Estimate For The Fundamental Solutions Of Heat Equation

The purpose of this paper is to learn Li-Yau Harnack inequality and the basic methods of estimate the fundamental solution of the linear parabolic equation. The work of Li-Yau introduce some methods and concepts which have basic im-portance.It includes the gradient estimate for heat equation, Harncak inequality and the Temporal and spatial distance with potential function and so on.These basic methods and concepts both have further applications in Ricci flow and av-erage curvature flow. In this paper,we mainly consider the estimate for positive solution u(x,t) of the parabolic equation on compact manifold M which has the form:We first give a gradient estimate for the solutions,then we get the Li-Yau Har-nack inequality about u(x,t) using this formula.At last,we get upper boundary for the fundamental solutions of heat equation using the Harnack inequality and monotonicity formula.