一类常微分方程的伯恩斯坦定理

On Bernstein'Theorem to a Class of Ordinary Differential Equations

本文对一类二阶常微分方程u″=exp(-u+1/2tu′),u=u(t)在一些条件下解的表达式进行探讨,若u′(0)=0,那么可得到方程的解一定是二次多项式形式,进而推动平均曲率流的自相似膨胀解的刚性定理这一新问题的研究进程。

For a class of second order ordinary differential equations u″=exp（-u＋1/2tu′）,u=u（t）,under some condition,the expressions of solutions of these equations are investigated.If u′（0）=0,it is shown that the solution of the equation has the form of a quadratic multinomial.This result may have a positive effect in finding out the rigidity theorem related to the self-similar expansion solution of the mean curvature flows.