Related papers: Differential Harnack Estimates for Time-dependent …
We show that a large class of stochastic heat equations can be approximated by systems of interacting stochastic differential equations. As a consequence, we prove various comparison principles extending earlier results. Among other things,…
We consider the linear heat equation on a bounded domain and on an exterior domain. We study estimates of any order derivatives of the solution locally in time in the Lebesgue spaces. We give a proof of the estimates in the end-point cases…
Let $(M^{n},g,e^{-\phi}dv)$ be a weighted Riemannian manifold evolving by geometric flow $\frac{\partial g}{\partial t}=2h(t),\,\,\,\frac{\partial \phi}{\partial t}=\Delta \phi$. In this paper, we obtain a series of space-time gradient…
In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem $$(P_\theta)\quad \left\{ \begin{array}{rcl} u_t+(-\Delta)^{s} u&=&\l\dfrac{\,u}{|x|^{2s}}+\theta u^p+ c…
Under general conditions we show an a priori probabilistic Harnack inequality for the non-negative solution of a stochastic partial differential equation of the following form d_tu = div (A\nabla u) + f (t, x, u;w) + g_i(t, x,…
We prove matrix and scalar differential Harnack inequalities for linear parabolic equations on Riemannian and K\"ahler manifolds.
We present pointwise gradient bounds for solutions to $p$-Laplacean type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic…
In the present paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds.
We prove the stronger version of Harnack's inequality for positive harmonic functions defined on the unit disc.
In this paper, we study the gradient estimates for the positive solutions of the weighted porous medium equation $$\Delta u^{m}=\delta(x)u_{t}+\psi u^{m}$$ on graphs for $m>1$, which is a nonlinear version of the heat equation. Moreover, as…
We prove an elliptic Harnack's inequality for a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version that has been proposed in stochastic game theory. This version…
This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon…
The paper considers a manifold $M$ evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on $M$. Among other results, we prove Li-Yau-type inequalities in this context. We…
We consider a class of generalized nonlocal $p$-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of…
We study inhomogeneous heat equation with inverse square potential, namely, \[\partial_tu + \mathcal{L}_a u= \pm |\cdot|^{-b} |u|^{\alpha}u,\] where $\mathcal{L}_a=-\Delta + a |x|^{-2}.$ We establish some fixed-time decay estimate for…
We prove a Li-Yau gradient estimate for positive solutions to the heat equation defined on a metric star graph $\mG$ given by the heat kernel formula. As consequence, we derive a Harnack estimate and a Liouville property for bounded…
We study the time regularity of local weak solutions of the heat equation in the context of local regular symmetric Dirichlet spaces. Under two basic and rather minimal assumptions, namely, the existence of certain cut-off functions and a…
This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the…
We study linear time fractional diffusion equations in divergence form of time order less than one. It is merely assumed that the coefficients are measurable and bounded, and that they satisfy a uniform parabolicity condition. As the main…
In this paper we explore the weak solution of a time-dependent inverse source problem and inverse initial problem for $q$-analogue of the heat equation. As an over-determination condition we have used integral type condition on…