Abstract: In this paper, the Bernstein theorem for the self-expansion solution of the Lagrangian mean curvature flow in the pseudo-European space is studied. Without loss of the generality, for a class of second order ordinary differential equations such as u″=F(u-$\frac{1}{2}$tu'),u=u(t), and under certain conditions, their solutions are investigated. If u'(0)=0, and the function F is an analytic function, it is shown that the solutions of the equations are quadratic polynomials. At the same time, the classical Bernstein theorem for the solution of a class of equations is proved by using a more concise and intuitionistic method for the first time, and then the study for the self-similarity solution of the pseudo-European space Lagrangian mean curvature is developed.

Key words: mean curvature flow, analytic solution, self-similar solution, Cauchy-Kowalev-skya theorem