Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II

Abstract

In this paper, we consider the mean curvature flow of entire Lagrangian graphs with initial data in the pseudo-Euclidean space, which is related to the special Lagrangian parabolic equation. We show that the parabolic equation \eqref{11} has a smooth solution $u(x,t)$ for three corresponding nonlinear equations between the Monge-Amp$\grave{e}$re type equation($\tau=0$) and the special Lagrangian parabolic equation($\tau=\frac{\pi}{2}$). Furthermore, we get the bound of $D^lu$, $l=\{3,4,5,\cdots\}$ for $\tau=\frac{\pi}{4}$ and the decay estimates of the higher order derivatives when $0<\tau<\frac{\pi}{4}$ and $\frac{\pi}{4}<\tau<\frac{\pi}{2}$. We also prove that $u(x,t)$ converges to smooth self-expanding solutions of \eqref{12}.