Related papers: A sharp Harnack bound for a nonlocal heat equation
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 prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation.
The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results…
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 establish Harnack's estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation. All results presented in this paper are provided together with quantitative estimates.
We aim to study nonnegative, global solutions to a general class of nonlocal parabolic equations with bounded measurable coefficients. First, we prove a Widder-type theorem. Such a result has previously been studied only for certain…
In this article, we study certain type of boundary behaviour of positive solutions of the heat equation on the upper half-space of $\R^{n+1}$. We prove that the existence of the parabolic limit of a positive solution of the heat equation at…
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 establish the Harnack inequality of nonnegative weak solutions to the doubly nonlinear mixed local and nonlocal parabolic equations. This result is obtained by combining a related comparison principle, a local boundedness…
We derive localized and global noncompact versions of Hamilton's gradient estimate for positive solutions to the heat equation on Riemannian manifolds with Ricci curvature bounded below. Our estimates are essentially optimal and…
In this article we establish the optimal $C^s$ boundary regularity for solutions to nonlocal parabolic equations in divergence form in $C^{1,\alpha}$ domains and prove a higher order boundary Harnack principle in this setting. Our approach…
A set of pointwise estimates are established for local solutions to nonlocal diffusion equations with a drift term. In particular, our Harnack estimates are the first ones for such equations, and our H\"older regularity refines certain…
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…
We prove that the (nonlocal) Marchaud fractional derivative in $\mathbb{R}$ can be obtained from a parabolic extension problem with an extra (positive) variable, as the operator that maps the heat conduction equation to the Neumann…
On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odinger operators -\Delta -V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, under Dirichlet…
A heat equation with uncertain domains is thoroughly investigated. Statistical moments of the solution is approximated by the counterparts of the shape derivative. A rigorous proof for the existence of the shape derivative is presented.…
We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation \begin{align*} \partial_t(|u|^{p-2}u) + (-\Delta_p)^s u = 0 \end{align*} for $p\in (1,\infty)$ and $s \in…
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 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…
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…