun. Finally, we give an application to the Finsler-Yamabe flow.     

非线性抛物型方程计算方法 值此周毓麟先生九十寿辰之际, 谨以本文表达对他的崇高敬意和真诚祝福

本文简要回顾非线性抛物型方程差分方法若干研究工作, 包括周毓麟先生在该研究方向取得的部分研究成果, 并对近年来相关的部分研究进展进行综述, 展望拟开展的研究工作.     

Harnack estimate for a semilinear parabolic equation

We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blow-up of the positive solutions and recover a classical Harnack inequality. We also obtain a result of Liouville type for the elliptic equation.     

A Differential Harnack Inequality for the Newell-Whitehead-Segel Equation

This paper will develop a Li-Yau-Hamilton type differential Harnack estimate for positive solutions to the Newell-Whitehead-Segel equation on R^n.We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation,including deriving a classical Harnack inequality and characterizing standing solutions and traveling wave solutions.     

Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let (M,g) be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equation