On the parabolic kernel of the Schrödinger operator

Abstract

In this paper, we will study parabolic equations of the typeon a general Riemannian manifold.The function q(x, t) is assumed to be C 2 in the first variable and C 1 in the second variable.In classical situations [20], a Harnack inequality for positive solutions was established locally.However, the geometric dependency of the estimates is complicated and sometimes unclear.Our goal is to prove a Harnack inequality for positive solutions of (0.1) (w 2) by utilizing a gradient estimate derived in w 1.The method of proof is originated in [26] and [8], where they have studied the elliptic case, i.e. the solution is time independent.In some situations (Theorems 2.2 and

Locations

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