Interior estimates of derivatives and a Liouville type theorem for Parabolic $k$-Hessian equations

In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_t{\sigma }_k(\lambda (D^{2}u))=\psi (x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_t{\sigma }_k(\lambda (D^{2}u))=1$ in ${\mathbb{R}}^n\times (-\mathrm{\infty },0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.