Positive Solutions Of Heat Equation Research Articles

Gradient estimates for positive solutions of heat equations under Finsler-Ricci flow

We establish first order gradient estimates for positive solutions of the heat equations under closed Finsler-Ricci flow with weighted Ricci curvature bounded from below. As an application, we derive the corresponding Harnack inequality. Our results generalize the previous known related results.

An Upper Bound for Hessian Matrices of Positive Solutions of Heat Equations

We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolved under the Ricci flow. These upper bounds seem to be the first general ones that match the well-known lower bounds which have been around for some time. As an application, we discover a local, time reversed Harnack type inequality for bounded positive solutions of the heat equation.

BMO martingales and positive solutions of heat equations

In this paper, we develop a new approach to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds.More precisely, we give some estimatesof the gradient of logarithm of a positive solution via the uniform bound ofthe logarithm of the solution. Moreover, we give a generalized version ofLi-Yau's estimate. Our proof is based on the link between PDE and quadraticBSDE. Our method might be useful to study some (nonlinear) PDEs.

Complete Classification of the Positive Solutions of Heat Equation with Super Critical Absorption

Abstract Let . We prove that any positive solution of (E) ∂tu − Δu + uq = 0 in ℝN × (0, ∞) admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity in ℝN (qʹ = q/(q − 1)) and absolutely continuous with respect to this capacity. If ν is a nonnegative Borel measure in ℝN with the above properties we construct a positive solution u of (E) with initial trace ν and we prove that this solution is the unique σ-moderate solution of (E) with such an initial trace. Finally we prove that every positive solution of (E) is σ-moderate.

Differential Harnack inequalities for heat equations with potentials under geometric flows

In the paper we consider a closed Riemannian manifold M with a time-dependent Riemannian metric g ij (t) evolving by ∂ t g ij = −2S ij , where S ij is a symmetric two-tensor on (M,g(t)). We prove some differential Harnack inequalities for positive solutions of heat equations with potentials on (M,g(t)). Some applications of these inequalities will be obtained.

Differential Harnack inequalities for heat equations with potentials under the Bernhard List’s flow

In the paper we prove some differential Harnack inequalities for positive solutions of heat equations with potentials when the metric is evolving by the Bernhard List’s flow. We also give some applications of these inequalities.

Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation

In the first part of this paper, we get new Li–Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci(M)⩾−k, k∈R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li–Yau–Hamilton differential Harnack inequality for heat kernels on manifolds with Ricci(M)⩾−k, which generalizes a result of L. Ni (2004, 2006) [20,21]. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.

Addenda to “The entropy formula for linear heat equation”

We add two sections to (8) and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of (8), which reveals the relation between the entropy formula, (1.4) of (8), and the well-known Li-Yau's gradient estimate. As a by-product we obtain the sharp estimates on 'Nash's entropy' for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau's gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n .I n the second section we derive a dual entropy formula which, to some degree, connects Hamilton's entropy with Perelman's entropy in the case of Riemann surfaces. 1. The relation with Li-Yau's gradient estimates In this section we provide another derivation of Theorem 1.1 of (8) and discuss its relation with Li-Yau's gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash's 'entropy quantity' − M H log Hd vin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of (9). Let u(x, t) be a positive solution to ∂ ∂t − � u(x, t) = 0 with M ud v= 1. We define