Related papers: Local estimates for positive solutions of porous m…
In this work we derive local gradient and Laplacian estimates of the Aronson-B\'enilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar…
The Aronson-B\'enilan gradient estimate for the porous medium equation has been studied as a counterpart to the Li-Yau gradient estimate for the heat equation. In this paper, we give the Aronson-B\'{e}nilan gradient estimates for the porous…
We consider the porous medium equation (PME) on a locally finite graph and identify suitable curvature-dimension (CD) conditions under which a discrete version of the fundamental Aronson-B\'enilan estimate holds true for positive solutions…
In this paper, we investigate some new local Aronson-B\'enilan type gradient estimates for positive solutions of the porous medium equation $$ u_{t}=\Delta u^{m}, $$ under Ricci flow. As application, the related Harnack inequalities are…
In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature operator. In particular, we derive Aronson-B\'enilan and Li-Yau-Hamilton type differential Harnack…
In this paper, we establish several local and global gradient estimates for the positive solution of Porous Medium Equations (PMEs) and Fast Diffusion Equations (FDEs). Our proof is probabilistic and uses martingale techniques.
In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where $m>1$, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for…
We consider a nearest neighbor, Lagrangian particle discretization of the one dimensional porous medium equation. We prove that the particle model satisfies a discrete analog of the celebrated Aronson-B\'enilan estimate, which we use to…
In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $$u_t=\Delta F(u),$$ with $F'(u) > 0$, on a complete Riemannian manifold with Ricci curvature bounded from…
We obtain new estimates for the solution of both the porous medium and the fast diffusion equations by studying the evolution of suitable Lipschitz norms. Our results include instantaneous regularization for all positive times, long-time…
We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=\Delta_\phi(u^p)$$ associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the…
Let $(M,g)$ be a complete non-compact Riemannian manifold with the $m$-dimensional Bakry-\'{E}mery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive…
We establish pathwise existence of solutions for porous media and fast diffusion equations with nonlinear gradient noise, in the full regime $m\in(0,\infty)$ and for any initial data in $L^2$. Moreover, if the initial data is positive,…
This article presents new gradient estimates for positive solutions to the nonlinear porous medium equation (NPME) in the context of smooth metric measure spaces. The diffusion operator here is the f-Laplacian and the gradient estimates of…
We derive an Aronson-B\'enilan / Li-Yau estimate in the JKO scheme associated to the porous-medium, heat, and fast-diffusion equations, in dimensions $1$ and $2$, and on simple domains (cubes, quarter-space, half-spaces, whole space, and…
We prove local higher integrability of the spatial gradient for solutions to obstacle problems of porous medium type in the fast diffusion case $m<1$. The result holds for the natural range of exponents that is known from other regularity…
In this paper, we consider bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type…
We study the elliptic version of doubly nonlinear diffusion equations on a complete Riemannian manifold $(M,g)$. Through the combination of a special nonlinear transformation and the standard Nash-Moser iteration procedure, some Cheng-Yau…
We consider on Riemannian manifolds solutions of the Leibenson equation \begin{equation*} \partial _{t}u=\Delta _{p}u^{q}. \end{equation*} This equation is also known as doubly nonlinear evolution equation. We prove gradient estimates for…
In the present paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds.