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Under general conditions we show an a priori probabilistic Harnack inequality for the non-negative solution of a stochastic partial differential equation of the following form d_tu = div (A\nabla u) + f (t, x, u;w) + g_i(t, x,…

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…

The weak Harnack inequality for $L^p$-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that H\"older…

We prove that ancient non-negative solutions to a fully anisotropic prototype evolution equation are constant if they satisfy a condition of finite speed of propagation and if they are both one-sided bounded, and bounded in space at a…

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…

This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\frac{1}{n}}$ class of Muckenhoupt…

In the context of a metric measure Dirichlet space satisfying volume doubling and Poincar\'e inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In…

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight.…

We extend some classical results dealing with boundary Harnack inequatilities to a class of quasilinear elliptic equations and derive some new estimates for solutions of such equations with an isolated singularity on the boundary of a…

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 consider degenerate fully nonlinear parabolic equations, which generalize the p-parabolic equation with $p>2$ to nondivergence form operators. We prove an intrinsic Harnack inequality for nonnegative solutions and a weak Harnack…

In this paper, the Harnack inequality result are established for a new class of the homogeneous nonlinear degenerate parabolic equations \begin{align*} div A(t,x,u,\nabla_x u)-\partial_t \vert u\vert^{p-2}u=0 \end{align*} on a bounded…

We prove matrix and scalar differential Harnack inequalities for linear parabolic equations on Riemannian and K\"ahler manifolds.

We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.

Harnack inequalities are useful qualitative tools for understanding the properties of partial differential equations. Originally discovered as a property of harmonic functions, Harnack inequalities have since been studied for solutions of…

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…

In this paper, we consider the finite time blow-up results for a parabolic equation coupled with superlinear source term and local linear boundary dissipation. Using a concavity argument, we derive the sufficient conditions for the…

We prove Harnack's inequality for bounded weak solutions to quasilinear second order elliptic equations with generalized Orlicz growth conditions. Our approach covers new cases of variable exponent and (p,q) growth conditions.

We derive the Strong Harnack inequality for a class of hypoelliptic integro-differential equations in divergence form. The proof is based on a priori estimates, and as such extends the first non-stochastic approach of the non-local…

We establish Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear anisotropic elliptic equations exhibiting non-standard growth conditions. A primary example of such operators is the degenerate anisotropic…