Related papers: Elliptic gradient estimates for a nonlinear heat e…
In this paper, we will establish an elliptic local Li-Yau gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp…
In this paper, by maximum principle and cutoff function, we investigate gradient estimates for positive solutions to two nonlinear parabolic equations under Ricci flow. The related Harnack inequalities are deduced. An result about positive…
In this paper we are concerned with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations. Firstly, we derive such estimate on a K\"{a}hler manifold with a fixed K\"{a}hler metric. Then we consider the estimate on K\"{a}hler…
We derive a local Gaussian upper bound for the $f$-heat kernel on complete smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery Ricci curvature, which generalizes the classic Li-Yau estimate. As applications, we…
In this article we derive gradient estimation for positive solution of the equation \begin{equation*} (\partial_t-\Delta_f)u = A(u)p(x,t) + B(u)q(x,t) + \mathcal{G}(u) \end{equation*} on a weighted Riemannian manifold evolving along the…
This article is devoted to the study of several estimations for a positive solution to a nonlinear weighted parabolic equation on a weighted Riemannian manifold. We therefore derive new Li-Yau type and Hamilton type gradient estimates…
In this paper, we consider a class of important nonlinear elliptic equations $$\Delta u + a(x)u\log u + b(x)u = 0$$ on a collapsed complete Riemannian manifold and its parabolic counterpart under integral curvature conditions, where $a(x)$…
We prove an $L^2$ estimate for the drift heat equation on a complete gradient shrinking Ricci soliton. This estimate has a time-dependent weight which is Gaussian in its spatial asymptotics. When transferred and scaled to an estimate for…
In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$\Delta_{V}u^{m}+\mu(x)u+p(x)u^{\alpha}=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-\'{E}mery curvature is…
In this paper, we consider gradient estimates for two type of nonlinear parabolic equations under the Ricci flow: one is the equation $$u_t=\Delta u+au\log u+bu$$ with $a,b$ two real constants, the other is $$u_t=\Delta u+\lambda…
The primary objective of this work is to establish pointwise gradient estimates for solutions to a class of parabolic nonlinear nonlocal measure data problems, expressed in terms of caloric Riesz potentials of the data. As a consequence of…
Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space $L^p$. Derivation…
In this paper, we explore the positive solutions to the Finslerian nonlinear equation $$\frac{\partial u}{\partial t} = \Delta^{\nabla u} u + au\log u + bu,$$ which is related to Ricci solitons and serves as the Euler-Lagrange equation to…
In this paper we extend a gradient estimate of R. Hamilton for positive solutions to the heat equation on closed manifolds to bounded positive solutions on complete, non-compact manifolds with $Rc \geq -Kg$. We accomplish this extension via…
We obtain a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. Namely, we consider solutions to $- L u = f(u)$, where $L$ is a linear uniformly elliptic operator and $f$ is…
We present pointwise gradient bounds for solutions to $p$-Laplacean type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic…
In this paper, we develop a new approach to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds. More precisely, we give some estimates…
We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending only on dimension and a lower bound for the…
This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation $u_t-\Delta u=u^p+M|\nabla u|^q$ in $\Omega\times I\subset \R^N\times \R$, where $M>0$, and $p,q>1$. We first establish the…
In this paper, we will address to the following parabolic equation $$ u_t=\Delta_fu + F(u) $$ on a smooth metric measure space with Bakry-\'{E}mery curvature bounded from below. Here $F$ is a differentiable function defined in $\mathbb{R}$.…