Related papers: A probabilistic Harnack inequality and strict posi…

Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form \[ \partial_t u = \mathrm{div} (A \nabla u) + f(t,x, u) + g_i (t,x,u) \dot{w}^i_t \] is almost surely H\"older continuous…

This paper aims to investigate a Harnack inequality for non-negative solutions of the normalized infinity Laplacian with nonlinear absorption and gradient terms. More specifically, we establish a Harnack inequality for non-negative…

This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to…

Let $u$ be a positive solution of the ultraparabolic equation \begin{equation*} \partial_t u=\sum_{i=1}^n \partial_{x_i}^2 u+\sum_{i=1}^k x_i\partial_{x_{n+i}}u \hspace{8mm} \mbox{on} \hspace{4mm} \mathbb{R}^{n+k}\times (0,T),…

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…

Let $u$ be a non-negative super-solution to a $1$-dimensional singular parabolic equation of $p$-Laplacian type ($1<p<2$). If $u$ is bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$, then it has a power-like decay…

We prove Harnack's type inequalities for bounded non-negative solutions of degenerate parabolic equations with $(p,q)$ growth $$ u_{t}-{\rm div}\left(\mid \nabla u \mid^{p-2}\nabla u + a(x,t) \mid \nabla u \mid^{q-2}\nabla u \right)=0,\quad…

In this paper, we investigate Harnack estimates for weak solutions to the following nonlocal equation: $$ \partial_t u = \Delta^{\alpha/2} u + b \cdot \nabla u + f, $$ where $\Delta^{\alpha/2}$ denotes the fractional Laplacian, $b$ is a…

Pathwise non-uniqueness is established for non-negative solutions of the parabolic stochastic pde $$\frac{\partial X}{\partial t}=\frac{\Delta}{2}X+X^p\dot W+\psi,\ X_0\equiv 0$$ where $\dot W$ is a white noise, $\psi\ge 0$ is smooth,…

We prove continuity and Harnack's inequality for bounded solutions to the equation $$ {\rm div}\big(|\nabla u|^{p(x)-2}\,\nabla u \big)=0, \quad p(x)= p + L\frac{\log\log\frac{1}{|x-x_{0}|}}{\log\frac{1}{|x-x_{0}|}},\quad L > 0, $$ under…

We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be…

Let $(M^{n},g)$ be a complete Riemannian manifold. In this paper, we establish a space-time gradient estimates for positive solutions of nonlinear parabolic equations $$\partial_{t}u(x,t)=\Delta u(x,t)+a u(x,t)(\log u(x,t))^b +…

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 $$\ppt f=\Delta f-f \ln f +Rf.$$…

We consider nonnegative solutions $u:\Omega\longrightarrow \mathbb{R}$ of second order hypoelliptic equations \begin{equation*} \mathscr{L} u(x) =\sum_{i,j=1}^n \partial_{x_i} \left(a_{ij}(x)\partial_{x_j} u(x) \right) + \sum_{i=1}^n b_i(x)…

We consider one-dimensional stochastic heat equation with nonlinear drift, $\displaystyle \partial_t u=\frac{1}{2}\Delta u+b(u)u+\sigma(u)\dot{W}(t,x)$, where $b:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function and…

This note is concerned with an important for modelling question of existence of solutions of stochastic partial differential equations as proper stochastic processes, rather than processes in the generalized sense. We consider a first order…

This article considers the stochastic partial differential equation \[ \left\{ \begin{array}{l} u_t = \frac{1}{2} u_{xx} + u^\gamma \xi u(0,.) = u_0 \end{array}\right. \] \noindent where $\xi$ is a space / time white noise Gaussian random…

We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem: \begin{align}\label{P} \left\{ \begin{array}{l} D_{0+}^\alpha u + h(t)f(u) = 0, \quad 0<t<1, \\[1ex] u(0)=u(1)=0,…

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 establish that the elliptic equation $\Delta u+f(x,u)+g(| x|)x\cdot \nabla u=0$, where $x\in\mathbb{R}^{n}$, $n\geq3$, and $| x|>R>0$, has a positive solution which decays to 0 as $| x|\to +\infty$ under mild restrictions on the…