Related papers: Differential Harnack Estimates for Time-dependent …
Based on gradient estimates for the heat equation by Hamilton, we discover a backward in time Harnack inequality for positive solutions on compact manifolds without further restrictions such as boundedness or vanishing boundary value for…
We study positive solutions to the heat equation on graphs. We prove variants of the Li-Yau gradient estimate and the differential Harnack inequality. For some graphs, we can show the estimates to be sharp. We establish new computation…
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…
A sharp double-sided Harnack bound is derived for positive solutions of a fractional order heat equation.
We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat…
We prove a differential Harnack inequality for the Endangered Species Equation, a nonlinear parabolic equation. Our derivation relies on an idea related to the parabolic maximum principle. As an application of this inequality, we will show…
In this paper, we obtain Li-Yau type gradient estimates with time dependent parameter for positive solutions of the heat equation that are different with the estimates by Li-Xu \cite{LX} and Qian \cite{Qi}. As an application of the…
In this paper, we derive several differential Harnack estimates (also known as Li-Yau-Hamilton-type estimates) for positive solutions of Fisher's equation. We use the estimates to obtain lower bounds on the speed of traveling wave solutions…
Let $(M,g(t))$ be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type $$\ppt f=\Delta f-f \ln f +Rf.$$…
This paper will develop a Li-Yau-Hamilton type differential Harnack estimate for positive solutions to the Newell-Whitehead equation on $\mathbb{R}^n$. We then use our LYH-differential Harnack inequality to prove several properties about…
We establish a connection between a sharp double-sided Harnack bound for positive solutions of a fractional heat equation and the circular geometry in higher dimensions. The present work extends and generalizes the results obtained in the…
In this paper we establish the Harnack inequality for globally positive local solutions to a general class of nonlocal in time subdiffusion equations in one space dimension, which includes time-fractional diffusion equations with time order…
We prove a Harnack inequality for functions which, at points of large gradient, are solutions of elliptic equations with unbounded drift.
In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time…
In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$. As applications, several parabolic Harnack inequalities are…
Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paper we…
We establish new Harnack estimates that defy the waiting-time phenomenon for global solutions to nonlocal parabolic equations. Our technique allows us to consider general nonlocal operators with bounded measurable coefficients. Moreover, we…
In this paper, we derive local and global Li-Yau type gradient estimates for the positive solutions of the CR heat equation on complete noncompact pseudo-Hermitian manifolds. As applications of the gradient estimates, we give a Harnack…
In this paper we prove first order differential Harnack estimates for positive solutions of the heat equation (in the sense of distributions) under closed Finsler-Ricci flows. We assume mild non-linearities (in terms of the Chern…
We use a Harnack-type inequality on exit times and spectral bounds to characterize upper bounds of the heat kernel associated with any regular Dirichlet form without killing part, where the scale function may vary with position. We further…