Study on the structure of nonnegative solutions for parabolic equations and the perturbation theory of elliptic operators

研究了黎曼流形上柱面抛物型方程非负解的结构，给出了常值函数1是相应椭圆算子的半小扰动的一般和最优条件下任意解的显式积分表示式，并给出了旋转对称黎曼流形上热方程的几何特征.此外，通过刻画和给出柯西问题非负解唯一的一个尖锐的充分条件，我们确定了旋转对称黎曼流形上热方程的非负解的结构。

We investigated the structure of nonnegative solutions to parabolic equations in cylinders on Riemannian manifolds, and gave explicit integral representation formulas for any solutions under the general and optimal condition that the constant function 1 is a semismall perturbation of the associated elliptic operator ; whose geometric characterization was also given in the case of the heat equation on rotationally symmetric Riemannian manifolds. Furthermore, by using the characterization and giving a sharp sufficient condition for the uniqueness of nonnegative solutions to the Cauchy problem, we determined the structure of nonnegative solutions to the heat equation on rotationally symmetric Riemannian manifolds.