Related papers: A sharp Harnack bound for a nonlocal heat equation
In this paper, we employ probabilistic techniques to derive sharp, explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator $$ \Delta^{\alpha/2}_v + v \cdot \nabla_x, \quad \alpha \in (0, 2),\ (x,v)\in {\mathbb…
H\"older estimates and Harnack inequalities are studied for fully nonlinear integro-differential equations under some mild assumptions. We allow the kernels of variable order and critically close to 2.
The present work is concerned with existence of positive solutions for a class of fractional equation involving a Kirchhoff term and singular potential.
In this paper we establish sharp weighted bounds (Buckley type theorems) for one{sided maximal and fractional integral operators in terms of one{sided $A_p$ characteristics. Appropriate sharp bounds for strong maximal functions, multiple…
Time fractional parabolic problem for p-Laplacian with double singular Hardy-type potential is considered. Comparison principle and appriory estimates for the weak solutions are proved. Existence of global weak solutions and finite-time…
The purpose of this paper is to prove local upper and lower bounds for weak solutions of semilinear elliptic equations of the form $-\Delta u= c u^p$, with $0<p<p_s=(d+2)/(d-2)$, defined on bounded domains of $\RR^d$, $d\ge 3$, without…
We prove the stronger version of Harnack's inequality for positive harmonic functions defined on the unit disc.
We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of well known two-sided Gaussian heat kernel…
We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =\Delta (u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative…
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…
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…
This paper deals with the blow-up properties of positive solutions to a system of two heat equations.
It is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain, with Dirichlet boundary conditions, are unique up to multiplication by a positive constant.
This paper considerers the problem of computing the value of a solution of the heat equation at a given point inside a bounded domain after the initial time. It is assumed that the initial value of the solution inside the domain (possibly…
We provide the classical Boundary Harnack principle in Lipschitz domains for solutions to two different linear uniformly elliptic equations with the same principal part.
The existence of a positive solution to a class of Choquard equations with potential going at a positive limit at infinity possibly from above or oscillating is proved. Our results include the physical case and do not require any symmetry…
Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which…
We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather…
We introduce a fractional stochastic heat equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize…
Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the…