Related papers: A sharp Harnack bound for a nonlocal heat equation

We derive a lower bound, independent of the initial condition, for the solution of the KPZ equation on the torus through its representation as the value function of a (conditional) stochastic control problem. With the same techniques, we…

In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations. We also set up some results about…

We study qualitative and quantitative properties of local weak solutions of the fast $p$-Laplacian equation, $\partial_t u=\Delta_{p}u$, with $1<p<2$. Our main results are quantitative positivity and boundedness estimates for locally…

We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in several uniformly local Zygumnd spaces.…

We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat…

We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all…

We establish two-sided Gaussian bounds for the fundamental solution of second-order parabolic operators in non-divergence form under minimal regularity assumptions. Specifically, we show that the upper and lower bounds follow from the local…

The variational heat equation is a nonlinear, parabolic equation not in divergence form that arises as a model for the dynamics of the director field in a nematic liquid crystal. We present a finite difference scheme for a transformed,…

In this short note, we present the sharp estimate for the existence of a unique solution for a Hadamard-type fractional differential equations with two-point boundary value conditions. The method of analysis is obtained by using the…

In this paper, we study quantitative spatial analytic bounds and unique continuation inequalities of solutions for fractional heat equations with an analytic lower order term on the whole space. At first, we show that the solution has a…

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.

This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms which…

We derive the heat equation for the thermal energy under diffusive space-time scaling for a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field.

Let $(X,d,\mu)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,\mu)$ by applying the parabolic Harnack inequality and the…

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for…

An inverse source problem for the heat equation is considered. Extraction formulae for information about the time and location when and where the unknown source of the equation firstly appeared are given from a single lateral boundary…

We prove sharp two-sided global estimates for the heat kernel associated with a Euclidean sphere of arbitrary dimension. This solves a long-standing open problem.

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…

We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $ \frac{\partial f}{\partial t}=\triangle f-(f^3-f)$ on a closed n-dimensional manifold. As a corollary we find a classical Harnack inequality.…

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and H\"older regularity hold for parabolic…